thesis tobias spalke

thesis tobias spalke
Dissertation
submitted to the
Combined Faculties for the Natural Sciences and for Mathematics
of the Ruperto-Carola University of Heidelberg, Germany
for the degree of
Doctor of Natural Sciences
Put forward by
Diplom-Physicist:
Tobias Friedrich Spalke
Born in:
Frankfurt am Main
Oral examination: 17th June, 2009
Application of
multiobjective optimization concepts
in inverse radiotherapy planning
Referees:
Prof. Dr. Uwe Oelfke
Prof. Dr. Wolfgang Schlegel
Sei geduldig mit allen Fragen in deinem Herzen,
und versuche, die Fragen an sich zu schätzen...
Rainer Maria Rilke
Zusammenfassung
Anwendung multikriterieller Optimierungskonzepte in der inversen Strahlentherapieplanung
Die Intensitätsmodulierte Strahlentherapie (IMRT) gehört zu den modernsten Methoden der
Krebsbehandlung. Aufgrund der hohen Zahl der Freiheitsgrade wird das inverse Problem der
IMRT als skalares Optimierungsproblem formuliert. Der konventionelle Ansatz einer Skalarisierung der Zielfunktion entspricht einem a priori Kompromiss zwischen den verschiedenen Planungszielen und kann zu einem zeitaufwändigen Planungsprozess führen. Neue multikriterielle
Optimierungsansätze versuchen, die Nachteile der konventionellen Planung zu umgehen. Dazu
wird eine Datenbank Pareto optimaler Bestrahlungspläne berechnet, aus der der Planer im
Anschluss den besten Plan auswählen kann. In dieser Arbeit wird ein neues multikriterielles
Planungssystem evaluiert. Wir untersuchen drei verschiedene auf der generalized equivalent
uniform dose (gEUD) basierende Modellierungsansätze und die Sensitivitäten der dazugehörigen Modellparameter. Für die Charakterisierung von ganzen Pareto optimalen Datenbanken
werden Qualitätsinidikatoren entwickelt und auf klinische Fälle angewandt. In restropektiven
Planungsstudien wird gezeigt, dass das neue Planungssystem die gleiche Planqualität wie das
klinische Referenzsystem erreicht und eine erhebliche Verringerung der Gesamtplanungszeit für
Prostatafälle erzielt werden kann. Im letzten Teil entwickeln wir eine Methode zum Auffinden
der zwingenden Kompromisse in einer Plandatenbank und verwenden Techniken der linearen
und nicht-linearen Dimensionsreduktion. Diese erlauben die aussagekräftige Visualisierung von
hochdimensionalen Paretofronten und eine Offenlegung der zugrundeliegenden Kompromisse
zwischen den beteiligten Planungszielen.
Abstract
Application of multiobjective optimization concepts in inverse radiotherapy planning
Intensity Modulated Radiotherapy (IMRT) belongs to the most advanced techniques in cancer
treatment. Due to the large number of degrees of freedom the inverse problem of IMRT is
formulated as a scalar optimization problem. The conventional scalarization approach of the
objective function represents an a priori trade-off between the planning goals and can lead to
time consuming planning process. New multiobjective approaches try to overcome the difficulties by a precalculation of a set of Pareto optimal treatment plans from which the planner
can subsequently choose the best suited plan. In this work we evaluate a new multiobjective
treatment planning system. We investigate three different generalized euqivalent uniform dose
(gEUD) based modeling approaches and study the sensitivities of the corresponding model parameters. Quality measures for entire Pareto optimal databases are developed and applied to
clincal cases. In retrospective planning studies we show that the new system is compatible with
a clinical reference treatment planning system and that the total planning time for prostate
cases can be significantly reduced. In the last part we develop a method to detect the imperative
trade-offs in a plan database and apply techniques from linear as well as non-linear dimensionality reduction. They allow meaningful visualizations of high dimensional Pareto fronts and an
insight into the underlying trade-offs between the planning goals.
ix
Contents
1 Introduction
2 Fundamentals
2.1 Mathematical Basics . . . . . . . . . . .
2.2 Pareto Optimality . . . . . . . . . . . . .
2.3 Intensity modulated Radiotherapy . . . .
2.3.1 Basic Concept . . . . . . . . . . .
2.3.2 Optimization in Radiotherapy - A
2.3.3 Notation and Terminology . . . .
2.4 Multiobjective Planning Systems . . . .
2.4.1 MIRA . . . . . . . . . . . . . . .
2.4.2 PGEN . . . . . . . . . . . . . . .
1
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General Framework
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3 Choosing Models and Parameters
3.1 Modeling Approaches . . . . . . . . . . . . . .
3.1.1 Motivation . . . . . . . . . . . . . . . .
3.1.2 Methods . . . . . . . . . . . . . . . . .
3.1.3 Results . . . . . . . . . . . . . . . . .
3.1.4 Discussion . . . . . . . . . . . . . . . .
3.1.5 Conclusions . . . . . . . . . . . . . . .
3.2 Quality Indicators for Efficient Sets . . . . . .
3.2.1 Motivation . . . . . . . . . . . . . . . .
3.2.2 Methods . . . . . . . . . . . . . . . . .
Cardinality and Hypervolume Indicator
Uniformity and Hyperspheres . . . . .
Coverage error . . . . . . . . . . . . .
3.2.3 Results . . . . . . . . . . . . . . . . . .
3.2.4 Discussion . . . . . . . . . . . . . . . .
3.2.5 Conclusion . . . . . . . . . . . . . . . .
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4 Clinical Application
4.1 Automatic Localization of a Reference Plan
4.1.1 Motivation . . . . . . . . . . . . . . .
4.1.2 Material and Methods . . . . . . . .
4.1.3 Results . . . . . . . . . . . . . . . . .
4.1.4 Discussion and Conclusion . . . . . .
4.2 Clinical Evaluation . . . . . . . . . . . . . .
4.2.1 Motivation . . . . . . . . . . . . . . .
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x
Contents
4.2.2
4.2.3
4.2.4
4.2.5
Methods . .
Results . . .
Discussion .
Conclusions
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5 Understanding Pareto Optimal Planning Databases
5.1 Sensitivities of the Efficient Set . . . . . . . . . . . . . . . . .
5.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Identifying the Main Trade-Offs in Pareto Optimal Databases
5.2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.2 Materials and Methods . . . . . . . . . . . . . . . . . .
Terminology . . . . . . . . . . . . . . . . . . . . . . . .
Principal Components Analysis . . . . . . . . . . . . .
Isomap Method . . . . . . . . . . . . . . . . . . . . . .
Phantoms and Clinical Cases . . . . . . . . . . . . . .
5.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . .
Phantoms . . . . . . . . . . . . . . . . . . . . . . . . .
Brain Case . . . . . . . . . . . . . . . . . . . . . . . .
Pancreas Case . . . . . . . . . . . . . . . . . . . . . . .
Effective Dimension . . . . . . . . . . . . . . . . . . . .
5.2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . .
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6 Summary and Outlook
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7 Acknowledgment
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Bibliography
81
1
1 Introduction
Radiotherapy is besides surgery the most important treatment option in oncology. The
central goal is to achieve a high tumor control probability while keeping the normal tissue
complication probability as low as possible. In order to achieve this goal one aims to deliver
a homogeneous high dose to the tumor while ensuring an adequate sparing of the organs at
risk (OARs) and the surrounding healthy tissue.
The development of Intensity Modulated Radiotherapy (IMRT) allows to create highly conformal dose distributions (Bortfeld, 2006), i.e. restricting the high dose areas to the target
volume, using multileaf collimators and high energetic photons. The task of deriving a set
of treatment parameters from a desired dose distribution is called the inverse problem of
IMRT. Because of the large number of degrees of freedom from the discretization of the
patient geometry into voxels and the beams into beamlets the problem is formulated as a
constrained optimization problem. If the irradiation geometry, i.e. number of beams and
the beam directions, is fixed the problem yields a fluence map optimization (Oelfke and
Bortfeld, 2001). The quality of a treatment plan is usually modeled by the value of a single
scalar objective function which has to be minimized. Consequently an entire treatment plan
is characterized by a single number.
The inverse planning of IMRT involves inherent trade-offs. The conflicting goals of a high
homogeneous target dose and the protection of healthy tissue have to be balanced against
each other. Traditionally all the different planning goals or objectives are aggregated into
the objective function, which is parameterized by weight factors that allow to influence the
outcome of the optimization and place emphasis on the individual goals. The selection
of the weight factors represents an a priori trade-off. As the sensitivity of the optimal
solution (therewith the dose distribution) with respect to the weight factors is not known
the planning process can become a cumbersome task, involving several human interactions
during the optimization to adjust the weights. In addition the weight factors have no clinical
meaning.
To overcome these difficulties multiobjective planning concepts for IMRT planning have been
proposed in the last years (see e.g. Yu (1997); Cotrutz et al. (2001); Küfer et al. (2000);
Thieke et al. (2002); Craft et al. (2006)). Within a multiobjective framework each planning
2
1 Introduction
goal is considered as an independent entity. The optimality criterion is generalized to Pareto
optimality. Pareto optimal plans cannot be improved in one objective without worsening
another. As a result a whole set of plans is optimal in a multiobjective sense, instead of only
one optimal plan for conventional planning. The set of all Pareto optimal plans constitutes
the Pareto front or the efficient set. During the multiobjective planning process a finite
number of Pareto optimal plans is calculated. These plans are stored in a plan database and
should capture the essence of the clinically relevant part of the Pareto front. Subsequently
the planner can choose the best suited plan from the database with the help of decision
support systems such as interactive navigation tools (Monz et al., 2008).
At present multiobjective planning systems are not yet introduced into clinical practice but
are still in a pre clinical stage. A general question is what are the realistic expectations of the
new class of planning systems? Furthermore new challenges arise by the increased amount
of information due to the Pareto optimal databases, because we are confronted with a whole
set of plans instead of only one optimal plan. So we are in a situation unknown to planners
in IMRT planning so far.
In this thesis we evaluate the potentials of a new multiobjective planning system MIRA. In
addition we develop methods to characterize the Pareto optimal databases and to efficiently
gain information that increases the understanding of a case and the planning process in
general.
This thesis is organized as follows. In chapter 2 we give a compact introduction to the basic
underlying mathematics and the inverse planning in IMRT. A basic knowledge in IMRT
treatment planning is assumed and the presentation is kept rather general. For more detailed
and practical sources the reader is referred to Webb (2001); Palta and Mackie (2003); J. Bille
(2002). The three remaining chapters are devoted to multiobjective modelling approaches
and quality measures for efficient sets (chapter 3), the clinical evaluation of the system
(chapter 4) and the extraction of information from Pareto optimal plan databases (chapter
5). The thesis concludes with a summary and an outlook.
3
2 Fundamentals
In this chapter we will first introduce the mathematical basic definitions which are employed
in the following. Then we will develop the concept of Pareto optimality. Afterwards we
give a short overview over standard and multiobjective radiotherapy planning. The chapter
concludes with the presentation of two prototypic multiobjective planning systems.
2.1 Mathematical Basics
In this section we will introduce the fundamental mathematical concepts and the basic notation which will be used throughout the subsequent chapters. In the beginning definitions and
notations are given and a general description of the multiobjective optimization framework
is presented.
One of the fundamental properties which is exploited in mathematical optimization and in
this thesis is convexity. The idea of convexity can apply to sets or functions. The according
definitions are as follows:
Definition 1 (Convexity of a set). Let C ⊂ Rn be a subset of the Euclidian space. C is
convex, if for any x1 , x2 ∈ C we have y = λx1 + (1 − λ)x2 ∈ C, with 0 ≤ λ ≤ 1
and
Definition 2 (Convexity of a function). A function f : dom f ⊂ Rn → R is convex, if it’s
domain dom f is a convex set and for any y, x ∈ dom f and any λ with 0 ≤ λ ≤ 1 we have
f (λx + (1 − λ)y) ≤ λf (x) + (1 − λ)f (y) .
(2.1)
We will also need the term of the convex hull conv(·) of a subset S ⊂ Rn , which describes
the “smallest” convex set containing S.
Definition 3 (Convex hull). Let S ⊂ Rn a subset of the Euclidian space. The convex hull
P
PN
of S is given as conv(S) = { N
i=1 αi · si | si ∈ S, N ∈ N,
i=1 αi = 1, αi ≥ 0}.
4
2 Fundamentals
So the convex hull consists of all finite convex combinations of elements from the subset S.
Another important concept that will be needed in the context of multiobjective optimization
is the one of a cone. A set K is called a cone, if for every x ∈ K and λ ≥ 0 we have λx ∈ K.
If K is also convex it is a convex cone, which means that with x1 , x2 ∈ K and λ1 , λ2 ≥ 0 for
the conic combination we have
λ1 x1 + λ2 x2 ∈ K.
(2.2)
Each proper cone induces a generalized inequality x1 4K x2 ⇔ x1 − x2 ∈ K (Boyd and
Vandenberghe, 2004). So x1 is smaller or equal to x2 with respect to “4K“ if the difference
lies in K. Proper cones will form the basis for the material presented in the following section
2.2.
We are often faced with the situation that we want to choose the ”best“ option out of a set
of alternatives or make the ”best“ decision for a set of policies. In order to put our choice
on a quantitative basis we usually have to model our preferences as a mathematical model
which has to be optimized. A scalar optimization problem has the general form
minimize fs (x)
subject to gi (x) ≤ 0, i = 1, . . . , m
(2.3)
where fs : dom fs ⊂ Rn → R is a scalar objective function which is to be minimized,
x ∈ Rn is the vector of optimization variables we have to choose and gi : Rn → R are
the corresponding constraint functions or just “constraints”. The constraint functions can be
further subdivided into equality and inequality constraints. According to the properties of the
objective and constraint functions an optimization problem can be classified. Well-known
examples would be linear programming (where the objective function and constraints are
linear functions of x), quadratic programming or general non-linear programming. For each
class of optimization problems there exists a variety of (mostly iterative) algorithms. The
classes can considerably differ in their computational complexity. In order to be tractable
many real world applications require a convex formulation.
One important theorem of convex programming states that if fs and gi are convex functions
and if the feasible set X = {x | gi (x) ≤ 0} is convex then all local optima are also global
optima (Bertsekas, 2004).
Considering terminology, each assignment of values to x is called a solution. If it satisfies the
constraints it is called a feasible solution and if the objective function has a local (global)
minimum at x∗ , it is referred to as a local (global) optimal feasible solution. For the sake of
simplicity we will however just use the term solution instead of optimal feasible solution.
2.2 Pareto Optimality
5
Often one is faced with more than one single objective to optimize. In this case a common
method is to include these objectives in one objective function by scalarization. In many
applications with more than one objective a simple scalar model of the problem is not appropriate (Zeleny, 1982). As there is no canonical way to balance the different objectives, we are
rather interested in considering all objectives independently. This leads to a substantially
new class of multiobjective optimization problems. The formulation the multiobjective program (MOP) where all k objectives have to be minimized simultaneously (Miettinen, 1999)
reads
minimize f (x) = (F1 (x), . . . , Fk (x))T
subject to gi (x) ≤ 0,
i = 1, . . . , m
(2.4)
f : dom f ⊂ Rn → Rk is now a vector valued function and each component Fi corresponds to
one objective. How a MOP can be solved and what the implications for the term optimality
are will be treated in the next section.
2.2 Pareto Optimality
In a standard scalar optimization problem each set of optimization variables is evaluated by
a scalar objective function fs : Rn → R. Thus each solution can be characterized by a single
number which reflects the quality of the respective solution. This implies that all solutions
can be compared by means of the standard order relation “≤”. In a multiobjective framework
the corresponding objective function is a vector valued function f : Rn → Rk , hence all
solutions are vectors in Rk . The absence of a total order in Rk makes it necessary to extend
the optimality criterion, because two solutions are not necessarily comparable anymore. This
leads to the definition of Pareto optimality. The concept of Pareto optimality was developed
by Vilfredo Pareto (Pareto, 1896) and first applied to problems in an economical context.
Formally we can define the optimality criterion as follows:
Definition 4 (Pareto optimality). Let y ∈ Y be a point with Y ⊂ Rk . y is Pareto optimal
or efficient if and only if there is no other point y0 ∈ Y with y0 4K y, where 4K is the partial
order relation induced by the proper cone K.
We will also say that a point is not dominated by another one, if it is efficient. The choice of
the cone K will in general affect the set of solutions which are regarded as efficient (Monz,
2006). In this work K will always be the convex cone of the positive orthant K = Rk+ , so
we skip the explicit notation of the cone and just write 4 instead of 4Rk+ . This leads to
a componentwise comparison between two solutions. That means if for all components i of
6
2 Fundamentals
two solutions y 1 and y 2 we have yi1 < yi2 then y 1 dominates y 2 . We can describe the efficient
solutions on the Pareto front also verbally (and a little less precisely):
A solution is Pareto optimal or efficient, if it cannot be improved in one objective without
worsening at least one other objective.
The set of all Pareto optimal points is called the Pareto front or the efficient set. Note that
two solutions are not necessarily comparable anymore. In particular all efficient solutions are
not comparable by the order relation “4“. The whole concept is illustrated in figure 2.1.
Figure 2.1: Efficient solution (green) on the Pareto front (blue) for a bicriterial problem. The
solution in red is dominated by the green one, because it lies in the red cone.
With the so defined extended optimality criterion the goal of an MOP becomes to find
the efficient set of the problem in equation 2.4. However, firstly the number of efficient
solutions is in general infinite, secondly for most application we are not interested in the
whole Pareto front, but rather in a part which includes only relevant trade-offs with respect
to the application. Therefore in a practical context we want to find a finite representation
of the Pareto front, which captures all relevant trade-offs.
2.3 Intensity modulated Radiotherapy
2.3.1 Basic Concept
The basic concept of IMRT can be illustrated by looking at the well known example by
Brahme (Brahme et al., 1982). A two dimensional circular OAR is surrounded by a ringshaped tumor volume (see figure 2.2). If we perform a rotation therapy by rotating a uniform
2.3 Intensity modulated Radiotherapy
7
radiation source around the central axis we can perfectly spare the OAR by a central block.
Due to the geometrical configuration the dose distribution in the target will not be homogeneous.
Figure 2.2: The example by Brahme with a rotating source and a centrally blocked beam. Only
a mudulated beam profile (dashed lines) leads to a homogeneous dose distribution in
the target. Figure from (Brahme et al., 1982).
The key insight is that a homogeneous distribution in the target can be achieved by inhomogeneous beam instensity profile. For simple geometries there exist analytical solutions
for the intensity profiles, for arbitrarily shaped targets the profiles usually have to be found
numerically.
The extension to three dimensions leads to two dimensional intensity maps. These maps are
discretized into the so called beamlets. Each beamlet represents a small subunit of the beam
and can be mudulated individually. Given a set of beams and the corresponding incident directions one searches for the spatially non-uniform intensity maps of all beams which (when
superimposed) lead to the desired dose distribution in the target and an adequate sparing
of the OARs.
Due to the physical reasons of a positive energy deposition and scattering effects IMRT can
obviously not achieve an ’ideal’ dose distribution with a perfect OAR sparing. So we always
have to accept a certain irradiation of the critical structures which has to be balanced against
the goal of a homogeneous target dose.
Note however, that if there would exist a radiation type with a negative dose contribution
we would be able to achieve any desired dose distribution. This fact was shown by Birkhoff
(Birkhoff, 1940), who proved that any black-and-white drawing can be obtained by a superposition of black (pencil) and white (ruler) lines of different thickness.
8
2 Fundamentals
2.3.2 Optimization in Radiotherapy - A General Framework
The optimization process in radiotherapy can be described in a general framework. From an
abstract point of view we can think of the inverse planning as a composition of two mappings,
relating three vector spaces (Kuefer et al., 2005). The first space is the so called parameter
space X, containing the optimization variables. All feasible points X in the parameter space
are then mapped into the design space, which represents the “system design” D resulting from
the chosen parameters. This design is then evaluated by a vector valued function and results
in set of feasible objective vectors F. Each component of one objective vector measures the
quality of one objective.
If we apply this general framework to IMRT we can identify the parameters with the beamlets
(see figure 2.3). This is a simplification because in general we would have to consider all
relevant treatment parameters including beam angles, beam energies, etc. leaving us with a
non convex problem (Bortfeld and Schlegel, 1993). However we will restrict ourselves to this
simplification and assume that we have a convex MOP (so the feasible set X , all objective
Functions Fi and constraints gi are convex). So each point in the parameters space can be
identified with a set of beamlets defining one treatment plan. This point is mapped into the
design space which corresponds to a dose calculation for the specific beamlets. The calculated
dosedistribution is finally evaluated by the respective objective functions and thus mapped
into the objective space. Each single objective function Fi is assigned to one structure of
interest, typically one function per OAR and one or two functions per target structure.
Figure 2.3: The tumor (red) is irradiated with high energetic photons from seven directions.
The irradiation fields with spatially non-uniform intensity maps are superimposed to
achieve the prescribed dose in the tumor and a sufficient sparing of crticial structures.
Our goal is to find all parameter vectors in the parameter space that correspond to efficient
2.3 Intensity modulated Radiotherapy
9
solutions in the objective space. For practical reasons we want to find at least a meaningful
representation of the efficient set from which the planner can then select the best suited
treatment plan for the patient.
Figure 2.4: All feasible solutions in X from the parameter space (left) are mapped to the design
space (middle). The set of efficient solutions is symbolized in red. The solutions in
the design space are evaluated by a vector valued function leading to a representation
in the objective space. Here the efficient solutions are located at the boundary of the
set F of feasible objective vectors.
We are only interested in Pareto optimal solutions, otherwise we could improve a solution
without worsening any other objective. The set of all Pareto optimal solutions constitutes the
Pareto frontier P or efficient set. The efficient set is a subset of the boundary of all feasible
solutions in the objective space. For a convex MOP it is connected (Miettinen, 1999) and
the set F is convex (Romeijn et al., 2004). Each efficient solution can be identified with a
k-dimensional objective vector in the objective space and will be denoted as a solution or
synonymously as a plan.
It is worth noticing that Pareto optimal solutions can be very different and the optimality of
a solution alone does not imply that the corresponding treatment plan is of clinical interest.
Giving no dose at all, thereby sparing all OARs and the normal tissue perfectly but having
no dose in the target volume, might be a Pareto-optimal ”treatment plan”. Therefore one
has to make sure, that the solutions in the objective space are located within a predefined
planning horizon, covering the range of clinical relevance.
2.3.3 Notation and Terminology
For the subsequent sections we will introduce the basic notation. We assume that the patient
geometry and the radiation field are discretized into N voxels and n beamlets. We will
consider a fluence map optimization, where the irradiation geometry is given, thus leaving
us with a convex optimization problem. The vector x ∈ Rn+ denotes all beamlets, and xi is
the intensity (weight) of beamlet i. The dose distribution d ∈ RN
+ can then be calculated
10
2 Fundamentals
×n
with the influence matrix D ∈ RN
via the linear relationship d(x) = D · x, where Dij
+
describes the dose from beamlet j deposited in voxel i under unit fluence (Gustafsson et al.,
1994).
The dose distribution d(x) is evaluated by a vector valued function f (d) ⊂ Rk where k is
the number of independent criteria or objectives, typically one per organ at risk and at least
one per target structure. So each set of beamlets x representing the optimization variables is
mapped from the n-dimensional parameter space X into the k-dimensional objective space
Y.
The fluence map optimization is formulated as a MOP, meaning that all individual criteria
have to be minimized simultaneously subject to the physical constraints xi ≥ 0 enforcing
non-negative fluence amplitudes. So each individual objective function Fi represents the
quality of the corresponding structure for a specific solution. All Fi as well as the feasible
set X with x ∈ X are assumed to be convex. The Pareto front will then be a connected
hypersurface in the objective space.
2.4 Multiobjective Planning Systems
So far there is no clinical usable multiobjective treatment planning system available. Standard treatment planning systems mostly rely on a single scalar objective function, resulting
in one optimal solution. If the solution doesn’t match the planner’s goals, he has to adapt
the objective function (that means either the weight factors representing the relative importance of a structure or the model itself) and start another optimization. Thus, the planning
of complex cases can become a time consuming process. A multiobjective planning system
aims to elegantly overcome these difficulties by providing the planner with a relevant part
of the efficient set. The efficient set can then be interactively explored in real time, helping
to select the best treatment plan for the patient. So the multiobjective treatment planning
system enables the planner to make an informed choice, and can be viewed as a decision
support system.
In the following section we will introduce the multiobjective planning system MIRA (Multiobjective Interactive Radiotherapy Assistant) developed by the ITWM-Kaiserslautern (Küfer
et al., 2006) which is the main system used for the computations. The computations in chapter 5.2 were done with another system called PGEN (Pareto Generator) (Craft et al., 2006)
which is briefly described afterwards.
2.4 Multiobjective Planning Systems
11
2.4.1 MIRA
The Multiobjective Interactive Radiotherapy Assistant MIRA is a pre-clinical multiobjective
treatment planning system consisting of two main parts: First, a nonlinear numeric solver,
which calculates a discrete representation of the Pareto front and stores the solutions in a
plan database. To meet the high computational effort an adaptive clustering strategy is
used, where groups of voxels are combined to clusters which are adaptively refined (Scherrer
et al., 2005). The second part is an interactive navigation tool (Monz et al., 2008) that
allows to explore the calculated plan database through an intuitive user interface. The user
can manipulate the individual objectives by sliders. An efficient interpolation mechanism
between the single solutions allows a real time update of the shown dose distribution and
dose volume histogram information (see figure 2.5). The input data for the MIRA system
is calculated within the clinical software environment of the DKFZ: all influence matrices
are calculated with the research version of the KonRad planning system (Nill, 2001) and the
patient contour data is generated within VIRTUOS (Höss et al., 1995).
The MIRA solver pursues a three phase strategy to generate a representative set of Pareto
optimal solutions:
ˆ In phase one a starting solution is calculated. The user will specify a set of treatment
goals for each planning structure. The solver then aims to satisfy as many goals as
possible to get a balanced starting solution. No fine-tuning is needed for the starting
solution because it’s only purpose is to define a clinically relevant domain within the
objective space. After the initial solution is calculated the planner defines tolerance
limits for each planning structure, thereby excluding possibly efficient but meaningless
solutions. This confines the area of interest on the Pareto front and establishes the
planning horizon of the problem.
ˆ In the second phase the planning horizon is filled with additional solutions. The aim
is to set up the corner posts of the area of interest, i.e. to calculate solutions enclosing
(and filling) the planning horizon. This is achieved by optimizing all subsets of planning
goals, while keeping the remaining objectives under the tolerance levels. As a result
we get 2k − 1 extreme compromises that span the planning horizon.
ˆ During the last phase additional intermediate solutions are calculated to improve the
approximation. For a systematic placement of intermedideate solutions the number
of intermediates would be too large. Therefore a fixed number of intermediates in
calculated and stochastically distributed within the planning horizon.
The efficient solutions are stored in a plan database. The database can be explored afterwards
by a Pareto-Navigation (Monz et al., 2008) to interactively select the best suited treatment
12
2 Fundamentals
plan (see figure 2.5).
Figure 2.5: Graphical user interface of the navigator. Each objective is represented by a slider
(left) and can be manipulated interactively. The corresponding dose distribution and
DVH information is updated in real time.
2.4.2 PGEN
The Pareto Generator PGEN also calculates a discrete representation of the efficient set.
It persues a strategy different from the MIRA system though. On the one hand a linear
modelling approach is chosen which allows for a linear program formulation and the use of
efficient numerical solvers. On the other hand the additional solutions are calculated differently. First a set of k anchor points is calculated for a case with k independent objectives,
by only considering exactly one objective per optimization. Then additional intermediate
solutions are calculated, whereas each solutions aims to improve the inner approximation of
the Pareto front as much as possible. This is done by making use of a sandwich-algorithm
based on lower and upper bounds of the Pareto front. As a price for the intelligent placement
of the additional solutions convex hull computations are used, which currently restricts the
system to problem dimensions of about eight.
13
3 Choosing Models and Parameters
First we investigate different modelling approaches and study the effect of the corresponding
model parameters. Then we will introduce quality indicators for efficient sets, from which
we can draw conclusions about the choice of the number of intermediate solutions.
3.1 Modeling Approaches
3.1.1 Motivation
To set up a computable optimization problem we have to build a model of the tumor and the
OARs, i.e. we must translate our clinical goals into mathematical equations which can be
handled by a numerical solver. In this section we will introduce the different mathematical
models implemented in the MIRA system.
The main open questions are: Which models should be chosen for which cases? Do some
models result in superior clinical treatment plans? What values should be chosen for the
model parameters? How do the solutions change if the model parameters are changed? Or
in other words, what is the sensitivity of the model parameters for the different optimization
problems?
3.1.2 Methods
Two of the considered models are based on the generalized equivalent uniform dose (gEUD),
which is a generalization of the EUD, introduced by Niemierko (1997). The EUD was
initially proposed for the modelling of target structures and then generalized to model targets
and OARs. For an inhomogeneous dose distribution the EUD describes the value of a
homogeneous dose distribution that would lead to the same biological effect as the previous
one. In that sense the gEUD is a biologically motivated objective, instead of pure physical
ones like e.g. the maximum dose of an OAR. The gEUD is given as
14
3 Choosing Models and Parameters
Nj
1 X p 1/p
gEUDp (d) = (
di ) ,
Nj i=1
(3.1)
where Nj is the number of voxels in the volume of interest (VOI) j. The organ parameter p
can be chosen according to the type of the OAR (serial or parallel or inbetween), because the
p-norm has the property that gEUD(d) → dmax for p → ∞ and gEUD(d) → dmean for p → 1,
with the mean dose dmean and the maximum dose dmax of the distribution. When modelling
target volumes with the gEUD the organ parameter is chosen to be negative p ≤ 0.
There are three different models under consideration.
1. In the first model (pq-model) each OAR is modeled by a combination of gEUDs.
FOAR (d) = µ · gEUDp (d) + (1 − µ)gEUDq (d)
(3.2)
with 0 ≤ µ ≤ 1. Mathematically speaking this is just a convex combination of a p- and
a q-norm. For target structures the under- and overdosage are considered separately
and penalized according to the functions
T
Flow
= ( N1j
PNj
p
i=1
max(dTlow − di , 0) T )1/pT
(3.3)
T
Fup
=
( N1j
PNj
i=1
max(di −
q
dTup , 0) T )1/qT
Here dup and dlow are the dose values from where on a violation of the under- or
overdosage is penalized.
2. In the second model (pt-model) each OAR is evaluated by a tail-penalty function
Fup
Nj
1 X
max(di − dup , 0)pt )1/pt
=(
Nj i=1
(3.4)
so only voxels which exceed the bound dup will contribute to the function. The target
is treated in the same way as in the pq-model.
3. The third model (std-model) uses for the OARs the same function class as the pqmodel. For the target however the target mean dose dmean is fixed and simultaneously
the standard deviation of the distribution is minimized:
Fstd
Nj
1 X
(di − dmean )2 )1/2
=(
Nj i=1
(3.5)
3.1 Modeling Approaches
15
Note that the std-model can also be given a biological motivation as the EUD for targets can
be approximated in a first order expansion (of the central moments) as EUD = dmean − 12 ασ 2
(U. Oelfke, 2002) with the radiobiological sensitivity parameter α and the standard deviation
σ.
For each objective upper bounds on the function values can be specified which restrict the
particular objective and ensure a reasonable extent of the planning horizon. In addition for
each OAR j with an gEUD objective one has to set an aspired value Fjasp , which represents the “ideal” function value one wishes to achieve for the corresponding structure. In
the optimization the relative differences of the OAR objectives from the aspired values are
minimized. Note that all functions are convex functions (given p ≥ 1 for all OARs), which
allows for a convex MOP formulation.
One difficulty in the comparison of the results from the different models is that the solutions
lie in different objective spaces which have no natural connection. Furthermore we are
always dealing with a whole set of plans. If we do comparisons of different treatment plans
there is no obvious way to determine which plans should be chosen (remember that efficient
solutions from one model are not comparable by definition). We therefore have to set up a
fixed reference point. So for clinical plan comparisons we will look at plans that offer the
same plan quality for the target and compare the remaining OARs.
Another way of comparing the results of the different models is to reevaluate a whole database
with the other model’s objective functions. That means one efficient set P2 is mapped from
it’s objective space Y2 to another one Y1 . It is obvious that the solutions from P2 will in
general not be efficient in Y1 . However, we can use this fact to quantify the changes in the
solutions induced by the change of the model. As the shape of the whole Pareto front will
change we can use the Hausdorff metric to give a meaning to the distance of the efficient sets
P1 and P2 . As we are always dealing with compact sets (finite sets of points), the Hausdorff
metric is guaranteed to be finite and is given by
dH (X, Y ) = max{ max min d(x, y), max min d(x, y) },
x∈X y∈Y
y∈Y x∈X
(3.6)
where X and Y denote subsets of the Euclidean space and d(·, ·) is chosen to be the ordinary
Euclidean metric. A graphical illustration is given in figure 3.1.
Strictly speaking we are interested in the distance of the efficient sets, but we do not have a
complete representation available, but discrete subsets D1,2 ⊂ P1,2 . If we use the finite sets
D1 and D2 of solutions as a basis for the computation of the Hausdorff metric the result will
depend on the actual placement of the points on the Pareto front. Hence we will use the
first order approximation (cf. section 3.2.2) of convex combinations, which is also used for
16
3 Choosing Models and Parameters
Figure 3.1: Schematic illustration of the Hausdorff distance. Solutions from another model (red)
are evaluated in the objective space. The Hausdorff distance between the first order
approximations is marked with an arrow.
the navigation. So we calculate dH (conv(D1 ), D2 ). The objectives will be scaled to 1 so that
the distances are not dominated by objectives with the highest order of magnitude.
Figure 3.2: The geometries of three cases. A horseshoe phantom case (left) with one OAR and
a horseshoe shaped target volume, a paraspinal tumor (paraspinal I) with one boost
volume (middle) and a small paraspinal case (paraspinal II) (right).
We will consider three different cases: a simple phantom case with a horseshoe shaped target
and a single OAR and two clinical paraspinal cases (see figure 3.2).
3.1.3 Results
We first consider the influence of the target parameters pT and qT which affect the penalization of the target under- and overdosage. Two starting solutions are calculated for the
horseshoe phantom with different settings for pT and qT . As shown in figure 3.3 and 3.4 the
influence of these parameters can differ depending on the actual position on the Pareto front.
In a region with a high target quality as in figure 3.3 the change of the parameters pT and qT
has a rather small influence on the shape of the target DVH, changing the minimum target
3.1 Modeling Approaches
17
dose dTmin by 3.9% and the maximum target dose by 5.1%. Note also that the maximum
OAR dose increases by 7.0% (from left to right) which means that the decrease in the target
quality could still be compensated for by improving the OAR through navigation. However,
if we investigate a different part of the Pareto front with a good OAR sparing (figure 3.4)
the influence of the parameter change on the target quality is stronger. The minimum target
dose decreases by 17.4% (from left to right), while the OAR changes by less than 1% and
the mean dose of the unclassified tissue increases by 1.6%
Figure 3.3: Dose volume histograms of the horseshoe phantom. The parameters pT and qT are
the only parameters which are modified. On the left pT = 1 and qT = 8, setting a
high priority on the target overdosage. On the right we have pT = 8 and qT = 1. The
= 30 Gy.
aspired objective value for the OAR was gEUDasp
8
Figure 3.4: DVHs of two solutions from two different databases that only differ in the setting of
the parameters pT and qT . On the right pT = 8 and qT = 1, setting a high priority
on the target underdosage. On the left we have pT = 1 and qT = 8. Here the aspired
objective value for the OAR was set to gEUDasp
= 10 Gy.
8
The effect of the organ parameter p in the pq-model is shown in figure 3.5. While p is
increasing from 2 to 8 (p = 2, 4, 8 from left to right), the maximum OAR dose decreases
18
3 Choosing Models and Parameters
Figure 3.5: Influence of the organ parameter p on the efficient solutions. All three solutions are
optimized with the pq-model and the organ parameter is varied from p = 2 to p = 8
(p = 2, 4, 8 from left to right). The maximum OAR dose decreases in total by 3.7%
(dOAR
max = 79.5, 77.6, 75.8).
by only 3.6%. Also, the shape of the DVH for the OAR stays nearly the same. The target
minimum and maximum doses as well as the mean dose are not affected to more than 1%.
The parameter with the largest effect on the optimal solution is the aspired objective value
asp
F asp . This fact is shown in figure 3.6 where a change from FOAR
= 10 Gy to 40 Gy results
in a change of maximum OAR dose of more then 40%.
Concerning the parameters for the pt-model the results are rather similar as for the pqmodel. The parameter pt which penalizes the tail of the distribution exceeding a dose value
of dup has only a small effect on the solution characteristic, whereas modifying dup results in
distinctive changes of the dose distribution (see figure 3.7).
The findings for the phantom studies also apply for the clinical cases. Two databases are
calculated with the pq-model for a paraspinal case, the planning horizon for the spine was
not restricted to have a larger extent of the database. The organ parameter for the spine
are p1 = 3 and p2 = 8 with a prescribed dose of 60 Gy for the target and 70 Gy for the
boost volume. In figure 3.8 two solutions from each of the databases are shown. For very
similar target distributions the maximum spine dose changes from 48 Gy to 42 Gy. Although
not clinically relevant, we see that the solutions with the best target coverage both have a
maximum spine dose of 63 Gy.
3.1 Modeling Approaches
19
Figure 3.6: Effect of changing the aspired function value for the OAR. The aspired function value
= 10 Gy (left) 30 Gy (middle) and 40 Gy
for the OAR is varied between gEUDasp
8
(right). The maximum OAR dose changes by more than 40%.
Figure 3.7: The horseshoe phantom is optimized with the pt-model. The first three solutions
correspond to different values of pt (pt = 2, 4, 8 from left to right), while in the
solution on the right dup was changed from 30 Gy to 10 Gy.
20
3 Choosing Models and Parameters
1
1
Spine p=8
Target
Boost
Cauda p=8
Spine p=3
Target
Boost
Cauda p=3
0.8
Rel. volume [%]
0.7
0.6
0.5
Spine p=3
Cauda p=8
0.6
Boost
Target
0.5
0.3
0.2
0.1
0.1
60
Dose [Gy]
80
100
0
0
120
Spine p=8
0.4
0.2
40
Target
0.7
0.3
20
Boost
0.8
0.4
0
0
Cauda p=3
0.9
Rel. volume [%]
0.9
20
40
60
Dose [Gy]
80
100
120
Figure 3.8: Influence of the parameter p for the spine. Two solutions with p = 3 (dashed lines)
and two solutions with p = 8 (solid lines) are shown. While the maximum dose of the
spine changes by 6 Gy on the left hand side there is nearly no difference on the right.
1
1
Stomach p=1
Target
Boost
Stomach p=8
Target
Boost
Stomach p=1
0.9
0.9
Target
Boost
0.8
0.8
Stomach p=8
Target
0.7
0.7
Rel. volume [%]
Rel. volume [%]
Boost
0.6
0.5
0.4
0.6
0.5
0.4
0.3
0.3
0.2
0.2
0.1
0
0
0.1
10
20
30
40
50
Dose [Gy]
60
70
80
90
100
0
0
20
40
60
80
100
Dose [Gy]
Figure 3.9: Influence of the parameter p for the stomach. All other OARs are virtually the same.
3.1 Modeling Approaches
21
To study the effect of the model parameters for a large organ which is not directly involved
in the trade-off with the target we investigate the influence of p for the stomach. As shown
in figure 3.9 the influence is marginal.
Figure 3.10: A paraspinal case is optimized with three different models. One solution from each
database is shown, the pq-model (left), the pt-model (middle) and the std-model
(right).
We now investigate the potentials of the different models. For the paraspinal case II three
databases are calculated with the three different models. The corresponding solutions from
each database show a few differences that are depicted in figure 3.10. The target quality
with regard to under- and overdosage improves from left to right, however, the esophagus
mean dose also increases. The maximum dose of the spine is below 45 Gy for all solutions,
the main differences between the DVH curves of the spine arises in the medium and low
dose regions, where the pq-model has significant lower dose values. The corresponding dose
distributions in figure 3.11 show that the solutions are not substantially different.
The Hausdorff distances dH for the horseshoe phantom and the paraspinal II case are shown
in table 3.1. Exemplarily the distances for all solutions are depicted in figure 3.12 for the
phantom case and the model comparison pt vs pq.
Figure 3.11: Dose distributions for the three optimized with the pq-model (left), the pt-model
(middle) and the std-model (right).
22
3 Choosing Models and Parameters
Table 3.1: Hausdorff distances dH = (conv(D1 ), D2 ) based on the first order approximation of the
efficient sets for the three models. The values in brackets correspond to the values of
maxy∈D2 minz∈conv(D1 ) d(y, z).
std vs pq
std vs pt
pt vs pq
Paraspinal 0.36 (0.11) 0.59 (0.09) 0.79 (0.11)
Phantom 0.64 (0.15) 0.82 (0.18) 0.61 (0.17)
0.7
conv(D1) to D2
0.6
D2 to conv(D1)
Hausdorff distance
0.5
0.4
0.3
0.2
0.1
0
0
20
40
60
80
Solution number
100
120
Figure 3.12: Maximal distances for the solutions from D2 to the convex hull of D1 (blue) and
from the convex hull of D1 to the solutions from D2 (red).
3.1 Modeling Approaches
23
3.1.4 Discussion
Concerning the sensitivity of the model parameters it was found that the influence of the
penalization parameters p, pt and pT is small for most of the cases. This is interesting
because in simple scalar optimization or approximation problems the choice of the penalty
parameters (or rather the norm) can have a significant effect on the optimal solutions and
the residuals. As the influence of p is small, the corresponding influence of µ is also of minor
importance. Although the ability of shaping the DVH curves via the penalization parameters
is limited, the influence depends on the particular solution. If the target has a homogeneous
dose distribution there is not much freedom of further manipulating the remaining planning
structures and the whole dose distribution is rather fixed and the actual value of the other
model parameters is not of great importance. If the target quality is relaxed however, the
effect of the parameters can increase.
The by far largest effect on the solution have the parameters for dup , dTup , dTlow , dmean and
F asp . They allow for large “horizontal” doseshifts in the DVHs and the corresponding dose
distributions are substantially affected. Similar results are reported by Krause et al. (2008)
where a 2d phantom case is studied by means of elasticity theory.
If the model parameters (like the parameter p) have an interpretation in a biological model,
one should obtain their actual values from fits to clinical data. None the less the question
remains what are the sensitivities of these parameters, to be able to answer questions such
as ”What is the effect on the solutions if one uses a “wrong” parameter value?” or ”Is
it important if the parameter value has a large uncertainty?”. Moreover in the pq-model
the parameters p and q are combined. The intention is to be able to shape the solutions
according to one’s own preferences. With this approach however the biological meaning of
the parameters vanishes and they become model parameters one has to choose.
Concerning the different optimization models, no model is found to be superior in general.
So there is no clinical dominance for all plans produced with one model when compared with
another one. Nevertheless there are cases where one model can produce favorable solutions
compared to another model. For example the pt-model better translates a relaxation of
the target quality into an reduction of high dose region for serial OARs near the target.
Whereas the pq-model also for high values of penalty parameters often uses the gained
freedom to reduce the intermediate and low dose regions of the OAR which are of minor
clinical relevance. This means that in the navigation process the high dose regions can lack
controllability. The obvious cure of increasing the parameter p fails, because p is restricted
due to numerical stability issues in the optimization algorithm. So for serial OARs there can
be a slight advantage when using the pt-model. Nevertheless, at some point the pt-modell
will not further improve the OAR, because the low dose regions don’t enter the objective,
24
3 Choosing Models and Parameters
whereas the pq-modell will also “see” the low dose regions and can therefore improve the
OAR still further. These situation will probably arise only for VOIs which are not involved
in the direct trade-off with the target volume.
The std-modell results in solutions similar to the pt-model. As the mean dose is fixed there
is only one quality indicator per target structure, so that under- and overdosage cannot be
controlled separately.
The computations of the Hausdorff distances show that there can occur considerable changes
in efficient sets with the change of models. That is the maximal change in a particular solution
can be pronounced. Although the practical meaning of dH might seem limited we can give
it an interpretation. If we obtain a particular value of dH = v after the change of the models
we get the following information: there are solutions in the second database that can differ
in one objective from the achievable values in the first database by up to v %. That happens
if the entire change occurs in the direction of one objective (an even split to all objective
√
directions would yield a change of v/ k respectively). Note that many solutions can still be
very similar, because we detect only the “maximal“ change in the underlying Pareto fronts.
This is also seen in figure 3.12 where many solutions change by less than 10%. No solutions
has a value of 0 however, which means that one cannot obtain the second solutions by a
convex combination of the first ones.
In general a more flexible modelling framework is desirable, where one can arbitrarily assign
functions to the respective planning structures. To improve the control over specific parts
of a DVH curve one possibility would be to include DVH-constraints, leading to a nonconvex optimization problem and a mixed-integer formulation (Halabi et al., 2006). Another
approach is to use the existing modelling framework and include a search for specific DVHinformation into the navigation process, which is a topic of ongoing research.
3.1.5 Conclusions
We studied the different modeling approaches for objective functions and analyzed the sensitivity of the corresponding model parameters. No modeling approach is found to be superior
in all cases. For serial OARs in a direct trade-off with the target the pt-model can be
advantageous. The penalization parameters in all models have little effect on the optimal
solutions, but the influence depends on the particular location on the Pareto front.
3.2 Quality Indicators for Efficient Sets
25
3.2 Quality Indicators for Efficient Sets
In this section we will introduce indicators which allow to evaluate the quality of an approximation of an efficient set.
3.2.1 Motivation
As the solution of a MOP is not unique one is faced with the problem of finding a good
(discrete) representation of the Pareto front. Since the actual structure of a database is not
known to the planner he doesn’t know if the underlying treatment plans form a good or less
good representation of the Pareto front on which he will base his decision. It is therefore
desirable to have additional information on the quality of the representation. We will define
several quality measures in the subsequent sections.
3.2.2 Methods
First we discuss the concepts to determine the approximation quality of efficient sets.
The number of efficient solutions for a convex MOP is in general infinite and there is no
closed form, describing the efficient set. Consequently we have to consider an approximation
of the Pareto front. Different approaches can be used:
ˆ The most elementary approach is the 0-th order approximation: a finite number of
Pareto optimal points is calculated and used as an approximation of the whole efficient
set.
ˆ The 1-st order methods use piecewise linear approximations. Inner approximations
use convex combinations of single efficient points (Chernykh, 1995) while outer approximations make use of supporting hyperplanes of the efficient set (Fruhwirth et al.,
1989).
ˆ Higher order approximations have been proposed (see Ruzika and Wiecek (2003)), but
will not be further discussed here.
In general the quality can be evaluated using three different criteria (Sayin, 2000):
1. Cardinality. The number of discrete points used to represent the connected Pareto
front is described by the cardinality. If the points are not wisely placed two representations can differ in their cardinality but carrying the same information about the
efficient set. In general we are interested in a small number of discrete points, which
26
3 Choosing Models and Parameters
can be calculated in a reasonable amount of time, without losing too much structural
information of the Pareto front.
2. Uniformity. Uniformity describes the distribution of the points in the objective space.
The points should cover the whole region of interest. Underrepresented areas or gaps
in the distribution resulting in large uncertainties of the Pareto front are undesirable.
Furthermore the distribution should not contain any clusters or overrepresented areas
including redundant points that don’t contribute to the information about the Pareto
front.
3. Coverage. For each point on the Pareto front there should be an element in the
approximation which represents this point reasonably well. So all areas of the Pareto
front have to be adequately covered, keeping the error introduced by the approximation
small.
These three issues are addressed in the the following sections.
Cardinality and Hypervolume Indicator
We will first discuss the cardinality, to address the question how many solutions have to
be calculated to obtain a good approximation of the Pareto front. As the Pareto front is
a hypersurface in the objective space the number of solutions needed for an appropriate
coverage will in general increase exponentially (Küfer et al., 2000) with the dimension k of
the problem. However if the shape of the Pareto front locally resembles a plane then a 1-st
order approximation with convex combinations of single solutions will yield a satisfactory
description. Hence less solutions would be needed. Large errors will only occur in areas with
a high “curvature”, that is where the tangent plane doesn’t resemble the Pareto front well.
The number of intermediate solutions to be calculated can be controlled directly within the
MIRA system. To address the question how many solutions have to be calculated to obtain
an adequate approximation of the Pareto front, we propose a measure based on convex hull
volumina. It makes use of the fact that every improvement in the approximation involves
an increase of the convex hull volume of all discrete points. Moreover an additional point
which adds no significant information (e.g. it is too close to an existing point or it is located
in a plane region of the Pareto front) will not contribute to the measure.
Definition 5 (Utopia point). The utopia point f ∗ ∈ Rk is defined as the vector of the global
minima Fi∗ of all individual objective functions Fi , so f ∗ = (F1∗ , ..., Fk∗ )T .
For a nontrivial multiobjective optimization problem the utopia point will (as the name
implies) never be reached.
3.2 Quality Indicators for Efficient Sets
27
Given a set of Pareto optimal points in the objective space we will define the hypervolume
indicator as follows:
Definition 6 (Hypervolume indicator). Let D be a set of discrete Pareto optimal points in
Rk and Di ⊂ D with |Di | ≥ k + 1 and f ∗ the utopia point. The hypervolume indicator Si is
defined as
vol(conv(Di ))
(3.7)
Si =
vol(conv(D ∪ f ∗ )
where conv(·) is the convex hull and vol(·) denotes the k-dimensional hypervolume.
Figure 3.13: Schematic picture of the hypervolume indicator.
The indicator is illustrated in figure 3.13. It allows to examine the improvement of the
approximation as a function of increasing number of solutions as follows: First we will set
D0 to be the set consisting of all extreme compromises, yielding a value for S0 . Then each
intermediate solution yi is successively added to Di−1 resulting in a sequence S0 , S1 , ..., SM
where M is the number of intermediate solutions in the database. So we have a monotone
increasing sequence of indicator values. Each indicator increase corresponds to an improvement of the discrete approximation, whereas a non-varying step means, that the additional
solution doesn’t add new information about the Pareto front. The range of the indicator is
between 0 and 1, whereas a value of zero corresponds to a flat Pareto front. We also note
that Si is shift invariant, in a sense that the absolute value of Si only depends on the shape
of the Pareto front and not on it’s location in the objective space. If the approximation can’t
be further improved then there won’t be a significant increase in Si anymore, resulting in a
saturation. Hence we have a necessary condition for an adequate number of solutions.
28
3 Choosing Models and Parameters
Uniformity and Hyperspheres
In this this section we will address the question of uniformity, as we are interested in a
uniform distribution of solutions in the objective space we have to be able to detect clusters
and gaps within the distribution.
Guarantying an evenly distributed set of solutions in higher dimensions is a non-trivial task.
For low dimensional problems methods have been proposed (Das, 1999; Messac et al., 2003),
but they cannot be easily extended to higher dimensions. Therefore the MIRA system uses a
stochastic mechanism for calculating intermediate solutions (Küfer et al., 2003) to approach
a satisfactory approximation also in higher dimensions.
As we are interested in a uniform distribution we have to be able to detect clusters and gaps.
Clusters will appear if there is no restriction to the minimal distance between two points so
that there are solutions that are too close together. To provide an insight into the spread of
the solutions for each solution the Euclidean distance to the nearest neighbour is calculated.
An alternative formulation would be: for each solution yi find the smallest hypersphere that
touches yi and exactly one other solution (implying that there are no solutions within the
interior of the hypersphere). The distribution of the radii of smallest hyperspheres enables
us to detect clusters in the distribution.
The exact value of the nearest neighbour distance for an ideal uniform distribution depends
on M and k. We can try to find a measure that is independent of M and k. For an even
√
distribution of M points on a k-dimensional unit-cube it is proportional to ( k M − 1)−1
(when M = mk q
with m ∈ N). Furthermore the volume of a standard n-simplex αn is given
n
as vol(αn ) = sn! n+1
(Sommerville, 1958), where s is the edge length of the simplex. For
2n
the standard n-simplex in Rn+1 defined be the vertices {e1 , . . . , en+1 } of the unit vectors ei
√
we have s = 2. So, assuming a flat Pareto front with M evenly distributed solutions as a
reference, we will calculate as a characteristic number the following:
Hs = d¯s · (
√
k−1
(k − 1)!
M − 1) · { √
·
2(k−1)
s
√
2(k−1)
(k − 1)! 1/k
k−1
}1/k = d¯s · (
M − 1) · { √
}
(3.8)
(k − 1) + 1
k
d¯s is the mean of the nearest neighbour values. For an ideal uniform distribution on a flat
Pareto front Hs will be 1. If the solutions tend to cluster Hs will decrease and eventually
approach 0.
Note that if there are some solutions having a significantly higher value in the nearest neighbour distance, we are also able to identify gaps in the discrete representation. However
an even nearest neighbour statistic is not sufficient to guarantee that there are no gaps in
3.2 Quality Indicators for Efficient Sets
29
the solution distribution (cf. the left distribution in figure 3.15 where one can remove any
solution and the nearest neighbour distribution will remain constant).
1
1
Solutions
Pareto front
Approximation
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0
0.2
0.4
0.6
0.8
Solutions
Pareto front
Approximation
1
0
0.2
0.4
0.6
0.8
1
Figure 3.14: Example of two dimensional efficient sets. On the left successive solutions (blue)
have the same arc length. The inner approximation (red) is close to the Pareto
front (black). On the right hand side the random placement of the same number
of solutions can result in large gaps between the approximation and the efficient
set. The corresponding indicator values for Hs are Hs = 1.09 (left) and Hs = 0.33
(right).
1.2
1.2
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
−0.2
−0.2
0
0.2
0.4
0.6
0.8
1
−0.2
1.2 −0.2
0
0.2
0.4
0.6
0.8
1
1.2
Figure 3.15: For the solutions above the largest hyperspheres are calculated. The gap in the distribution on the right results in a large hypersphere. For the hypersphere indicators
we have Hl = 0.00 (left) and Hl = 1.38 (right).
To detect all gaps another indicator is needed. Similar to the indicator concept above for
each solution yi we will now look at the largest hypersphere touching yi and exactly one
other solution, while having no other solutions in the interior of the hypersphere (cf. figure
3.16). We can set up the following discrete optimization problem to find the corresponding
hyperspheres. Let the cij denote the distances between solutions yi and yj , and yijm =
(yi + 12 (yj − yi )) = 12 (yi + yj ) the middle point of a hypersphere connecting yi and yj , then we
30
3 Choosing Models and Parameters
Figure 3.16: Example of a two dimensional Pareto front in three dimensions. Hyperspheres for
three arbitrary solutions are shown. The even distribution results in very similar
spheres (left) with Hs = 1.15 and Hl = 0.03 while for the stochastically placed
solutions the size of the radii can differ (Hs = 0.65 and Hl = 0.53).
can introduce the binary decision variables xij ∈ {0, 1} indicating if there is a hypersphere
placed at yijm . For each i we then solve
maximize
f (xij ) =
subject to
P
P
j
cij · xij
xij = 1
(select one hypersphere)
xjj = 0
∀j
(non degenerated)
2
1
m 2
(yl − yij ) ≥ ( 2 cij ) · xij ∀j, l (empty interior)
xij ∈ {0, 1}
(binary).
j
(3.9)
Note that both the objective function and all constraints are linear in the decision variables,
leading to a simple integer program.
To characterize the distribution of the hypersphere radii we calculate
Hl =
σl
.
d̄l
(3.10)
also known as the coefficient of variation. d̄l is the mean of the radii of the hyperspheres and
σl is the corresponding standard deviation. Hl will approach zero for an even distribution.
Any gaps will increase Hl (as long as d̄l is not too close to zero) and separated solutions can
be identified.
3.2 Quality Indicators for Efficient Sets
31
Only the approach based on the largest hyperspheres guarantees to find all underrepresented
areas on the Pareto front.
Coverage error
An important information about the approximation quality is the coverage error. It measures
the distance between the efficient set P and the approximation D. Commonly the measure
is based on the maximum orthogonal line segment from the approximation to P (cf ref
covError). In general we can define the coverage error ε as:
ε = max min d(y, z).
y∈P z∈D
(3.11)
Here, d(·, ·) denotes the Euclidean metric, P is the efficient set and D denotes the approximation. This definition doesn’t depend on the order of the approximation. In the subsequent
section though, we will assume the 1-st order approximation based on convex combinations
of single efficient solutions, as used with the MIRA system.
As the Pareto front is in general unknown, we cannot calculate ε directly, but have to consider
an upper bound on the coverage error.
We will now describe a procedure to calculate an upper bound on the coverage error for a
given set of discrete efficient points:
1. Triangulation. First we will calculate convex hull of all points, resulting in a polyhedron
consisting of a set of facets and the corresponding normal vectors. All normal vectors
have to be aligned in the same way, pointing outwards (or inwards) the convex hull.
2. Determine the efficient facets. The triangulation will contain facets that are dominated
by others and therefore don’t belong to the approximation of the efficient set. We will
reject all facets whose normal vector has positive components because they are in
general not efficient (see proof 1 below). In this way we can determine all efficient
1
Proof: Suppose we have an efficient facet with at least one positive component in the corresponding normal
vector. Without loss of generality we assume all normal vectors n are adjusted that they point outwards
the convex hull. For each facet with the vertices yjvertex we look at a point on the facet, for example the
Pk
middle point given as ymid = j=1 k1 yjvertex . A direction s of a step that stays on the facet is determined
by the condition nf acet · s = 0. If all components of nf acet are negative then a non-trivial direction
will include positive and negative components to fullfill the above condition, thereby conform to the
requirement Pareto optimality of the facet. If on the other hand there is a normal vector which has at
least one strictly positive component (nf acet )j > 0 , then we can choose a direction s̃ with (s̃)j < 0 and
(s̃)l < 0 (for a suited index l) and zero for all other components which satisfies the orthogonal condition
above. As we are in the middle of the facet there is a t > 0 (t small enough) for which ymid + t · s is still
on the facet. Thus we found a step that stays on the facet and improves two objective functions without
worsening another one, contradicting the assumption of Pareto optimality. 32
3 Choosing Models and Parameters
facets of the triangulation.
3. Identify an upper bound. The utopia point of each facet and the facet itself define a
simplex. Due to the convexity of the MOP this simplex consists of upper and lower
approximations of the Pareto front, so the front will entirely be contained in the simplex. The simplex has an analytical description, consequently we are able to calculate
an upper bound of ε.
4. Restrict the bounds. As the lower bound introduced by the utopia point of each facet is
a very rough estimate, we can narrow the lower bounds by neighbouring hyperplanes.
Once again we can exploit the convexity of the MOP. The simplex can be restricted
by the cut of neighbouring hyperplanes defined by adjacent facets, whereas adjacent
facets are facets which share at least one vertex with the current facet.
5. Calculate an upper bound. We can now set up a simple linear program to calculate an
upper bound of ε for each facet. The program reads
minimize
f (y) = −nTf · y
subject to I · y f ∗
nTf · y ≤ bf
A · y br
(dominated by utopia point)
(bounded by inner approximation)
(restrictions)
(3.12)
Here I denotes the k × k identity matrix, y is the variable point within the simplex. nf
describes the normal vector of the current facet and bf is the corresponding scalar which
stands for the distance to the origin of the hyperplane defined by the facet. The rows of the
matrix A contain the normal vectors of the neighbouring hyperplanes used to restrict the
simplex and the components of the vector br are the corresponding distances to the origin of
these hyperplanes. The linear program is easily solved with the LP-solver GLPK (Makhorin,
2009).
3.2.3 Results
We consider a phantom case with two OARs one target volume and the unclassified tissue (cf.
chapter 4 figure 4.5) which is optimized with the pq-model, resulting in five objectives. The
hypervolume indicator in figure 3.18 shows a large jump after solution 200, which means that
a solution adding substantial information on the Pareto front is found. The upper bound on
the relative coverage error is ≈ 12% while the mean coverage error is ≈ 0.1%. Figure 3.19
shows the corresponding distributions for the nearest neighbour and the hyperspheres.
Another database for the same phantom case with 800 solutions results in a steady curve for
3.2 Quality Indicators for Efficient Sets
33
Figure 3.17: The simplex (bluish) formed by a facet and the utopia point for that facet is restricted
by the hyperplane (red) defined by a neighbouring facet.
0.7
0.14
0.6
0.12
0.5
0.1
Rel. coverage error
Rel. Hypervolume
the hypervolume indicator (figure 3.20). Here the main shape of the Pareto front is already
captured at the beginning and the approximation gradually improves.
The corresponding coverage error for the database is ≈ 11% and the mean coverage error is
≈ 0.8%. Figure 3.21 shows the hypervolume indicator of the phantom case for two databases
with 800 and 1200 solutions. After 600 solutions another plan that extends the approximation
is found for the second database.
0.4
0.3
0.08
0.06
0.2
0.04
0.1
0.02
0
0
100
200
300
400
Solution Nr.
500
600
700
Target
OAR1
Skin
OAR2
Target upper
0
0
200
400
600
No. of facet
800
1000
Figure 3.18: The hypervolume indicator for the phantom with two OARs and a database of 615
solutions (left). The upper bound of the coverage error for the database is shown on
the right.
We consider the results for a clinical case are shown for a prostate case optimized with the
pq-model resulting in a eight-dimensional trade-off. The hypervolume indicator and the
corresponding bound on the coverage error is shown in figure 3.22. The database consists of
112 solutions and a value of over 0.7 for Si is already reached at the solution i = 38. The
corresponding for the database is ≈ 19% with a mean coverage error of ≈ 1.8%.
3 Choosing Models and Parameters
0.35
0.7
0.3
0.6
Diameter of largest Hypersphere
Euclidean distance to nearest neighbour
34
0.25
0.2
0.15
0.1
0.05
0.5
0.4
0.3
0.2
0.1
0
0
100
200
300
400
Solution Nr.
500
600
0
0
700
100
200
300
400
No. of Solution
500
600
700
Figure 3.19: Distributions of the nearest neighbour distances (left) and the largest hyperspheres
(right) for the phantom case. The corresponding indicator values are Hs = 0.69 and
Hl = 1.25.
0.12
0.8
Target
OAR1
Skin
OAR2
Target upper
0.7
0.1
Rel. coverage error
Rel. Hypervolume
0.6
0.5
0.4
0.3
0.08
0.06
0.04
0.2
0.02
0.1
0
0
100
200
300
400
500
Solution Nr.
600
700
800
0
0
900
100
200
300
400
No. of facet
500
600
700
0.8
0.8
0.7
0.7
0.6
0.6
Rel. Hypervolume
Rel. Hypervolume
Figure 3.20: Hypervolume indicator of the phantom case with a database of 800 solutions. The
upper bound of the coverage error for the database is shown on the right.
0.5
0.4
0.3
0.5
0.4
0.3
0.2
0.2
0.1
0.1
0
0
100
200
300
400
500
Solution Nr.
600
700
800
900
0
0
200
400
600
800
Solution Nr.
1000
1200
1400
Figure 3.21: Hypervolume indicator for two different databases with 800 and 1200 solutions for
the same phantom case.
3.2 Quality Indicators for Efficient Sets
35
0.9
GTV
0.25
CTV
0.8
PTV
Rectum
0.2
0.7
Rel. coverage error
Rel. Hypervolume
Bladder
0.6
0.5
0.4
0.3
0.2
GTV up
0.15
CTV up
PTV up
0.1
0.05
0.1
0
0
20
40
60
Solution Nr.
80
100
0
0
120
20
40
60
80
100
No. of facet
0.045
0.2
0.04
0.18
0.035
Diameter of largest Hypersphere
Euclidean distance to nearest neighbour
Figure 3.22: Hypervolume indicator (left) and the upper bounds on the coverage error (right) for
a prostate database.
0.03
0.025
0.02
0.015
0.01
0.005
0
0
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
20
40
60
Solution Nr.
80
100
120
0
0
20
40
60
No. of Solution
80
100
120
Figure 3.23: For a prostate case with a 110 solutions the nearest neighbour distribution of the
solutions (left, Hs = 0.59) and the distribution of the largest hyperspheres (right,
Hl = 1.07) are shown.
36
3 Choosing Models and Parameters
3.2.4 Discussion
The introduced measure Si allows for an analysis of the appropriate number of solutions
for a given database. It was found that the number of solutions needed for an appropriate
approximation (regarding Si ) can differ from case to case. It also became evident that
solutions adding substantial information to the Pareto front can be found in a late phase of
the database calculation (cf. figure 3.21). Such a situation would also occur if suddenly a
new solution outside the initial planning horizon is produced. For the case in figure 3.21 the
jump could be anticipated because Si only had a value of ≤ 0.5 and another database for
the same case had the value of Si ≥ 0.7. Although Si is of course case dependent and related
to the curvature of the Pareto front, in most analyzed cases we find a value of Si ≈ 0.7 or
above.
We also see due to the non-varying steps that many solutions in the later calculation phase
don’t add new information. Thus a lot of computing time goes into the calculation of
solutions which turn out to be not relevant for the approximation. This is in contrast to the
PGEN algorithm which tries to steadily improve the approximation based on convex hull
information. But this property of the MIRA system comes as a price one has to pay for the
stochastical placement of intermediate solutions, that on the other hand allows to handle
high dimensional cases. Although the computation time is not a crucial limiting factor at
the moment there is still a potential to speed up the database generation.
As the computation of Si involves convex hull calculations it can’t be applied to arbitrary
dimensions but is restricted to dimensions up to about ten. However a large class of cases is
covered within this dimensional range and more complex cases can always be simplified by
grouping some individual structures.
For many studied clinical cases the approximation is improving fast in the beginning phase
of the database generation. That means a moderate number of solutions (in the order of
100) can reproduce the main characteristic of the Pareto front also for high dimensional
cases. In the later phase however the approximation improves at a much slower rate so that
many solutions are needed (several hundred for high dimensional cases) to achieve a reliable
representation of the Pareto front.
The introduced hypersphere-based methods allow for a characterization of the distribution
of all Pareto optimal points. They show general underlying characteristics for the studied
cases.
Firstly, the solutions tend to cluster, which can be seen by the large fraction of solutions with
a low nearest neighbour value and which is confirmed in the largest hypersphere distribution,
where also a large fraction of solutions shows significant lower values in the hypersphere radii
(although they try they cannot grow). Note that if there are kinks in the Pareto front (which
can happen), then these points will be found prominently by weighted scalarization methods
3.2 Quality Indicators for Efficient Sets
37
with several different settings of weight factors, hence many solutions will be located at these
kinks.
Secondly, there are often isolated solutions that can be identified from the nearest neighbour
statistic. These turn out to be the extreme compromises which is also apparent in the
navigation process where there can suddenly occur large jumps towards the extremal value
of an objective during the navigation.
Thirdly, there are large gaps as seen from the maximal values of the hypersphere diameters
that are much higher (up to a factor of 5) than the maximum nearest neighbour values.
The indicators Hs and Hl summarize these findings in a single number. Hs is never close to 1
and Hl is never close to 0. Although the exact values of the indicators lack of interpretability
they allow for a quick description of the distribution characteristics and a comparison of
different databases.
The calculation of upper bounds for the coverage error yields a direct statement of the
approximation quality. The mean upper bound on the coverage error for each structure
tends to zero for low dimensional (k = 3 to 5) cases. Single facets however can still exhibit
a large error. For high dimensional (k = 7 and above) cases with a moderate number of
solutions there can occur still large values for the coverage error, but we have to keep in
mind that these are values for single facets which might never be visited in the navigation
and moreover it is an upper bound so the actual error will always be smaller.
3.2.5 Conclusion
The presented quality measures allow for a detailed characterization of a Pareto optimal set
of plans. They provide an insight into the approximation quality of the Pareto front. This
additional information can detect weaknesses of the discrete approximation which would
otherwise not be available in the calculation and navigation process, thus offering a quality
assurance of the output of the MIRA system. Furthermore an appropriate choice of the
number of intermediate solutions can be derived. Finally different databases even with
different objective functions can be compared.
39
4 Clinical Application
In this chapter we evaluate the practical potentials of the planning system. Section 4.1
describes a way to locate a clinical reference plan from another planning system within
a Pareto optimal database. In section 4.2 we perform retrospective planning studies and
illuminate the practical benefits and limitations for the system.
4.1 Automatic Localization of a Reference Plan
4.1.1 Motivation
In a multiobjective treatment planning framework we are confronted with a whole set of
efficient solutions instead of one single optimal solution in conventional planning. A representative discrete subset of the efficient set is stored in a database and is presented to the
treatment planner who then chooses the best suited compromise. This raises the question
how a conventionally obtained reference plan from an accredited planning system relates to
the solutions in the database of the new multiobjective system. In general it is important
that the solutions in the database are situated within a clinically relevant domain, preferably
around the reference plan, which represents a trusted option. Therefore it is beneficial to
have a quick confirmation if a reference plan is contained in the database. In this section we
describe how to automatically identify a reference plan within a database of Pareto optimal
solutions.
4.1.2 Material and Methods
Solving the MOP results in a database of Pareto optimal treatment plans with 2k − 1 + M
solutions. In addition we have a clinically reference plan from another planning system,
which was conventionally optimized with different objectives by a planner. To relate the
reference plan to the database we can proceed as follows. For each reference plan that is
characterized by its dose distribution there exists a representation in the objective space
40
4 Clinical Application
with respect to the set of objective functions Fi . The coordinates in the objective space are
given by the values of the individual objective functions. The question if a reference plan is
“contained” in a stored database can also be put in the following way: Is it possible to find
a convex combination of solutions from the database that approximates the reference plan
reasonably well? As the combination lies within the convex hull, feasibility of the new plan
is ensured. Formally we can set up the constrained optimization problem:
minimize
f (λ) = k
subject to
P
P
i
λi = 1
λi ≥ 0
λi · yi − rk2
(4.1)
i=1
Where λi are interpolation coefficients, yi are the coordinate vectors of the solutions in the
database and r is the coordinate vector of the reference plan and the norm denotes the
Euclidean norm. The constraints make sure that the combination stays within the convex
hull. If the reference plan is contained in the convex hull then the minimal value of 4.1 is
zero. Equation 4.1 is easily transformed into an Quadratic Program in standard form and
can then be solved by any Quadratic-Program-solver.
For a clinical test case of a paraspinal tumor (see figure 4.1) a discrete representation of the
Pareto front was calculated with the MIRA software and stored in a database. Structures
considered in the optimization were the spinal cord, the esophagus and the target using
the pq-model, consequently there are four independent evaluation criteria and the objective
space is four dimensional. For this case a corresponding plan was optimized by the clinical
routine version of KonRad, which uses a common quadratic objective function, and chosen
as the reference plan. Afterwards the KonRad plan was evaluated by the multiobjective
functions and thus mapped into the objective space. The obtained coordinate vector of the
reference plan was then used to solve the Quadratic Program of equation 4.1.
4.1.3 Results
A database of 27 Pareto optimal plans was generated and stored. The dose volume histograms of the different solutions which illustrate the different trade-offs are depicted in
figure 4.1.
The Quadratic Program for determining the interpolation coefficients terminated with a value
of zero, indicating that the reference plan is representable through a convex combination of
the solutions in the database. Five interpolation coefficients were found to be non zero (cf.
table 1) so only five solutions were necessary to generate the reference plan.
4.1 Automatic Localization of a Reference Plan
41
Figure 4.1: Geometry of the vois for the paraspinal case (left) and the DVHs from the Pareto
optimal database.
Table 4.1: Values of the interpolation coefficients and the corresponding solution numbers after
solving the quadratic program.
Value of interpolation coefficient λi
Number of solution
0.016
1
0.001 0.702
2
10
0.052 0.229
14
25
The reference plan and the solution generated with the interpolation coefficients were compared in VIRTUOS. The corresponding DVHs are shown in figure 4.2. Although the reference
plan and the solutions in the database were optimized with different optimization models, the
resulting DVHs were found to be very similar, indicating that the clinically relevant domain
is covered by the database. The convex combination of solutions yields a good approximation
of the reference plan.
4.1.4 Discussion and Conclusion
Although the method is based on objective values it is capable to generate new solutions
which are very similar to the reference plan, provided it is contained in the database. For
simple cases we are mostly able to generate a convex combination of solutions which resemble
the reference plan. If the reference plan is rather different the method will return a non-zero
value. This happens if either the planning horizon of the database is not well placed and
the clinically relevant domain is not covered, or if due to the different optimization models
the solutions are so different that they will not match. In the first case we can use this as
an indicator that the planning horizon should be extended and the database should be filled
with additional solutions. In the second case one could alternatively search in the design
space instead of the objective space (e.g. by considering a Chebichev problem which can be
42
4 Clinical Application
Figure 4.2: The KonRad reference plan compared to the the convex combination of five solution
from the database (dashed lines).
formulated as a linear program).
The method can be used to analyze the quality and the benefits of the MIRA system via
clinical evaluation. It can assist the quality assurance for the system and allows for a quick
identification if a reference plan is contained in a database or if an enhancement of the
planning horizon is needed.
4.2 Clinical Evaluation
43
4.2 Clinical Evaluation
4.2.1 Motivation
The objective of the studies is to evaluate the clinical potential of the MIRA system and to
compare the plan quality with the clinically approved KonRad planning system. On the one
hand one has to assure that the MIRA systems meets the KonRad standards. On the other
hand one also has to verify that cases conventionally planned with the KonRad system are
competitive to the multiobjective planning system which takes into account all objectives
simultaneously.
4.2.2 Methods
Pareto optimal plan databases for three different cases are calculated with the MIRA system: a phantom case, a paraspinal case and different prostate cases. They are navigated
and compared with a KonRad treatment plan which was conventionally optimized with a
quadratic objective function. The navigation process is done within VIRTUOS which offers
an interface to incorporate the navigation principle from MIRA (Tesarczyk, 2006). For the
prostate cases we also consider a larger retrospective study with ten patients. Besides the
plan quality we examine the planning efficiency, i.e. the overall planning time the planer has
to spend to arrive at the actual treatment plan. For IMRT of prostate cancer the macroscopic prostate is defined as GTV, a 5 mm margin excluding bladder and rectum is added to
obtain the CTV, and another 5 mm are added for the PTV. A total median dose of 78 Gy
is delivered in 2 Gy fractions to the GTV with 6 MeV photons using a setup of 7 coplanar,
equidistant beams.
4.2.3 Results
The phantom case is optimized with the pt-model and compared to a reference plan from
KonRad. We find that the plan quality for both planning systems is rather similar. The
DVHs for three different solutions from the database are shown in figure 4.4. The different
treatment options available are easily accessible with the MIRA system, e.g. both OARs can
be traded independently. Also the cost of the important trade-offs (e.g. target quality vs.
OAR maximum doses or countour mean dose) can be explored interactively.
As shown in figure 4.6 (left) the plan quality for the pt-model is similar to KonRad. For
another plan of the same phantom optimized with the std-model we can find solutions in the
44
4 Clinical Application
Figure 4.3: Incorporation of the navigation principle into the routine planning system VIRTUOS
at the DKFZ.
database which are slightly superior to KonRad, both maximum doses of the OARs as well
as the mean dose for the contour are smaller, while the target quality is virtually identical.
Figure 4.4: Three different trade-offs from a database compared to a KonRad plan. On the left
the sparing of both OARs is achieved by a worse target coverage. In the middle the
coverage is improved and both OARs are inferior in return. On the right both OARs
are decoupled and can be traded independently.
For the prostate case the plans are similar for both systems, again different trade-off options
are available with the MIRA system (see figure 4.7).
In the planning study with ten prostate cases, MIRA could process the clinical data sets
without further modification. Defining the optimization parameters so that the resulting
database really included the clinical relevant part of the solution space took approx. 1 hour
for the first patient. For all 9 remaining patients the same parameters could be used as a
template. The unattended database generation took approx. 3 hours per patient (between
100 and 150 IMRT plans, voxel resolution 2.6 mm3 , bixel size 1 cm2 ). The best IMRT plan
was found by interactive navigation in approx. 5 minutes. Realtime interpolation of plans
4.2 Clinical Evaluation
45
Figure 4.5: Dosedistributions of a KonRad plan (left) and a plan from the database (right) corresponding to the left DVH in figure 4.4.
Figure 4.6: The plan from a database optimized with the pt-model is similar to the KonRad plan
(left). In a database optimized with the std-model a clinically slightly superior plan
compared to KonRad can be found.
46
4 Clinical Application
ensured that the navigation was smooth and that far more than the 100-150 precalculated
plans were accessible. Within the approx. 5 mins, the planner could also experience the
sensitivity to changes of the plan, e.g. how much a certain improvement to the GTV coverage
increased the dose to the rectum. In KonRad the total planning time was 30-60 minutes per
patient during which the planner was occupied, and the planning process was not interactive
because each single optimization run took 1-3 minutes. (Considering the uncertainty due to
different dose calculation algorithms, the plan quality eventually achieved by KonRad and
MIRA was virtually identical for each of the 10 patients. One example is shown in figure
4.7.) The results are summarized in tables 4.2 and 4.3.
Table 4.2: Results for the prostate planning study. The range covered by the database is shown
and the values of the corresponding clinical reference plan are reported in brackets.
case
Max. rectum dose [Gy]
Rectum volume > 68 Gy [%]
Mean bladder dose [Gy]
Max. bladder dose [Gy]
GTV volume < 70.2 Gy [%]
1
2
62-94 (77) 64-72 (78)
0-15 (2)
0-1 (3)
78-86 (80) 67-82 (78)
27-59 (42) 17-24 (24)
0-13 (3)
0-7 (1)
3
4
5
73-76 (75) 69-79 (77) 70-89 (79)
0-2 (2)
2-39 (3)
1-20 (2)
77-80 (77) 81-85 (80) 77-93 (77)
18-30 (36) 17-31 (23) 26-52 (42)
0-8 (2)
0-9 (1)
0-2 (2)
Table 4.3: Results for the prostate planning study.
case
Max. rectum dose [Gy]
Rectum volume > 68 Gy [%]
Mean bladder dose [Gy]
Max. bladder dose [Gy]
GTV volume < 70.2 Gy [%]
6
7
68-90 (76) 68-76 (73)
0-12 (2)
0-4 (1)
89-91 (79) 78-94 (82)
33-51 (44) 18-32 (33)
0-7 (2)
0-8 (3)
8
9
10
70-110 (83) 57-79 (77) 69-75 (72)
0-22 (3)
0-3 (2)
0-5 (1)
87-93 (85) 78-81 (81) 79-91 (83)
18-28 (26) 12-28 (29) 22-35 (30)
0-11 (3)
1-17 (1)
0-9 (2)
For a paraspinal case (cf. chapter 3 figure 3.2) we find a MIRA database plan that fails to
match the target quality in both target structures. This indicates two different problems.
Firstly, the selection of the planning horizon. As the values for the planning horizon have
to be specified in the form of objective values they are not necessarily intuitive, that means
it can be difficult to guarantee certain desirable solutions in the database as it is difficult to
relate objective values to DVH-shapes. Secondly, the normalization problem plays a role, as
all plans are normalized to the median target dose, which introduces a perturbation of the
efficient set. In the worst case an improvement through the navigation slider will not result
in a real improvement for that plan (see discussion of this chapter).
4.2 Clinical Evaluation
47
Figure 4.7: DVHs for a prostate case. Two solutions from a database are compared to the KonRad
plan.
Figure 4.8: For a paraspinal case a clinical inferior plan can be found in the database (left). Both
OARs exibit larger maximum doses and the coverage of the boost volume is inferior.
For a prostate case there can occur hot spots (right) in the normal tissue.
48
4 Clinical Application
4.2.4 Discussion
The studies of the phantom cases showed that the MIRA system is able to produce treatment plans which are competitive with the clinical reference system. In some cases there can
even be a slight superiority of the MIRA system. This situation can occur if the optimization algorithms use different stopping criteria, as the Gradient- or Newton-based descent
algorithms are not run to optimality and the stopping criteria are always arbitrary.
In the prostate planning study it could be demonstrated that MIRA is able to reduce the time
for the treatment planning process when many cases are considered. After finding a starting
configuration of the optimization parameters it was possible to calculate all subsequent plans
with the same parameter setting as a template. So all databases then where located in a
suitable part of the Pareto front and could be generated without further user interaction
leading to a significant reduction of planning time. However, the initial parameter setting
has to be found manually, because the values of the parameters (like the aspired objective
values and the planning horizon) can be non intuitive. E.g. to guarantee that solutions
with an gEUD value of 40 Gy for an OAR are contained in a database the corresponding
parameter for the aspired value could be 10 Gy. As the sensitivity of the solution with respect
to the parameters is not constant the search for a suitable set of starting parameters can
also take several optimization runs, but there is no need to fine-tune the solution, because
the only goal is to roughly position the database on the relevant part of the Pareto front.
(If we could automatically generate a starting solution that satisfies all planning goals, we
would have revolutionized the conventional planning procedure and there would be no real
need for MCO). So for prostate cases and other standard case classes templates can be used
to generate the databases, for more complex cases like head and neck cancer several runs
for the starting solutions might be necessary before the calculation of the database can take
place.
We will shortly discuss the normalization problem in the following. In some cases we find that
there are clinically inferior solutions in the database. This seems paradoxical at first, because
the solutions are Pareto optimal and the model accurately embodies the clinical goals then
there should be no clinical inferior plans. The problem originates from the normalization
to the median target dose, which is done for all clinical treatment plans (in some hospitals
in the US other DVH-point normalizations to 90 % of the target dose are routinely used).
The normalization to a DVH-point destroys the convexity of the Pareto front as all solutions
will in general have different median target doses. That means that efficient solutions can
loose their property of Pareto optimality after the normalization (depending on the scaling
properties of the functions), but they are the solutions the planner chooses from. In the worst
case a navigation step which improves a particular objective can turn out be be worse in
4.2 Clinical Evaluation
49
that objective after the normalization. This is an inherent problem because the solutions are
only Pareto optimal with respect the particular objective functions which allow for different
median doses. Note that also in the std-model this normalization effect can occur, because
the median and the mean target dose can still differ if the distribution has a high non-zero
skewness (the third centralized moment of the distribution), which is often the case. As the
median and the skewness of a distribution are non-convex functions they cannot be easily
incorporated into the optimization framework which is essentially based on convexity, for
example to restrict the skewness of the dosedistribution for target. This problem normally
doesn’t attract attention in conventional planning, because there is potentially only one
normalization for the optimal solution which is then used as the optimal one, independent of
the actual scaling. Although this problem is still unsolved it only occurs for some solutions
and is not emerging for the whole range of the efficient set.
To maintain the dose conformality in the target the unclassified tissue can be compromised
leading to severe hot spots in the dose distribution along the beam entrance directions. The
hot spots can be partly controlled by navigating the objective of the unclassified tissue. In
some cases however the local characteristic of an hot spot is not controllable via the global
objective value for the whole structure.
4.2.5 Conclusions
The MIRA system is able to produce IMRT plans that are competitive with the clinical reference system. There is still some “parameter tweeking”, that means a database generation
with relevant plans cannot be fully automized and might still need some interaction in the
beginning (although the parameters have clinically interpretable meaning in contrast to the
conventional weight factors). As demonstrated in the prostate study, a significant reduction
in human planning time can be achieved in the long run.
The whole planning process becomes interactive which is a large benefit, thus the planner
can focus his time on choosing a plan instead of designing it.
For a first clinical application however an accredited quality assurance is necessary. The
normalization problem can hinder the practical navigation process and the planner actively
has to check for hot spots.
51
5 Understanding Pareto Optimal
Planning Databases
In this chapter we will present methods that are enable us to gain an understanding of the
efficient planning databases as a whole. In section 5.1 we develop a method that allows us to
identify the “imperative trade-offs” in a database. Section 5.2 presents ways to identify the
main planning options and the relations between the participating planning structures.
5.1 Sensitivities of the Efficient Set
5.1.1 Motivation
Although one can navigate the Pareto optimal databases and get a realtime response from
the system it can be difficult to overlook high dimensional trade-offs. Evaluating a five
dimensional trade-off for example can become quite difficult if many objectives change at
the same time. As visualization techniques are usually limited to dimensions ≤ 3 it is
desirable to have some easily accessible and interpretable information about the content of
a Pareto optimal database of an arbitrary dimension.
Getting information on the extension of a database is easily achieved by inspecting the
extreme compromises. We can also get some information on the distribution of the solutions
by looking at scatter plots from the projection of the solutions (see figure 5.1 and 5.2),
however the information can be difficult to judge. The matter of interest is the sensitivity of
the problem in the objective space, i.e. what is the change in the objectives, if one particular
objective is improved by a certain amount (so what does the planner get in return for giving
something). If we would have an analytical expression for the efficient set we could employ
methods from differential geometry and look at the tangent space of the Pareto front. But
firstly the mathematical apparatus would become quite complex and secondly we only have
a discrete representation of the efficient set and any smooth approximation to it contains
unknown approximation errors. For this reason we develop a simple procedure to study the
52
5 Understanding Pareto Optimal Planning Databases
OAR1
Target
Skin
OAR2
Target upper
1
0.8
0.6
1
Rel. Indicator values
0.5
0
1
0.95
0.9
1
0.8
0.6
1
0.5
0
0.6
0.8
1
0
0.5
1
0.9
0.95
1
0.5
1
0
0.5
1
Rel. Indicator values
Figure 5.1: Scatter plots of the projected solution for a phantom case with 500 solutions. The
figure (i, j) in the matrix corresponds to the projection of all solutions onto the j, iplane. The diagonal (i, i) shows the histograms of the solution when projected onto
the axis of objective i.
Bladder
Rectum
GTV
CTV
PTV
Contour
1
0.9
0.8
1
0.8
0.6
Rel. Indicator values
1
0.5
0
1
0.5
0
1
0.5
0
1
0.8
0.6
0.8
0.9
1 0.6
0.8
1
0
0.5
1
0
0.5
1
0
0.5
1 0.6
0.8
1
Rel. Indicator values
Figure 5.2: Scatter plots of the projected solution for a prostate case.
5.1 Sensitivities of the Efficient Set
53
sensitivity of the Pareto front. The approach uses successive hyperplane restrictions of the
efficient set.
5.1.2 Methods
We will investigate how the objectives change if we improve one particular objective k0 . Let
P be the efficient set. We assume without loss of generality that the range of all objectives
is between 0 and 1, so we can set up the following halfspace equation
hi = {y | ek0 · y ≤ αi , 0 ≤ αi ≤ 1}
(5.1)
where ek0 is the unit vector in the direction of the objective k0 , y denotes a vector in the
objective space and 0 ≤ αi ≤ 1.
Now consider a decreasing sequence of αi . For each αi we will look at the intersection of
the efficient front and the halfspace hi ∩ P thereby restricting the efficient front more and
more. At each step for all points in the intersection we will determine the minimal achievable
objective value in each objective. These values can be plotted against alpha and allow for a
visual inspection of the sensitivity of the efficient front. The concept is illustrated in figure
5.3.
We will demonstrate the method for three different cases, including a simple horseshoe
phantom case with three objectives (cf. chapter 3 figure 3.2) a clinical prostate case with six
objectives and a more complex head and neck case with eight independent objectives (see
figure 5.4).
Figure 5.3: For the efficient set P two restricting hyperplanes are shown.
54
5 Understanding Pareto Optimal Planning Databases
Figure 5.4: Geometries of two clinical cases of a prostate case (left) and a head and neck case
(right) with the corresponding CTs.
5.1.3 Results
As a first simple phantom case we consider the three dimensional horseshoe phantom case
optimized with the std-model. Figure 5.5 shows the dependency of the target and unclassified
tissue objectives (std-deviation and mean dose) from the OAR objective (gEUD8 ). The
unclassified tissue is very little affected and also shows a small range of objective values over
the whole database. The obvious “opponent“ of the OAR is the target. We can see how the
target has to change if the OAR is improved further. For values of alpha between 1 and
≈ 0.7 the dependency of the target is on the OAR is only moderate. If the OAR is improved
further the target objective value now has to increase rapidly.
Figures 5.6 and 5.7 show the sensitivities for a prostate case optimized with the std-model
when the bladder or the rectum are improved. In both cases the GTV, PTV and CTV have
very similar curves due to the geometric formation of these structures. For the rectum the
trade-off is imperative over the range of the whole efficient set, that means that the target
quality has to be deteriorated whenever the rectum is improved. The target curves for the
bladder are flat in the middle which shows that the trade-off is not imperative, so there exists
at least one solution with good objective values for the bladder and the target at the same
time. Both figures also reveal that the bladder and the rectum are not direct opponents but
can be improved simultaneously (then of course at the cost of the target).
The sensitivities for a head and neck case with eight different objectives are shown in figure
5.8. The objective values of the spine are restricted. Except for small values of alpha, all
curves of the other objectives are nearly flat. This means that the trade-off between the spine
and the other objective is not imperative, but there is a freedom of choice which structure
is traded against the spine.
5.1 Sensitivities of the Efficient Set
55
Indicator −OAR
1
0.8
0.8
Min. ind. value
Min. ind. value
Indicator −TARGET_1
1
0.6
0.4
0.2
0.6
0.4
0.2
0
0.4
0.6
0.8
0
0.4
1
0.6
alpha
0.8
1
alpha
Indicator −SKIN
1
Min. ind. value
0.8
0.6
0.4
0.2
0
0.4
0.6
0.8
1
alpha
Figure 5.5: For the horseshoe phantom with three objectives the sensitivities for the objectives is
shown when the OAR (red) is restricted.
Indicator −BLASE
Indicator −REKTUM
1
Min. ind. value
Min. ind. value
1
0.5
0
0.8
0.85
0.9
0.95
alpha
Indicator −GTV
0
0.8
1
0.5
0.85
0.9
0.95
alpha
Indicator −PTV
1
0.9
0.95
alpha
Indicator −contour_1
1
0.85
1
Min. ind. value
Min. ind. value
0.9
0.95
alpha
Indicator −CTV
0.5
0
0.8
1
1
0.5
0
0.8
0.85
1
Min. ind. value
Min. ind. value
1
0
0.8
0.5
0.85
0.9
alpha
0.95
1
0.5
0
0.8
0.85
0.9
alpha
0.95
1
Figure 5.6: The sensitivities of the objectives for a prostate case where the objective values for
the bladder are restricted.
56
5 Understanding Pareto Optimal Planning Databases
Indicator −BLASE
Indicator −REKTUM
1
Min. ind. value
Min. ind. value
1
0.5
0
0.75
0.8
0.85
0.9
alpha
Indicator −GTV
0.95
0
0.75
1
0.5
0.8
0.85
0.9
alpha
Indicator −PTV
0.95
0.95
1
0.8
0.85
0.9
0.95
alpha
Indicator −contour_1
1
1
Min. ind. value
Min. ind. value
0.85
0.9
alpha
Indicator −CTV
0.5
0
0.75
1
1
0.5
0
0.75
0.8
1
Min. ind. value
Min. ind. value
1
0
0.75
0.5
0.8
0.85
0.9
alpha
0.95
0.5
0
0.75
1
0.8
0.85
0.9
alpha
0.95
1
Figure 5.7: Restriction of the objective values of the rectum and the corresponding sensitivities
for the other objectives.
Min. ind. value
Indicator −HIRNSTAMM
0.5
0.6
0.8
alpha
Indicator −HAUTCONTOUR
1
1
0.5
0
0.2
0.4
0.6
0.8
alpha
Indicator −PAROTIS_LINKS
1
1
0.5
0
0.2
0.4
0.6
0.8
alpha
Indicator −CTV
1
1
0.5
0
0.2
0.4
0.6
alpha
0.8
1
Min. ind. value
0.4
Min. ind. value
0
0.2
Min. ind. value
Min. ind. value
Min. ind. value
Min. ind. value
Min. ind. value
Indicator −RUECKENMARK
1
1
0.5
0
0.2
0.4
0.6
0.8
alpha
Indicator −PAROTIS_RECHTS
1
1
0.5
0
0.2
0.4
0.6
0.8
alpha
Indicator −GTV
1
1
0.5
0
0.2
0.4
0.6
alpha
Indicator −LK
0.8
1
0.4
0.6
alpha
0.8
1
1
0.5
0
0.2
Figure 5.8: A head and neck case with eight independent objectives. The sensitivities of the
objectives is shown when the spine (red curve) is restricted.
5.1 Sensitivities of the Efficient Set
57
5.1.4 Discussion
For simple cases we find the obvious relations between the volumes of interest which are also
evident from the geometry of the cases (for example whenever an OAR is close to a target
structure it will be an obvious “opponent” of the target quality). But with the presented
method we are now able to answer more sophisticated questions. E.g. are the rectum and
the bladder real opponents in a prostate case (so are we only shifting dose from one side of
the target to the other)? Which objective has a stronger effect on the target quality? And
“where” in a database is this effect strong?
For the presented (and other) prostate cases we see that the bladder and the rectum are no
direct opponents, and the influence of the rectum on the target is larger then for the bladder.
However the bladder becomes a strong opponent of the target at a particular point of the
objective values. So it is possible to identify different “regimes” for each objective. In each
regime there can be different correlations between the objectives.
For the eight dimensional head and neck case we find that the curves are much more flat
over a large extend of the database. Note that the plots do not necessarily represent the
change in another objectives when the criterion k0 is improved, because this change is of
course dependent on the chosen path on the Pareto front. In fact the plots show what is
maximal achievable for each objective when k0 is improved. Note also that a flat curve does
not contradict the convexity of the set F or the Pareto optimality of the solutions (it would
in two dimensions, where the method just reproduces the Pareto front). It only states that
the trade-offs between the structures is not imperative.
We see that with an increasing complexity of a case the trade-offs can become more “independent” and further planning options become available.
Two things make the whole concept easy to grasp. First no sophisticated mathematical
formalism is involved, second one has to compare only two things at the same time.
5.1.5 Conclusion
We presented a simple procedure to visualize the sensitivity of an efficient set of an arbitrary
dimension. The plots for each indicator show what is maximal achievable for each objective
when a particular objective k0 is improved. So we can reveal if a certain trade-off between
two structures is really imperative.
The analysis of the cases also shows that for simple cases the objectives can be strongly
correlated which restricts the variety of the planning options in contrast to more complex
cases where there exists more freedom in the plan design.
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5 Understanding Pareto Optimal Planning Databases
5.2 Identifying the Main Trade-Offs in Pareto Optimal
Databases
5.2.1 Motivation
The shape of the Pareto surface and therefore the structure of a database is not known a
priori and depends on the geometry of the clinical case and the formulation of the optimization problem. In the course of a better understanding and an accurate characterization of
multiobjective databases there are several arising questions:
ˆ What are the main underlying trade-offs?
ˆ What is the complexity for a given database and how we can quantify it?
ˆ What is the variety of planning options available?
We will address these questions by studying the efficient set in the space of IMRT ”beamlets”, which is a high-dimensional vector space. We are interested to see if we can ”boil
down” the results by use of dimensionality reduction methods to gain a better understanding of the underlying tradeoffs. Two different methods for dimensionality reduction will
be investigated: principal components analysis (linear) and the (nonlinear) isomap method
(Tenenbaum et al., 2000). Both methods will reveal insight into the topology of the efficient
set and will increase the understanding of the underlying trade-offs present in the database
of a specific case.
5.2.2 Materials and Methods
Terminology
We consider an IMRT fluence map optimization with fixed beam angles. The optimization
problem is formulated as a convex MOP, meaning the feasible set of the beamlets is convex
and all objectives for the different anatomical structures are convex functions. For OARs
we use the max and mean model (Thieke et al., 2002), and for targets we use linear ramp
functions (Craft et al., 2007). One treatment plan is defined by the values of all beamlets.
This plan can be interpreted as a point in high dimensional beamlet space (in which the
dimension equals the number of beamlets). So a plan will be often referred to as a point and
the set of plans in the database as the data.
5.2 Identifying the Main Trade-Offs in Pareto Optimal Databases
59
Principal Components Analysis
Principal components analysis (PCA) is a well known method for multidimensional data
reduction. It finds an ordered set of orthogonal directions along which the data primary lies.
Each direction is a one dimensional subspace and defines one principal component (PC). The
dimensionality reduction is achieved by considering only a small number of subspaces. The
PCs are the eigenvectors of the covariance matrix of the data and are sorted with respect to
the corresponding decreasing eigenvalues. The first PC (the one with the largest eigenvalue)
offers the largest variance of the data points when projected on this subspace. In other
words, the data primarily lie along this direction. The computation of the PCs was done
with MATLAB using the function svd to compute the singular value decomposition. PCA
belongs to the class of linear spectral dimensionality reduction methods, where it is assumed
the data lie on a linear manifold.
Isomap Method
In general the efficient set does not necessary lie on a linear manifold but instead is a
curved structure. The feasible set in beamlet space is convex and the efficient set will depending on the formulation of the optimization problem- lie on boarder of this convex set.
This motivated us to investigate a method capable of describing nonlinear low dimensional
structures embedded in a higher dimensional space, such as the isomap method. Figure 5.9
shows an example of a curved set, where PCA fails to detect the true dimensionality but the
isomap method is successful.
Isomap allows for an estimate of the manifold dimensionality (see 5.2.2) and for low dimensional representation of the data. It is able to detect nonlinear characteristics in the data.
As input N datapoints in n dimensional space are used, which represent a sampling of the
underlying manifold.
The isomap algorithm consists of four parts (for a detailed description see Tenenbaum et al.
(2000)). First a distance matrix ∆ is calculated from the N points. Each entry ∆ij is the
Euclidean distance between points i and j. Second, a number k of nearest neighbours is
chosen and a weighted graph G is constructed using the underlying data as nodes. For
each node, k edges to the k nearest neighbors (distance between nodes is given by ∆ij ) are
created and these edges are weighted by the Euclidean distances. Third, geodesic distances
between every pair of points is estimated by solving a shortest path problem between the
two points in the graph G. These geodesic distances take the geometry of the manifold
into account and incorporate the nonlinear shape of the manifold. They approximate the
distance between the points on the manifold. Last, a simple metric multi dimensional scaling
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5 Understanding Pareto Optimal Planning Databases
b)
a)
c)
Figure 5.9: Example of a one dimensional curve embedded in three dimensions and the directions
of the PCs (a). The PCA yields three nonzero eigenvalues (b) and is not able to
discover the true dimensionality due to the non-linear structure. The isomap method
unambiguously reveals the true dimension (c).
is performed with the geodesic distances as inputs. The multidimensional scaling finds, for a
set of pairwise similarities between objects, a spatial low dimensional configuration of points,
whose distances best resemble these similarities. As a result we get a p < n dimensional
approximation of the data, which is referred to as a p-dimensional embedding. For p ≤ 3
the output can be visualized by plotting the isomap coordinate values of each point. The
MATLAB implementation of isomap provided by Tenenbaum was used for the calculations.
The isomap method is a graph based dimensionality reduction method.
A standard way of evaluating the quality of a low dimensional approximation is to plot the
remaining variance, which is calculated with the help of eigenvalues Haerdle and Simar (2003)
and has a range from zero to one. If all of the data can be explained with a small number
of components then the remaining variance will be close to zero and additional components
do not add any significant information.
The plots of the remaining variance allow for an estimate of the manifold dimension which
we will call the effective dimension. We have to determine the point in the curve where it
stops decreasing. At this point additional dimensions do not add much explanatory power
and this number gives the estimate of the effective dimension. The more pronounced the
kink is, the better is the estimate. Only one curve with a specific nearest neighbor number
k is used for the estimate. A typical set of curves for different values of k is shown in Figure
5.2 Identifying the Main Trade-Offs in Pareto Optimal Databases
61
5.23, where one sees that for a broad range of values for k, the effective dimension is stable.
It should be noted that the dimension of the PCA and the isomap method relate to two different description concepts, in analogy to Euclidean and Riemann geometry. The dimension
for a PC approximation describes the dimension of the subspace formed by the considered
PCs, while the effective dimension in the isomap method is a better representation of the
number of degrees of freedom needed to describe the data. As an analogy one can think of
the generalized coordinates in Lagrangian mechanics.
Phantoms and Clinical Cases
We demonstrate the techniques on a two dimensional horseshoe phantom with multiple OARs
(see Figure 5.10) and two clinical cases. All phantom cases have five equidistant beams with
16 bixels per beam and a total number of 2300 voxels. For all OARs the mean dose is chosen
as the respective objective while the target and the unclassified tissue are modeled with linear
ramp functions (for the target the ramp function penalizes dose under the prescription value,
and for the unclassified tissue the ramp function penalizes doses above a certain level). The
prescribed dose for the target is 70 Gy. Hard constraints of 60 Gy minimum dose are set
for the target, as well as 50 Gy maximum dose constraints for all OARs and 70 Gy for the
unclassified tissue.
Two clinical examples, a brain and a pancreas case, are also considered. The prescribed target
doses are 59.4 Gy for the brain case and 50.4 Gy for the pancreas case. Hard constraints for
the minimum target doses are set to 55 Gy and 45 Gy, respectively. In the optimization of the
brain case five independent objectives are considered: target under- and overdosage, brain
stem, optical nerve and the chiasm. The pancreas case was optimized with respect to five
different objectives: target underdosage, stomach, both kidneys, liver and the unclassified
tissue. For all OARs the mean dose was considered in the optimization. All databases are
calculated with PGEN (Craft et al., 2006). The phantom databases contain 160 solutions for
the first case (one OAR) and 107 solutions for the second (two OARs), and both clinical cases
have 60 solutions, which is well over the number of cases required for a close representation
of the Pareto surface (Craft and Bortfeld, 2008).
5.2.3 Results
Phantoms
The first phantom case has one OAR (OAR1), so four objectives are optimized independently:
the OAR, target under- and overdosage, and the unclassified tissue. A PCA is performed
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5 Understanding Pareto Optimal Planning Databases
Figure 5.10: Geometry of the phantoms. OAR1 and OAR2 are used for the isomap and PCA
analysis, whereas up to four OARs are added to study the effective dimension.
and, to display the trade-offs exposed by the principal tradeoffs, new points are created along
the directions of the first and second PC (see Figures 5.11 and 5.12). Along the first PC
the main trade-off is between OAR1 and the unclassified tissue, while the target coverage
is nearly unchanged. In the direction of the second PC both the OAR and the unclassified
tissue are now balanced against the target coverage.
The second phantom has one additional OAR (OAR2), which yields five independent objectives. The PCA shows that both OARs are effected along PC1 and PC2 (see Figure 5.13 and
5.14). In the direction of PC1 the OAR1 increases in the mean dose while OAR2 is spared
more and more. Also the target underdosage slightly improves. Along the second PC both
OARs show a decreasing mean dose while the unclassified tissue dose increases. It is also
possible to decouple the two OARs by combining both PCs.
The phantom with one OAR is analyzed by the isomap method. An estimate of the dimensionality is obtained by looking for the “elbow” in the graph of the remaining variance.
Clearly, a two dimensional structure is suggested (Figure 5.15).
Next the isomap method is applied to the phantom with two OARs. For this phantom the
isomap method also yields an effective dimension of two. The corresponding 2d embedding
in Figure 5.16 shows that all inherent trade-offs can still be captured by considering only
two degrees of freedom. Clearly two main trade-off directions can be identified. They can
be interpreted as the two diagonals of the plot. In the upper left corner the best solutions
for OAR2 are located. In opposition to this in the lower right are solutions with a good
unclassified tissue sparing and no overdosage in the target. The other diagonal consists of
5.2 Identifying the Main Trade-Offs in Pareto Optimal Databases
63
PC 1
[Vol %]
[Gy]
Figure 5.11: Solutions along the first PC for the phantom with one OAR. The prescribed dose of
70 Gy is marked.
PC 2
Figure 5.12: Solutions along the second PC for the phantom with one OAR.
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5 Understanding Pareto Optimal Planning Databases
Figure 5.13: Solutions along the first PC for the phantom with two OARs.
Figure 5.14: Solutions along the second PC for the phantom with two OARs.
5.2 Identifying the Main Trade-Offs in Pareto Optimal Databases
65
0.5
PCA
Isomap k=16
Residual variance
0.4
0.3
0.2
0.1
0
0
2
4
6
8
Dimensions considered
10
Figure 5.15: The remaining variance for the PCA and the isomap method is shown for the phantom with one OAR.
the extremes of the best solutions for the OAR1 (upper right) and solutions with no target
underdosage. Alternatively we can interpret the vertical direction as OAR sparing vs target
quality and the horizontal direction as OAR1 vs OAR2.
Brain Case
The brain case is optimized with respect to five objectives: target under- and overdosage,
brain stem, chiasm and the optical nerve. A PCA is performed. PC1 shows the main
trade-off between the optic nerve and the brain stem while the target quality remains almost
constant. The second PC balances the overdosage in the target against the optic nerve and
the chiasm.
The brain case shows an effective dimension of two (see Figure 5.19), thus only two degrees
of freedom are needed to describe the data.
Pancreas Case
For the pancreas case five objectives are considered including target underdosage, stomach,
liver, both kidneys and the unclassified tissue. A PCA shows that the dominating trade-off
is between the liver together with the stomach and the two kidneys together with a slight
target underdosage. In the second PC the kidneys are not coupled anymore. The right
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5 Understanding Pareto Optimal Planning Databases
Figure 5.16: Two dimensional embedding of all solutions of the phantom case with two OARs.
The anchor point solutions for both OARs, the target under- and overdosage (ctvcold, ctv-hot) and the unclassified tissue (ut) are labeled.
5.2 Identifying the Main Trade-Offs in Pareto Optimal Databases
Figure 5.17: DVHs of the brain case for solutions along the first and the second PC.
67
68
5 Understanding Pareto Optimal Planning Databases
Figure 5.18: Dose distributions of the brain case along the first PC (lower left to upper right)
and the second PC (upper left to lower right).
0.8
Remaining Variance
0.7
0.6
PCA
Isomap k=4
0.5
0.4
0.3
0.2
0.1
0
0
2
4
6
8
No of Dimensions considered
10
Figure 5.19: The remaining variance of the brain case for the PCA and the isomap method.
5.2 Identifying the Main Trade-Offs in Pareto Optimal Databases
69
kidney and the stomach are balanced against the left kidney and the liver. Variations in the
third PC are less pronounced. The most prominent trade-off is observed between the left
kidney and the liver together with the right kidney.
PC 1
PC 2
Figure 5.20: DVHs of the pancreas case for the first and second PC.
The analysis of the pancreas case results in an effective dimension of three (cf. Figure 5.23).
So two dimension are not sufficient to describe the database, but rather three dimensions
are needed for this case. Nevertheless we can get an insight by looking at the 2d-embedding
shown in Figure 5.24. Dimension one distinguishes between vertical (right) and horizontal
(left) dose entry channels. Due to the geometry of the organs the corresponding tradeoff is
observed between the liver and the kidneys. The second dimension improves the stomach
when moving downwards, either by distributing dose horizontally (left side) or vertically
(right side). So good stomach solutions can involve good kidney sparing or good liver sparing.
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5 Understanding Pareto Optimal Planning Databases
Figure 5.21: The DVHs for the third PC.
PC
1
2
PC
67
0
Figure 5.22: Dose distributions of the pancreas case along the first PC (lower left to upper right)
and the second PC (upper left to lower right).
5.2 Identifying the Main Trade-Offs in Pareto Optimal Databases
71
0.7
Remaining Variance
0.6
0.5
0.4
PCA
Isomap k=2
Isomap k=4
Isomap k=8
Isomap k=12
0.3
0.2
0.1
0
0
2
4
6
8
No of Principal Component considered
10
Figure 5.23: The remaining variance of the pancreas case for the PCA and the isomap method.
Figure 5.24: The two dimensional embedding of the pancreas case.
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5 Understanding Pareto Optimal Planning Databases
Effective Dimension
Can we understand why the cases differ in the effective dimension, or even predict it? The
four phantoms containing between one and four OARs are considered. We study how the
effective dimension changes when new objectives are added. As shown in table 5.1 the
effective dimension can increase with additional OARs. The corresponding Spearman rankcorrelation matrices are shown in figure 5.25.
Table 5.1: Effective dimension and corresponding indicator values for all cases. Another case
(p7b) is included in which OAR3 and OAR4 are moved towards OAR2.
case
objectives
p4
4
p5
5
p6
6
p7
7
p7b
7
brain
5
pancreas
5
negative entries
3
6
8
11
9
5
6
neg. entries / objectives
0.75
1.2
1.33
1.57
1.28
1
1.2
p4 1 oar:
OAR1 CTVc
p5 2 oars:
CTVh
OAR1 OAR2 CTVc CTVh Uncl.
Uncl.
p6 3 oars:
p7 4 oars:
OAR1 OAR2 OAR3 OAR4 CTVc CTVh Uncl.
OAR1 OAR2 OAR3 CTVc CTVh Uncl.
effective dimension
2
2
3
4
3
2
3
Figure 5.25: Spearman rank correlation coefficients of the phantom cases.
A simple way to approach the effective dimension is to look at Spearman correlation matrices
5.2 Identifying the Main Trade-Offs in Pareto Optimal Databases
73
Effective Dimension
in the objective space. The complexity of a database should also be reflected in the entries
of the correlation matrices. If two VOIs have a high positive correlation in their objectives
then this decreases the degrees of freedom of a database: these VOIs cannot be traded
independently, this compromise is not accessible. Therefore the number of negative entries
in the correlation matrix is expected to be related to the effective dimension of the database,
which we indeed see in Table 5.1 and in Figure 5.26.
4
3
2
1
0.8
1
1.2
1.4
1.6
No. of negative correlation entries / no. of objectives
Figure 5.26: The effective dimension as a function of the indicator value.
5.2.4 Discussion
With PCA the main trade-off directions can be identified and relations between VOIs become comprehensible. Few (2-3) PCs are appropriate to describe the main variations in a
database.
The isomap method, better suited for data that inherently exists on a nonlinear manifold,
reveals a low dimensional structure of the efficient set in beamlet space for all cases. It
should be noted that the effective dimension is only an estimate for the true dimension of
the manifold Tenenbaum et al. (2000). This insight into the topology of the efficient set is
not possible by looking at the PCA alone, which tends to overestimate the dimensionality
for nonlinear data.
Both methods allow for a systematic analysis of a database. The isomap method achieves
better estimates on the true dimensionality and can produce meaningful visualizations. The
direction of a single isomap dimension is not constant in beamlet space, but will depend
on the current value of the isomap projection. With regard to further practical proceedings
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5 Understanding Pareto Optimal Planning Databases
the PCA is advantageous. As each PC defines a fixed direction in beamlet space one could
e.g. easily navigate along the PCs and thereby explore the whole range of the database. If
the nonlinearity of the efficient set is less pronounced the differences in both methods will
become smaller.
The effective dimension can change with added objectives. An increase in dimension stands
for an increase in complexity. If the additional VOI does not change the efficient set much the
effective dimension will stay constant. The presented indicator derived from the correlation
matrices should give a reasonable guess for the effective dimension for a wide range of cases.
The pancreas case has a higher effective dimension than the brain case and the two phantom
cases. This can be understood by geometry of the organs. If the organs are large and
separated from each other the dose distributions can largely differ in shape. Hence the
dimension of the efficient set in beamlet space is likely to be higher. Similarly, we see a
decrease in the effective dimension in the phantom case p7b in Table 5.1 when we move the
critical structures closer together.
The low effective dimension revealed by the isomap method also supports the findings that
the number of plans needed for an accurate covering of the Pareto front is small and does
not necessary increase exponentially Craft and Bortfeld (2008). The correlation between
different VOIs and the imposed constraints in the optimization problem restricts the feasible
set and therefore also the efficient set.
The answers to our questions from the introduction are as follows:
ˆ Main trade-offs: PCA allows to identify the main trade-offs in a database. The distribution of the data is captured by the main PCs and each PC exposes one key trade-off.
The 2d visualization of the database provided by the isomap method enables one to
see the compromises and understand the relation between different VOIs.
ˆ Complexity: The effective dimension can serve as a measure of complexity. If it is
small, e.g. two, then it is possible to visualize the whole database, by laying out the
plans in a 2D figure, see e.g. Figure 5.16. If the effective dimension stays unchanged
after adding an objective (or other actions that potentially increase complexity) this
indicates that the additional plans can still be understood with the same number of
degrees of freedom. The newly added plans can be inserted in the existing embedding
of plans and a smooth transition between new and old plans is possible. If on the other
hand the effective dimension increases after adding a new OAR this means that the
only way to achieve an satisfactory embedding is to extend the dimension and move
some points in this direction.
5.2 Identifying the Main Trade-Offs in Pareto Optimal Databases
75
ˆ Variety: The variety of available planning options can be judged by considering the
database visualizations and by looking at the effective dimension. The 2d visualization
allows one to grasp the whole range of the database and also the transitions between
the different plans. The higher the effective dimension the more degrees of freedom are
contained in the database. The different planning options are also reflected in the first
PCs. Their effect on the dose distribution reveals the extent of the database and thus
the main planning options. Although local fine tunings are certainly possible the first
PCs describe the primary characteristics of the data for a specific case.
5.2.5 Conclusion
We presented two methods of dimensionality reduction for the analysis of multiobjective
databases. They provide the basis to extract the main trade-offs from a database. For
each individual case opposing structures and the available planning options contained in a
database can be revealed. It is possible to visualize the efficient set in a meaningful way even
for high dimensional databases with many objectives (providing literally an insight into the
Pareto surface). The gained information can be helpful in the planning process and improve
understanding of a case.
77
6 Summary and Outlook
Summary
In this thesis we have presented and evaluated a new prototypical treatment planning system
which uses a multiobjective approach. We further on developed methods that allow to
systematically extract information from an efficient set, increasing the understanding of a
Pareto optimal database as a whole.
We evaluated different modelling frameworks and studied the influence of the corresponding
modelling parameters on the resulting solutions. Furthermore we introduced a quantitative
measure which allows to determine the change of the efficient set due to a change in the
model framework, so different efficient sets become comparable.
To determine the quality of a finite approximation of a high dimensional Pareto surface we
developed a set of indicators. These indicators yield additional information on the approximation quality and can assist the quality assurance for the MIRA system. The indicators
are of general nature and are not restricted to the models presented here but can be applied
to characterize any efficient set.
In retrospective planning studies we examined the practical feasibility of the new multiobjective approach. We could demonstrate that the system is able to produce clinically
meaningful plans. For a first clinical application a certified quality assurance has to be
carried out. Furthermore, the normalization problem has to be dealt with.
In the last part we have introduced several methods to obtain a better understanding of the
Pareto optimal databases and the underlying cases. These methods use successive hyperplane restrictions of the efficient set and linear as well as non-linear dimensionality reduction
techniques. The revealed low dimensional structure of the efficient set in the beamlet-space
allows for meaningful visualizations of high dimensional Pareto fronts and an insight into
the main trade-offs of a case.
As a consequence of this thesis the developer as well as the practial planner will benefit from
the achievement of this work. The developing side can directly use the presented methods
to assist the quality assurance in the further development process. The planner is provided
78
6 Summary and Outlook
with additional systematic information on the case that supports him in choosing the best
suited treatment plan for the patient. Alltogheter we are confident that a multiobjective
treatment planning environment will soon be established in clinical practice to allow for an
efficent and flexible decision support in inverse radiotherapy planning.
Outlook
For future developments there are still a number of open questions. The problem of the
beam angle optimization is at present separated from the subsequent planning process. The
efficient set of the whole planning problem though, would include plans with different beam
angle setups. Besides the computational challenges in generating the databases there have
to be found ways how to practically navigate these databases which are not convex anymore
and the navigation conceptually relies on this property. It remains to be seen if the potential
non-convex effects play a role in a practical context.
To extend the control of the dose distribution in a particular regions of a structure one
would have to incorporate local objectives or one could modify the search of the databases
to navigate DVH-points instead of values of a whole objective.
It is possible to relate the multiobjective framework with some other current developments
in the field of radiotherapy planning.
The inclusion of several biological models is already implicitly achieved as the efficient sets
are equivalent if the biological model is expressible as a monotone function of the gEUD.
Many other biological models can potentially be included via the convex reformulation of
objectives.
Some research has focused on the inclusion of uncertainties in treatment planning leading to
robust optimization. As the resulting formulations are also convex optimization problems,
the extension to a robust multiobjective framework is also possible. The databases could
either contain robust plans or the robustness could even be included as an independent
controllable meta objective.
A challenging topic might be the connection with adaptive planning where the treatment
plan is adapted on the basis of new information like additional imaging information before
a treatment. If effective ways can be found to adapt a whole plan database due to the new
gained information, using (some of) the previous knowledge on the efficient set, has to be
seen.
79
7 Acknowledgment
Einer Dissertation geht immer eine lange Lebensphase des Lernens voraus. Obwohl nicht
viel Physik in dieser Arbeit durchscheint, so empfinde ich sie doch als mein eigentliches
wissenschaftliches zu Hause. Ich bin dankbar die Möglichkeit gehabt zu haben, dieses
faszinierende und grundlegende Feld zu studieren.
Viel des Gelernten erfährt erst mit der Zeit die eigene Wertschätzung und gelangt in eine
gedankliche Form, die eine Erfassung der Dinge in einem größeren Zusammenhang ermöglicht
(und ich glaube, dieser Prozess ist - zum Glück - noch nicht am Ende). Ich möchte all denen
danken, die darauf Bedacht waren, meinen physikalischen und kritischen Verstand zu schärfen und mich auf meinem wissenschaftlichen Weg bis hier her begleitet haben. Dies gilt in
gleichem Maße für Lehrende und Mentoren wie auch für Weggefährten und Freunde.
Im Bezug auf diese Arbeit möchte ich an erster Stelle Uwe Oelfke sehr herzlich danken,
für die Betreuung dieser Arbeit und das Vertrauen, welches er mir stets entgegengebracht
hat. Er hat mich immer in meinem Tun unterstützt, sich für mein fachliches Weiterkommen
eingesetzt und mich durch seine Betonung der physikalischen Hintergründe für das Gebiet
der Medizinischen Physik begeistern können. Herrn Wolfgang Schlegel danke ich für die
Bereitschaft, die Arbeit als Zweit-Gutachter zu bewerten und für seine angenehme Leitung
der Medizinischen Physik Abteilung des DKFZs.
Desweiteren bin ich all denen zu Dank verpflichtet, die sich ebenfalls für das Gelingen dieser
Arbeit eingesetzt haben. Dazu gehört Christian Thieke, der sich in vielen Gesprächen stets
motivierend und mit hilfreichen Ratschlägen einbracht hat. Herzlich möchte ich mich auch
bei den Kooperationspartnern vom ITWM in Kaiserslautern Alexander Scherrer, Michael
Monz, Phil Süss und Karl-Heinz Küfer bedanken, die durch Ihre Entwicklungsarbeit dieses
Projekt erst ermöglichten, sich immer mit großer Hilfsbereitschaft und Unterstützung einbrachten und mir viele Geheimnisse der multikriteriellen Optimierung vermitteln konnten.
Eine sehr schöne und anregende Zeit in Boston konnte ich in der Gruppe von Thomas
Bortfeld verbringen. Ihm gilt mein aufrichtiger Dank für seine weitsichtige Betreuung, das
Teilen vieler interessanter Ideen und seine Großzügigkeit, mit der er mir stets begegnet ist.
Eine hervorragende und bereichernde Zusammenarbeit verdanke ich David Craft, aus der
viele Ergebnisse und neue Ideen entstanden.
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7 Acknowledgment
Ich habe die Arbeitsbedingugen und die Atmosphäre am DKFZ immer als sehr gut und
kollegial erlebt. Bei meinen Zimmerkollegen, meiner Arbeitsgruppe und vielen weiteren
Kollegen bedanke ich mich herzlich.
Ohne eine Finanzierung ist das Verfassen einer Dissertation kaum möglich. Für die breite
finanzielle Unterstützung durch die Deutsche Krebshilfe, den DAAD und das DKFZ möchte
ich mich anerkennend bedanken.
Letzlich befinden sich die tragenden Säulen immer jenseits des Fachlichen. Ich bin all meinen
Freunden dankbar für die gemeinsame Zeit, die vielen heiteren Momente und Ihre ehrliche
Freundschaft! US, DPR, AJK, MA, BKK, DWB, AK, UG, KO, ...
Mein abschließender und tief empfundener Dank gilt Euch, lieber Vater, liebe Mutter und
liebe S.
81
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85
List of Figures
2.1
2.2
2.3
2.4
2.5
3.1
3.2
3.3
3.4
3.5
Efficient solution (green) on the Pareto front (blue) for a bicriterial problem.
The solution in red is dominated by the green one, because it lies in the red
cone. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The example by Brahme with a rotating source and a centrally blocked beam.
Only a mudulated beam profile (dashed lines) leads to a homogeneous dose
distribution in the target. Figure from (Brahme et al., 1982). . . . . . . . . .
The tumor (red) is irradiated with high energetic photons from seven directions. The irradiation fields with spatially non-uniform intensity maps are
superimposed to achieve the prescribed dose in the tumor and a sufficient
sparing of crticial structures. . . . . . . . . . . . . . . . . . . . . . . . . . . .
All feasible solutions in X from the parameter space (left) are mapped to
the design space (middle). The set of efficient solutions is symbolized in red.
The solutions in the design space are evaluated by a vector valued function
leading to a representation in the objective space. Here the efficient solutions
are located at the boundary of the set F of feasible objective vectors. . . . .
Graphical user interface of the navigator. Each objective is represented by
a slider (left) and can be manipulated interactively. The corresponding dose
distribution and DVH information is updated in real time. . . . . . . . . . .
my short caption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The geometries of three cases. A horseshoe phantom case (left) with one OAR
and a horseshoe shaped target volume, a paraspinal tumor (paraspinal I) with
one boost volume (middle) and a small paraspinal case (paraspinal II) (right).
Dose volume histograms of the horseshoe phantom. The parameters pT and
qT are the only parameters which are modified. On the left pT = 1 and qT = 8,
setting a high priority on the target overdosage. On the right we have pT = 8
and qT = 1. The aspired objective value for the OAR was gEUDasp
= 30 Gy.
8
DVHs of two solutions from two different databases that only differ in the
setting of the parameters pT and qT . On the right pT = 8 and qT = 1, setting
a high priority on the target underdosage. On the left we have pT = 1 and
qT = 8. Here the aspired objective value for the OAR was set to gEUDasp
=
8
10 Gy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Influence of the organ parameter p on the efficient solutions. All three solutions are optimized with the pq-model and the organ parameter is varied
from p = 2 to p = 8 (p = 2, 4, 8 from left to right). The maximum OAR dose
decreases in total by 3.7% (dOAR
max = 79.5, 77.6, 75.8). . . . . . . . . . . . . . .
6
7
8
9
12
16
16
17
17
18
86
List of Figures
3.6
3.7
3.8
3.9
3.10
3.11
3.12
3.13
3.14
3.15
3.16
3.17
3.18
3.19
3.20
3.21
Effect of changing the aspired function value for the OAR. The aspired function value for the OAR is varied between gEUDasp
= 10 Gy (left) 30 Gy (mid8
dle) and 40 Gy (right). The maximum OAR dose changes by more than 40%.
The horseshoe phantom is optimized with the pt-model. The first three solutions correspond to different values of pt (pt = 2, 4, 8 from left to right), while
in the solution on the right dup was changed from 30 Gy to 10 Gy. . . . . . .
Influence of the parameter p for the spine. Two solutions with p = 3 (dashed
lines) and two solutions with p = 8 (solid lines) are shown. While the maximum dose of the spine changes by 6 Gy on the left hand side there is nearly
no difference on the right. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Influence of the parameter p for the stomach. All other OARs are virtually
the same. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A paraspinal case is optimized with three different models. One solution from
each database is shown, the pq-model (left), the pt-model (middle) and the
std-model (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Dose distributions for the three optimized with the pq-model (left), the ptmodel (middle) and the std-model (right). . . . . . . . . . . . . . . . . . . .
Maximal distances for the solutions from D2 to the convex hull of D1 (blue)
and from the convex hull of D1 to the solutions from D2 (red). . . . . . . . .
Schematic picture of the hypervolume indicator. . . . . . . . . . . . . . . . .
Example of two dimensional efficient sets. On the left successive solutions
(blue) have the same arc length. The inner approximation (red) is close to
the Pareto front (black). On the right hand side the random placement of the
same number of solutions can result in large gaps between the approximation
and the efficient set. The corresponding indicator values for Hs are Hs = 1.09
(left) and Hs = 0.33 (right). . . . . . . . . . . . . . . . . . . . . . . . . . . .
For the solutions above the largest hyperspheres are calculated. The gap in the
distribution on the right results in a large hypersphere. For the hypersphere
indicators we have Hl = 0.00 (left) and Hl = 1.38 (right). . . . . . . . . . . .
Example of a two dimensional Pareto front in three dimensions. Hyperspheres
for three arbitrary solutions are shown. The even distribution results in very
similar spheres (left) with Hs = 1.15 and Hl = 0.03 while for the stochastically
placed solutions the size of the radii can differ (Hs = 0.65 and Hl = 0.53). . .
The simplex (bluish) formed by a facet and the utopia point for that facet is
restricted by the hyperplane (red) defined by a neighbouring facet. . . . . . .
The hypervolume indicator for the phantom with two OARs and a database
of 615 solutions (left). The upper bound of the coverage error for the database
is shown on the right. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Distributions of the nearest neighbour distances (left) and the largest hyperspheres (right) for the phantom case. The corresponding indicator values are
Hs = 0.69 and Hl = 1.25. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Hypervolume indicator of the phantom case with a database of 800 solutions.
The upper bound of the coverage error for the database is shown on the right.
Hypervolume indicator for two different databases with 800 and 1200 solutions
for the same phantom case. . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
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21
21
22
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29
29
30
33
33
34
34
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List of Figures
3.22 Hypervolume indicator (left) and the upper bounds on the coverage error
(right) for a prostate database. . . . . . . . . . . . . . . . . . . . . . . . . .
3.23 For a prostate case with a 110 solutions the nearest neighbour distribution of
the solutions (left, Hs = 0.59) and the distribution of the largest hyperspheres
(right, Hl = 1.07) are shown. . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
Geometry of the vois for the paraspinal case (left) and the DVHs from the
Pareto optimal database. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The KonRad reference plan compared to the the convex combination of five
solution from the database (dashed lines). . . . . . . . . . . . . . . . . . . .
Incorporation of the navigation principle into the routine planning system
VIRTUOS at the DKFZ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Three different trade-offs from a database compared to a KonRad plan. On
the left the sparing of both OARs is achieved by a worse target coverage. In
the middle the coverage is improved and both OARs are inferior in return.
On the right both OARs are decoupled and can be traded independently. . .
Dosedistributions of a KonRad plan (left) and a plan from the database (right)
corresponding to the left DVH in figure 4.4. . . . . . . . . . . . . . . . . . .
The plan from a database optimized with the pt-model is similar to the KonRad plan (left). In a database optimized with the std-model a clinically
slightly superior plan compared to KonRad can be found. . . . . . . . . . . .
DVHs for a prostate case. Two solutions from a database are compared to
the KonRad plan. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
For a paraspinal case a clinical inferior plan can be found in the database
(left). Both OARs exibit larger maximum doses and the coverage of the boost
volume is inferior. For a prostate case there can occur hot spots (right) in the
normal tissue. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Scatter plots of the projected solution for a phantom case with 500 solutions.
The figure (i, j) in the matrix corresponds to the projection of all solutions
onto the j, i-plane. The diagonal (i, i) shows the histograms of the solution
when projected onto the axis of objective i. . . . . . . . . . . . . . . . . . . .
Scatter plots of the projected solution for a prostate case. . . . . . . . . . . .
For the efficient set P two restricting hyperplanes are shown. . . . . . . . . .
Geometries of two clinical cases of a prostate case (left) and a head and neck
case (right) with the corresponding CTs. . . . . . . . . . . . . . . . . . . . .
For the horseshoe phantom with three objectives the sensitivities for the objectives is shown when the OAR (red) is restricted. . . . . . . . . . . . . . .
The sensitivities of the objectives for a prostate case where the objective values
for the bladder are restricted. . . . . . . . . . . . . . . . . . . . . . . . . . .
Restriction of the objective values of the rectum and the corresponding sensitivities for the other objectives. . . . . . . . . . . . . . . . . . . . . . . . .
A head and neck case with eight independent objectives. The sensitivities of
the objectives is shown when the spine (red curve) is restricted. . . . . . . .
87
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44
44
45
45
47
47
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52
53
54
55
55
56
56
88
List of Figures
5.9
5.10
5.11
5.12
5.13
5.14
5.15
5.16
5.17
5.18
5.19
5.20
5.21
5.22
5.23
5.24
5.25
5.26
Example of a one dimensional curve embedded in three dimensions and the
directions of the PCs (a). The PCA yields three nonzero eigenvalues (b) and
is not able to discover the true dimensionality due to the non-linear structure.
The isomap method unambiguously reveals the true dimension (c). . . . . .
Geometry of the phantoms. OAR1 and OAR2 are used for the isomap and
PCA analysis, whereas up to four OARs are added to study the effective
dimension. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Solutions along the first PC for the phantom with one OAR. The prescribed
dose of 70 Gy is marked. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Solutions along the second PC for the phantom with one OAR. . . . . . . .
Solutions along the first PC for the phantom with two OARs. . . . . . . . .
Solutions along the second PC for the phantom with two OARs. . . . . . . .
The remaining variance for the PCA and the isomap method is shown for the
phantom with one OAR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Two dimensional embedding of all solutions of the phantom case with two
OARs. The anchor point solutions for both OARs, the target under- and
overdosage (ctv-cold, ctv-hot) and the unclassified tissue (ut) are labeled. . .
DVHs of the brain case for solutions along the first and the second PC. . . .
Dose distributions of the brain case along the first PC (lower left to upper
right) and the second PC (upper left to lower right). . . . . . . . . . . . . .
The remaining variance of the brain case for the PCA and the isomap method.
DVHs of the pancreas case for the first and second PC. . . . . . . . . . . . .
The DVHs for the third PC. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Dose distributions of the pancreas case along the first PC (lower left to upper
right) and the second PC (upper left to lower right). . . . . . . . . . . . . .
The remaining variance of the pancreas case for the PCA and the isomap
method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The two dimensional embedding of the pancreas case. . . . . . . . . . . . . .
Spearman rank correlation coefficients of the phantom cases. . . . . . . . . .
The effective dimension as a function of the indicator value. . . . . . . . . .
60
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64
64
65
66
67
68
68
69
70
70
71
71
72
73
89
List of Tables
3.1
4.1
4.2
4.3
5.1
Hausdorff distances dH = (conv(D1 ), D2 ) based on the first order approximation of the efficient sets for the three models. The values in brackets correspond
to the values of maxy∈D2 minz∈conv(D1 ) d(y, z). . . . . . . . . . . . . . . . . .
Values of the interpolation coefficients and the corresponding solution numbers
after solving the quadratic program. . . . . . . . . . . . . . . . . . . . . . . .
Results for the prostate planning study. The range covered by the database is
shown and the values of the corresponding clinical reference plan are reported
in brackets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Results for the prostate planning study. . . . . . . . . . . . . . . . . . . . . .
Effective dimension and corresponding indicator values for all cases. Another
case (p7b) is included in which OAR3 and OAR4 are moved towards OAR2.
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