Evaluation of the adjoint equation based algorithm for elasticity imaging

Evaluation of the adjoint equation based algorithm for elasticity imaging
INSTITUTE OF PHYSICS PUBLISHING
PHYSICS IN MEDICINE AND BIOLOGY
Phys. Med. Biol. 49 (2004) 2955–2974
PII: S0031-9155(04)77153-5
Evaluation of the adjoint equation based algorithm for
elasticity imaging
Assad A Oberai1, Nachiket H Gokhale1, Marvin M Doyley2
and Jeffrey C Bamber3
1 Department of Aerospace and Mechanical Engineering, Boston University, Boston,
MA 02215, USA
2 Thayer School of Engineering, Dartmouth College, and Department of Radiology,
Dartmouth-Hitchcock Medical Center, Lebanon, NH, USA
3 Joint Department of Physics, Royal Marsden Hospital and Institute of Cancer Research, Sutton,
Surrey, UK
E-mail: [email protected]
Received 3 March 2004
Published 17 June 2004
Online at stacks.iop.org/PMB/49/2955
doi:10.1088/0031-9155/49/13/013
Abstract
Recently a new adjoint equation based iterative method was proposed for
evaluating the spatial distribution of the elastic modulus of tissue based on the
knowledge of its displacement field under a deformation. In this method the
original problem was reformulated as a minimization problem, and a gradientbased optimization algorithm was used to solve it. Significant computational
savings were realized by utilizing the solution of the adjoint elasticity equations
in calculating the gradient. In this paper, we examine the performance of this
method with regard to measures which we believe will impact its eventual
clinical use. In particular, we evaluate its abilities to (1) resolve geometrically
the complex regions of elevated stiffness; (2) to handle noise levels inherent
in typical instrumentation; and (3) to generate three-dimensional elasticity
images. For our tests we utilize both synthetic and experimental displacement
data, and consider both qualitative and quantitative measures of performance.
We conclude that the method is robust and accurate, and a good candidate for
clinical application because of its computational speed and efficiency.
1. Introduction
Elastography is an emerging new imaging technique with applications in detecting breast and
other cancers (Ophir et al 1999, Chenevert et al 1998, Bamber et al 2002, Garra et al 1997,
Weaver et al 2001, Skovoroda et al 1995, Sumi et al 1995), atherosclerosis (Ryan and Foster
1997, de Korte et al 1997, 1998, 1999, 2000a, 2000b, Cespedes et al 2000), and deep vein
0031-9155/04/132955+20$30.00 © 2004 IOP Publishing Ltd Printed in the UK
2955
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A A Oberai
thrombosis (Emelianov et al 2002). This technique is based on the premise that the elastic
modulus (Young’s or shear modulus) of different kinds of tissue may be sufficiently different
so as to distinguish them in an elastic image. In particular, this may be true for certain kinds
of cancers, which tend to be stiffer (often by a factor of ten times) than the surrounding tissue
(see Wellman et al (1999) for example).
To arrive at an image of the elastic modulus of tissue the following steps are performed
(here we consider elastography via ultrasound only). (1) An ultrasound image of the specimen
in an undeformed state is recorded. (2) The specimen is deformed and another image (in
practice, a series of images) is recorded. (3) The pre- and post-deformation images are
registered to yield a displacement field in the entire specimen. (4) This displacement field is
either (a) numerically differentiated to yield a strain image or (b) used in solving a problem
to evaluate the spatial variation of the elastic modulus. We refer to the problem in step 4b,
that of determining the elastic modulus based on the knowledge of the displacement field, as
the inverse elasticity problem. This is in contrast to the forward elasticity problem, which
involves determining the displacement field from the knowledge of the elastic modulus.
When a strain image is produced (that is step 4a is performed), it is assumed that at any
given spatial location the elastic modulus is inversely proportional to strain, and hence this
image may be interpreted as an image of the compliance (reciprocal of stiffness). It is worth
noting that this interpretation is valid only for simple modulus and stress distributions. In
order to construct an accurate elasticity image the inverse problem must be solved (that is step
4b must be performed). However, the solution of this problem is challenging and requires far
greater computational effort. In this paper, we describe and evaluate an efficient technique for
solving this problem.
Broadly speaking, there are two approaches that are utilized for calculating the
elastic modulus once the displacements are known. In the direct approach, equations
of equilibrium for a linear elastic solid are interpreted as an equation for the elastic
modulus (Raghavan and Yagle 1994, Skovoroda et al 1995, Sumi et al 1995, Plewes et al
2000, Bishop et al 2000). This leads to a partial differential equation for the elastic modulus
in which the known strain fields (strains are the spatial derivatives of displacements) appear
as coefficients. While this approach promises to be the most efficient solution of the
inverse problem, it suffers from the following drawbacks: (1) its accuracy and resolution
are determined by the least accurate component of the displacement field. In ultrasound,
this is the component of displacement perpendicular to the axis of the transducer (the lateral
component), and is typically an order of magnitude less accurate than the component along
the axis of the transducer (the axial component). (2) It utilizes strains and their derivatives,
and hence requires differentiating noisy displacement data which may negatively impact its
accuracy. (3) It implicitly assumes a degree of continuity for strain and elastic modulus
distributions, and as a result has difficulties in coping with sharply varying elastic properties.
Iterative methods overcome the drawbacks of direct methods, but tend to be
computationally intensive. In an iterative method the inverse problem is posed as a
minimization problem. The quantity to be minimized, called the functional, is a measure
of the difference between the measured and a predicted displacement field. The predicted
displacement field is determined from an assumed elastic modulus distribution by solving the
equations of equilibrium. The goal is to find an elastic modulus distribution that produces a
predicted displacement field that is closest to the measured displacement field.
To date, most iterative methods for solving the inverse elasticity problem have relied
on some form of the Gauss–Newton algorithm (see, for example, (Kallel and Bertrand 1996,
Van Houten et al 1999, Doyley et al 2000)). This algorithm involves evaluating a Jacobian
matrix. This matrix represents changes in the displacement field at a given location due to
Evaluation of the adjoint equation based algorithm for elasticity imaging
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a small change in the elastic modulus at an independent location. The computational costs
associated with forming this matrix are equivalent to solving the forward elasticity problem
as many times as there are unknown elastic parameters. For a 50 × 50 pixel elasticity image,
this number (denoted by N in this paper) is equal to 2.5 × 103 . Typically, N ≈ 103 –105 , and
these costs may become computationally prohibitive.
In Oberai et al (2003), we proposed a new iterative method which did not rely on
the Jacobian matrix. This method was based on utilizing a gradient-based optimization
algorithm (any of the quasi-Newton algorithms) and using the adjoint elasticity equations
to efficiently calculate the gradient. The predominant computational costs for this method
were comparable to solving two forward elasticity problems, independent of N, the number
of elastic parameters. In the same reference, we tested the performance of this method using
synthetically generated displacement data for two-dimensional circular elastic inclusions, and
found it to be promising. In this paper, motivated by the eventual clinical application of this
method, we consider additional, more realistic tests. First, using synthetic data we evaluate
its ability to discern geometrically complex regions of elevated stiffness. Second, using
displacement data acquired on a tissue-mimicking phantom, we evaluate its performance in
the presence of instrumentation-induced noise. Finally, since in practice modulus distributions
and deformation patterns are never perfectly two dimensional, we consider its performance in
generating three-dimensional elasticity images.
The layout of the remainder of this paper is as follows. In the following section, we
state forward and inverse elasticity problems, and derive the method used to solve the inverse
problem. Our derivation of the method is different from Oberai et al (2003), and perhaps
simpler, as it does not use the concept of a Lagrangian to derive the adjoint elasticity equations.
Thereafter, we present results that validate the performance of the proposed method for twoand three-dimensional problems and tissue phantom data. We end with concluding remarks
and directions for future work.
2. Numerical method
In this section, we first consider forward and inverse elasticity problems. Then we describe an
efficient adjoint equation based method for solving the inverse problem. When compared with
Oberai et al (2003), we provide an alternate and simpler derivation of this method. Thereafter
we compare it with other iterative methods. Finally, we provide guidelines for choosing the
regularization parameter that appears in this method.
Throughout the description we denote a scalar by a lowercase symbol, a vector by a
lowercase bold symbol and a matrix, by an uppercase bold symbol. For example, the symbols
a, a and A represent a scalar, a vector and a matrix, respectively. Elements of a vector a are
denoted by ai , and elements of the matrix A are denoted by Aij .
2.1. The forward problem
Let u represent the nodal vector of displacements corresponding to a finite element solution
of the elasticity problem. The equation for determining u is given by
n
Kij (µ)uj = fi .
(1)
j =1
This equation is referred to as the forward elasticity problem. It involves determining u,
given the vector of material properties (µ), and appropriate boundary conditions. In the above
equation
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A A Oberai
(i) Indices i and j vary from 1 to n, where n is the total number of displacement unknowns
in the model.
(ii) The stiffness matrix K depends on µ, the vector containing the nodal values of the shear
modulus. The total number of parameters used to represent the shear modulus (length of
the vector µ) is N.
(iii) The vector f which appears on the right-hand side contains contributions from prescribed
traction and displacement boundary conditions.
(iv) The precise definitions of K and f depend on the discretization scheme used to solve
the problem. In this study, we have used the finite element formulation for a nearly
incompressible linear elastic solid described in Hughes (2000). For details on its
implementation the reader is referred to this reference and to Oberai et al (2003). This
formulation provides a stable numerical method for evaluating the response of nearly
incompressible materials, that is materials for which the Lame parameter λ is large. In
all examples considered in this paper we have set λ = 106 µ, where the components of µ
are O(1).
Given a description of the material properties and the appropriate boundary conditions,
(1) may be solved to determine the displacement field. Symbolically this solution may be
written as
n
Kj−1
(2)
uj =
i (µ)fi .
i=1
In the above equation K −1 , sometimes referred to as the compliance matrix, is the inverse of
the matrix K. Note that Kij−1 is not the reciprocal of Kij .
2.2. The inverse problem
We now consider the inverse elasticity problem. We assume that we have a partial knowledge
of the vector of measured displacements. The complete vector is denoted by um . We assume
that we do not know all components of um everywhere in the domain, instead we only know
certain components on a subset of the domain. Thus instead of um , we know Dum , where D
is an n × n diagonal matrix with ones in the diagonal for rows corresponding to the known
displacements, and zeros for rows corresponding to unknown displacements. As mentioned
above, the signal-to-noise ratio for lateral displacements is usually very much poorer than that
for the axial displacements, to the point that lateral displacement estimates may contribute
negligible information to the reconstruction and including them may make the reconstruction
noisier. For all examples considered in this study, the matrix D is therefore selected to
represent the fact that we know the axial component of the displacement everywhere in the
domain, and do not know the lateral component anywhere.
The inverse elasticity problem may be stated as follows: given some components of
the vector of the measured displacement field, denoted by Dum , and appropriate boundary
conditions, determine the corresponding shear modulus distribution µ. This problem may be
recast into the following minimization problem:
Find the vector µ, that minimizes the functional π given by
n
N
α 1 m
ui − um
D
u
+
M
D
−
u
µK RKJ µJ
π(µ) =
ki
kl
lj
j
i
j
2 i,j,k,l=1
2 K,J =1
(3)
where u is the solution to the forward elasticity problem (i.e. it satisfies (2)).
Observe that the dependence of π on µ is explicit, due to the second term on the RHS of
(3), and implicit through the dependence of u on µ via (2).
Evaluation of the adjoint equation based algorithm for elasticity imaging
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The matrix M that appears in (3) is symmetric and positive definite. On a uniform
grid, it may be approximated by the identity matrix. In that case, it is easy to see that the
first term in (3) is a measure of the discrepancy in the predicted and measured displacement
fields. This term is zero when the two fields are identical. The second term on the RHS of
(3) is a regularization term. It is included to smooth the solution in the presence of noise in
measurements. The scalar α, often called the Tikhonov parameter, represents the degree of
regularization. For a description of the relevance of this term and the choice of α, the reader is
referred to section 2.5 and to Colton and Kress (1998) and Isakov (1998). The regularization
matrix R in (3) is typically symmetric and positive semi-definite. In this paper, we choose R
to approximate the Laplacian, in which case the regularization term penalizes large variations
in the spatial gradient of the shear modulus.
2.3. Solution of the inverse problem
The minimization problem described above may be solved using one of a large number
of optimization algorithms. These algorithms are often classified based on the amount
of information they require at each iteration. For example, some algorithms, such as
genetic algorithms, require only the value of the functional π at every iteration. Some
algorithms require the value of the functional and the gradient vector g (defined below),
which contains the derivative of the functional with respect to the optimization parameters.
These include algorithms such as steepest descent and most quasi-Newton algorithms. Some
algorithms, such as Newton’s iterations, require the value of the functional, the gradient and
the Hessian matrix H, which contains the second derivative of the functional with respect
to the optimization parameters. As might be expected, algorithms which utilize the Hessian
have higher asymptotic convergence rates than algorithms which utilize the gradient, which
in turn converge faster than algorithms which require only the functional. The interested
reader is referred to Gill et al (2000) for a detailed description of these algorithms. In our
study, following Oberai et al (2003), we restrict our attention to algorithms which require
the functional and the gradient. Several well-known methods such as steepest-descent, and
other quasi-Newton methods such as BFGS (Broyden–Fletcher–Goldfarb–Shanno) and DFP
(David–Fletcher–Powell) (see Gill et al (2000) for descriptions) belong to this category. In
implementing these methods the main computational cost is associated with determining
the gradient vector. In the following paragraphs we first describe a straightforward, though
computationally inefficient algorithm for calculating the gradient vector. Thereafter we derive
an efficient, adjoint equation based approach.
The components of the gradient vector g, are given by
∂π
I = 1, . . . , N.
(4)
gI =
∂µI
The straightforward way of computing g is to differentiate π in (3) with respect to µ. This
yields
gI =
n
N
∂uj
ui − um
D
M
D
+
α
µK RKI
ki kl lj
i
∂µI
i,j,k,l=1
K=1
I = 1, . . . , N
(5)
where we have assumed that M and R are all symmetric. This expression for gI requires ui
(given by (2)), and ∂uj /∂µI . To calculate ∂uj /∂µI we differentiate (1) with respect to µI .
This yields the following relation:
n
∂uj
∂Kmp
=−
Kj−1
up
I = 1, . . . , N
(6)
m
∂µI
∂µI
m,p=1
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A A Oberai
−1
where Kj−1
m are components of K , inverse of the matrix K. This expression for evaluating
∂uj /∂µI involves N solutions
of the forward elasticity problem, each with a different righthand side given by fm = np=1 (∂Kmp /∂µI )up , I = 1, . . . , N. Thus the straightforward way
of evaluating the gradient involves computing ui and ∂uj /∂µI , and using these in (5). This
approach requires N + 1 solves with the forward elasticity operator: N for calculating ∂uj /∂µI
and one for calculating ui . For a large number of elastic modulus parameters N ≈ 103 –105 ,
the cost of performing these solves grows rapidly and renders the method described above
computationally prohibitive.
A significant reduction in this cost may be made if it is recognized that the term ∂uj /∂µI
is needed only to evaluate the first term on the right-hand side of (5). Thus utilizing (6) in (5)
we arrive at an alternative definition of g, viz.,
n
N
∂Kmp
gI =
wm
up + α
µK RKI
∂µI
m,p=1
K=1
I = 1, . . . , N
(7)
where w is obtained by solving
n
m=1
n
ui − um
Kmj wm = −
i Dki Mkl Dlj .
(8)
i,k,l=1
So now to calculate g we first evaluate u by solving (1), then evaluate w by solving (8), and
use these in (7). This leads to a method that requires two solves of the forward elasticity
problem, independent of the number of material parameters N.
Due to the symmetry of the tensor of elastic properties, the equations for linear elasticity
are self-adjoint and as a result for most numerical methods K T = K. Thus equation (8),
which involves the transpose of K is transformed to
n
m=1
n
Kj m w m = −
ui − um
i Dki Mkl Dlj
(9)
i,k,l=1
which involves K.
The final algorithm to determine g, which we shall call algorithm 1, is as follows:
Algorithm 1.
(i) Determine u by solving (1).
(ii) Determine w by solving (9).
(iii) Determine g by using u and w in (7).
Remark. The equation for w (i.e. (8)) involves the transpose of the matrix that appears in
the equation for u (i.e. (1)). If a matrix is viewed as an operator that acts on vectors, then the
transpose of the matrix represents the adjoint of the operator. Indeed, the connection between
the operator in the equation for w and the operator in the equation for u, can be established at
several levels, and in every case the operator on w, is the adjoint of the operator on u. Hence
algorithm 1 is referred to as the adjoint equation based algorithm. These ideas are explored in
more detail in Oberai et al (2003).
2.4. Cost of solving the inverse problem
In this section, we estimate the computational costs of solving the inverse problem using three
different iterative methods: (1) A gradient-based method where the adjoint approach is used
to calculate the gradient (the proposed approach); (2) A gradient-based method where the
straightforward approach is used to calculate the gradient and (3) the Gauss–Newton method
Evaluation of the adjoint equation based algorithm for elasticity imaging
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z
1
x
n
1/2
1
x
n
1/3
y
y
n
n
n
2/3
Figure 1. Node-numbering scheme for the two- and three-dimensional problems. The node
number increases by 1 in the x direction and by n1/2 in the y direction for the two-dimensional
case. For the three-dimensional case it increases by 1 in the x direction, by n1/3 in the y direction
and by n2/3 in the z direction.
used in Kallel and Bertrand (1996), and Doyley et al (2000). We determine the dependence
of the cost of performing one iteration of these methods on the number of optimization
variables (N).
We first estimate the cost of solving the forward elasticity problem, as it forms a major
component of the algorithms we have considered. This cost depends on the method used
to solve the resulting system of linear equations (Gaussian elimination or Krylov subspace
methods such as conjugate gradients) and on the dimensionality of the problem (two or three
dimensions), and hence must be determined accordingly. In order to simplify the comparison
we consider the problem on a square domain in two dimensions and a cubical domain in three
dimensions. We denote by N the total number of shear modulus parameters and by n the total
number of displacement unknowns. The number of displacement unknowns per edge is n1/2
and n1/3 in two and three dimensions, respectively. Note that n is approximately equal to 2N
in two dimensions and to 3N in three dimensions.
Gaussian elimination. We first consider the use of Gaussian elimination in solving the
system of linear equations. Recall that the cost of an LU factorization for a banded, n × n
matrix, is O(b2 n), where b is the bandwidth of the matrix (see Golub and Van Loan (1996)
for example). Using the node numbering scheme depicted in figure 1, we conclude that the
average bandwidth b, scales as n1/2 in two dimensions and n2/3 in three dimensions. Further,
as pointed out earlier, n scales linearly with N. Thus b = O(N 1/2 ) in two dimensions and
b = O(N 2/3 ) in three dimensions, and the cost of an LU factorization is O(N 2 ) in two
dimensions and O(N 7/3 ) in three dimensions. The cost of a back-substitution is O(bn) (see
Golub and Van Loan (1996)). Using the estimates developed above this cost is O(N 3/2 ) and
O(N 5/3 ) in two and three dimensions, respectively.
Krylov subspace methods. Next we consider the use of a Krylov subspace method for
solving the resulting system of linear equations. This class of methods includes solvers such
as conjugate gradients, GMRES and QMR (Saad 1995). For an efficient Krylov subspace
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Table 1. Estimate of leading-order computational costs per iteration for various iterative methods.
Method
Gradient-based
(adjoint approach)
Gradient-based
(straightforward approach)
Gauss–Newton
Gaussian elimination
two dimensions
Gaussian elimination
three dimensions
Krylov subspace
methods
O(N 2 )
O(N 7/3 )
O(cN )
O(N 5/2 )
O(N 8/3 )
O(cN 2 )
O(N 3 )
O(N 3 )
O(cN 2 )
method the number of iterations to convergence (denoted by c) is independent of, and much
smaller than, the dimension of the matrix (n). In addition, at each iteration major computational
costs are incurred in computing the product of the given coefficient matrix with a vector. For a
sparse matrix (such as the one considered here), this cost is O(n), or O(N ) (since n = O(N )).
Thus the cost of solving one system of equations is O(cN ).
With these estimates in hand, we are ready to compute the cost of performing one iteration
with the algorithms selected for solving the inverse problem.
1. Gradient-based algorithm with the adjoint approach. This approach requires the solution
of two forward elasticity problems per iteration. If Gaussian elimination is used to solve
these problems, it requires one LU factorization followed by two back substitutions.
Hence the costs are O(N 2 ) + O(N 3/2 ) in two dimensions and O(N 7/3 ) + O(N 5/3 ) in three
dimensions. On the other hand, if a Krylov subspace method is used the costs are O(cN )
(see table 1 for a summary).
2. Gradient-based algorithm with the straightforward approach. The straightforward
approach of calculating the gradient involves N + 1 solves per iteration. If Gaussian
elimination is used to solve these systems, it would require one LU factorization followed
by N + 1 back substitutions. Thus the costs for this algorithm are O(N 5/2 ) + O(N 2 ) in
two dimensions and O(N 8/3 ) + O(N 7/3 ) in three dimensions. If, on the other hand, a
Krylov subspace method is used, the costs are O(cN 2 ).
3. Gauss–Newton type methods. Using these method updates to the shear modulus, denoted
by µ, are obtained by solving the following equation:
(J T J + R)µ = r,
(10)
where r is the residual based on the previous estimate, R is a regularization matrix and
J is the Jacobian matrix. The entries of this matrix are given by
∂uj
Jj I =
.
(11)
∂µI
To evaluate this matrix, (6) is utilized, which involves N solves of the forward elasticity
problem. This implies that at each iteration N + 1 solves are required. Thus the
costs of computing the matrix J are the same as for computing the gradient using
the straightforward approach. However in this method, in addition, the linear system of
equations given by (10) has to be solved at each iteration. In this system the coefficient
matrix is dense (bandwidth b = O(N )), hence the costs of solving this system using
Gaussian elimination scale as O(b2 N ) = O(N 3 ). These costs scale as O(cN 2 ) when an
iterative solver is used because the cost of computing a matrix–vector product is O(N 2 )
since the matrix is dense. Thus the total costs per iteration for this algorithm, when
Gaussian elimination is used, are O(N 3 ) + O(N 5/2 ) + O(N 2 ) in two dimensions and
O(N 3 ) + O(N 8/3 ) + O(N 7/3 ) in three dimensions. If on the other hand Krylov subspace
methods are used these costs are O(cN 2 ).
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Remark.
(i) It is clear from the preceding analysis (see table 1) that leading-order costs for the gradientbased method with the adjoint approach are smaller than the other two methods. From
this table we also conclude that perhaps the most efficient iterative approach to solving
the inverse problem involves using the adjoint approach in conjunction with a Krylov
subspace method.
(ii) It is worth noting that the difference in the costs between the straightforward approach
and the adjoint approach to calculating the gradient is the cost of N − 1 back substitutions,
which is a non-negative number for N 1. Thus the costs associated with the adjoint
approach are guaranteed to be less than those for the straightforward approach for any
non-trivial problem (N 2).
(iii) In defence of the Gauss–Newton method it should be pointed out that it utilizes an
approximation to the Hessian that becomes increasingly accurate as the iterative solution
approaches the true solution. Hence it may converge faster than quasi-Newton methods
that utilize only the gradient. Thus even though the costs per iteration for this method are
high, the total number of iterations may be smaller.
2.5. Choice of the regularization parameter
We now describe a principle for determining the regularization parameter α. For examples
considered in this paper, the matrix R in (3) is an approximation to the Laplacian operator.
This choice penalizes large gradients in the shear modulus field, and results in solutions which
are smooth. In order to determine the regularization parameter α, it is assumed that the
measured displacement Dum has a known level of noise. For our purpose, we quantify the
noise in the system as
= D(um − ū)/Dum (12)
where ū is the measurement with no noise, and for a vector v, the symbol v =
i vi vi is
used to denote its magnitude. Then, in accordance with the theory of residues due to Morozov
(see Colton and Kress (1998), Isakov (1998) for details), α should be chosen to be the largest
real number that yields a predicted shear modulus µ and a corresponding displacement vector
u, which satisfy
D(um − u) = CD(um − ū)
(13)
where C ≈ 1. That is the difference between the measured and predicted displacement
fields is approximately equal to the difference between the measured and the ideal, noiseless
displacement field. The rationale behind this choice is described next.
First, it is important to recognize the connection between α and D(um − u). For this
we turn to (3). We observe that α controls the relative importance of the regularization term
in the functional. While a large value of α implies that the optimal solution µ will be regular,
it also means that the quantity D(um − u) may not be small. On the other hand, a small
value of α may result in a solution for which D(um − u) is small, but the solution (that is
µ) is not necessarily regular or smooth.
Second, using the triangle inequality we expand the total error D(ū − u) (which is
unknown, since ū is unknown) as follows:
D(ū − u) D(um − u) + D(um − ū).
(14)
In (14), the first term on the right-hand side represents error in the reconstruction, while the
second term represents error in the measurement. In the previous paragraph, we described
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A A Oberai
how α affects the first term of this equation. Morozov’s principle (13) states that by selecting
α appropriately we should allow a level of smoothness in the final solution that makes these
two terms more or less equal. In other words, while increasing α adds to the smoothness
of the problem, and hence makes it easier to solve, to increase it beyond a point where the
reconstruction error is much greater than the measurement error is counterproductive because
in that case the reconstruction error dominates the total error. Also decreasing α to make the
reconstruction error much smaller than the measurement error at the expense of the loss of
stability of the numerical problem is not desirable, because the total error is always bounded
from below by the measurement error.
We now describe how (13) may be used to estimate C when working with experimentally
acquired displacement data. The term on the left-hand side of (13), which is a measure of
the reconstruction error is easily calculable as both Dum and Du are known. The term on
the right-hand side, that is D(um − ū), which is a measure of the measurement error, can
only be estimated. For example, if for a given measurement system the ratio of signal to
noise (1/) as defined in (12) is known, then this term is given by D(um ), and can be
calculated. Once this is done, we can obtain an estimate of C, and adjust α accordingly until
we obtain a desirable value of C.
In cases where is not known, Morozov’s principle described above cannot be used and
a popular choice is to use the L-curve (see Vogel (2002) for a description of this method).
3. Validation
In this section, we evaluate the performance of the method developed in the previous
section on certain benchmark problems. We have designed these problems so as to test
the eventual clinical applicability of the proposed method. In the first problem, using
synthetically generated displacement data, we test the ability of the proposed method to
reconstruct geometrically irregular regions of elevated stiffness. In the second problem, by
using displacement data acquired for a tissue-mimicking phantom we test its performance in
the presence of instrumentation induced noise. In the third problem, we test its ability to
generate three-dimensional elasticity images.
We solve the benchmark problems using a limited memory BFGS algorithm (Zhu et al
1997).
The computer program that implements this algorithm is obtained from
http://www-fp.mcs.anl.gov/otc/Tools/LBFGS-B. We evaluate the gradient at each
iteration using the adjoint equation based approach described in the previous section
(algorithm 1). We solve the forward and the adjoint elasticity problems using the finite
element method. In BFGS, as in any quasi-Newton method, within each iteration several calls
are made to the routine that calculates the functional and the gradient. We term each such call
a sub-iteration. Thus 30 sub-iterations correspond to 30 evaluations of the functional and the
gradient and in our case, 30 solves of the forward and adjoint problems.
The BFGS algorithm is terminated once it is found that the relative drop in the functional
over the last five iterations, measured by
π(µi ) − π(µi−5 )
π(µi−5 )
(15)
is less than 0.01. Here µi is the guess for the shear modulus distribution after i iterations.
The input parameters for the benchmark problems are listed in table 2. These include the
size of the finite element mesh, number of parameters used to represent the shear modulus and
the noise level () in the displacement (see (12) for the definition of ). Note that it is possible
to compute exactly only for those cases where the noiseless displacement field is known, that
Evaluation of the adjoint equation based algorithm for elasticity imaging
2965
Table 2. Input parameters for the example problems.
Problem
Mesh
Unknowns N
Noise Level Inclusion I
Inclusion II
Tissue phantom
Three-dimensional inclusion
127 × 95
127 × 95
30 × 30
30 × 30 × 30
12, 161
12, 161
931
29, 791
1.0%
1.0%
3.0%
1.0%
Table 3. CPU time for solving the example problems.
Problem
CPU time per
sub-iteration (s)
Approximate
total time (min)
Inclusion I
Inclusion II
Tissue phantom
Three-dimensional inclusion
16.1
16.1
0.8
85.8
10.7
10.7
0.5
57.2
is, the measured displacement field is generated synthetically. For the tissue phantom example
considered, this number has been estimated. All modulus reconstructions were performed on
a PC, with a single 1 MHz Intel Pentium processor. A homogeneous modulus distribution was
used as an initial guess in all cases. In table 3, we have listed the CPU time (in s) it took to
perform a single sub-iteration for each problem. Since most of the problems converged after
approximately 40 sub-iterations, the total time for each reconstruction can be estimated by
multiplying the time per sub-iteration by 40.
3.1. Two dimensional problems with synthetic displacement data
In Oberai et al (2003), the authors have tested the adjoint equation based method for identifying
simple two-dimensional circular regions of elevated stiffness. In this section we consider
complex shapes, which are more likely to occur in tissue. We generate synthetic displacement
data by solving the forward elasticity problem corresponding to a known target modulus
distribution using the finite element method, and then adding Gaussian white noise to the
displacement field. Similar experiments with more realistic models for noise (Chaturvedi et al
1998) will be performed in the future. In order to maintain consistency with experimental
conditions, we utilize only the axial component of the displacement field in the reconstruction.
A schematic of the two-dimensional problem is shown in figure 2. The length of the
sides Lx = Ly = 1 unit. The boundary conditions are as follows: on the x2 = 0 edge, we
prescribe a displacement of 0.05 units in the x2 direction and no traction in the x1 direction on
the x2 = Ly edge, we prescribe zero displacement in the x2 direction and no traction in the x1
direction and on the x1 = 0 and x1 = Lx edges, we prescribe zero traction in both x1 and x2
directions. To prevent rigid translation of the body, we constrain the centre point not to move
in the x1 direction. As mentioned earlier, only the axial (x2 ) component of displacement is
assumed to be known. Further it is assumed that the value of the shear modulus along the
x2 = 0 edge is fixed, but unknown. The uniqueness of this inverse elasticity problem under
these conditions has been examined in Barbone and Bamber (2002), where it is shown that
the reconstructed modulus is unique up to the value prescribed on the x2 = 0 edge.
In the first example we consider a single, bean-shaped inclusion. The ratio of the shear
modulus of the inclusion to the background is 5.3, which could be representative of ductal
carcinoma in situ in glandular tissue in the breast under a strain of 15% (Wellman et al 1999).
2966
A A Oberai
X2
(0,Ly )
Ω
(Lx ,0)
X1
APPLIED VERTICAL DISPLACEMENT
Figure 2. Schematic of the example problem.
We test the effectiveness of the proposed methodology in recovering the shape and contrast
of the inclusion. The elastic modulus is represented by 12 161 parameters, leading to an
inverse problem with a large number of unknowns. The target shear modulus distribution
is shown in figure 3(a). Figure 3(b) represents the reconstructed modulus in the absence of
noise, and figure 3(c) displays the reconstructed modulus with = 1.0%. From these images
we conclude that the reconstruction has accurately captured the shape of the inclusion. In
figure 3(d), the value of the shear modulus as a function of distance along A–A is shown. We
conclude that the contrast as well as the sharp boundary of the inclusion have been accurately
recovered.
As a second example, we consider the same inclusion but with a thin finger-like protrusion
(see figure 4(a)). The ratio of the shear modulus of the inclusion to the background is increased
to 8.6, which represents the upper range of modulus contrast of cancer in glandular tissue in
the breast under a strain of 15% (Wellman et al 1999). We are interested in determining if
the thin protrusion is resolved in the reconstruction. As in the previous problem the elastic
modulus is represented by 12 161 parameters. In figures 4(a)–(d), we note that the finger-like
protrusion is well resolved in the reconstruction without noise, whereas it is somewhat faded
in the presence of noise. An interesting feature of the noiseless reconstruction is an ‘artefact’
in the form of a streak running through the protrusion. We believe that this streak is absent
from the reconstruction with noise because of the effect of the regularization term, which
would suppress any feature with a sharp gradient. For the noiseless reconstruction, we have
found that this streak persists even when the functional drops to 4 × 10−7 times its initial value.
In this case, the two distinct modulus distributions (the target distribution without the streak,
and the reconstructed distribution with the streak) with displacement fields that are identical
up to numerical precision, highlight the ill-posedness of the inverse elasticity problem, even
when it possesses a unique solution.
Evaluation of the adjoint equation based algorithm for elasticity imaging
2967
'
(a)
(b)
10
9
Reconstruction with noise
8
Shear modulus
7
Target
6
5
4
3
2
1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Distance
(c)
(d)
Figure 3. Shear modulus reconstructions for the bean-shaped inclusion. (a) Target shear modulus
distribution; (b) reconstruction with no noise; (c) reconstruction with = 1%; (d) modulus along
A–A .
3.2. Tissue phantom
In this example, we reconstruct the shear modulus distribution of a tissue-mimicking phantom
using the adjoint equation based method. The goal is to evaluate the performance of this method
in the presence of noise induced by the instrumentation. We first describe the processes of
phantom fabrication and displacement measurement, and thereafter present the shear modulus
reconstruction results.
3.2.1. Phantom fabrication. An elasticity phantom (70 mm (axial) by 70 mm (lateral) by
90 mm (elevational)) containing a single cylindrical inclusion (15 mm diameter by 90 mm
long) was manufactured from porcine skin gelatin (Type A, approximately 175 bloom, Sigma
2968
A A Oberai
B
B'
(a)
(b)
15
Shear modulus
Target
Reconstruction
with noise
10
5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Distance
(c)
(d)
Figure 4. Shear modulus reconstructions for the bean-shaped inclusion with a thin protrusion.
(a) Target shear modulus distribution; (b) reconstruction with no noise; (c) reconstruction with
= 1%; (d) Shear modulus along B–B .
Chemical Co, MO), distilled water (18 MW), formaldehyde (Sigma Chemical Co, MO) and
ethylenediamine tetraacetic acid. The surrounding background tissue and inclusion were
manufactured from 6% and 24% by weight gelatin, respectively. Polyethylene granules
(∼119 mm, 2% by weight) were added to both materials to create an acoustic scattering
structure to enable ultrasonic tracking of medium displacements.
3.2.2. Data acquisition. Imaging was performed using the experimental elastographic
imaging system described in Doyley et al (2000), consisting of an ACUSON 128 XP
commercial ultrasound scanner which was equipped with a 7.5 MHz (L7) linear transducer
array and an external intermediate frequency (IF) interface which provided direct access to
Evaluation of the adjoint equation based algorithm for elasticity imaging
2969
the raw analogue IF echo signal and three TTL control signals (i.e. start of frame, start of line
and a 10 MHz internal pixel clock).
The phantom was compressed by 2% of its undeformed height (i.e. 70 mm) in steps of 0.2%
between two large compression plates (150 mm2 ) using a computer controlled mechanical
deformation system. Constraining the motion of the phantom in the lateral dimension by
employing a U-shape clamp minimized lateral decorrelation effects. The upper and lower
surfaces of the phantom were lubricated with corn oil to simulate a slip boundary condition.
For the purpose of this investigation the scanner was set to operate with dynamic receive
focus and a single transmit focus that was positioned approximately in the centre of the
phantom (i.e. a depth of 35 mm). All IF echo frames were digitized to 10 bits at a sampling
frequency of 20 MHz using a MV-1000 frame grabber (Mu-tech Corp., MA) and stored on an
external Pentium based computer for offline-processing.
3.2.3. Displacement estimation. The digitized IF echo frames were converted to RF and
tracked for displacement estimation as described in Doyley et al (2000). Axial displacement
images were computed by performing a two-dimensional cross-correlation analysis on
consecutive pairs of RF echo frames using a 0.75 mm × 2.9 mm window length by width,
which was shifted by 50% in both the axial and lateral directions. The displacement images
were averaged to produce a composite displacement image.
3.2.4. Shear modulus reconstruction. The schematic of the problem is given in figure 2, with
Lx = 38 mm and Ly = 70 mm. The boundary conditions are the same as in section 3.1. The
details of the problem appear in tables 2 and 3. For this problem was estimated to be 3%.
This estimate was obtained by making ultrasound measurements and displacement estimations
on a homogeneous phantom. It was assumed that the noise level does not change significantly
for the inhomogeneous case. Since this is a relatively small problem with 931 unknowns, it
took about 30 s to calculate the converged modulus distribution. The reconstructed modulus
is shown in figure 5. We observe that the location of the inclusion is captured accurately and
the artefacts are minimal. In figure 6 the shear modulus distribution along C–C is shown. We
observe that when compared with synthetic data, the reconstruction appears to be smeared.
This is due to a larger value of the regularization parameter. The ratio of the shear modulus
of the inclusion to the background is seen to be in good agreement with the actual value
of 4.2, obtained by performing an independent test with an INSTRON mechanical testing
machine.
3.3. Three-dimensional inclusion
Finally, we assess the performance of the adjoint equation based method in reconstructing a
three-dimensional modulus distribution. The motivation for this problem lies in recognizing
that purely two-dimensional displacement and modulus distributions are only approximations
to the actual state, which is almost always three dimensional. In many practical cases, this
approximation may in fact be completely invalid. With this background, we ask the following
question: given the axial component of the displacement on a three-dimensional grid, can the
proposed method reconstruct an inherently three-dimensional modulus distribution?
To answer this question we consider a target shear modulus distribution in the form of a
spherical inclusion which is ten times stiffer than its surroundings. The domain of the problem
is a cube of unit dimension. The top face of the cube is compressed by 5% and no displacement
is allowed in the lateral directions. The bottom face is held fixed, with no displacement in any
direction. All other faces (the side faces) are traction free. The ‘measured’ displacement data
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A A Oberai
Figure 5. Reconstructed shear modulus for the tissue-mimicking phantom.
5.5
5
4.5
Shear modulus
4
3.5
3
2.5
2
1.5
1
0.5
0
5
10
15
20
25
30
35
40
Distance
Figure 6. Shear modulus along C–C for the tissue-mimicking phantom.
are generated using the target modulus distribution in a finite element program, and by adding
Gaussian noise to the resulting displacement field. The level of noise is given by = 1%.
In the reconstruction algorithm, only the axial component of the displacement field (along the
Evaluation of the adjoint equation based algorithm for elasticity imaging
2971
Figure 7. (a) Target shear modulus on the y = 0 plane. (b) Reconstruction shear modulus on the
y = 0 plane. (c) Target shear modulus on the x = 0 plane. (d) Reconstruction shear modulus on
the x = 0 plane.
compressive direction) is utilized. For this problem, it is not clear whether a unique solution
exists, and as a result, we are not guaranteed that the solution to which we converge is the only
possible solution.
The target and reconstructed shear modulus along two mutually perpendicular planes
passing through the centre are shown in figures 7(a)–(d). We observe that the spherical shape
2972
A A Oberai
15
Target
Shear modulus
Reconstruction with noise
10
5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
distance
Figure 8. Shear modulus along D–D for the three-dimensional problem.
15
Shear modulus
Target
Reconstruction with noise
10
5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
distance
Figure 9. Shear modulus along E–E for the three-dimensional problem.
of the inclusion is accurately captured in the reconstruction. The values of the shear modulus
along D–D and E–E are shown in figures 8 and 9, respectively. From these plots we conclude
that the shape and the contrast of the reconstructed inclusion are in good agreement with the
target.
4. Conclusions
In this paper, we have considered the adjoint equation based algorithm proposed in Oberai
et al (2003). We have presented an alternative, simpler derivation of this algorithm. We have
tested its performance with regard to several clinically relevant parameters:
(i) We have evaluated its ability to reconstruct regions of elevated stiffness that are
geometrically irregular. In this regard, we have found that it can handle complex shapes
with relative ease, however, it may fail to reproduce small regions of contrast.
Evaluation of the adjoint equation based algorithm for elasticity imaging
2973
(ii) Using tissue-mimicking phantom data, we have tested its performance in the presence
of noise levels that are typical in experiment. We have found that it reproduces elastic
contrast with reasonable accuracy.
(iii) Using synthetically generated displacement data, we have demonstrated its ability to
generate three-dimensional elasticity images.
(iv) We have estimated the computational costs for this algorithm and compared them with
those of other popular iterative methods. We have found that the costs for the proposed
algorithm scale with a smaller power of the number of unknown parameters.
In general, we have found the algorithm to be accurate, robust and efficient. In future studies
we propose to use it for reconstructing stiffness distributions using clinical ultrasound data.
In this case, as pointed out in Barbone and Gokhale (2004), the information contained in the
displacement field corresponding to a single deformation state will not be sufficient to yield
a unique reconstructed modulus distribution. Following the ideas developed in Barbone and
Gokhale (2004), we will utilize two distinct deformations (compression and shear) to look for
unique answers. The extension of the adjoint equation based method to accommodate this
will be developed.
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