# Solution of inverse problems in elasticity imaging using the adjoint method ```INSTITUTE OF PHYSICS PUBLISHING
INVERSE PROBLEMS
Inverse Problems 19 (2003) 297–313
PII: S0266-5611(03)54272-1
Solution of inverse problems in elasticity imaging
Assad A Oberai1 , Nachiket H Gokhale1 and Gonzalo R Feijóo2
1 Department of Aerospace and Mechanical Engineering, Boston University, 110 Cummington
Street, Boston, MA 02215, USA
2 Sandia National Laboratories, MS 9405, Livermore, CA 94551, USA
Received 1 October 2002, in final form 10 December 2002
Published 7 February 2003
Online at stacks.iop.org/IP/19/297
Abstract
We consider the problem of determining the shear modulus of a linearelastic, incompressible medium given boundary data and one component of
the displacement field in the entire domain. The problem is derived from
applications in quantitative elasticity imaging. We pose the problem as one of
minimizing a functional and consider the use of gradient-based algorithms to
solve it. In order to calculate the gradient efficiently we develop an algorithm
based on the adjoint elasticity operator. The main cost associated with this
algorithm is equivalent to solving two forward problems, independent of
the number of optimization variables. We present numerical examples that
demonstrate the effectiveness of the proposed approach.
1. Introduction
Elasticity imaging is a relatively new and promising technique in medical imaging. It relies
on using the difference in elastic modulus of tissues to distinguish them. The basic steps are
(1) subject the specimen to a deformation,
(2) measure the displacement field in the entire domain and
(3) compute the elastic modulus (usually the shear modulus) by solving an inverse problem.
The displacement field may be determined using either ultrasound (see  for an example)
or nuclear magnetic resonance (NMR) (see  for an example). In the case of ultrasound, a
speckle image of the specimen in the undeformed state is recorded. Thereafter, the specimen
is deformed and another speckle image is recorded. These images are registered to yield
the displacement field. Typically, the speckle images, and hence the displacement field, are
obtained under static conditions, and effort is made to keep the deformations small. As a
result, the displacements are governed by the equations of equilibrium of an incompressible,
linear-elastic solid undergoing small, quasi-static deformation. On the other hand, when using
NMR, displacements are calculated from the phase of the measured magnetic field and are
required to be time harmonic. Thus the governing equations are time harmonic. In this study,
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we consider the case of quasi-static deformations, although the algorithms developed may be
easily extended to the time-harmonic case.
Once the displacement field is determined, the inverse elasticity problem is solved in
one of two ways. The first is the so-called direct method (see [3–7] for applications). This
method is based on interpreting the equations of equilibrium as a single, linear, hyperbolic
partial differential equation for the shear modulus in which strains appear as known ‘material
parameters’. This equation is solved with appropriate boundary data to reconstruct the shear
modulus. While the direct approach is appealing in that the inverse problem is reduced to a
forward problem, it suffers from the following drawbacks:
(1) calculation of the strain field requires differentiating a noisy, measured displacement field
and the process of numerical differentiation degrades the accuracy of the method and
(2) the boundary data required to render the equation for the shear modulus well posed are
often not known.
The second method for solving the inverse elasticity problem is based on recasting the
problem as a non-linear optimization problem (see [8–10] for applications). In this approach, a
distribution of shear modulus that minimizes the difference between the measured and predicted
displacement fields is sought. The predicted displacement field is required to satisfy the
appropriate elasticity equations. In relation to the direct method this approach is more robust,
and it addresses the drawbacks described in the previous paragraph. However, it tends to be
computationally expensive.
In this study we develop and implement efficient algorithms for solving the non-linear
optimization problem resulting from the inverse elasticity problem. In particular, we consider
the problem of finding that distribution of shear modulus which minimizes the L 2 norm of the
difference between the measured and predicted displacement fields, subject to the constraint
that the predicted displacement field satisfy the equations of equilibrium for an incompressible,
linear-elastic solid undergoing small, quasi-static deformations. In discretizing this problem,
we represent the shear modulus using N piece-wise linear finite-element shape functions whose
nodal values are the parameters of the inverse problem. We consider the use of gradient-based
algorithms and show that a straightforward calculation of the gradient requires N solves of the
forward elasticity problem. This cost is computationally prohibitive for typical values of N
(103 ). To circumvent this difficulty, we develop and implement a new algorithm based on
the adjoint elasticity operator which requires only two solves (independent of N) to compute
the gradient. Similar ideas involving the use of adjoint equations in solving optimization
problems have been considered in areas such as impedance tomography , electromagnetic
imaging , acoustic shape identification  and optimal shape design in aerodynamics .
The format of the paper is as follows. In section 2, we describe the forward elasticity
problem. In section 3, we define the inverse elasticity problem and consider its solution using
gradient based methods. In section 4, we present a straightforward method for computing the
gradient, and in section 5 we present a more efficient approach based on the adjoint elasticity
operator. In section 6, we present numerical examples, and close with concluding remarks in
section 7.
2. Forward elasticity problem
In this section we define the equations that the measured and predicted displacement fields
are assumed to satisfy. These describe the equilibrium of an isotropic, linear-elastic solid
undergoing small, quasi-static deformations. We present the strong formulation first and
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then an equivalent weak or variational formulation. The weak formulation is subsequently
approximated using Galerkin’s method.
2.1. Strong formulation
The strong form of the forward problem is given as follows: given
(i) the Lamé parameters, λ(x) and µ(x), in the entire spatial domain ,
(ii) the traction vector h(x) on a part of the boundary denoted by h and
(iii) the prescribed displacement vector r (x) on a part of the boundary denoted by r ,
find the displacement and pressure fields denoted by u(x) and p(x), respectively, that satisfy
the equations of equilibrium for a linear, incompressible material given by
∇ · (− p1 + µ(∇ u + ∇ uT )) = 0,
in (1)
∇ · u + p/λ = 0,
(− p1 + µ(∇ u + ∇ uT ))n = h,
in on h
(2)
(3)
u = r,
on r .
(4)
In the equations above, ⊂ R , represents the interior of a body whose boundary is
∂ = h ∪ r , n is the unit outward normal on ∂ and the superscript T denotes the transpose
of a tensor. Equation (1) represents the balance of linear momentum within an elastic solid.
The term that appears within the divergence operator is the stress tensor. For an isotropic
material, Lamé’s parameters are related to the more familiar Young modulus (E) and Poisson
ratio (ν) via µ = E/(2(1 + ν)) and λ = ν E/((1 + ν)(1 − 2ν)). Note that µ is also referred to
as the shear modulus. Equation (2) is a convenient way of modelling materials that are nearly
incompressible. In this equation for a finite value of pressure p, the limit λ → +∞ implies
∇ · u = 0, which is the incompressibility condition. In our formulation we approximate this
limit replacing λ(x) by λ(x) = βµ(x) with β chosen to be a large constant (in our numerical
calculations β = 106 ). Equation (3) specifies the tractions on a part of the domain denoted
by h , and equation (4) specifies the displacement on the other part r . Note that for a well
posed forward problem the union of h and r must be the entire boundary ∂.
s
2.2. Weak formulation
An equivalent weak formulation of (1)–(4) for the incompressible limit is the following: find
U = (u, p) ∈ S , such that
A(W , U ; µ) = (w, h)h ,
∀W = (w, q) ∈ V
(5)
where A(·, ·; µ) and (·, ·)h are bilinear forms defined as
qp
µ
T
T
(∇ w + ∇ w ) : (∇ u + ∇ u ) −
− (∇ · w) p − q(∇ · u) d, (6)
A(W , U ; µ) ≡
βµ
2
(w, h)h ≡
w · h d.
(7)
h
In (6) the symbol : is used to represent the inner product between two tensors. That is, for
two tensors, T̄ and T̂ , T̄ : T̂ = T̄i j T̂i j , where the sum on i and j is implied. The weighting
function space V and the trial solution space S are defined as
V = {(w, q)|wi ∈ H 1 (), w = 0 on r , q ∈ L 2 ()}
(8)
S = {(u, p)|u i ∈ H (), u = r on r , p ∈ L 2 ()}.
(9)
1
Note that the bilinear form A(·, ·; µ) depends on the shear modulus µ.
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The weak form of the problem may be derived from the strong form by
(1) multiplying (1) and (2) with the weighting functions w and q respectively,
(2) integrating the sum over the entire domain and
(3) performing integration by parts and then making use of the boundary conditions.
It may also be obtained directly by evaluating the conditions for minimizing the total potential
energy. In this case, the potential energy is the total elastic strain energy minus the work
done by the external body forces and boundary tractions. For the derivation of the weak
formulation by minimizing the potential energy the reader is referred to material under the
theorem of virtual work in . For the equivalence of strong and weak formulations the
2.3. Galerkin’s approximation
In order to solve (5) numerically we introduce finite-dimensional subspaces associated with
the weighting and trial solution function spaces. These are denoted by V h ⊂ V and S h ⊂ S
respectively. Throughout this manuscript a superscript h denotes the finite-dimensional
(numerical) counterpart of a continuous quantity. To obtain the Galerkin approximation to
the weak formulation we introduce a function Rh = (r h , 0) ∈ S h , where r h is a specified
function in that approximates the prescribed Dirichlet data r on r . Thereafter we set
U h = V h + Rh and derive an equation to determine V h ∈ V h . This leads to the Galerkin
approximation: find V h = (v h , ph ) ∈ V h , such that
A(W h , V h ; µh ) = (wh , h)h − A(W h , Rh ; µh ),
∀W h = (wh , q h ) ∈ V h ;
(10)
note that the superscript h is used to denote finite-dimensional approximation of quantities that
appear in the weak formulation. The split U h = V h + Rh is typical of a finite-element solution
to the weak form. In this case the function Rh is a known function that is constructed from
finite-element shape functions with nodes on the Dirichlet boundary. The contribution from
Rh appears in the right-hand side of the algebraic equations resulting from the finite-element
discretization.
As part of the proposed algorithm to solve the inverse problem, we solve (10) using the
finite-element method. We represent uh , wh and µh as a linear combination of continuous,
piece-wise bilinear finite-element shape functions. For example, µh is given by
µh =
N
µi φi (x),
(11)
i=1
where N is the number of nodes in the finite-element model and φi represents a typical
bilinear finite-element shape function. We have chosen the same shape functions to represent
displacements and shear modulus in order to simplify the programming requirements. In
future work other choices will be considered. We represent ph using discontinuous, piecewise constant shape functions. This choice allows for the pressure degrees of freedom to be
statically condensed from the element matrix, thereby greatly simplifying the overall solution
process (see  for a discussion).
3. Inverse elasticity problem
In the forward problem described in the previous section we were given the material properties
and boundary data and were asked to calculate the displacement field. In the inverse problem
the situation is reversed. We are now given the displacement field (or a related measurement)
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and the boundary data and are asked to calculate the material properties. While this situation is
typical of inverse problems in other fields such as ultrasound tomography and seismic imaging,
the inverse elasticity problem is distinct in that the measured data is known on a significant
subset of the domain, and not on a lower-dimensional manifold.
The inverse problem is defined as follows: given boundary data r and h, and the
measurement T (um ) in , find the distribution of shear modulus µ(x) and a displacement
field u, such that (1)–(4) hold, and T (um ) − T (u) = 0. Here T : Rs → Rs is a second-order
tensor with Cartesian components Ti j , and um represents the measured displacement field.
This problem may be posed as a minimization problem as follows: find µ(x) such that the
functional
1
α
π(µ) = T (u) − T (um )2 + µ2b
(12)
2
2
is minimized subject to the constraint that u satisfies the forward elasticity problem. In the
above equation · 2 = (·, ·) represents the L 2 norm in , and · b represents an appropriate
norm chosen for regularizing the solution. We assume that the norm is derived from an
appropriate inner product, that is · 2b = (·, ·)b . The parameter α is the Tikhonov parameter
and is chosen according to the theory of residues due to Morozov (see [17, 18] for example).
Remark.
(i) The tensor T is introduced to account for the fact that the measurement may be a linear
function of the displacement field. For example, in the case when ultrasound is used to
measure the displacement field, the resolution in the direction perpendicular to the axis
of the transducer is poor, and only one component of the displacement field is measured.
Thus
0 0
,
(13)
[T ] =
0 1
where it is assumed that the transducer is aligned along the x 2 -direction.
(ii) The solution to the inverse problem defined above is not unique. To render it unique either
the value of the shear modulus must be known on a part of the boundary, or more than
one linearly independent, measured displacement field must be specified. The reader is
referred to [19, 20] for a detailed discussion. In the examples considered, we have ensured
uniqueness by specifying the shear modulus on a part of the boundary.
(iii) In practice when solving the minimization problem, π(µ) is replaced by π(µh ), which
takes into account the finite-dimensional approximation of all the fields appearing in π.
Thus the discretized minimization problem reads as follows: find µh (x) such that the
functional
1
α
(14)
π(µh ) = T (uh ) − T (um )2 + µh 2b
2
2
is minimized subject to the constraint that uh is the numerical solution (i.e. the solution
of (10)). Note that the first term in this expression now evaluates the difference between
the predicted numerical solution and the measured data.
We consider the solution of this problem using a class of optimization algorithms that
require the value of the functional and its derivative (also called the gradient) at each
iteration. Several quasi-Newton algorithms such as steepest descent, BFGS (Broyden–
Fletcher–Goldfarb–Shanno) and DFP (Davidson–Fletcher–Powell) fall under this category
(see  for a description of these algorithms).
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4. Straightforward calculation of the gradient
We consider the straightforward approach to evaluating the gradient in the continuous and
discrete cases. In the latter case, we estimate the cost of this computation.
4.1. Continuous case
We define the differential of a functional in the usual way. That is, for a functional ρ(v) which
depends on v, the differential in the direction δv is given by
d
ρ(v + δv) ,
(15)
δρ = Dv ρ · δv =
d
=0
where Dv ρ is the derivative or the gradient of the functional. Using this definition in (12) we
have
δπ = Dµ π · δµ = (T (δ u), T (u − um )) + α(δµ, µ)b ,
(16)
where the differential δ u represents a change in the displacement u corresponding to a change
in shear modulus from µ to µ + δµ. It is evaluated by differentiating (5). This yields the
following: find δ U ∈ V such that
A(W , δ U ; µ) = −Dµ A(W , U ; µ) · δµ,
∀W ∈ V ,
where δ U = (δ u, δp), U is the solution to (5), and the right-hand side is given by
1
qp
T
T
δµ (∇ w + ∇ w ) : (∇ u + ∇ u ) +
Dµ A(W , U ; µ) · δµ =
d.
2
βµ2
(17)
(18)
In deriving (17), we have made use of the linearity of the form A(W , U ; µ) on U ,
i.e. DU A(·, U ; ·) · δ U = A(·, δ U ; ·).
4.2. Discrete case
In the discrete case, µ is approximated by µh and π(µ) by π(µh ). Thus the appropriate
differential is given by
δπ = Dµh π · δµh = (T (δ uh ), T (uh − um )) + α(δµh , µh )b ,
(19)
where the differential δ u is evaluated by differentiating (10). This yields
h
A(W h , δ V h ; µh ) = −Dµ A(W h , U h ; µh ) · δµh ,
∀W h ∈ V h ,
(20)
where δ V h = (δ uh , δp h ), and U h is the solution to (10).
Let µh be represented by (11). Then it is clear that the scalars µi , i = 1, . . . , N are the
parameters of the inverse problem, and that any variation in µh is represented by
δµh =
N
δµi φi (x).
(21)
i=1
Using (11) in (19) and (20) we have
δπ = G · δ µ,
(22)
where G ∈ R N is the gradient vector whose components are given by
G i = Dµh π · φi = (T (δ uh ), T (uh − um )) + α(φi , µh )b .
(23)
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In the above equation, for each i , δ uh is obtained by solving
A(W h , δ V h ; µh ) = −Dµ A(W h , V h ; µh ) · φi ,
W h ∈ Vh,
(24)
where δ V = (δ u , δp ), and V is the solution to (10). Using (23), the following algorithm
may be used to calculate the gradient vector G.
h
h
h
h
Algorithm 4.1.
(i) Solve (10) to evaluate V h .
(ii) For i = 1, . . . , N
(a) solve (24) to evaluate δ V h ;
(b) calculate G i using (23).
The above algorithm requires N + 1 solves (one in step (i) and N in step (ii)) of the
size of a forward elasticity problem. For typical values of N 103 , this represents a large
computational cost.
In this section we propose an efficient algorithm for calculating the gradient vector. This
approach is based on using the adjoint of the bilinear form A(·, ·; µ).
5.1. The continuous case
We begin by introducing the Lagrangian L,
1
α
(25)
L(U , W , µ) = T (u) − T (um )2 + µ2b + A(W , U ; µ) − (w, h)h ,
2
2
where W = (w, q) plays the role of a Lagrange multiplier. The differential of L is given by
δ L = DU L · δ U + DW L · δ W + Dµ L · δµ.
(26)
In the above equation, setting DW L · δ W = 0, ∀δ W ∈ V , yields the following equation for
U:
A(δ W , U ; µ) = (δ w, h)h ,
∀δ W ∈ V ,
(27)
which implies that U satisfies the original elasticity problem (5). For this value of U , from (12)
and (25) we conclude that L(U , W , ·) = π(·), ∀W , and consequently for a given δµ
δ L = δπ.
(28)
Thus, we may use the expression for δ L to evaluate δπ. The expression for δL can be simplified
considerably if DU L · δ U = 0, ∀δ U ∈ V . This condition yields the following equation for
W:
A(W , δ U ; µ) = −(T (δ u), T (u − um )),
∀δ U ∈ V .
(29)
This may be written as
A∗ (δ U , W ; µ) = −(T (δ u), T (u − um )),
∀δ U ∈ V ,
(30)
∗
where A (·, ·; µ) is the adjoint of the bilinear form A(·, ·; µ). Using (6) it is easily verified
that A(·, ·; µ) is self-adjoint, that is A∗ (V1 , V2 , µ) = A(V1 , V2 , µ) ∀V1 , V2 ∈ V , and hence
the above equation reduces to
A(δ U , W ; µ) = −(T (δ u), T (u − um )),
∀δ U ∈ V .
(31)
With U and W given by (27) and (31) respectively, from (26) and (28) we have
δπ = δ L = Dµ A(W , U ; µ) · δµ + α(µ, δµ)b .
(32)
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Remark.
(i) The equation for determining U , that is (27), is the same as (5), the forward elasticity
problem. This equation is often referred to as the primal problem.
(ii) The equation for the Lagrange multiplier W , that is (31), is of the same type as the primal
problem and is driven by the difference in the predicted and measured data. This equation
is referred to as the dual problem. The strong form associated with the problem may be
obtained by performing integration by parts on (31) and is given by
∇ · (−q1 + µ(∇ w + ∇ wT )) = −T T T (u − um ),
∇ · w + q/λ = 0,
(−q1 + µ(∇ w + ∇ wT ))n = 0,
w = 0,
in (33)
in on h
(34)
(35)
on r .
(36)
Thus the dual displacement w and the dual pressure field q are determined by solving the
original elasticity equations forced by the difference between the predicted and measured
displacements (see (33), (34)). The boundary conditions for these fields (35) and (36) are
the homogeneous counterparts of the boundary conditions for the primal field.
5.2. Discrete case
Repeating the derivation in the previous section with π(µ) replaced by π(µh ), which is the
discrete approximation to π(µ), we have
δπ = Dµ A(W h , U h ; µh ) · δµh + α(µh , δµh )b ,
(37)
where U h = V h + Rh , V h satisfies
A(δ W h , V h ; µh ) = (δ wh , h)h − A(δ W h , Rh ; µh ),
∀δ W h ∈ V h , (38)
and W h satisfies
A(δ U h , W h ; µh ) = −(T (δ uh ), T (uh − um )),
∀δ U h ∈ V h .
(39)
Once again the gradient may be explicitly expressed as a finite-dimensional vector. This is
accomplished by utilizing (21) in (37)–(39) to arrive at
δπ = G · δ µ.
(40)
The components of the gradient vector are given by
G i = A(W h , U h ; φi ) + α(µh , φi )b ,
(41)
where U is given by (38) and W is given by (39). Note that (38) is an equation for calculating
V h , and that U h may be obtained from V h by the relation U h = V h + Rh . Thus, using (41),
the following algorithm may be used to compute G.
h
h
Algorithm 5.1.
(i) Solve (38) to evaluate U h .
(ii) Solve (39) to evaluate W h .
(iii) Use U h and W h in (41) to compute G.
Solution of inverse problems in elasticity imaging using the adjoint method
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Remarks.
(i) Algorithm 5.1 involves two solves (one in step (i) and one in step (ii)) of the forward
elasticity problem to compute G. This in contrast to the N + 1 solves required in
algorithm 4.1.
(ii) Previous studies aimed at solving the inverse elasticity problem by treating it as an
optimization problem have utilized a regularized version of the Gauss–Newton algorithm
(see [8–10] for example). This algorithm involves evaluation of the so-called Jacobian
matrix, a calculation that is embedded in step (ii) of algorithm 4.1. The cost associated
with this algorithm also scales as N + 1 forward solves per iteration.
(iii) Interesting connections can be drawn between the algorithms proposed in [11, 12] and
our approach. For the case of a single measurement, the algorithms proposed in these
studies are equivalent to using steepest descent instead of a quasi-Newton algorithm with
algorithm 5.1 to calculate the gradient.
6. Numerical examples
In this section we present numerical examples that demonstrate the effectiveness of the
proposed approach. In all examples the gradient is computed using algorithm 5.1. To ensure
that the shear modulus remains bounded away from zero in the entire domain,a dummy variable
ξ is used in the optimization algorithm. The shear modulus µ is related to ξ via
µ = 1 + ξ 2.
(42)
The expression µ = 1 + |ξ | has also been tried and found to yield similar results. However we
prefer to use (42) since the derivative of µ = 1 + |ξ | is not defined at ξ = 0.
6.1. Problem description
A schematic of the example problem is shown in figure 1. The domain = (0, L) × (0, L),
where L = 10 units. On the x 2 = 0 edge, u 2 = 0.1 and h 1 = 0; on the x 2 = 10 edge, u 2 = 0
and h 1 = 0, and on the x 1 = 0, 10 edges, h 1 = 0 and h 2 = 0, where u i and h i are components
of the displacement and traction vectors u and h, respectively. To prevent rigid translation of
the body in the x 1 -direction the centre point of the domain is constrained not to move in the
x 1 -direction. It is assumed that only the x 2 -component of displacement is measured. Thus the
only non-zero entry of the tensor T is T22 = 1. The shear modulus is assumed to be constant
on the x 2 = 0 edge. The uniqueness of this problem for the incompressible case has been
proven in .
For the inverse problem, we generate the ‘measured’ displacement field by solving the
forward elasticity problem with the target material field, and then adding white Gaussian noise
to the displacements. We consider three levels of noise corresponding to = 3, 1 and 0.3%,
where is given by
m
= u m
2 − ū · e2 /u 2 .
(43)
In the above equation ū is the displacement field with no noise. Note that values of = 3,
1 and 0.3% correspond approximately to a signal to noise ratios (SNR) of 30, 40 and 50 dB,
respectively. It is believed that an SNR of 40 dB is typical for systems that use ultrasound to
measure displacements (see  for example).
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X2
(0,L)
Ω
(L,0)
X1
APPLIED VERTICAL DISPLACEMENT
Figure 1. Schematic of the example problem.
Table 1. Choice of the regularization parameter.
Noise level, Regularization parameter, α
3 × 10−2
2.5 × 10−6
1 × 10−2
2.5 × 10−7
3 × 10−3
2.5 × 10−8
6.2. Regularization parameter
We choose the norm for the regularization term to be the L 2 norm on . That is, · b = · .
In choosing the regularization parameter α, we assume that the measured solution u m
2 has a
known level of noise . According to the theory of residues due to Morozov, α should be
chosen to be the largest real number that allows
h
m
m
u m
2 − u · e2 = Cu 2 − ū · e2 = Cu 2 ,
(44)
for the optimal solution. In the equation above, C is a number that is√of order 100 , and is
greater than unity. In practice, we have found that a value of C ≈ 1/ 2 provides the best
results. Values much larger than this over-regularize the solution. Based on this criterion the
values of α used for three different noise levels are shown in table 1. From this table, it is clear
that, within the range considered, α ∼ 2 .
6.3. Reconstructions
We perform all reconstructions using the limited memory BFGS algorithm . The
computer program that implements this algorithm was downloaded from http://wwwfp.mcs.anl.gov/otc/Tools/LBFGS-B. In BFGS, as in any quasi-Newton method, within each
iteration several calls may be made to the routine that calculates the functional and the gradient.
Solution of inverse problems in elasticity imaging using the adjoint method
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Figure 2. Shear modulus reconstructions for a specimen with a single inclusion on a coarse mesh.
Each such call is termed a sub-iteration. Thus 25 sub-iterations correspond to 25 evaluations
of the functional and the gradient, and in our case 25 solves of the primal and dual problems.
In all cases we terminate the optimization algorithm at the smallest iteration number n for
which (π(µn ) − π(µn−5 ))/π(µn−5 ) < 0.01. The total number of sub-iterations is labelled
n s . The system of linear equations resulting from the finite-element discretization of the
primal and dual problems is solved using conjugate gradient iterations with an incomplete LU
preconditioner (see  for example). The subroutines to perform these solves are part of the
PETSC library . All calculations are performed on a Linux workstation with a 1.7 GHz
Pentium processor.
6.3.1. Single inclusion on a coarse mesh. First we consider the problem of a single inclusion
in a uniform, soft matrix. The contrast, defined as the ratio of the modulus of the inclusion
to the surrounding medium, is five. At 0.05% nominal strain, this ratio is typical of ductal
carcinoma in situ surrounded by glandular tissue in the human breast (see  for example).
The problem is solved on a 30 × 30 quadrilateral finite-element mesh. This leads to 931
parameters in the inverse problem. Their break-up is as follows. There are 961 nodes in the
domain, out of which 31 were on the x 2 = 0 edge. It is assumed that the distribution of the
shear modulus on this edge is known (uniform), however its value is not. Thus the 31 nodal
values on the x 2 = 0 edge are represented by a single parameter, leading to 961 − 31 + 1 = 931
total parameters.
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6
5
4
3
2
1
0
0
1
2
3
4
5
6
7
8
9
10
Figure 3. Shear modulus along A–A .
Table 2. Output data for reconstructions.
Reconstruction
Sub-iterations (n s )
Wall clock time (s)
Error () (%)
mesh, = 3%
302 mesh, = 1%
302 mesh, = 0.3%
1002 mesh, = 3%
1002 mesh, = 1%
1002 mesh, = 0.3%
1502 mesh, = 3%
1502 mesh, = 1%
1502 mesh, = 0.3%
17
24
55
20
32
41
22
28
58
14
20
46
261
418
535
882
1122
2324
39
23
13
40
23
14
40
26
15
302
In figure 2, the target distribution and three reconstructions of µ corresponding to different
levels of noise are shown. To highlight the contrast, different grey scale maps are used within
each plot. We observe that the shape of the inclusion is recovered accurately in all cases. In
figure 3, the shear modulus along the line A–A is compared. In this figure, differences among
the three reconstructions are apparent. With 3% noise the shear modulus of the inclusion is
underestimated by about 40%,with 1% noise this value is 20%, and with 0.3% noise it is smaller
still. These observations are quantified by calculating the L 2 error in the reconstructions,
namely = µtarget − µpredicted /µtarget . The values of , listed in table 2, are consistent
with the observations made for figure 3.
The reduced accuracy of the reconstructed contrast ratio in the high-noise-level case can
be ascribed to the higher value of regularization parameter used in that case. A higher value
of α enhances the role of the regularizing term leading to solutions with lower L 2 norm of the
Solution of inverse problems in elasticity imaging using the adjoint method
309
Figure 4. Shear modulus reconstructions for a specimen with a single inclusion on a fine mesh.
reconstructed field. Regularization terms based on other norms will have different effects. For
instance, for the H 1 norm, we would expect reconstructions with smeared-out boundaries, but
more accurate contrast ratios. We are currently exploring possibilities for what might be an
optimal norm for regularizing the inverse elasticity problem.
6.3.2. Single inclusion with a fine mesh. We consider the same inclusion on a finer mesh
of 150 × 150 quadrilateral finite elements. This represents a relatively large inverse problem
with 22 651 parameters.
In figure 4, the target distribution and three reconstructions of µ with different levels of
noise are shown. As for the previous case, different grey scale maps are used for each plot.
From the figure it is clear that for all levels of noise the shape of the inclusion is recovered
accurately, though for the 3.0 and 1.0% cases the circular inclusion is somewhat elongated in
the x 1 -direction. This artefact is absent from the reconstruction with 0.1% noise. In figure 5,
the shear modulus along the line B–B is shown. Trends observed for the coarse mesh are seen
to persist. These results are validated by calculating , the L 2 error in the reconstructions (see
table 2).
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6
5
4
3
2
1
0
0
1
2
3
4
5
6
7
8
9
10
Figure 5. Shear modulus along line B–B .
6.3.3. Multiple inclusions. Finally, we considerer two inclusions, each of different size and
contrast. This problem is relevant to the imaging of human breast where fibroadenoma has
been found to have a smaller contrast when compared with ductal carcinoma in situ (see ).
We solve this problem on a 100 × 100 mesh with a total of 10 101 parameters.
The target shear modulus and three reconstructions are shown in figure 6. As before,
different grey scale maps are used for each plot. From this figure it is clear that the location of
both inclusions is recovered accurately for all cases; however, their shapes are recovered most
accurately for data with least noise. In addition, an artefact in the form of a strip of material
with lower contrast is seen running through the smaller of the two inclusions for cases with
lower noise. It is found that this streak does not vanish even when the convergence criterion is
tightened. Tests are being performed to estimate the minimum level of noise in the measured
data required to remove this artefact. In figure 7, the shear modulus along the line C–C is
plotted. In all cases the relative magnitude of the shear modulus of the two inclusions is
recovered fairly accurately; however, the contrast with respect to the surrounding medium is
most accurate for the reconstruction with least noise. The L 2 error in reconstruction is listed
in table 2, and is consistent with the problems with single inclusions. In addition, from data
presented in this table we draw the following conclusions.
(i) For a given level of noise in measured data the error in reconstruction is independent of
the mesh size for the problem.
(ii) The number of sub-iterations to convergence increases with decreasing noise level. This
is because the functional drops by a greater factor for cases with less noise and hence the
optimization algorithm does more work in these cases.
(iii) For all cases, the overall time taken to solve the problem is reasonable. For = 1%,
which corresponds to the level of noise in a typical measurement, this value is about half
Solution of inverse problems in elasticity imaging using the adjoint method
311
Figure 6. Shear modulus reconstructions for specimen with two inclusions.
a minute for the 302 mesh, 7 min for the 1002 mesh and about 20 min for the 1502 mesh.
These results are obtained on a single 1.7 GHz Pentium processor.
7. Conclusions
We have developed and implemented an efficient formulation to solve the inverse elasticity
problem. The main features of this approach are the use of a gradient based algorithm and
efficient computation of the gradient using the adjoint equations. Using this formulation we
have solved model problems with noise and studied the effect of regularizing terms. We
have found the proposed algorithm is fast and yields accurate results for typical values of
noise (SNR ≈ 40 dB) attained in experiments. Future work includes the application of this
methodology to problems with multiple displacement measurements, time-harmonic fields and
three-dimensional reconstructions.
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5.5
Target
5
0.3% Noise
Shear Modulus
4.5
1% Noise
4
3.5
3%
Noise
3
2.5
2
1.5
1
0
5
10
Distance along Diagonal
15
Figure 7. Shear modulus along C–C .
Acknowledgments
The authors would like to acknowledge P E Barbone for many helpful and stimulating
discussions on quantitative elasticity imaging and A Dwitama for help in writing the finiteelement code used to solve the forward problem. The third author of this paper is supported
by Sandia, a multiprogram laboratory operated by the Sandia Corporation, a Lockheed Martin
Company, for the United States Department of Energy under contract DE-ACO4-94AL85000.
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