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
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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
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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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(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).
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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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−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
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z
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x
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1
x n
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y
y
n
n
n
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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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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 m = D(um − u)/Du ¯
(12)
where u ¯ 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 − u)
(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(u ¯ − u) (which is unknown, since u ¯ is unknown) as follows: ¯ D(u ¯ − u) D(um − u) + D(um − u).
(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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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 ¯ which is a measure of the measurement error, can the right-hand side, that is D(um − u), 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
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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).
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(0,Ly )
Ω
(Lx ,0) X1
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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.
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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
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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
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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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Figure 5. Reconstructed shear modulus for the tissue-mimicking phantom.
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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
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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
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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.
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(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. References Bamber J C, Barbone P E, Bush N L, Cosgrove D O, Doyley M M, Fuechsel F G, Meaney P M, Miller N R, Shiina T and Tranquart F 2002 Progress in freehand elastography of the breast IEICE Trans. Inf. Syst. 85 5–14 Barbone P E and Bamber J C 2002 Quantitative elasticity imaging: what can and cannot be inferred from strain images Phys. Med. Biol. 47 2147–64 Barbone P E and Gokhale N H 2004 Elastic modulus imaging: on the uniqueness and nonuniqueness of the elastography inverse problem in two dimensions Inverse Problems 20 283–96 Bishop J, Samani A, Sciaretta J and Plewes D B 2000 Two dimensional MR elastography with linear inversion reconstruction: methodology and noise analysis Phys. Med. Biol. 45 2081–91 Cespedes E I, de Korte C L and van der Steen A F W 2000 Intraluminal ultrasonic palpation: assessment of local and cross-sectional tissue stiffness Ultrasound Med. Biol. 26 385–96 Chaturvedi P, Insana M F and Hall T J 1998 2-D companding for noise reduction in strain imaging IEEE Trans. Ultrason. Ferroelectr. Freq. Control 45 179–91 Chenevert T L, Skovoroda A R, O’Donnell M and Emelianov S Y 1998 Elasticity reconstructive imaging by means of stimulated echo MRI Magn. Reson. Med. 39 482–90 Colton D and Kress R 1998 Inverse Acoustic and Electromagnetic Scattering Theory 2nd edn (Berlin: Springer) de Korte C L, Cespedes E I and van der Steen A F W 1999 Influence of catheter position on estimated strain in intravascular elastography IEEE Trans. Ultrason. Ferroelectr. Freq. Control 46 616–25 de Korte C L, Cespedes E I, van der Steen A F W and Lancee C T 1997 Intravascular elasticity imaging using ultrasound: feasibility studies in phantoms Ultrasound Med. Biol. 23 735–46 de Korte C L, Pasterkamp G, van der Steen A F W, Woutman H A and Bom N 2000a Characterization of plaque components with intravascular ultrasound elastography in human femoral and coronary arteries in vitro Circulation 102 617–23 de Korte C L, van der Steen A F W, Cespedes E I and Pasterkamp G 1998 Intravascular ultrasound elastography in human arteries: initial experience in vitro Ultrasound Med. Biol. 24 401–8 de Korte C L, van der Steen A F W, Cepedes E I, Pasterkamp G, Carlier S G, Mastik F, Schoneveld A H, Serruys P W and Bom N 2000b Characterization of plaque components and vulnerability with intravascular ultrasound elastography Phys. Med. Biol. 45 1465–75 Doyley M M, Meaney P M and Bamber J C 2000 Evaluation of an iterative reconstruction method for quantitative elasticity Phys. Med. Biol. 45 1521–40 Emelianov S Y, Chen X, O’Donnell M, Knipp B, Myers D, Wakefield T W and Rubin J M 2002 Triplex ultrasound: elasticity imaging to age deep venous thrombosis Ultrasound Med. Biol. 28 757–67 Garra B S, Cespedes I, Ophir J, Spratt S, Zuurbier R A, Magnant C M and Pennanen M F 1997 Elastography of breast lesions: initial clinical results Radiology 202 79–86 Gill P E, Murray W and Wright M H 2000 Practical Optimization 12th edn (London: Academic) Golub G H and Van Loan C F 1996 Matrix Computations 3rd edn (Philadelphia: Johns Hopkins University Press)
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Hughes T J R 2000 The Finite Element Method—Linear Static and Dynamic Finite Element Analysis (Mineola, NY: Dover) Isakov V 1998 Inverse Problems for Partial Differential Equations 1st edn (New York: Springer) Kallel F and Bertrand M 1996 Tissue elasticity reconstruction using linear perturbation method IEEE Trans. Med. Imaging 15 299–313 Oberai A A, Gokhale N H and Feijoo G R 2003 Solution of inverse problems in elasticity imaging using the adjoint method Inverse Problems 19 297–313 Ophir J, Alam S K, Garra B, Kallel F, Konofagou E, Krouskop T and Varghese T 1999 Elastography: ultrasonic estimation and imaging of the elastic properties of tissues Proc. of the Institution of Mechanical Engineers Part H-J. Eng. Med. 213 203–33 Plewes D B, Bishop J, Samani A and Sciaretta J 2000 Visualization and quantification of breast cancer biomechanical properties with magnetic resonance elastography Phys. Med. Biol. 45 1591–610 Raghavan K R and Yagle A E 1994 Forward and inverse problems in elasticity imaging of soft tissues IEEE Trans. Nucl. Sci. 41 1639–48 Ryan L K and Foster F S 1997 Ultrasonic measurement of differential displacement strain in a vascular model Ultrasonic Imaging 19 19–38 Saad Y 1995 Iterative Methods for Sparse Linear Systems 1st edn (Boston, MA: PWS Publishing Company) Skovoroda A R, Emelianov S Y and O’Donnell M 1995 Tissue elasticity reconstruction based on ultrasonic displacement and strain images IEEE Trans. Ultrason. Ferroelectr. Freq. Control 42 747–65 Sumi C, Suzuki A and Nakayama K 1995 Estimation of shear modulus distribution in soft tissue from strain distribution IEEE Trans. Biomed. Eng. 42 193–202 Van Houten E E W, Paulsen K D, Miga M I, Kennedy F E and Weaver J B 1999 An overlapping subzone technique for MR-based elastic property reconstruction Magn. Reson. Med. 99 779–86 Vogel C R 2002 Computational Methods for Inverse Problems (Philadelphia: SIAM) Weaver J B, Van Houten E E W, Miga M I, Kennedy F E and Paulsen K D 2001 Magnetic resonance elastography using 3D gradient echo measurements of steady-state motion Med. Phys. 28 1620–8 Wellman P, Howe R H, Dalton E and Kern K A 1999 Breast tissue stiffness in compression is correlated to histological diagnosis Technical Report Harvard BioRobotics Laboratory, Division of Engineering and Applied Sciences, Harvard University Zhu C, Byrd R H and Nocedal J 1997 L-BFGS-B: algorithm 778: L-BFGS-B, FORTRAN routines for large scale bound constrained optimization ACM Trans. Math. Softwa. 23 550–60
