10.2.1 The Inverse Algorithm

The inversion strategy uses a quasi‐Newton optimization algorithm (L‐BFGS‐B 28, 29) coupled with a nonlinear finite element code. The objective function is given by the sum of a data matching term and total variation (TV) regularization terms 11, 15:

(10.1)

In the equation above is the ‐th measured displacement field, is the corresponding predicted displacement field which satisfies the equations of equilibrium for a hyperelastic material, is a diagonal matrix whose values are selected to emphasize the displacement components that are measured accurately, and is a weight related to the amplitude of . It is selected so that measurements made at disparate levels of strain contribute in equal measure to the data‐matching terms. The material parameters are denoted by and , which denote the shear modulus at zero strain and nonlinear parameter distributions, respectively. Finally, and are small constants that ensure the differentiability of the regularization terms at the origin; and are the regularization parameters associated with and , respectively. These parameters determine the relative importance of the data matching and the regularization terms. We note that a change in the material parameters leads to a change in the predicted displacements which in turn leads to a chance in the functional . The goal is to find the distribution of the material parameters that minimizes .

We model the material (gel) as an incompressible hyperelastic material. The stress‐strain behavior is thus completely determined by the strain energy density function given by

(10.2)

In the equation above is the shear modulus at small strains and represents the rate of increase of stiffness with strain. In addition and are the invariants of the two‐dimensional Cauchy‐Green strain tensor . We note that the expression for the strain energy density we have used is the restriction of a simplified Veronda‐Westmann model 15, 30 to plane stress.

At every iteration the quasi‐Newton algorithm requires the gradient vector which represents the change in corresponding to a change in or at a spatial location. This vector is computed efficiently by utilizing the adjoint elasticity equations and a continuation strategy in material parameters 15, 16.

10.2.2 Phantom Description and RF Data Acquisition

A phantom containing four spherical inclusions was manufactured using previously reported ultrasound phantom materials with nonlinear elastic properties 17. The methods for manufacturing phantoms with coplanar spherical inclusions were also previously reported 18, 19. Sketches of the phantom identifying the background layers and individual targets are provided in Figure 10.1. The assembled phantom is a 100 mm cube. Each spherical target is 10 mm diameter. They are separated by 30 mm (center‐to‐center) and are placed 35 mm (center to phantom edge) from the top and each lateral side of the phantom.

When the phantom components were manufactured, a test cylinder of the same material was also cast to provide identical material with simple geometry to independently test the stress–strain behavior of the material. Mechanical tests were performed with an EnduraTEC ELF 3200 (Bose‐EnduraTEC Systems Corporation, Minnetonka, MN, USA) using a 1 kg load cell and platens larger than the sample surface. Each test cylinder was stored in oil to prevent desiccation, and that oil was used to lubricate the platens for the mechanical testing. During the mechanical testing, the upper platen was first lowered to make contact with the test cylinder. Then, the platen was lowered further (at 0.04 mm/s) to the center of the cyclic deformation range used for material characterization and the platen remained at that deformation for 5 s. A 1 Hz oscillatory compressive load (ranging from 1% to 25% strain) was applied for 5 s before starting data acquisition to precondition the sample 20, 21. Following the preconditioning step, force and displacement data were acquired for the randomized range of displacement magnitudes for each phantom material. Thus, the stress–strain curve of each tissue‐mimicking material (2 backgrounds and 4 inclusions) was obtained and subsequently fitted to the modified Veronda‐Westmann model to estimate the small strain shear modulus and the nonlinear parameter (see Eq. 10.2).

During phantom experiments, radiofrequency (RF) data were acquired using a Siemens SONOLINE Antares (Siemens Medical Solutions USA, Inc, Malvern, PA) clinical ultrasound system with a linear array ultrasound transducer (Siemens VFX9‐4) pulsed at 8.89 MHz. Details of the URI and bandwidth of the Siemens system are given by Brunke et al. 22. A large (approximately 15 cm by 15 cm) compression plate was attached to the ultrasound transducer to ensure nearly uniaxial deformations of the phantom. More specifically, ultrasound RF data were acquired for every 1.5% increment deformation (after the first few frames of 0.5% strain) up to approximately 20% (with respect to its height). Further details regarding this phantom are reported by Pavan et al. 23.

10.2.3 Displacement Estimation

To obtain large deformations (approximately 20%) required for nonlinear elasticity imaging, we first tracked motion sequentially through multiple frames similar to the multi‐compression technique 24. In the first step, offline motion tracking using RF echo data acquired from the nonlinear tissue‐mimicking phantom was performed between each pair of two adjacent RF echo frames [e.g. from the ‐th frame to the ( )‐th frame] using a modified block‐matching algorithm 25. This modified block‐matching algorithm combines smoothness‐constrained motion tracking with predictive search 26. Specifically, this modified block‐matching algorithm first estimated reliable motion within a small neigborbood to obtain high quality displacement estimates in a sparse grid (e.g. every 5 mm). Then, these reliable high quality displacements were used as seeds to perform predictions for its immediate neighbors. During this process, a small (approximately 0.5 mm [height] by 1 mm [width]) tracking kernel was first used to obtain integer displacements. Then, sub‐sample displacement estimates (necessary for modulus inversion) were obtained with quadratic interpolations. Collectively, this algorithm significantly reduces the occurance of large motion tracking errors between two adjacent RF frames. Once all displacement fields between adjacent RF echo frames were obtained, we mapped all displacement estimates in this sequence (from the first frame to the ‐th frame) to the coordinate system of the first echo frame by using B‐spline interpolations.

10.3.1 Description of the Forward Problem

As part of our iterative strategy a forward elasticity problem is solved at every iteration of the optimization algorithm. For the boundary conditions in the forward problem, on every edge, the axial component of the displacement was set equal to the measured displacement while in the lateral direction a traction‐free state was assumed. This implied perfect slip on the axial (the top and the bottom) edges and zero normal traction on the lateral (left and right) edges. This choice was motivated by the experimental set‐up and the fact that the measured lateral displacements were too noisy to be imposed strongly. The effect of other choices for boundary conditions was also explored and is described in Section 10.4.4. The phantom was assumed to be in a state of plane stress, which is appropriate since the phantom was not confined in the elevational direction.

10.3.2 Options for the Optimization Strategy

The material parameters and were determined sequentially. That is, first was reconstructed using measurements at small deformation while keeping fixed at 0.1. Thereafter was reconstructed using two displacement fields at large strains while was held fixed at the previously computed value.

Several tests (not reported here) demonstrated that the choice of the initial value for and did not impact the final reconstructions provided that the initial field(s) were in a reasonable range. In this chapter, the initial fields were chosen to be uniform with a value of and . We would like to emphasize that even though the final results are not influenced by the initial field, the rate of convergence is: not surprisingly, the closer the initial choice is to the final solution, the faster the convergence.

The TV regularization used in Eq. (10.1) is a linear function of the magnitude of the gradient of the fields and consequently for a given contrast in material properties it will tend to reduce the mean value of the field. Additionally, the shear modulus is recovered only to a multiplicative constant as we use only displacement data in our inverse problem. Therefore it is necessary to specify a lower bound for the optimization algorithm. In this chapter, the shear modulus and nonlinear parameter are constrained as follows: and . Note that we ensured that the upper bound was never obtained for both and .

The convergence criterion plays an important role in determining the final answer. The L‐BFGS‐B algorithm implements two stopping criteria: stop when

1. the relative change in the objective function value is less than a prescribed value, or
2. when value of the ‐norm of the projected gradient is smaller than a prescribed value.

In our experience the iterations terminated due to the first criterion and we set the tolerance to be , where is the machine precision. Choosing such a small value for the tolerance leads to results that are convincingly converged. A slightly higher value could have been chosen to reduce the number of iterations.

10.3.3 Shear Modulus Images

The shear modulus was reconstructed using the displacement measured with an overall strain of approximately 1.5%. The results are shown in Figure 10.2 and the values of the regularization parameters appear in Table 10.1. The robustness of this strategy was verified by using one or more displacement fields with overall strain in the range of to in order to recover the shear modulus. It was found that this choice did not significantly change the results.

Parameter	set 1	set 2	set 3	set 4
	7e‐5	4e‐5	4e‐5	1e‐5
	6e‐3	6e‐3	6e‐3	3e‐3
	4e‐4	4e‐4	7e‐5	1e‐4
	7e‐3	7e‐3	7e‐3	7e‐3

10.3.4 Nonlinear Parameter Images

The nonlinear mechanical behavior can only be observed for rather large deformations and therefore it is appropriate to use deformation states where the strain is around . Figure 10.3 shows the non linear parameter fields obtained when frames 30 (with overall strain) and 36 (with overall strain) are used. Note that, since these strains are close to each other, the weights in Eq. ( 10.1) are identical and equal to unity. As for the shear modulus, the values of the regularization parameters are given in Table 10.1.

10.3.5 Axial Strain Images

The displacement data used in generating the shear modulus and nonlinear parameter images was also used to generate axial strain images. This lead to two strain images for each section of the phantom, at approximately and overall strain which were evaluated using the gradient filter within Paraview 31. These images are shown in Figures 10.4 and 10.5. We observe that at small strain the strain contrast changes somewhat from one section of the phantom to another. Also, for a given section it changes rather dramatically from small to large strains.

10.4.1 Analysis of the Shear Modulus Distributions

The shear modulus reconstructions show that every data set contains a circular inclusion in the lower part of the image. We observe that the shape and the sharp boundary of these inclusions have been recovered very accurately. The diameter of these inclusions, measured along the axial and lateral directions, is listed in Table 10.2, where we have also listed the manufactured diameter of these inclusions. We observe very good agreement too, with the maximum error of about .

	incl. 1	incl. 2	incl. 3	incl. 4	Manufactured
Lateral diameter (mm)	9.60	9.38	9.28	9.21	10.0
Axial diameter (mm)	10.87	10.94	10.74	10.61	10.0
Area ( )	81.7	76.8	75.8	76.5	78.5

Source	incl. 1	incl. 2	incl. 3	incl. 4	backg. 1/ backg. 2
Recontructed	2.11	1.86	3.02	4.09	1 / 1.2
Measured	2.83	2.27	3.54	5.26	1 / 1.16
Reconstructed	0.35	2.6	3.8	2.9	3.4 / 2.7
Measured	2.2	6.5	4.9	4.5	4.7 / 4.3

With only measured displacement data, the shear modulus can be determined up to a multiplicative constant and therefore it is interesting to focus on the contrast between the inclusion and the background. The predicted and measured values are compared in Table 10.3. The measured values are obtained as described in Section 10.2.2. That is, a cylinder of each material type was compressed to obtain the uniaxial stress–strain curve, and that curve was matched to the analytical curve for the Veronda‐Westmann strain energy density function in order to determine and .

We observe that in all cases we have underestimated the contrast by about 20%. It is well known that the TV regularization results in diminishing contrast as is apparent here. Further, based on our previous experience with noisy data, a factor of 20 % appears reasonable.

We note that there is a small gradient in the shear modulus reconstruction in the axial direction. In particular the top portion of the phantom appears to be stiffer than the lower portion. We have observed the same effect in all sections and in axial strain images. This is consistent with how the phantom was manufactured in that the top portion of the background was slightly stiffer than the bottom portion (see Section 10.2.2). It is remarkable that the reconstructions are able to pick up this small effect.

10.4.2 Analysis of the Nonlinear Parameter Images

The images for the nonlinear parameter for the sections of the phantoms are shown in Figure 10.3. Here we can clearly see an inclusion only in the first section. For the other sections the inclusion is not clearly seen. This observation is consistent with the actual tissue phantom, where the background material and the inclusion material for inclusions 2, 3, and 4 was constructed to have the same nonlinear behavior. It is remarkable that our reconstructions recover this fact. Additionally, the independent mechanical measurements suggest that top background is slightly more nonlinear than bottom portion; this trend is properly captured in our reconstructions.

The measured and predicted values of the nonlinear parameter are presented in Table 10.1. We observe that, unlike with the values for the shear modulus, there is greater discrepancy between the measured and predicted values. This may be attributed to the following causes:

1. In order to quantitatively recover and to compare the reconstructed and the measured values, as well as to compare the reconstructed values of across different, phantom sections, it is necessary to utilize displacement measurement at two, significantly different, levels of finite strain. In our reconstructions we have utilized displacements at and overall strain. These may not be sufficiently different so as to measure quantitatively.
2. Using the formula derived in Gokhale et al. 15, see Equation (56) therein, we conclude that we require a strain of about in the inclusion in order to measure accurately. At overall strain we are close to, but still less than, this number. Thus the amount of total strain may not be sufficient to accurately estimate for the given problem.

We also note that the images for display more artifacts than the corresponding images for . This may be explained by recognizing that in general the measured displacements are less sensitive to changes in than in 16, which makes recovering from displacement measurements more sensitive to noise. In addition we note that is recovered sequentially, that is it is based on utilizing a reconstructed (and hence noisy) distribution. Therefore, in addition to the noise in the measured displacements, it is also influenced by errors in the recovered distribution.

Finally, we note that strain images can be misleading when interpreting nonlinear elastic effects. A quick glance at Figures 10.4 and 10.5 might convince the reader, that relative to the background, the first two inclusions soften with strain. However, a look at the measured and reconstructed values of reveals that this not the case for the second inclusion. Only the first inclusion behaves this way, a fact that is correctly captured in the reconstruction for .

10.4.3 Effect of Varying Regularization Parameters

The total variation regularization terms are the last two terms in Eq. ( 10.1). In these terms and are parameters that determine the importance of the regularization term relative to the data matching term while and are parameters that ensure that the integrand of the regularization terms is twice‐differentiable when or . The idea is to select to be small enough so that they do not influence the final result, and at the same time provide the problem with enough smoothness so that quasi‐Newton methods can be used to solve it.

We have thoroughly investigated the effect of varying and on the corresponding reconstructions. In Figure 10.6 we present one such study for reconstructing the nonlinear parameter of the first specimen. We select three different values of (that differ from each other by a factor of ) while keeping all other parameters fixed. We observe that changes in the reconstructions, especially between the two images corresponding to the smaller values of , are minimal, indicating that we have reached an asymptotically small value for . We also observe that reducing adversely effects the smoothness of the problem which translates to an increase in the total number of iterations to convergence.

The effect of varying the regularization parameter is shown in Figure 10.7, where we plot the reconstructed images for for section 1 at three different values of . We note that the image on the right (Figure 10.7c) is clearly under‐regularized and displays strong variations even within the homogeneous background region. The center image (Figure 10.7b) gets rid of this artifact while preserving the boundary of the inclusion. The image on the left (Figure 10.7a) has a still higher regularization parameter (around 2.5 times that for the center image). We observe that this image is similar to the center image, although the distribution for appears to have shifted to a lower overall value. This is to be expected as this change corresponds to lowering the total variation of .

10.4.4 Effect of Boundary Conditions on Lateral Edges

While solving the forward problem we are required to specify a boundary condition in each direction on every edge. In the axial direction the measured displacements are used as boundary conditions on every edge. If we also use the measured displacement in the lateral direction as boundary data we obtain the shear modulus image for section 1 shown in Figure 10.8b. In this image we can clearly see artifacts along the edges due to the noisy lateral displacement estimates that are imposed strongly through the boundary conditions. These artifacts are somewhat mitigated by using a smoothed version of the lateral displacements, as seen in the image in Figure 10.8c. The smoothed displacement field on the boundary is obtained by linearly interpolating between 24 carefully selected points on the boundary. A point is included in this set if its lateral displacement is close to the average lateral displacement on the edge (so as to preclude outliers) and if it is sufficiently far from other points that have already been included in the selected set. Perhaps the best (in terms of a homogeneous background) reconstruction is obtained when homogeneous lateral traction is imposed (see Figure 10.8c). We note that this latter boundary condition is appropriate for our data since the phantom was not constrained along the lateral direction as they were compressed. This is the boundary condition we have used in the reconstructions shown in Figures 10.2 and 10.3.