math.OCOct 2, 2026

A hybrid CNN-adjoint optimization framework for reconstruction of viscoelastic tissue properties in magnetic resonance elastography

Authors: Anwesa Dey, Johann Rudi, Elena Cherkaev

Organizations: Department of Mathematics, The University of Utah, address USA. · Department of Mathematics, Virginia Tech, address USA. · Department of Mathematics, The University of Utah, address, address, USA.

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Abstract

Magnetic resonance elastography (MRE) is a noninvasive imaging modality for quantifying the viscoelastic properties of soft tissues from shear wave propagation. Recovering the complex-valued shear modulus from measured displacement fields leads to a severely ill-posed inverse problem, particularly in the presence of noise and limited boundary excitations. We investigate adjoint-based optimization, convolutional neural network (CNN) reconstruction, and a hybrid framework combining both approaches. The forward model is based on a scalar form of the modified stationary Stokes system with a complex shear modulus. We establish well-posedness of the forward problem, existence of minimizers, and first-order optimality conditions for the adjoint-based formulation, and implement a nonlinear conjugate-gradient method with Armijo line search. While PDE-constrained optimization can accurately refine coefficient reconstructions, its performance depends strongly on initialization. We therefore construct a two-dimensional CNN that maps complex-valued displacement measurements to spatially varying complex shear modulus fields and provides rapid, informative initial reconstructions. The proposed hybrid method uses the CNN reconstruction to initialize the adjoint-based optimization, yielding faster convergence and improved accuracy. The CNN is trained on coefficient fields containing individual perturbations and tested on both individual and previously unseen combined configurations. Numerical experiments demonstrate that the CNN generalizes to these more challenging configurations, while subsequent PDE-constrained optimization further refines the reconstructed coefficient. These results demonstrate the potential of combining data-driven initialization with physics-based optimization for efficient and accurate MRE reconstruction.

Figures & tables

Figure 1: Representative complex-valued shear modulus A(x,y)A(x,y) containing Gaussian perturbations in its real and imaginary components.
Figure 2: Real and imaginary parts of the displacement field u(x,y)u(x,y) obtained by solving the forward PDE for the Gaussian shear modulus shown in Figure 1 .
Figure 3: Representative complex-valued shear modulus A(x,y)A(x,y) containing mollified elliptical disc-shaped perturbations in its real and imaginary components.
Figure 4: Real and imaginary parts of the displacement field u(x,y)u(x,y) obtained by solving the forward PDE for the mollified elliptical disc-shaped shear modulus shown in Figure 3 .
StageOperationOutput size
Input–48×48×248\times 48\times 2
Encoder 1Conv2D (3×3)(3\times 3) , 32 filters, ReLU48×48×3248\times 48\times 32
Downsampling 1Conv2D (3×3)(3\times 3) , 32 filters, stride 2, ReLU24×24×3224\times 24\times 32
Encoder 2Conv2D (3×3)(3\times 3) , 64 filters, ReLU24×24×6424\times 24\times 64
Downsampling 2Conv2D (3×3)(3\times 3) , 64 filters, stride 2, ReLU12×12×6412\times 12\times 64
Latent representation2 Conv2D (3×3)(3\times 3) , 128 filters, ReLU12×12×12812\times 12\times 128
Table 1: Architecture of the 2D convolutional neural network used to generate the initial coefficient reconstruction.
 MeanStd.MedianMin.Max.
Re⁡(A)\operatorname{Re}(A)6.4693×10−26.4693\times 10^{-2}3.0304×10−23.0304\times 10^{-2}5.8194×10−25.8194\times 10^{-2}1.4715×10−21.4715\times 10^{-2}2.1719×10−12.1719\times 10^{-1}
Im⁡(A)\operatorname{Im}(A)6.6093×10−26.6093\times 10^{-2}2.7720×10−22.7720\times 10^{-2}6.0846×10−26.0846\times 10^{-2}1.8937×10−21.8937\times 10^{-2}1.9950×10−11.9950\times 10^{-1}
AA9.4681×10−29.4681\times 10^{-2}3.5716×10−23.5716\times 10^{-2}8.8932×10−28.8932\times 10^{-2}3.1948×10−23.1948\times 10^{-2}2.6949×10−12.6949\times 10^{-1}
Table 2: CNN reconstruction errors over the 500500 test samples in Experiment I.
Figure 5: Distribution of the CNN reconstruction errors over all 500500 test samples in Experiment I. The test dataset contains both Gaussian and mollified-ellipse coefficient fields.
Figure 6: Real and imaginary components of the internal displacement field udu_{d} corresponding to a representative complex-valued mollified-ellipse perturbation. These displacement data form the two input channels to the CNN.
Figure 7: Representative CNN prediction for a complex-valued mollified-ellipse test sample from Experiment I. The top row compares the true and CNN-predicted real components of A(x,y)A(x,y) , while the bottom row compares the corresponding imaginary components.
Figure 8: Real and imaginary components of the internal displacement field udu_{d} corresponding to a representative complex-valued Gaussian perturbation. These displacement data are supplied to the CNN as the two input channels for predicting the corresponding complex-valued coefficient field.
Figure 9: Representative CNN prediction for a complex-valued Gaussian test sample from Experiment I. The top row compares the true and CNN-predicted real components of A(x,y)A(x,y) , while the bottom row compares the corresponding imaginary components. The CNN prediction is subsequently used as the initial guess for the PDE-constrained optimization stage of the hybrid reconstruction framework.
Figure 10: Real component of the representative Gaussian test sample shown using separate color scales for the true coefficient and the CNN prediction. The individually scaled CNN prediction reveals the weak spatial variations that are not readily visible when the common color scale in Figure 9 is used.
Parametermollified elliptical discGaussian
Data-fidelity weight α\alpha2020
Regularization parameter β\beta1×10−71\times 10^{-7}5×10−65\times 10^{-6}
Maximum NCG iterations kmax⁡k_{\max}10002000
Initial line-search step size11
Armijo parameter1×10−41\times 10^{-4}1×10−41\times 10^{-4}
Backtracking reduction factor0.80.8
Table 3: Optimization parameters used for the representative mollified-ellipse and Gaussian test samples in Experiment I.
Figure 11: End-to-end hybrid reconstruction for a representative complex-valued mollified-ellipse test sample from Experiment I. The left column shows the true coefficient, the middle column shows the CNN-generated initial estimate, and the right column shows the coefficient reconstructed after PDE-constrained optimization. The top and bottom rows correspond to the real and imaginary components of A(x,y)A(x,y) , respectively.
Figure 12: End-to-end hybrid reconstruction for a representative complex-valued Gaussian test sample from Experiment I. The left column shows the true coefficient, the middle column shows the CNN-generated initial estimate, and the right column shows the coefficient reconstructed after PDE-constrained optimization. The top and bottom rows correspond to the real and imaginary components of A(x,y)A(x,y) , respectively.
 CNN initial estimateHybrid reconstructionError decrease (%)
∥Re⁡(A)−Re⁡(Ad)∥L2\|\operatorname{Re}(A)-\operatorname{Re}(A_{d})\|_{L^{2}}1.005910×10−11.005910\times 10^{-1}3.075439×10−23.075439\times 10^{-2}69.4369.43
∥Im⁡(A)−Im⁡(Ad)∥L2\|\operatorname{Im}(A)-\operatorname{Im}(A_{d})\|_{L^{2}}1.365835×10−11.365835\times 10^{-1}2.752220×10−22.752220\times 10^{-2}79.8579.85
∥A−Ad∥L2\|A-A_{d}\|_{L^{2}}1.696279×10−11.696279\times 10^{-1}4.127111×10−24.127111\times 10^{-2}75.6775.67
Table 4: Reconstruction errors for the representative mollified-elliptical disc-shaped test sample before and after the hybrid reconstruction framework.
 CNN initial estimateHybrid reconstructionError decrease (%)
∥Re⁡(A)−Re⁡(Ad)∥L2\|\operatorname{Re}(A)-\operatorname{Re}(A_{d})\|_{L^{2}}1.2582×10−11.2582\times 10^{-1}7.3615×10−27.3615\times 10^{-2}41.4941.49
∥Im⁡(A)−Im⁡(Ad)∥L2\|\operatorname{Im}(A)-\operatorname{Im}(A_{d})\|_{L^{2}}4.3595×10−24.3595\times 10^{-2}4.1262×10−24.1262\times 10^{-2}5.355.35
∥A−Ad∥L2\|A-A_{d}\|_{L^{2}}1.3316×10−11.3316\times 10^{-1}8.4390×10−28.4390\times 10^{-2}36.6236.62
Table 5: Reconstruction errors for the representative Gaussian test sample before and after the hybrid reconstruction framework.
Figure 13: Distribution of the CNN reconstruction errors over all 500500 test samples using internal displacement data contaminated with 1%1\% relative noise. The CNN is trained using noiseless displacement data and is applied directly to the noisy measurements without retraining.
 MeanStd.MedianMin.Max.
Re⁡(A)\operatorname{Re}(A)7.1454×10−27.1454\times 10^{-2}3.2029×10−23.2029\times 10^{-2}6.5777×10−26.5777\times 10^{-2}1.5458×10−21.5458\times 10^{-2}2.3168×10−12.3168\times 10^{-1}
Im⁡(A)\operatorname{Im}(A)7.1907×10−27.1907\times 10^{-2}2.8521×10−22.8521\times 10^{-2}6.7771×10−26.7771\times 10^{-2}2.0490×10−22.0490\times 10^{-2}1.9553×10−11.9553\times 10^{-1}
AA1.0377×10−11.0377\times 10^{-1}3.6707×10−23.6707\times 10^{-2}9.7522×10−29.7522\times 10^{-2}4.2700×10−24.2700\times 10^{-2}2.9684×10−12.9684\times 10^{-1}
Table 6: Summary statistics of the CNN reconstruction errors for the 1%1\% noisy test data.
ParameterGaussianmollified elliptical disc
Data-fidelity parameter α\alpha20202020
Regularization parameter β\beta1×10−51\times 10^{-5}1×10−61\times 10^{-6}
Maximum number of iterations kmax⁡k_{\max}200200150150
Initial line-search step size1111
Armijo parameter10−410^{-4}10−410^{-4}
Step-size reduction factor0.80.80.80.8
Table 7: Optimization parameters used for the reconstruction of the Gaussian and mollified-ellipse inclusions with 1%1\% relative noise.
Figure 14: Reconstruction of a representative Gaussian coefficient from internal displacement data contaminated with 1%1\% relative noise. The left, middle, and right columns show the true coefficient, the CNN-generated initial estimate, and the final hybrid reconstruction, respectively. The top and bottom rows correspond to the real and imaginary components of A(x,y)A(x,y) .
Reconstruction errorCNN initializationHybrid reconstructionError decrease (%)
∥Re⁡(A)−Re⁡(Ad)∥L2(Ω)\|\operatorname{Re}(A)-\operatorname{Re}(A_{d})\|_{L^{2}(\Omega)}6.291202×10−26.291202\times 10^{-2}6.229779×10−26.229779\times 10^{-2}0.98%0.98\%
∥Im⁡(A)−Im⁡(Ad)∥L2(Ω)\|\operatorname{Im}(A)-\operatorname{Im}(A_{d})\|_{L^{2}(\Omega)}5.586334×10−25.586334\times 10^{-2}5.436543×10−25.436543\times 10^{-2}2.68%2.68\%
∥A−Ad∥L2(Ω)\|A-A_{d}\|_{L^{2}(\Omega)}8.413462×10−28.413462\times 10^{-2}8.268383×10−28.268383\times 10^{-2}1.72%1.72\%
Table 8: Reconstruction errors for the representative Gaussian coefficient using displacement data contaminated with 1%1\% relative noise.
Figure 15: Reconstruction of a representative mollified-ellipse coefficient from internal displacement data contaminated with 1%1\% relative noise. The left, middle, and right columns show the true coefficient, the CNN-generated initial estimate, and the final hybrid reconstruction, respectively. The top and bottom rows correspond to the real and imaginary components of A(x,y)A(x,y) .
Reconstruction errorCNN initializationHybrid reconstructionError decrease
∥Re⁡(A)−Re⁡(Ad)∥L2(Ω)\|\operatorname{Re}(A)-\operatorname{Re}(A_{d})\|_{L^{2}(\Omega)}7.901710×10−27.901710\times 10^{-2}7.279785×10−27.279785\times 10^{-2}7.87%7.87\%
∥Im⁡(A)−Im⁡(Ad)∥L2(Ω)\|\operatorname{Im}(A)-\operatorname{Im}(A_{d})\|_{L^{2}(\Omega)}1.040895×10−11.040895\times 10^{-1}6.924087×10−26.924087\times 10^{-2}33.48%33.48\%
∥A−Ad∥L2(Ω)\|A-A_{d}\|_{L^{2}(\Omega)}1.306841×10−11.306841\times 10^{-1}1.004680×10−11.004680\times 10^{-1}23.12%23.12\%
Table 9: Reconstruction errors for the representative mollified-ellipstical disc coefficient using displacement data contaminated with 1%1\% relative noise.
 MeanStd.MedianMin.Max.
Re⁡(A)\operatorname{Re}(A)1.279967×10−11.279967\times 10^{-1}4.694296×10−24.694296\times 10^{-2}1.202852×10−11.202852\times 10^{-1}4.829479×10−24.829479\times 10^{-2}3.416799×10−13.416799\times 10^{-1}
Im⁡(A)\operatorname{Im}(A)1.290662×10−11.290662\times 10^{-1}4.606712×10−24.606712\times 10^{-2}1.234582×10−11.234582\times 10^{-1}4.722147×10−24.722147\times 10^{-2}3.733799×10−13.733799\times 10^{-1}
AA1.852892×10−11.852892\times 10^{-1}5.509064×10−25.509064\times 10^{-2}1.760132×10−11.760132\times 10^{-1}8.382436×10−28.382436\times 10^{-2}4.479576×10−14.479576\times 10^{-1}
Table 10: CNN reconstruction errors over the 500500 combined Gaussian and mollified-ellipse test samples in Experiment II.
Figure 16: Distribution of the CNN reconstruction errors over all 500500 test samples in Experiment II. Each real and imaginary component of the coefficient contains both a Gaussian and a mollified-ellipse perturbation.
Figure 17: Real and imaginary components of the internal displacement field udu_{d} corresponding to a representative test coefficient containing combined Gaussian and mollified-ellipse perturbations. These displacement data form the two input channels to the CNN.
Figure 18: Representative CNN prediction for a combined Gaussian and mollified-ellipse test sample from Experiment II. The top row compares the true and CNN-predicted real components of A(x,y)A(x,y) , while the bottom row compares the corresponding imaginary components. Both components contain one Gaussian and one independently generated mollified elliptical disc.
ParameterValue
Data-fidelity weight α\alpha20
Regularization parameter β\beta1×10−71\times 10^{-7}
Maximum NCG iterations kmax⁡k_{\max}2500
Initial line-search step size1
Armijo parameter1×10−41\times 10^{-4}
Backtracking reduction factor0.8
Table 11: Optimization parameters used throughout Experiment II.
Figure 19: End-to-end hybrid reconstruction for a representative combined Gaussian and mollified-ellipse test sample from Experiment II. The left column shows the true coefficient, the middle column shows the CNN-generated initial estimate, and the right column shows the coefficient reconstructed after PDE-constrained optimization. The top and bottom rows correspond to the real and imaginary components of A(x,y)A(x,y) , respectively.
 CNN initial estimateHybrid reconstructionError decrease (%)
∥Re⁡(A)−Re⁡(Ad)∥L2\|\operatorname{Re}(A)-\operatorname{Re}(A_{d})\|_{L^{2}}1.097275×10−11.097275\times 10^{-1}5.953433×10−25.953433\times 10^{-2}45.7445.74
∥Im⁡(A)−Im⁡(Ad)∥L2\|\operatorname{Im}(A)-\operatorname{Im}(A_{d})\|_{L^{2}}7.435254×10−27.435254\times 10^{-2}4.500365×10−24.500365\times 10^{-2}39.4739.47
∥A−Ad∥L2\|A-A_{d}\|_{L^{2}}1.325459×10−11.325459\times 10^{-1}7.463019×10−27.463019\times 10^{-2}43.7043.70
Table 12: Reconstruction errors for the representative combined Gaussian and mollified-ellipse test sample before and after the hybrid reconstruction framework.
Figure 20: End-to-end hybrid reconstruction for a representative test sample with spatially overlapping Gaussian and mollified-ellipse perturbations. The left column shows the true coefficient, the middle column shows the CNN-generated initial estimate, and the right column shows the coefficient obtained after PDE-constrained optimization. The top and bottom rows correspond to the real and imaginary components of A(x,y)A(x,y) , respectively. This example provides a more challenging generalization test because the two inclusion types occupy overlapping spatial regions, while the CNN was trained only on the individual shape families.
 CNN initial estimateHybrid reconstructionError decrease (%)
∥Re⁡(A)−Re⁡(Ad)∥L2\|\operatorname{Re}(A)-\operatorname{Re}(A_{d})\|_{L^{2}}8.751159×10−28.751159\times 10^{-2}5.142576×10−25.142576\times 10^{-2}41.2441.24
∥Im⁡(A)−Im⁡(Ad)∥L2\|\operatorname{Im}(A)-\operatorname{Im}(A_{d})\|_{L^{2}}1.562200×10−11.562200\times 10^{-1}1.140009×10−11.140009\times 10^{-1}27.0327.03
∥A−Ad∥L2\|A-A_{d}\|_{L^{2}}1.790614×10−11.790614\times 10^{-1}1.250632×10−11.250632\times 10^{-1}30.1630.16
Table 13: Reconstruction errors for the representative test sample with overlapping Gaussian and mollified-ellipse perturbations.
Figure 21: Distribution of the CNN reconstruction errors over all 500500 combined-shape test samples using internal displacement data contaminated with 1%1\% relative noise. The CNN is trained using noiseless individual Gaussian and mollified-ellipse samples and is applied to the noisy combined-shape measurements without retraining.
Figure 22: Reconstruction of a representative combined-shape coefficient from internal displacement data contaminated with 1%1\% relative noise. The real component contains spatially separated Gaussian and mollified-ellipse inclusions, whereas the inclusions partially overlap in the imaginary component. The left, middle, and right columns show the true coefficient, the CNN-generated initial estimate, and the final hybrid reconstruction, respectively. The top and bottom rows correspond to the real and imaginary components of A(x,y)A(x,y) .
Reconstruction errorCNN initializationHybrid reconstructionError decrease (%)
∥Re⁡(A)−Re⁡(Ad)∥L2(Ω)\|\operatorname{Re}(A)-\operatorname{Re}(A_{d})\|_{L^{2}(\Omega)}1.229673×10−11.229673\times 10^{-1}8.524580×10−28.524580\times 10^{-2}30.6830.68
∥Im⁡(A)−Im⁡(Ad)∥L2(Ω)\|\operatorname{Im}(A)-\operatorname{Im}(A_{d})\|_{L^{2}(\Omega)}1.193186×10−11.193186\times 10^{-1}6.553128×10−26.553128\times 10^{-2}45.0845.08
∥A−Ad∥L2(Ω)\|A-A_{d}\|_{L^{2}(\Omega)}1.713414×10−11.713414\times 10^{-1}1.075230×10−11.075230\times 10^{-1}37.2437.24
Table 14: Reconstruction errors for the representative combined-shape coefficient using displacement data contaminated with 1%1\% relative noise.

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