Estimation of in vivo constitutive parameters of the aortic wall using a machine learning approach

Abstract

The patient-specific biomechanical analysis of the aorta requires the quantification of the in vivo mechanical properties of individual patients. Current inverse approaches have attempted to estimate the nonlinear, anisotropic material parameters from in vivo image data using certain optimization schemes. However, since such inverse methods are dependent on iterative nonlinear optimization, these methods are highly computation-intensive. A potential paradigm-changing solution to the bottleneck associated with patient-specific computational modeling is to incorporate machine learning (ML) algorithms to expedite the procedure of in vivo material parameter identification. In this paper, we developed an ML-based approach to estimate the material parameters from three-dimensional aorta geometries obtained at two different blood pressure (i.e., systolic and diastolic) levels. The nonlinear relationship between the two loaded shapes and the constitutive parameters are established by an ML-model, which was trained and tested using finite element (FE) simulation datasets. Cross-validations were used to adjust the ML-model structure on a training/validation dataset. The accuracy of the ML-model was examined using a testing dataset.

Keywords: constitutive parameter estimation; machine learning; neural network.

Figure 1.

The proposed machine learning (ML) approach.

Figure 2.

Datasets projected in 3D material parameter subspaces. The convex hull is plotted in the 3D subspaces for illustrative purpose.

Figure 3.

Sampling the SSM parameter spaces.

Figure 4.

Systolic aorta shapes corresponding to some representative sets of SSM parameters. The shapes are color-coded with curvature values.

Figure 5.

The procedure to generate aorta geometries at systole and diastole. The number in the parenthesis indicates the testing dataset.

Figure 6.

The neural network for mapping the shape codes to the material parameters. The green dots represents the input layer, and the blue dots represent the softplus units in the hidden layers and the output layer of the neural network.

Figure 7.

Adjusting the network structure using the leave-one-out (LOO) cross-validation.

Figure 8.

Evaluating the accuracy using the testing dataset.

Figure 9.

The actual and predicted material parameters. Each point was plotted using its actual value as horizontal x-coordinate and the ML-predicted value as the vertical y- coordinate. A perfect straight line (y=x) indicates perfect prediction, and any deviation from the straight line indicates prediction errors.

Figure 10.

The actual and predicted stress-stretch curves for the best ((a), (b) and (c)), median ((d), (e) and (f)) and worst cases ((g), (h) and (i)).

Figure 11.

MSE loss function for training and testing using softplus and other units.