Using dropout based active learning and surrogate models in the inverse viscoelastic parameter identification of human brain tissue

Abstract

Inverse mechanical parameter identification enables the characterization of ultrasoft materials, for which it is difficult to achieve homogeneous deformation states. However, this usually involves high computational costs that are mainly determined by the complexity of the forward model. While simulation methods like finite element models can capture nearly arbitrary geometries and implement involved constitutive equations, they are also computationally expensive. Machine learning models, such as neural networks, can help mitigate this problem when they are used as surrogate models replacing the complex high fidelity models. Thereby, they serve as a reduced order model after an initial training phase, where they learn the relation of in- and outputs of the high fidelity model. The generation of the required training data is computationally expensive due to the necessary simulation runs. Here, active learning techniques enable the selection of the "most rewarding" training points in terms of estimated gained accuracy for the trained model. In this work, we present a recurrent neural network that can well approximate the output of a viscoelastic finite element simulation while significantly speeding up the evaluation times. Additionally, we use Monte-Carlo dropout based active learning to identify highly informative training data. Finally, we showcase the potential of the developed pipeline by identifying viscoelastic material parameters for human brain tissue.

Keywords: active learning; human brain tissue; neural network; parameter identification; surrogate model.

FIGURE 1

Rheological scheme of the implemented generalized Maxwell model with one Maxwell element. Each nonlinear spring represents a hyperelastic one-term Ogden model characterized by the shear modulus μ and the nonlinearity α. η denotes the viscosity of the dashpot element.

FIGURE 2

Experimental data of a human brain tissue specimen from the frontal cortex.

FIGURE 3

(A) Signal flow through an LSTM cell. σ denotes the sigmoid activation function. (B) Recurrent neural network architecture consisting of three layers with long short-term memory (LSTM) cells and two dropout layers in between. The input vector x contains the constitutive parameters as well as strain and time while the output h is the nominal stress P and shear stress τ. The internal cell state is stored in the state vector a and the subscript n denotes the timestep.

FIGURE 4

Correlation between the variance obtained by Monte Carlo dropout sampling and the mean absolute error (MAE) where both are normalized by their standard deviation over all samples. Results are shown for 100 points in the material parameter space and the same time series input (shear and stretch) that were created using Poisson Disk sampling.

FIGURE 5

Comparison between the value as well as the standard deviation of the determination coefficient R ² for adding new points to the training data set when they are selected randomly or via active learning, e.g., those with the highest estimated variance.

FIGURE 6

Comparison of metamodel and finite element simulation output in terms of nominal and shear stress for the same time series input (stretch and shear stress) as well as identical material parameters.

FIGURE 7

Parameter identification results for experimental data of human brain tissue from the frontal cortex. The finite element simulation output is shown for the final identified parameter set with α _∞ = −16, μ _∞ = 162 Pa, η ₁ = 13,949 Pa⋅s, α ₁ = −18, μ ₁ = 398 Pa.

FIGURE 8

History of parameter values during the optimization using the hybrid approach with the surrogate and finite element model combined (hybrid) as well as only the finite element model. Function evaluations contain all model evaluations including finite difference approximations of the gradient as well as failed optimization steps. The cpu times are measured using the perf_counter function of the Python module time.