Combined Estimation of Material Parameters and Unloaded Configurations 

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Literature Review

For solving the inverse problem in hyperelasticity, different methods have been proposed. First of all, it is necessary to consider a re-parameterization of the weak form of the forward problem of finite elasticity as a solution method for the inverse incompressible problem. Many numerical approaches have been proposed for solving forward problems in incompressible finite elasticity. Most commonly, a mixed formulation is assumed with independent fields for displacements, pressure, and sometimes the volumetric deformation[7].
The methods used in [12] has been implemented for a problem which contains fluid like what is discussed in the next chapter [11] as in Figure 1.2. The concept of the inverse problem in hyperelasticity is what is proposed in [2].
Figure 1.2: Illustration of Sellier’s iterative method for identifying a stress-free reference configu-ration in biomechanical boundary value problems [11].
The concept of the inverse problem in hyperelasticity is what is proposed in [2].
As discussed in the last chapter, the inverse problem can be solved for the multi-dimensional problems if there are as many data points as needed. The same problem can be solved for the lung, but there are other aspects of this problem: the existence of the thorax and the porosity in the lung.
In [9], the lung’s inverse problem by considering the porosity and poromechanics theory for large deformations has been discussed hyperelastic potential reproducing the volumetric response of the pulmonary mixture to a change of pressure has been studied.
Figure 1.3: Schematic of the reference configuration and the deformed configuration with the associated local quantities, volumes and porosity [9].
A similar method has been discussed for the heart. The proposed method is embedded within the continuum theory of fictitious configurations and uses a fixed-point iteration on the geometry itself [3].
The motivation to work on the lung is its importance on human health as there are many pulmonary diseases. For example, a chronic disease in which collagen fibers accumulate into interstitial tissue leads to thickening, stiffening, and damage of alveolar walls. This disease remains poorly understood, poorly diagnosed, and poorly treated and represents a real clinical challenge. In interaction with data such as medical imaging, mechanical modeling-based tools could help clinicians in classifying patients and thus deciding on the treatment options [8]. For modeling the lung from the CT images, images registration tools have been implemented [4].
In the lung problem, the lung is in contact with the thorax, so the boundary problem by consid-ering the contact between the lung and thorax should be considered in the boundary condition problem [10]. In [5], image processing has been implemented for the heart problem for extracting ventricular strain data from MR images.
Figure 1.4: Kinematics of pre-strained biological systems. Pre-strain maps the stress-free refer-ence configuration onto the residually stressed but mechanically unloaded configuration [3].

Types of the Problems Discussed

This chapter will start with the most simple problem, which is the linear spring. This problem has a one-dimensional configuration, which is the length of the spring. And also, there is stiffness for this spring as its material parameter.
Assumptions for obtaining the unknown material parameter and unloaded configuration is that the spring’s length under an applied force has been measured. So, two applied forces with the corresponding deformed length are needed two obtain these two unknowns.
Different methods for solving this problem are implemented and generalized for a nonlinear spring and a three-dimensional problem, which is the Rivlin cube. For Rivlin cube, different material laws have been studied, and the more practical one is introduced. All the different methods have been implemented in the tube problem with the n-dimension configuration, and the more efficient one is introduced.

Materials and Methods

In mechanical systems, we may face problems in which the initial (unloaded) state is known, plus the material parameters. In this situation, the configuration under a specific load can be determined.
There are some unknown parameters, while these parameters like deformed configuration in these kinds of problems are known. Moreover, this deformed configuration is typically measured from the imaging data. So, there would be a cost function like Equation (1.1), which shows that the parameter µ is the actual value when making the cost function minimum, which is equal to zero.
µ ˘ argmin‰ 1 Z›mes (U(µ) ¡U mes)2 ¾
So, the goal is to minimize the cost function, which varies from problem to problem. Never-theless, the critical point is that the suggested cost function needs at least as many terms as the unknown parameters. There are different ways to minimize such a cost function, containing unknown parameters such as the material parameters or the unknown configurations. The unknown configuration depends on the problem dimension; for instance, it is one for a linear spring and three for a cube with three different dimensions.

COMBINED ESTIMATION OF MATERIAL PARAMETERS AND UNLOADED CONFIGURATIONS

Introduction

In this chapter, the inverse problem has been discussed for different models, and different methods have been implemented for solving the nonlinear equations. The inverse problem contains two aspects of the unloaded configuration and material parameter estimation, estimated together. Before going to the inverse problem concept, the forward problem has been explained. In the end, the minimization of the cost function has been discussed and methods of doing it.

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Models

Materials discussed in this paper are linear and nonlinear spring, Rivlin cube, and a tube. Each of them has its material parameters and coordinates, which are discussed in the following.

Linear Spring

A linear spring is the most straightforward problem which can be discussed here. The linear spring has only a stiffness coefficient and a length, which changes by applying a force. In general, the linear spring law can be written as below: (2.1) E ude f ¡ uunde f ˘ F

Nonlinear Spring

For the linear spring, there was only one material parameter, which was the young modulus. For the nonlinear spring, there are two parameters for the nonlinear spring, Three data points are needed for solving the inverse problem and material parameters for this geometry as there are two parameters as material parameters. The solving method is a bit different here, though the CMA1 algorithm has been used again. For solving this problem with three data points and three unknown parameters, the solving process can be started from a known configuration with the applied force, different values for E0 and †1 can be guessed, which are the material parameters.
There are three deformed configurations with measured forces. So, the solving process can be started with one of the known configurations and with the guessed material parameters, and the unloaded configuration can be computed by it. There are an unloaded configuration and two measured forces with which two corresponding loaded configuration can be computed. Here, the cost function is defined as calculating the difference between the measured deformed configuration and the calculated ones. The minimization of the cost function concerning the material parameters are shown in the Figure 2.1

Rivlin Cube

What is mentioned simply for spring can be generalized for a 3D cube, which has three different dimensions, and any vector force can be applied on each aspect with different values that can deform the shape of the cube. If we applied force on a different aspect of such a cube, there would be deformation on each side.

Tube

Here a 2D tube has been discussed, which has an inner radius and an outer radius. This problem is symmetric and has a constant young modulus. The input of the problem is the tube inner pressure, which makes inflation of the tube.
This 2D problem is multi-dimensional in that it would be parameterized by the finite element method, and the node displacements would be the configuration of the problem. There is the young modulus as the material parameter for the tube problem and the unloaded configuration. There are some time steps, which are the configurations of the tube for different values of the inner pressure. In Figure 2.2, the first and last loaded configuration has been shown.

Forward Problem

This kind of problem is not very common to solve. The meaning of the forward problem is that we change the model configuration and see what forces correspond to the new configuration. In the real problem, this is rarely happening, or it may only be calculated to see the force needed to obtain the ideal configuration. In the following, the forward problem is demonstrated for the models discussed in this paper.

A Method for Solving Nonlinear Problems

For the hyperelastic model the stress tensor ¾(‚) should be equal to the applied pressure P, but as an initial guess like ‚0 we have a residual which is the difference between the stress tensor and the applied pressure vector: (2.12) R0 ˘ ¾(‚0)¡P
What has been done here is that the cost function should be minimized, which means that its gradient should become zero. So, the newton method has been done on the derivative of the cost function.

Linear Spring

For a linear spring, The forward problem is such that the stiffness coefficient, applied force, and the undeformed length is known, and the only unknown is the deformed length, which can be obtained by the following relation: (2.20) ude f ˘ uundef ¯ F

Rivlin Cube

What is mentioned simply for spring can be generalized for a 3D cube, which has three different dimensions, and any force vector can be applied on each face with different values that can deform the shape of the cube. If a force is applied to different aspects of such a cube, there would be deformation on each side. For example, the mentioned force can be applied on a cube

Table of contents :

1 Introduction 
2 Combined Estimation of Material Parameters and Unloaded Configurations 
2.1 Models
2.1.1 Linear Spring
2.1.2 Nonlinear Spring
2.1.3 Rivlin Cube
2.1.4 Tube
2.2 Forward Problem
2.2.1 A Method for Solving Nonlinear Problems
2.2.2 Linear Spring
2.2.3 Rivlin Cube
2.3 Inverse Problem
2.3.1 Methods for Minimizing a Cost Function
2.3.2 Methods for Computing Gradients and Hessians
2.3.3 Unloaded Configuration Estimation
2.3.4 Material Parameter Estimation
2.3.5 Combined Estimation of the Unloaded Configuration and Material Parameters
2.4 Finite Differences Approximation of the Derivative and Hessian Matrix
2.4.1 Computing the Gradient of Some Functions
3 Lung Mechanical Modeling 
3.1 Introduction
3.1.1 Lung Mesh
3.1.2 Thorax Mesh
3.1.3 Lung Porosity
3.1.4 Thorax Displacement Field
4 Discussion and Ongoing Research 
Bibliography 

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