Material Parameter Identification and Inverse Problems in Soft Tissue Biomechanics

Modelling the mechanical behaviour of soft tissues like tendon, ligament, skin and cartilage requires a knowledge of the structural and mechanical properties of the constitutive elements. These tissues have a hierarchical architecture from the molecular to the macroscopic scale, and are composites of different molecular building blocks. Here we first review the structure of the proteins and polysaccharides comprising such tissues. We then consider the structure and mechanical properties of two prototypical soft tissues: tendons and heart valves. An overview of their structure is followed by a description of the known mechanical behaviour of these tissues. Consideration is given to the role of different constituent components in mechanical response, structural anisotropy and testing methods which can probe mechanical deformation at multiple levels.

In this chapter, we are interested in the constitutive equations used to model macroscopically the mechanical function of soft tissues. After reviewing some basics about nonlinear finite strain constitutive relations, we present recent developments of experimental biomechanics and inverse methods aimed at quantifying constitutive parameters of soft tissues. A focus is given to in vitro characterization of hyperelastic parameters based on full-field data that can be collected with digital image correlation systems during the experimental tests. The specific use of these data for membrane-like tissues is first illustrated through the example of bulge inflation tests carried out onto pieces of aortic aneurysms. Then an inverse method, based on the principle of virtual power, is introduced to estimate regional variations of material parameters for more general applications.

Measuring the mechanical properties of soft tissues presents three interlinked problems. First, we must carry out experimental measurements to quantify the actual behaviour of the tissue. Second, we need to represent this by some kind of mathematical model, which typically has to be solved using numerical techniques such as the finite element (FE) method. Third, we need to find the parameter values in the model that best match the experiment and to quantify the uncertainty in the resulting material properties. Experimental measurements present numerous difficulties in comparison with conventional engineering materials and care is needed in the choice of test method, sample selection and preparation, calibration and interpretation of the results. Typically an optical technique may be needed to measure the deformation, such as digital image correlation (DIC). FE models of soft tissues are inherently difficult to solve because of their extreme nonlinearity and the typical stiffening behaviour with increasing deformation which leads to numerical instabilities. Possible ways to reduce convergence problems and increase the reliability of these models are discussed. The most common method to find the parameter values that match an experiment is to use an optimisation algorithm to try to find the parameters that best match the experimental results. However this is slow and there is no way of knowing whether the best parameters have been found or what range of other values could also be compatible with the experiment. A better approach is to generate a statistical emulator that predicts the result of the model and then to evaluate a wide range of parameter values in order to find the range of values that could be compatible with the experiment. This gives revealing insights into the uncertainty of the procedure and the validity of the final results.

The present chapter reviews vessel wall histology and summarizes relevant continuum mechanical concepts to study mechanics-induced tissue damage. As long as the accumulated damage does not trigger strain localizations, the standard nonpolar continuum mechanical framework is applicable. As an example, a damage model for collagenous tissue is discussed and used to predict collagen damage in the aneurysm wall at supra-physiologic loading. The physical meaning of model parameters allow their straight forward identification from independent mechanical and histological experimental data. In contrast, if damage accumulates until the material’s stiffness looses its strong ellipticity, more advanced continuum mechanical approaches are required. Specifically, modeling vascular failure by a fracture process zone is discussed, such that initialization and coalescence of micro-defects is mechanically represented by a phenomenological cohesive traction separation law. Failure of ventricular tissue due to deep penetration illustrates the applicability of the model. Besides appropriate continuum mechanical approaches, laboratory experiments that are sensitive to constitutive model parameters and ensure controlled failure propagation are crucial for a robust parameter identification of failure models.

This chapter describes the main features of standard tests for a mechanical characterisation of biological materials, like uniaxial, biaxial and shear tests. After that, the inverse, mixed experimental/numerical methods will be introduced as a tool to create more freedom in the design of experiments and to make the transition from ex vivo testing to in vivo testing possible. A short introduction to the algorithms that can be used to minimise the difference between the experimental results and the numerical results will be discussed, followed by two practical examples applied to skin. The chapter finishes with a comparison between the advantages and disadvantages of in vivo and ex vivo testing.

This chapter aims at illustrating how patient-specific models of human organs and soft tissues can be implemented into FE packages. First is addressed the question of the generation of patient-specific FE models compatible with the clinical constraints. Then is discussed the calibration of the material properties, with choices that should be done between calibrations based on ex vivo or in vivo tissues loadings. The example of computer-assisted maxillofacial surgery is addressed and results based on patients’ data are provided.