Identifiability of tissue material parameters from uniaxial tests using multi-start optimization

Abstract

Determining tissue biomechanical material properties from mechanical test data is frequently required in a variety of applications. However, the validity of the resulting constitutive model parameters is the subject of debate in the field. Parameter optimization in tissue mechanics often comes down to the “identifiability” or “uniqueness” of constitutive model parameters; however, despite advances in formulating complex constitutive relations and many classic and creative curve-fitting approaches, there is currently no accessible framework to study the identifiability of tissue material parameters. Our objective was to assess the identifiability of material parameters for established constitutive models of fiber-reinforced soft tissues, biomaterials, and tissue-engineered constructs and establish a generalizable procedure for other applications. To do so, we generated synthetic experimental data by simulating uniaxial tension and compression tests, commonly used in biomechanics. We then fit this data using a multi-start optimization technique based on the nonlinear least-squares method with multiple initial parameter guesses. We considered tendon and sclera as example tissues, using constitutive models that describe these fiber-reinforced tissues. We demonstrated that not all the model parameters of these constitutive models were identifiable from uniaxial mechanical tests, despite achieving virtually identical fits to the stress-stretch response. We further show that when the lateral strain was considered as an additional fitting criterion, more parameters are identifiable, but some remain unidentified. This work provides a practical approach for addressing parameter identifiability in tissue mechanics.

Introduction

Knowledge of material properties is valuable to study complex physio-mechanical tissue function, to monitor pathophysiological changes, and to characterize tissue-engineered constructs [1], [2], [3], [4], [5]. Despite the widespread use of parameter optimization when fitting experimental data, there are important limitations to this technique arising from the uniqueness (or lack thereof) of the fitted parameters. The lack of uniqueness is problematic because it hinders the ability to measure a tissue's mechanical properties and limits the usefulness of the measured parameters in finite element modeling and other applications. Therefore, a systematic and practical understanding of this limitation in parameter optimization is needed.

Several optimization approaches have been used in tissue mechanics, yet their success is a subject of debate. Nonlinear least-squares optimization (NLSQ) is perhaps the most commonly used approach. However, NLSQ methods are prone to local minima traps and depend strongly on the initial guesses used in the fitting algorithm; further, when fitting parameters of highly complex, nonlinear, and anisotropic tissues, uniqueness (or identifiability) is another hurdle [6]. These problems limit the value of the resulting fitted parameters. Global optimization algorithms, such as genetic algorithms, particle swarm, and simulated annealing, have been used in tissue mechanics to avoid local minima trap with variable success [5], [7], [8].

Parameter optimization in tissue mechanics is often a problem in the “identifiability” or “uniqueness” of constitutive model parameters [9], [10]. A general definition for identifiability is provided in the classic text by Walter and Pronzato [10]. Several studies have previously addressed the identifiability of tissue material parameters. For example, Hartmann and Gilbert used the determinant of the Hessian matrix to study identifiability of the two parameters (bulk and shear modulus) describing an elastic material using analytical and finite element solutions [11]; in another study, Akintunde and co-workers used rank deficiency of the Fisher information matrix to study uncertainty and identifiability in murine patellar tendon stress-strain responses [12]. Several other studies have also addressed aspects of characterization of material parameters by using both phenomenological [13], [14], and micromechanical [15] modeling approaches. However, despite advances in formulating complex constitutive relations and curve-fitting approaches, there is currently no accessible framework to assess the identifiability of tissue material parameters.

The objective of this study was to assess the identifiability of material parameters for established constitutive models used to describe the anisotropic and nonlinear response of fiber-reinforced soft tissues, biomaterials, and tissue engineered constructs and establish a generalizable procedure for other applications. We used a numerical approach and focused on several commonly-performed canonical experiments: uniaxial tension and unconfined compression. We used a Monte-Carlo-type multi-start optimization approach [16], [17] that Safa and co-workers recently implemented to study poroelasticity and inelasticity of tendon [18], [19]. This method enables exploration of the search space around the initial guess and reveals parameter sets that produce the same mechanical response. We show that while some parameters are identifiable, many are not, even though nearly perfect fits to the stress-stretch response can be achieved. We further show that when we impose a second fitting criterion, namely lateral strain, more parameters are identifiable, but some are still not.