Full length articleIdentifiability of tissue material parameters from uniaxial tests using multi-start optimization
Graphical abstract
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.
Section snippets
Overview
An overview of the methods is shown in Fig. 1. We first specified constitutive models for nonlinear isotropic and fiber-reinforced anisotropic materials (Section 2.2, Table 1). We numerically implemented these models using the kinematics of uniaxial tension and compression with traction-free lateral boundary conditions (Section 2.3). We next used representative baseline material parameters from sclera in compression and tendon in tension to simulate the baseline “experimental” tissue
Stress fitting results
All the acceptable fitted responses obtained by the optimization of the axial stress (Criterion 1; Eq. (5)) closely matched the baseline (“experimental”) stress response (Fig. 2). The number of acceptable solutions found through optimization for the IsoComp model was 600/600, i.e., all 600 random initial guesses resulted in fits that satisfied Criterion 1. The corresponding statistics for the IsoTens, XIsoComp, XIsoTens, and OrthoComp models were 252/600, 482/600, 577/600, and 481/600,
Discussion
We here investigated the identifiability of constitutive relationship parameters from uniaxial tension and compression mechanical testing of several types of materials: isotropic, transversely isotropic, and orthotropic. These materials and the associated uniaxial testing are relevant to a wide range of applications in fibrous soft tissues, biomaterials, and tissue-engineered constructs. We used baseline material properties relevant for the sclera in compression and for tendon in tension and
Declaration of Competing Interest
The authors declare no conflicts of interest.
Acknowledgments
NIBIB-NIH R01 EB002425, NEI-NIH R01 EY025286, and the Georgia Research Alliance.
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