Bayesian inference of constitutive model parameters from uncertain uniaxial experiments on murine tendons

Abstract

Constitutive models for biological tissue are typically formulated as a mixture of constituents and the overall response is then assembled by superposition or compatibility. This ensures the stress response of the biological tissue to be in the range of a given constitutive relationship, guaranteeing that at least one parameter combination exists so that an experimental response can be sufficiently well captured. Another, perhaps more challenging, problem is to use constitutive models as a proxy to infer the structure/function of a biological tissue from experiments. In other words, we determine the optimal set of parameters by solving an inverse problem and use these parameters to infer the integrity of the tissue constituents. In previous studies, we focused on the mechanical stress-stretch response of the murine patellar tendon at various age and healing timepoints and solved the inverse problem using three constitutive models, i.e., the Freed-Rajagopal, Gasser-Ogden-Holzapfel and Shearer in order of increasing microstructural detail. Herein, we extend this work by adopting a Bayesian perspective on parameter estimation and implement the constitutive relations in the tulip library for uncertainty analysis, critically analyzing parameter marginals, correlations, identifiability and sensitivity. Our results show the importance of investigating the variability of parameter estimates and that results from optimization may be misleading, particularly for models with many parameters inferred from limited experimental evidence. In our study, we show that different age and healing conditions do not correspond to statistically significant separation among the Gasser-Ogden-Holzapfel and Shearer model parameters, while the phenomenological Freed-Rajagopal model is instead characterized by better indentifiability and parameter learning. Use of the complete experimental observations rather than averaged stress-stretch responses appears to positively constrain inference and results appear to be invariant with respect to the scaling of the experimental uncertainty.

Keywords: Bayesian inference; Constitutive models for murine patellar tendons; Parameter sensitivity and identifiability; Uncertainty quantification.

Figure 1:

Relation between the Shearer transition stretch λ* and a transition stretch based on the stress-stretch curve derivative. For an increasing alignment angle ψ, the large transition stretch affects the extension of the linear portion of the stress-stretch curve.

Figure 2:

Mean value and 5–95% percentiles for Gelman-Rubin diagnostic across all parameters and age-healing groups.

Figure 3:

Learning factors for GOH, SHR (arbitrary and zero alignment angles) and FR.

Figure 4:

Number of experimental data points at various stretch levels, for the same healing group (3 weeks) at age 120 days (a), 270 days (b) and 540 days (c).

Figure 5:

Changes in the parameter median value and 10–90% percentile due to injury and age for the GOH model with ξ ≠ 0.

Figure 6:

Changes in the parameter median value and 10–90% percentile due to injury and age for the GOH model with ξ = 0.

Figure 7:

Changes in the parameter median value and 10–90% percentile due to injury and age for the SHR model with ψ ≠ 0.

Figure 8:

Changes in the parameter median value and 10–90% percentile due to injury and age for the SHR model with ψ = 0.

Figure 9:

Changes in the parameter median value and 10–90% percentile due to injury and age for the FR model.

Figure 10:

GOH, SHR and FR representative model parameters computed through optimization and maximum a posteriori estimates from MCMC.

Figure 11:

Assimilated and experimental stress-stretch response for the GOH model. The experimental stress at various ages has been slightly shifted on the stretch axis to improve clarity.

Figure 12:

Assimilated and experimental stress-stretch response for the SHR model. The experimental stress at various ages has been slightly shifted on the stretch axis to improve clarity.

Figure 13:

Assimilated and experimental stress-stretch response for the FR model. The experimental stress at various ages has been slightly shifted on the stretch axis to improve clarity.

Figure 14:

GOH parameter correlations from MCMC under average (a) and complete (d) experimental data. The parameter correlations for the SHR (b, e) and FR (c, f) models are also shown.

Figure 15:

Parameter correlations under zero alignment angle. GOH parameter correlations from MCMC under average (a) and complete (b) experimental data. The parameter correlations for the SHR (c, d) are also shown.

Figure 16:

Representative marginal distributions for the standard deviation of the experimental stress σ_ϵ in the SHR model.

Figure 17:

Fisher Information matrix eigenvalues for GOH and SHR models. Models with zero and no­zero alignment angles are compared in (a) to emphasize the change in local identifiability promoted by suppressing the alignment angles. The null eigenvectors are also illustrated in (b, c) for the GOH and SHR models, respectively.

Figure 18:

Average global and local sensitivity indices for the GOH model and associated 10%−90% percentile confidence across age-healing groups. Model parameters are identified using the complete dataset of uniaxial tests.

Figure 19:

Average global and local sensitivity indices for the GOH model and associated 10%−90% percentile confidence across age-healing groups. Model parameters are identified using the complete dataset of uniaxial tests and with zero alignment angle ξ.

Figure 20:

Average global and local sensitivity indices for the SHR model and associated 10%−90% percentile confidence. Model parameter are identified using the complete dataset of uniaxial tests.

Figure 21:

Average global and local sensitivity indices for the SHR model and associated 10%−90% percentile confidence. Model parameter are identified using the complete dataset of uniaxial tests and with zero helix angle.

Figure 22:

Average global and local sensitivity indices for the FR model and associated 10%−90% percentile confidence across age-healing groups. Model parameter are identified using the complete dataset of uniaxial tests.