Identification of constitutive materials of bi-layer soft tissues from multimodal indentations.

The personalisation of finite element models is an important problem in the biomechanical fields where subject-specific analyses are fundamental, particularly in studying soft tissue mechanics. The personalisation includes the choice of the constitutive law of the model's material, as well as the choice of the material parameters. In vivo identification of the material properties of soft tissues is challenging considering the complex behaviour of soft tissues that are, among other things, non-linear hyperelastic and heterogeneous. Hybrid experimental-numerical methods combining in vivo indentations and inverse finite element analyses are common to identify these material parameters. Yet, the uniqueness and the uncertainty of the multi-material hyperelastic model have not been evaluated. This study presents a sensitivity analysis performed on synthetic indentation data to investigate the identification uncertainties of the material parameters in a bi-material thigh phantom. Synthetic numerical data, used to replace experimental measurements, considered several measurement modalities: indenter force and displacement, stereo-camera 3D digital image correlation of the indented surface, and ultrasound B-mode images. A finite element model of the indentation was designed with either Ogden-Moerman or Mooney-Rivlin constitutive laws for both materials. The parameters' identifiability (i.e. the possibility of converging to a unique parameter set within an acceptable margin of error) was assessed with various cost functions formulated using the different synthetic data sets. The results underline the need for multiple experimental modalities to reduce the uncertainty of the identified parameters. Additionally, the experimental error can impede the identification of a unique parameter set, and the cost function depends on the constitutive law. This study highlights the need for sensitivity analyses before the design of the experimental protocol. Such studies can also be used to define the acceptable range of errors in the experimental measurement. Eventually, the impact of the evaluated uncertainty of the identified parameters should be further investigated according to the purpose of the finite element modelling.

Identifiability of soft tissue constitutive parameters from in-vivo macro-indentation.

Reliable identification of soft tissue material parameters is frequently required in a variety of applications, particularly for biomechanical simulations using finite element analysis (FEA). However, determining representative constitutive laws and material parameters is challenging and often comprises a bottleneck that hinders the successful implementation of FEA. Soft tissues exhibit a nonlinear response and are commonly modeled using hyperelastic constitutive laws. In-vivo material parameter identification, for which standard mechanical tests (e.g., uniaxial tension and compression) are inapplicable, is commonly achieved using finite macro-indentation test. Due to the lack of analytical solutions, the parameters are commonly identified using inverse FEA (iFEA), in which simulated results and experimental data are iteratively compared. However, determining what data must be collected to accurately identify a unique parameter set remains unclear. This work investigates the sensitivities of two types of measurements: indentation force-depth data (e.g., measured using an instrumented indenter) and full-field surface displacements (e.g., using digital image correlation). To eliminate model fidelity and measurement-related errors, we employed an axisymmetric indentation FE model to produce synthetic data for four 2-parameter hyperelastic constitutive laws: compressible Neo-Hookean, and nearly incompressible Mooney-Rivlin, Ogden, and Ogden-Moerman models. For each constitutive law, we computed the objective functions representing the discrepancies in the reaction force, the surface displacement, and their combination, and visualized them for hundreds of parameter sets, spanning a representative range as found in the literature for the bulk soft tissue complex in human lower limbs. Moreover, we quantified three identifiability metrics, which provided insights into the uniqueness (or lack thereof) and the sensitivities. This approach provides a clear and systematic evaluation of the parameter identifiability, which is independent of the selection of the optimization algorithm and initial guesses required in iFEA. Our analysis indicated that the indenter's force-depth data, despite being commonly used for parameter identification, was insufficient for reliably and accurately identifying both parameters for all the investigated material models and that the surface displacement data improved the parameter identifiability in all cases, although the Mooney-Rivlin parameters remained poorly identifiable. Informed by the results, we then discuss several identification strategies for each constitutive model. Finally, we openly provide the codes used in this study, to allow others to further investigate the indentation problem according to their specifications (e.g., by modifying the geometries, dimensions, mesh, material models, boundary conditions, contact parameters, or objective functions).