Implementation of Passive and Active Material Behaviour of Cardiac Tissue into the Finite Element Software ABAQUS via a User Subroutine

Abstract

Intro - The rapidly developing technology of cardiac finite element modelling aims to improve heart failure treatment by quantifying stresses acting in the cardiac tissue. Cardiac finite element modelling may improve heart failure treatment by providing more insight in the pathophysiology, enabling a patient-specific assessment and improving the efficiency of clinical trials and medical devices. An essential part of the cardiac finite element model is capturing the mechanical behaviour of the heart, including its active behaviour (contraction). This mechanical behaviour is captured by means of a material model. However, the active behaviour of the heart is not readily available in material models provided by commercial finite element software like Abaqus/CAE. Consequently, such active material model needs to be added in a user subroutine manually. This is a time-consuming task, prone to errors. Additionally, the availability of the documentation is limited, and in literature the methodology of the implementation is oftentimes not disclosed.

Thesis goal - The goal of this master-thesis is to implement a combination of the passive and active mathematical material models in a commercial finite element platform. This was done by implementing the combined material models in user subroutine UMAT that can be used in FEA-software Abaqus. Reproducibility of the UMAT is ensured by a detailed documentation. Also, the UMAT is provided in the supplementary material.

Methods - The UMAT consists of the Holzapfel Ogden constitutive law as passive component, and the Time Varying-Elastance contstitutive law as active component, which are combined by means of the active stress approach. Additionally, the UMAT requires computation of the elasticity tensor, which is computed by means of a numerical formulation. The incorporation of the passive and active constitutive laws in the UMAT was verified by means of multiple test-cases. The outcomes of the UMAT were compared to an analytical solution and a benchmark user subroutine in the form of UANISOHYPER_INV. Lastly, reality-check test-cases were carried out by comparing the UMAT outcomes to the results from similar test-cases found in literature.

Verification results - Verification of the implemented material models showed good agreement with the analytical computed solution of equibiaxial extension, equibiaxial compression and isometric contraction test-cases, as all cases showed an MAPEmax or APEmax error lower than 1%. Shear test-case results of the UMAT showed some bigger APEmax values (maximal 17%) with relation to the analytical solution, possibly caused by numerical errors during the elasticity tensor computation. Results of the reality-check cases showed similar trends to the mechanical experiments done on cardiac tissue on which these cases are based.

Conclusion - The public availability of the implemented passive and active material models in the user subroutine UMAT, which is working reasonably well according to conducted verification, forms a significant step forwards in the field of cardiac finite element modelling. The current work can be further extended by the incorporations of compressibility during contraction and viscoelasticity in the material models. This thesis provides the basis for future projects in the field of cardiac finite element modelling including an active material model. Also, the provided implementation of material models may aid in the implementation of other mathematical material models in a user subroutine like UMAT.