On the AIC-based model reduction for the general Holzapfel-Ogden myocardial constitutive law

Abstract

Constitutive laws that describe the mechanical responses of cardiac tissue under loading hold the key to accurately model the biomechanical behaviour of the heart. There have been ample choices of phenomenological constitutive laws derived from experiments, some of which are quite sophisticated and include effects of microscopic fibre structures of the myocardium. A typical example is the strain-invariant-based Holzapfel-Ogden 2009 model that is excellently fitted to simple shear tests. It has been widely used and regarded as the state-of-the-art constitutive law for myocardium. However, there has been no analysis to show if it has both adequate descriptive and predictive capabilities for other tissue tests of myocardium. Indeed, such an analysis is important for any constitutive laws for clinically useful computational simulations. In this work, we perform such an analysis using combinations of tissue tests, uniaxial tension, biaxial tension and simple shear from three different sets of myocardial tissue studies. Starting from the general 14-parameter myocardial constitutive law developed by Holzapfel and Ogden, denoted as the general HO model, we show that this model has good descriptive and predictive capabilities for all the experimental tests. However, to reliably determine all 14 parameters of the model from experiments remains a great challenge. Our aim is to reduce the constitutive law using Akaike information criterion, to maintain its mechanical integrity whilst achieving minimal computational cost. A competent constitutive law should have descriptive and predictive capabilities for different tissue tests. By competent, we mean the model has least terms but is still able to describe and predict experimental data. We also investigate the optimal combinations of tissue tests for a given constitutive model. For example, our results show that using one of the reduced HO models, one may need just one shear response (along normal-fibre direction) and one biaxial stretch (ratio of 1 mean fibre : 1 cross-fibre) to satisfactorily describe Sommer et al. human myocardial mechanical properties. Our study suggests that single-state tests (i.e. simple shear or stretching only) are insufficient to determine the myocardium responses. We also found it is important to consider transmural fibre rotations within each myocardial sample of tests during the fitting process. This is done by excluding un-stretched fibres using an "effective fibre ratio", which depends on the sample size, shape, local myofibre architecture and loading conditions. We conclude that a competent myocardium material model can be obtained from the general HO model using AIC analysis and a suitable combination of tissue tests.

Keywords: Akaike information criterion (AIC); Biaxial tests; Holzapfel–Ogden (HO) constitutive law; Myocardial mechanical tests; Reduced HO models; Simple shear tests; Uniaxial tests.

Fig. 1

a A sketch of all six possible shear modes, $\mathbf{f}$, $\mathbf{s}$, and $\mathbf{n}$ denote the fibre, sheet, and normal direction, respectively. (ij) refers to shear in the j direction within the ij plane, where $i\ne j\in \{\text{f, s, n}\}$. b A sample with fibres (red dash lines), which is stretched along the two orthogonal directions (MFD and CFD) in fibre-normal plane during a biaxial test; c uniaxial tension tests along the MFD and CFD; $f_1$ and $f_2$ are the loading force along the MFD and CFD. L is the initial length of specimen, and ${\lambda }_1$ and ${\lambda }_2$ are stretch ratios

Fig. 2

The recorded image in a biaxial tensile specimen in Ahmad et al. (2018) (a). The four white markers in the centre of the experimental sample in a are also shown in b, in which the solid rectangle represents the initial shape; and the deformed shape is shown in dashed lines. $f_1$ and $f_2$ are the loading forces in MFD and CFD

Fig. 3

a The fibre direction varies along the thickness of myocardium. b The effective area (blue) when the fibre direction is $\theta$ under uniaxial loading in the MFD test. The collagen fibres (red dot line) within the region enclosed by the two blue dashed lines are defined as effective fibres that are stretched both sides. The effective fibre ratio is defined by rectangle area dividing blue effective area

Fig. 4

a The 3D FE bi-ventricle mesh geometry with a pressure boundary condition applied to the left ventricle inner surface (red surface). The pressure linearly increases from 0 to 4 mmHg in a period of 0.5s. b The myofibre distribution in the ventricle wall, which rotates from epicardium to endocardium (${60}^{\mathit{o}}$ to $-{60}^o$)

Fig. 5

Comparison of the fitting results with and without considering fibre effective ratio ($\alpha$). a Fitting the general HO model to Dokos’s data, b–d the differences in uniaxial, biaxial and simple shear tests in Ahmad’s data

Fig. 6

Comparison of the first P–K stress, including shear (red) and not including shear (blue) using same strain energy function. a is for Sommer et al. biaxial test and a minimum shear angle of $6^{\mathit{o}}$ is introduced. Below $6^{\mathit{o}}$ there is no good fit, above it is not supported by Sommer et al. experiments. b is for Ahmad et al. biaxial test, corresponding to Fig. 8e

Fig. 7

Comparison between descriptive ability of the general HO and the HO2009 models for the three experimental studies. a Dokos’s simple shear tests; b and c Sommer’s biaxial tension and simple shear tests; d–f Ahmad’s uniaxial, biaxial tension and simple shear tests

Fig. 8

Descriptive capability of reduced HO models. a Change of $\eta$ when dropping the terms associated with the invariants for the different three experiments. The fitting results for the HO-D model (b), and c–d the HO-S model and e–g the HO-A model

Fig. 9

The differences of FE bi-ventricle model using the HO2009, HO-A and general HO models for Ahmad et al. data. a The pressure–volume curve in diastolic filling, b the displacement differences between the general HO and HO2009 models and c the displacement differences between the general HO and HO-A models

Fig. 10

$\delta$ values that are computed according to Algorithm 1, where the cases whose average (avg) $\delta \ge 0.8$ are marked in red. a In Dokos et al. experiments, case 25 ((fs) + (fn) + (ns)) is the optimal case which has few tests whilst meeting the criterion, b the corresponding fitting curves using case 25. In Sommer et al. experiments, case 20 ((1:1) + (nf)) is the optimal case as shown in c, and the corresponding fitting curves are shown in d and e. f is for Ahmad et al. experiments, case ALL is the only one which satisfies the criterion. The other cases are corresponding to certain combinations to be discussed in the text

Fig. 11

Stress distribution when fibre direction is ${10}^{\circ }$ in uniaxial tensile along MFD as shown in Fig. 3. The green area enclosed by the two dashed lines is the effective area with higher stress, whilst the blue area (the right bottom and left upper corners ) is the ineffective area with much lower stress

Fig. 12

The rest four loading protocols [(1:0.75), (0.75:1), (1:0.5) and (0.5:1)] for Sommer et al. biaxial tests. a Comparison of the first P–K stress including shear (solid lines) or not (dash lines) as in Fig. 6a. b The comparison between the general HO model and HO2009 model in Fig. 7b. c Fitting results using the reduced HO model (HO-S) in Fig. 8c. d The simulated results according to optimized combinations (1:1)+(nf) in Fig. 10e

Fig. 13

Only using biaxial tests in Sommer et al. data, we compute $\delta$ by fitting the HO-S model to one individual stretch ratio and predicting the remained experimental data from other stretch ratios. Only (1:1) and (1:0.75) meet $\delta \ge 0.8$, whilst (1:0.5) and (0.5:1) have $\delta \le 0.4$