# A modified Holzapfel-Ogden law for a residually stressed finite strain model of the human left ventricle in diastole

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## Abstract

In this work, we introduce a modified Holzapfel-Ogden hyperelastic constitutive model for ventricular myocardium that accounts for residual stresses, and we investigate the effects of residual stresses in diastole using a magnetic resonance imaging–derived model of the human left ventricle (LV). We adopt an invariant-based constitutive modelling approach and treat the left ventricular myocardium as a non-homogeneous, fibre-reinforced, incompressible material. Because in vivo images provide the configuration of the LV in a loaded state even in diastole, an inverse analysis is used to determine the corresponding unloaded reference configuration. The residual stress in this unloaded state is estimated by two different methods. One is based on three-dimensional strain measurements in a local region of the canine LV, and the other uses the opening angle method for a cylindrical tube. We find that including residual stress in the model changes the stress distributions across the myocardium and that whereas both methods yield qualitatively similar changes, there are quantitative differences between the two approaches. Although the effects of residual stresses are relatively small in diastole, the model can be extended to explore the full impact of residual stress on LV mechanical behaviour for the whole cardiac cycle as more experimental data become available. In addition, although not considered here, residual stresses may also play a larger role in models that account for tissue growth and remodelling.

## Keywords

Residual stress Left ventricle Finite strain Finite stress## 1 Introduction

Even in the absence of externally applied loads, soft tissues generally are not stress-free. The stresses that remain after all external loads have been removed are termed *residual stresses*. Residual stresses are present in a large variety of biological tissues and result from tissue growth and remodelling occurring over the life span of the tissue (Bovendeerd et al. 1994).

Fung (1993) and Vaishnav and Vossoughi (1987) appear to have been the first to consider residual stresses in soft tissues. They found that when a length of artery is excised from a body, the artery contracts. Thus, in vivo, arteries are stretched (i.e. subject to a large axial deformation) and tethered (i.e. held in place) by the surrounding tissue. However, although an excised artery is not subject to any axial load or to any traction on its inner and outer surfaces, it is not unstressed; rather, there remains a residual stress distribution across the arterial wall. The existence of these residual stresses may be demonstrated by the so-called *opening angle experiment*, first proposed by Chuong and Fung (1986), in which a short length of artery in the form of a ring is cut radially. The ring springs open to form an open sector, thus indicating the presence of a compressive circumferential stress in the inner part of the wall of the ring and a tensile circumferential stress in the outer part. The studies by Han and Fung (1996) and by Liu and Fung (1989) suggest that most of the residual stress is released by a single cut. In contrast, Vossoughi et al. (1993), Greenwald et al. (1997), and Schulze-Bauer et al. (2002) have shown that a single cut is not sufficient to relieve all the residual stress in arterial walls. Residual stresses have an important influence on the mechanical response of the artery under physiological conditions. It is believed that residual stress tends to reduce the stress concentration at the inner arterial wall (Chuong and Fung 1986), and it has been speculated that residual stresses are distributed so that the stress distributions across the arterial wall layers are more uniform at physiological pressures (Rachev and Greenwald 2003).

Finite strain and stress analyses of the left ventricular wall can further our understanding of the heart in health and disease. Despite extensive studies of residual stress in arteries, there have been relatively few studies on residual stress and strain in the myocardium. Residual stresses could have an influence on the dynamics of the left ventricle (LV) and on transmural stress and strain distributions. To date, however, most ventricular mechanics models have assumed that the unloaded configuration is a stress-free configuration (Mirsky 1973; Demiray 1976a, b; Bovendeerd et al. 1994; Wang et al. 2013). Indeed, it is an ongoing challenge to recover residual stress in myocardial tissues both theoretically and experimentally.

Omens and Fung (1990) studied residual strains in rat LV by measuring the opening angles of equatorial LV slices. They discovered that equatorial rings opened into arcs with a mean opening angle of about \(45^\circ \). Once ischaemic contracture had set in, they observed a continual increase in the opening angle, up to approximately \(180^\circ \), that was associated with a dramatic increase in specimen stiffness. Residual strains were found to be negative (compressive) in the endocardium and positive (tensile) in the epicardium. Rodriguez et al. (1993) studied the effects of residual stress on the transmural sarcomere length (SL) distributions in the equatorial region of the rat LV. Upon comparing the distributions of SL between the unloaded but residually stressed state and the stress-free state, they found that the SL was uniform across the wall in the stress-free state; however, in intact tissue, there was a significant decrease in SL from epicardium to endocardium. This gradient is believed to offset the opposing gradient in sarcomere extension during filling, thus leading to a more uniform transmural distribution of SL at end diastole and hence more uniform development of systolic force. Summerour et al. (1998) used the opening angle method to estimate residual strains in equatorial slices of normal and ischaemic rat LV. They did not observe obvious differences in the residual stress distributions between normal and ischaemic myocardium. Costa et al. (1997) performed in vitro experiments using biplane radiography in which columns of beads implanted in the mid-anterior free wall of the canine LV determined transmural distributions of the three-dimensional residual strains. To date, no experimental studies have been reported on the residual stress and strain distributions in human myocardium.

Using analytical and numerical modelling, Guccione et al. (1991) studied the passive mechanics of the canine LV using a thick-walled cylindrical LV model in which the myocardium is treated as an incompressible hyperelastic material. They further assumed that the LV is transversely isotropic and may be described by a four-parameter Fung-type model and that the residual stress is isotropic. They looked at the effects of residual strain and stress on the circumferential stress distributions when using opening angles. Nevo and Lanir (1994) carried out a similar study on the influence of residual strain on the diastolic function of the LV using a structural model. Nash (1998) assumed an initial strain field based on the experimental data of Omens and Fung (1990) and used the finite element method and the *pole-zero* constitutive law (Hunter et al. 1997) to develop an anatomically accurate mathematical model of canine LV. Their computational results, which covered the whole cardiac cycle, showed that the effects of residual strain in the LV are small. Guccione et al. (2001) also used the finite element method to estimate residual stresses during ventricular volume reduction surgery. When they evaluated the impact of the residual stress on ventricular function, they also found that the effects of these stresses are small. Taber (1991) simulated the beating left ventricle using a nonlinear laminated thick-shell model with residual strains and discovered that the residual strains in the LV significantly affect the peak fibre stress and transmural stress gradients near the beginning of systole. Because of the transmural changes in fibre orientation in the LV, these effects are not as large as they are in arteries.

Although most studies on the role of residual stresses in the LV myocardium suggest that the effects of these stresses are smaller than those in arteries, mathematically, it is appropriate to evaluate the actual stress with residual stress contribution included. In addition, accounting for residual stresses may be important in models that describe tissue growth and remodelling because the growth and remodelling processes may themselves act to set up and to maintain residual stresses. Recently, Holzapfel and Ogden (2009) proposed a structure-based constitutive model of ventricular myocardium that accounts for the locally orthotropic tissue microstructure by expressing the strain-energy function using fibre-based material invariants. In the incompressible case, their strain-energy function has eight material parameters with relatively clear physical meanings. Moreover, this model satisfies convexity and strong ellipticity properties that are important both mathematically and physically. In this work, we adopt the approach of Shams et al. (2011), who describe a general invariant-based method for incorporating residual stresses into hyperelastic material models, to extend the Holzapfel-Ogden constitutive law to account for residual stresses.

In previous work, we developed a three-dimensional computational model of the human LV that is derived from non-invasive magnetic resonance imaging (MRI) data (Wang et al. 2013). This anatomically realistic model has a rule-based fibre structure and, in our earlier study, used the original structure-based constitutive model of Holzapfel and Ogden (2009) (i.e. without residual stresses). In this work, we use the method introduced by Shams et al. (2011) to modify the Holzapfel-Ogden law so that the residual stresses are taken into account. This constitutive model, along with our MRI-based anatomic model, is then used to investigate the influence of residual stress on the mechanical behaviour of the LV in diastole.

## 2 Constitutive laws for the passive myocardium

### 2.1 Constitutive law based on stress-free configuration

### 2.2 Constitutive law for the residually stressed myocardium

^{1}If the traction on the boundary \(\partial B_r\) of \(B_r\) vanishes, then \(\varvec{\tau }\) is referred to as the residual stress, which is necessarily non-uniform and symmetric and satisfies the equilibrium equation (Johnson and Hoger 1995; Shams et al. 2011):

Notice that (11) tells us nothing about the specific form of \(\varvec{\tau }\), which could arise through various routes, including residual strain with respect to the zero-stress configuration, and which could possibly be determined by a constitutive law that is different from (3).

*simple approach*to including residual stresses, we consider small strains, so that \(\mathbf{B} \approx \mathbb{I }\). Hence, \(\varvec{\Sigma } \mathbf{B}\approx \varvec{\Sigma }\) and \(\mathbf{B}\varvec{\Sigma }\approx \varvec{\Sigma }\) in (9). Consequently, we include only the \(W_6\) term. For simplicity, we also assume that \(W_6\) is constant. Then, for consistency with (12)\(_2,\,W_6=1/2\). Under these assumptions along with (12), the modified strain-energy function \(W\) becomes

*extended approach*, we also specifically include the term \(2W_7 (\varvec{\Sigma } \mathbf{B}+\mathbf{B}\varvec{\Sigma })\) in (9), assuming \(W_7\) is constant. Similar to the simple approach, consistency with (12)\(_2\) requires \(W_6=1/4\) and \(W_7=1/8\). The strain-energy function \(W\) is

The material parameters from Wang et al. (2013)

\(a\) (kPa) | \(b\) | \(a_{\mathrm{f}}\) (kPa) | \(b_{\mathrm{f}}\) | \(a_{\mathrm{s}}\) (kPa) | \(b_{\mathrm{s}}\) | \(a_{\mathrm{fs}}\) (kPa) | \(b_{\mathrm{fs}}\) |
---|---|---|---|---|---|---|---|

0.236 | 10.81 | 20.04 | 14.15 | 3.72 | 5.16 | 0.41 | 11.3 |

### 2.3 Estimate of the residual stress

It is important to note that the zero-stressed reference configuration \(B_0\) defined here is fictitious, as the deformations inducing residual stresses are probably not compatible, and probably involve microscopic phenomena such as cell rotations. In particular, residual stresses likely arise from motions that cannot be described within the standard framework of continuum mechanics. With this in mind, we propose two methods to estimate the residual stress for the LV in \(B_r\) with respect to \(B_0\). (The method used to determine the unloaded configuration \(B_r\) is described below in Sect. 2.5.) One approach to estimating the residual stress is to use the measured residual strain field of Costa et al. (1997), and the other is to use the opening angle method.

#### 2.3.1 Residual stress estimated from data of Costa et al. (1997)

Note the residual strain generated from this method is not compatible because the average strain components were used. In addition, the experimental data are only from along the mid-anterior free wall of the canine LV. The residual stress tensor thus constructed is only suited for a cylindrical section of the LV.

#### 2.3.2 Residual stress estimated by the opening angle method

### 2.4 The finite element model

The geometry of the human LV model is derived from MR imaging of a healthy volunteer (male, age 28) acquired at the end of diastole, which is identified by the peak of the R-wave from the subject’s ECG. The computational approach that we adopt to model the passive mechanics of the LV is based on the classical pressure-dilatation-displacement three-field formulation commonly used to overcome locking problems exhibited by purely displacement-based finite element formulations of incompressible elasticity. We use the decoupled volumetric-isochoric formulation of finite elasticity and decompose the strain-energy function into volumetric and isochoric parts, with the incompressibility ensured using a Lagrange multiplier method. Detailed descriptions of the generation of the geometry and fibre structure, as well as the finite element procedure, were described previously (Wang et al. 2013). The LV model is discretized with 48,050 hexahedral elements and 53,548 nodes. Validations using different meshes and elements were performed in our previous work (Wang et al. 2013).

To constrain the motion of the model, the longitudinal displacement of the base and the circumferential displacement of the epicardial wall at the base are set to zero. The remainder of the left ventricular wall, including the apex, is left free. A pressure load, generally varying from 0 to 8 mmHg, is applied on the endocardial surface. Such loads are typical physiological end-diastolic pressures.

### 2.5 The unloaded configuration

An essential requirement of the finite strain model is an unloaded reference geometry. Since the end-diastolic pressure is not zero, this requires the determination of unloaded reference geometries that are different from the imaged ones (Walker et al. 2005; Sermesant and Razavi 2010). For this purpose, we use an inverse displacement algorithm (Bols et al. 2011), which is especially easy to implement within a finite element framework. Other similar methods to determining the unloaded geometry include methods that compute a multiplicative decomposition of the deformation gradient tensor (Rodriguez et al. 1994; Aguado-Sierra et al. 2011). The objective of the iterative scheme is to find the unloaded reference geometry that minimizes the difference between the measured end-diastolic geometry and the computed geometry when the unloaded model is inflated to the measured end-diastolic pressures.

The parameters are chosen to be: \(\alpha =85^\circ \), along with \(R_i=2.5~\text{ cm },\,R_o=4.2~\text{ cm }\), and \(\lambda _z=1\).

### 2.6 Finite strain in the LV model with residual stress

## 3 Results

### 3.1 Residual stress estimates

#### 3.1.1 Residual stress from Costa’s residual strain measurements

#### 3.1.2 Residual stresses by the opening angle method

#### 3.1.3 Early diastole

Stress means and standard deviations along the endocardial and epicardial surfaces at an endocardial loading pressure of \(3\) mmHg

\(\sigma _\text{ ff }\) (kPa) | \(\sigma _\text{ ss }\) (kPa) | \(\sigma _\text{ nn }\) (kPa) | |
---|---|---|---|

| |||

Without \(\varvec{\tau }\) | \(2.9\pm 1.6\) | \(-0.33\pm 0.32\) | \(-0.15 \pm 0.35\) |

With \(\varvec{\tau }\) (Costa et al. data) | \(2.9\pm 1.6\) | \(-0.34\pm 0.31\) | \(-0.17 \pm 0.35\) |

With \(\varvec{\tau }\) (opening angle) | \(2.8\pm 1.5\) | \(-0.35\pm 0.30\) | \(-0.20 \pm 0.35\) |

Difference (Costa et al. data) | \(-0.01\pm 0.07\) | \(-0.004\pm 0.01\) | \(-0.01 \pm 0.02\) |

Difference (opening angle) | \(-0.16\pm 0.15\) | \(-0.02\pm 0.05\) | \(-0.04 \pm 0.06\) |

| |||

Without \(\varvec{\tau }\) | \(1.23 \pm 0.77\) | \(0.04 \pm 0.08\) | \(0.11 \pm 0.13\) |

With \(\varvec{\tau }\) (Costa et al. data) | \(1.28 \pm 0.8\) | \(0.04 \pm 0.08\) | \(0.11 \pm 0.13\) |

With \(\varvec{\tau }\) (opening angle) | \(1.31 \pm 0.8\) | \(0.06 \pm 0.08\) | \(0.16 \pm 0.14\) |

Difference (Costa et al. data) | \(0.05\pm 0.09\) | \(0.0\pm 0.02\) | \(0.0\pm 0.02\) |

Difference (opening angle) | \(0.08\pm 0.11\) | \(0.02\pm 0.03\) | \(0.05 \pm 0.04\) |

Both the direct strain measurement and the opening angle method show that there are some non-negligible differences in the stress distribution when residual stress is included, particularly near the epicardial region. As discovered by Guccione et al. (1991), Taber (1991), Nash (1998), and others, residual stress tends to reduce the total stress in the sub-endocardium and to increase the stress in the sub-epicardium, therefore releasing the stretch of SL during loading. This is particularly evident for the residual stress resulting from the opening angle method, as in Fig. 12. The results also demonstrate that both methods of estimating the residual stress give quantitatively similar results.

To see whether the simpler approach (i.e. considering only the \(I_6\) term in the strain-energy function with residual stresses) is sufficiently accurate for these analyses, we also compare the results obtained by the simple approach to results obtained by the extended approach (i.e. in which the effects of the \(I_7\) term are also included). Comparisons between the two approaches are shown in Figs. 11 and 13. It is clear that the results from both approaches are almost identical. We conclude that the simpler model is sufficient for this case.

#### 3.1.4 Late diastole

#### 3.1.5 Pressure–volume relations

*P*–

*V*loops are often less sensitive to detailed mechanical changes. A similar finding was reported by Eriksson et al. (2012), who found that the non-negligible role of heterogeneity in structural fibre/sheet orthotropy is not reflected in the

*P*–

*V*curves generated by models with or without heterogeneity.

## 4 Discussion

We have modelled the effects of residual stresses on LV mechanics using a modified Holzapfel-Ogden model. Applications of this model show that the residual stress in the human LV could make non-negligible changes in the stress distributions at physiological pressures in diastole. Since estimates of the residual stress are necessarily limited by the lack of three-dimensional strain data in the LV, the results presented herein are therefore primarily illustrative, and may not give the full scale of the impact of residual stress in the LV. With the given assumptions, however, we have not found any changes in the pressure–volume curve in diastole when incorporating residual stresses into our model.

In determining the unloaded configuration, we have made two further assumptions. One is that the unloaded configuration is in equilibrium with the addition of the residual stresses obtained from limited experimental data or by simplified opening angle methods. In practice, the residual stresses computed by either approach do not strictly satisfy the condition (5), and hence the addition of residual strains results in a shift in the unloaded configuration. It may be important to determine the unloaded configuration and the residual stress distributions simultaneously; however, doing so is beyond the scope of the present work. In addition, because we have access only to limited experimental data on residual stresses or strains, it is not clear that the significant additional effort required to determine the unloaded configuration and residual stress states simultaneously is well justified. The other assumption is that the fibre structure may be regenerated using the rule-based algorithm in the unloaded configuration. This last assumption is made because it is difficult to track the fibre structure during the inverse analysis; regenerating the fibre structure simplifies the implementation significantly.

In an effort to find the simplest way to include the residual stresses, we have shown that residual stress in the myocardium can be represented accurately by a modified Holzapfel-Ogden model. This modified model can readily be applied to similar problems in which the residual stress tensor has the principal directions in the fibre coordinate system. For more general problems, our results suggest that the contribution from a single \(I_6\) term is sufficient to model the residual stress.

Overall, the residual stress has some effects, particularly on the fibre stress and cross-fibre stress, although the extent of this seems to be less significant later in diastole when pressure is higher. However, it does not necessaily follow that the impact of the residual stress would be further decreased in systole, when the pressure reaches its peak. This is because the additional coupling terms of residual stress to larger strains and fibre directions will come into play, which are ignored in the present study. When a more general theory similar to that of Ogden and Singh (2011) is employed, the effect of the residul stress on the whole cardiac cycle may be more significant.This may also be the case when more detailed experimental measurements of residual strain are available for the whole LV, including the apex. In addition, we remark that the presence of residual stress may have a significant impact on myocardial growth and remodelling, which have not been modelled here. Indeed, the growth and remodelling processes may themselves act to establish and to maintain residual stresses.

## 5 Conclusions

In this paper, we proposed a modified Holzapfel-Ogden model for the myocardium in diastole which includes a basic contribution from residual stress. This work follows on the theoretical framework of Holzapfel and Ogden (2009) and of Shams et al. (2011). We found that using only one extra invariant, \(I_6\), in the strain-energy function is sufficiently accurate for the finite strain models considered herein. The modified constitutive model allows the residual stress to be considered without too much additional computational effort. Using this constitutive law, we carried out stress and strain analyses using a MRI-derived human left ventricular model. The residual stress is applied to the unloaded configuration, which is determined by an inverse displacement analysis. Two different methods of estimating the residual strain are considered. One is from the three-dimensional strain measurements of canine LV, and the other uses the opening angle method. Each of these methods involves simplifications and can only estimate the residual stress in the cylindrical section of the LV, which could underestimates the magnitude of the true residual stress field inside the LV. Our results show that even the simplified residual stress representations, in which the coupling terms with fibre directions are ignored, could make non-negligible changes in the stress distribution in the human LV model at physiological pressures in diastole. Further experimental data are required to investigate fully the effects of residual stresses and to determine the most suitable constitutive laws for capturing the residual strain and stress in the LV.

## Footnotes

## Notes

### Acknowledgments

We gratefully acknowledge support from the UK Engineering and the Physical sciences Research Council (grant EP/I02990) and the Excellence Exchange Project funded by the Xi’an Jiaotong University, as well as support from the British Heart Foundation, Medical Research Scotland, the Chief Scientist Office, Edinburgh Mathematical Society, and the Scottish Funding Council. B.E.G. was supported in part by American Heart Association Scientist Development Grant 10SDG4320049 and by National Science Foundation Awards DMS 1016554 and OCI 1047734.

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