Page 51 - Artificial Intelligence for Computational Modeling of the Heart

Chapter 1 Multi-scale models of the heart for patient-specific simulations 21

the fiber, along the sheets or in the fiber-sheet plane normal. For
all these conditions, the stress-strain relationship is exponential.
Based on these considerations, the HO energy function writes:

a
ψ = exp[b(I 1 − 3)]
2b
a i
2
+ exp[b i (I 4i − 1) ]− 1
2b i (1.17)
i={f,s}
a fs 2
+ exp(b fs I 8fs ) − 1 .
2b fs
Here a, b, a i , b i , a fs , b fs are tissue parameters with i ={f,s},the
a parameters having dimension of stress, whereas the b param-
eters are dimensionless. Furthermore, the following deformation
T
T
invariants are defined: I 4i = i Ci, with i ={f,s},and I 8fs = f Cs.
To model near incompressible materials like the heart it is cus-
2
tomary to add the term d 1 (J − 1) , with d 1 homologous to a bulk
modulus. Note that the second and third terms of the energy func-
tion (Eq. (1.17)) are null when the tissue is under compression,
i.e. I 4i < 1, since fibers do not show resistance to compression.
In summary, the HO model is governed by nine parameters (Ta-
ble 1.1), obtained from the experiments reported in [113].
A simplified version of the HO model is often chosen, where
the contributions of the fiber sheets are neglected. Indeed, current
imaging limitations do not allow in-vivo assessment of myocar-
dial fibers, and even less fiber sheets. Furthermore, the impact of
the sheets on the overall cardiac function is still not well under-
stood. Therefore, the transverse isotropic version of the Holzapfel–
Ogden model is often preferred for patient-specific simulations,
having less parameters to identify. The modified strain energy
function becomes:

a a f
2
ψ = exp[b(I 1 − 3)]+ exp[b f (I 4f − 1) ]− 1 .
2b 2b f

Table 1.1 Parameters and default values of the hyper-elastic model of cardiac myocardium
proposed by Holzapfel–Ogden (values from [113]).

a (kPa) b a f (kPa) b f a s (kPa) b s a fs (kPa) b fs d 1
0.496 7.209 15.193 20.417 3.283 11.176 0.662 9.466 496