Page 98 - Artificial Intelligence for Computational Modeling of the Heart

68 Chapter 2 Implementation of a patient-specific cardiac model

where Z is the strain-displacement matrix [104], V 0 is the volume
of the element at time t = 0, and S is the SPK tensor expressed in
terms of the Cauchy stress tensor as

S = JF −1 T a − T p F −T . (2.17)
The next sections describe how each force is computed. In [226]
the authors provide a more detailed description of the algorithm,
including the computation of the strain-displacement operator Z
from the finite element discretization.

2.3.1 Passive stress component
As introduced in section 1.3.1, the simplified, transverse iso-
tropic Holzapfel–Ogden model energy function writes:
a a f 2 2
ψ = exp[b(I 1 − 3)]+ exp[b f (I 4f − 1) ]− 1 + d 1 (J − 1) .
2b 2b f
(2.18)
2
The term d 1 (J − 1) models incompressibility. The parameter d 1
is therefore equivalent to a bulk modulus. Deriving the invari-
T
ants with respect to C gives ∂I 1 /∂C = Iand ∂I 4f /∂C = ff .Asa
result, the Cauchy stress writes as follows, where we introduced a
stiffness parameter β to simplify model personalization (see sec-
tion 2.5.3):
a 2 T

T p = β exp[b(I 1 − 3)] I + a f exp[b f (I 4f − 1) ] ff
2
+ d 1 (J − 1)JC −1 . (2.19)

2.3.2 Active stress component
In this implementation, a simplified, phenomenological model
is preferred for its computational efficiency while being globally
fidel to the underlying mechanisms of myocyte contraction [45].
TheactivestressT a is controlled by a switch function u(t),re-
lated to the activation time, here denoted t d , and the repolariza-
tion time, denoted t r , computed by the cardiac electrophysiol-
ogy model (section 2.2,Fig. 2.20). When the cell is depolarized
(t d ≤ t< t r ), u(t) is constant and is equal to the contraction rate
+k AT P . This variable relates to the rate of ATP consumption by
the sarcomere that fuels the CICR mechanism of the myofila-
ments (see section 1.3.2), as well as the number of cross-bridges
recruited to perform the contraction. When the cell is repolarized
(t r ≤ t< t d + CL, CL is the cycle length of the myocyte), u(t) is