Page 123 - Artificial Intelligence for Computational Modeling of the Heart

Chapter 2 Implementation of a patient-specific cardiac model 93

The parameters to estimate are the maximum active stress
τ 0 , the stiffness coefficient β, the contraction and relaxation
rates k AT P and k RS respectively, for both left and right ventri-
cles. They form the parameter vector θ = (τ 0,LV ,β LV ,k AT P,LV ,
k RS,LV ,τ 0,RV ,β RV ,k AT P,RV ,k RS,RV ). The parameters are first ini-
tialized to default values obtained from the literature, and then
optimized by minimizing the cost function ψ:

#
ψ(Ω m ,Ω c ) = λ i · D(Ω m,i ,Ω c,i ) (2.39)
i=1..6
where the indices m and c denote “measured” and “computed”
respectively. D is a distance function. For scalar parameters,
2
D(a,b) = (a − b) . For vectorial parameters, D(a,b) =||a − b|| L 2 ,
the L 2 norm. The λ i are weighting coefficients, with λ = (3,2,1,1,2,
2,3,2,1,1,2,2) to improve optimization performance by increas-
ing the weights of the most reliable features, as observed experi-
mentally.
The pulmonary vein pressure is also estimated during the per-
sonalization of the biomechanical model. This is achieved in two
steps. First, an initial computation is performed with generic pa-
rameters prior to biomechanical personalization. The pulmonary
vein parameter of the model is adjusted by adding the difference
between measured and computed atrial pressure at beginning
of diastole. After biomechanical personalization, the pulmonary
vein is further adjusted if needed.
It is worth noting the cost function must be calculated after
three or more heart cycles to minimize the effects of the tran-
sient regime. As a result, estimating biomechanical parameters
can quickly become time consuming. Even if one heart cycle takes
only 2–3 minutes to compute, personalizing a cardiac model could
still take few hours to converge due to the large number of itera-
tions needed by traditional gradient-free optimization methods. It
is therefore crucial to develop computational methods that allow
either fast simulation or fast parameter estimation, or both.

2.6 Summary
Nowadays, modelers and computer scientists can leverage a
wide variety of numerical discretization schemes to solve the
complex equations that govern the laws of cardiac function. This
chapter presents a specific implementation strategy, targeting
fast, multi-scale and personalized modeling. Other methodolo-
gies are possible of course, as it can be seen in the literature. Some
approaches are more optimized for computer graphics, favoring