Page 225 - Artificial Intelligence for Computational Modeling of the Heart

198 Chapter 6 Additional clinical applications

where A inlet and r inlet are the inlet area and radius respectively,
and ν is the kinematic viscosity of blood.
2. For each of the computed flow rates, the pressure drop P
is estimated using the Young-Tsai model. If the maximum
P (corresponding to the largest Reynolds number) is larger
than the physiological threshold of 60 mmHg, all the Reynolds
numbers are scaled down until the maximum P becomes
60 mmHg.
3. If any of the computed flow rates is larger than 32 L/min, the
computation corresponding to that flow rate is not performed.

6.2.2.3 Pressure drop model for aortic coarctation
The pressure drop model initially proposed by Young and Tsai
[438] is formulated as a function of flow rate containing a linear
and a quadratic term that account for the viscous and turbulent
effects respectively. In its initial form, the Young-Tsai model is for-
mulated as follows:
2

μK v ρp 4 A 0 2
P YT = 3 Q + 2 − 1 Q , (6.8)
2πr 2A A s
0 0
2

L a A 0
K v = p 1 , (6.9)
r 0 A s
L a = p 2 L s + p 3 r s , (6.10)
where, μ is the dynamic viscosity of blood, r 0 and r s are the healthy
and stenosed radius respectively, A 0 and A s are the healthy and
stenosed areas, L s is the stenosis length and Q is the flow rate.
Finally, the factors p 1 ...p 4 are regression parameters, and were
determined by fitting the model to experimental data. Young and
Tsai performed the experiments on idealized, in vitro, stenosed
vessel geometries by applying different flow rates. The numerical
values for these parameters are given in Table 6.3.
Our work aims at improving the original Young-Tsai model
by specializing it for aortic flow conditions. Thus, we explored
three strategies: (i) a fine-tuning of the regression parameters, (ii)
adding additional terms to Eq. (6.8) and finally (iii) coupling the
resulting optimized model with a deep neural network.
The resulting pressure drop model was formulated based on
Eq. (6.8) and three additional terms as follows:

P = P YT + p 5 P convective + p 6 E + p 7 B, (6.11)

where, p 5 ,p 6 and p 7 are the parameters that determine the contri-
bution of the additional terms to the total pressure drop.