Page 222 - Artificial Intelligence for Computational Modeling of the Heart

Chapter 6 Additional clinical applications 195

Figure 6.7. Fitting the surface model ¯r(z,φ) to the points corresponding to the patient-specific anatomical model.

typically present immediately downstream from the CoA. The
2
(1−s 1 cos (φ/2)) factor of the first exponential term models the
CoA eccentricity, i.e. if the factor s 1 isnonzerotheCoAisno
longer axi-symmetric.
The procedure described above is applied only for defining the
radius as a function of centerline position and angle, i.e. it pro-
vides no information about the centerline of the vessel (curvature,
torsion, etc). An additional model function is employed for the
vessel centerline, consisting of a three-dimensional Bezier curve
containing 5 control points:

5

C(t) = b i,5 (z)c i ,z ∈[0,1], (6.6)
i=1
where θ cl =[c 1 ,...,c 5 ] are the control points, representing the pa-
rameters to be optimized.
The model fitting process is formulated as a least squares prob-
lem, and the cost function is minimized using the optimize pack-
age in the Scipy Python library [443]. The fitting process is run
separately for the surface and the centerline model, using the
patient-specific anatomical models, so that a set of θ surf ace and
θ cl parameters are obtained for each given patient-specific model.
To fit the centerline model, the given points should be normal-
ized, centered in the origin, and have the same orientation. Oth-
erwise the parameter values vary significantly from case to case,
therefore making it impossible to generate realistic synthetic cen-
terlines. Besides the parameters described above, an additional
parameter θ scale is required for de-normalization, i.e. to scale the
synthetic anatomical models to physiological ranges. The θ scale
parameters are computed for each given patient-specific model in
the pre-processing stage, when the meshes are scaled. Thus, the fi-
nal parameter list θ consists of the surface parameters θ surf ace ,the
centerline parameters θ cl and the scaling factor θ scale .
The synthetic models are generated using Eq. (6.3), and (6.6)
by randomly choosing the θ parameters in the value ranges deter-