Page 96 - Artificial Intelligence for Computational Modeling of the Heart

66 Chapter 2 Implementation of a patient-specific cardiac model

In both cases, special care has to be taken if integration and ob-
servation surface are identical (i.e., matrices P BB ,P HH and G HH ),
in which case partitioning into a close region (triangles around
the integration point) and a distant region (all other triangles,
Fig. 2.19) is reasonable, as it simplifies computation significantly.
Finally, we define the transformation matrix

Z BH =[P BB − G BH G −1 P HB ] −1 [G BH G −1 − P BH ]
HH HH
which allows us to compute body surface potentials directly from
the extracellular potentials of the heart by means of a single matrix
multiplication: φ b = Z BH φ e .

2.2.4.1 ECG calculation
Based on the body surface potentials, which are computed for
each vertex at the torso mesh, we compute the standard Einthoven
and Goldberger limb leads (I, II, III, a VR ,a VL ,a VF )aswellastheWil-
son precordial leads (V 1 –V 6 ). To avoid interpolation on the torso
mesh, the electrode positions can be chosen to coincide with ver-
tex positions. The following ECG features are automatically de-
tected as in [223]:
QRS duration QRSd: For numerical stability, the QRS complex
is detected using the depolarization times computed by
the tissue level electrophysiology model. Assuming one full
heart cycle is computed, QRSd = max x a(x) − min x a(x).The
depolarization (or activation) times a are obtained as the
points in time when the potential first exceeds the change-
over voltage:

a(x) = argmin {v(x,t) ≥ v gate }
t
Electrical Axis EA: For the limb leads I and II, the peak ampli-
tudes h I and h II are computed by summing up the ampli-
tudes of R and S waves in the respective leads. The electrical
axis is then calculated using the formula EA = arctan 2h II −h I .
√
3h I
2.3 Biomechanics modeling

The next modeling component is cardiac biomechanics. The
goal is to efficiently solve Newton’s law of motion for the my-
ocardium (Eq. (2.15)), given patient-specific anatomy and electro-
physiology:

M ¨ u + D ˙ u + K(u)u = f a + f p + f b (2.15)