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8.4 WHOLE HEART CYCLE MODELING 151

employed to compute the interpolants and subsequently the PODI coefficients as per Eq. (8.24). In order to recover
the full-order solution from the coefficient, Eq. (8.23) is then employed, after which the mean vector v used in
calculating the mean centered ensemble matrix U i is added.
• Postprocessing and validation: Once the PODI calculations are completed, the results are saved and viewed by
postprocessing software called GiD (CIMNE International Center for Numerical Methods in Engineering). For the
validation process, the error in the PODI calculation is then compared with the full-scale simulation solution of the
problem at hand using EFG or FEM again, which is assumed to be the exact solution. The error calculation is given as
the ‘ 2 error norm:

k U PODI U EFG=FEM k
: (8.33)
ε ‘ 2 norm ¼ EFG=FEM
k U k

8.4.3 Numerical Examples

With the cardiac PODI algorithm introduced, its performance is illustrated with two representative examples. The
first example consists of an LV with varying preload, that is end-diastolic volume. Subsequently, another more chal-
lenging example is considered, which is a biventricle (BV) model with varying end-diastolic and end-isovolumetric
contraction pressures.

8.4.3.1 Human Left Ventricle Example
A patient-specific LV is extracted from MRI images. The geometry is discretized using 2659 tetrahedral elements
and 745 nodes. The values of the material constants used in Eq. (8.2) are as given in Table 8.2 while the stress scaling
coefficient, A, and the compressibility controlling penalty factor, A comp , are fixed to 0.46 and 100 kPa, respectively. The
active stress parameters of Eq. (8.7) are listed in Table 8.3. For the three-element WK model, R a and R p are calibrated as
1
3
8
7
3
1.28 10 Pa s m 3 and 1.0 10 Pa s m , while C is determined as 4.0 10 9 m Pa . As the Dirichlet boundary
condition, the base of the LV is fixed in a direction along the z-axis, the LV’s long axis, while the endo and epicardium
1
5
line on the base surface are subjected to an elastic spring force of stiffness 1 10 Nm , emulating the connection of
the heart to the major blood vessels. The Neumann boundary condition, which consists of a surface pressure applied to
the endocardium surface, is the parameter used to create the PODI database. This is done by varying the magnitude of
the surface pressure from 1.0 to 2.0 kPa. Using an increment of 0.1 kPa, a database with a size of 11 datasets is created
using the FEM. Across all datasets, the end-isovolumetric contraction pressure is set to 5.5 kPa and the end-
isovolumetric relaxation pressure to 0.25 kPa. The energy conservation level is set to 99.999%. The time standardiza-
tion process discretizes the timeline of each cardiac phase with 200 time points, equally spaced.
Following the creation of the database, the PODI calculation is now undertaken. Six datasets are mobilized. These
are the ones associated with an end-diastole pressure of 1.2, 1.3, 1.4, 1.6, 1.7, and 1.8, respectively. The POVs and POMs
of the end-diastolic fiber strain data are plotted in Figs. 8.6 and 8.7, respectively. The magnitude of the POVs are, as
expected, exponentially distributed across the POMs.
In terms of computation time, the PODI calculation is carried out within 53 s while the full-scale simulation requires
858 s, which represents a speed-up by a factor 16. The calculation is found to be affected by two processes. The first one
is reading the datasets into the computer memory, which takes about 14.5 s, while the temporal standardization needs
about 23.8 s. The reduced order calculation requires 4.9 s while the postprocessing stage is 9.7 s. The computation
times of the PODI, especially the temporal standardization, and the postprocessing stage can still be further reduced
if fewer time points were used to discretize the cardiac timeline. Yet, the computational speed of PODI can be

TABLE 8.2 Coefficients of a i , Converted From Usyk et al. [82]
Parameter a 1 a 2 a 3 a 4 a 5 a 6
Coefficient 6.00 5.00 9.00 12.00 12.00 6.00

TABLE 8.3 Active Contraction Parameters of the Human Left Ventricle
T max Ca 0 Ca max B l 0 t 0 m b
0
120 kPa 4.35 μmol 4.35 μmol 4.75 μm 1 1.58 μm 220 ms 1.0489 s μm 1 1.429 s

I. BIOMECHANICS