Page 180 - Advances in Biomechanics and Tissue Regeneration

176 8. TOWARDS THE REAL-TIME MODELING OF THE HEART

a particular phase of the cardiac cycle such that a standardized timeline can be established and the datasets can be
subsequently standardized via temporal PODI calculations. Two examples are considered to demonstrate and study
the performance of the developed time standardization scheme. The calculation times of the whole PODI process,
including temporal and parametric calculations, are all below 1 min. This represented a speed-up of 15 times or more
when compared to their full-scale simulation counterparts, even if the time required for reading the datasets is
included, which accounted for more than 25% of the PODI calculation time. The performance could be further
improved if the datasets are prestandardized before being stored in the database. The results obtained are found
error range of
to be very accurate, as the PODI solutions for displacement, stress, and strain fields are within an ε ‘ 2
only 0.022–0.032. It is, therefore, demonstrated that this PODI approach is able to capture the heart’s behavior for vary-
ing hemodynamics in terms of preload and postload.
Second, in order to facilitate patient-specific modeling with arbitrary heart anatomies, two DOFS methods are pre-
sented in this work. They are found to solve the problem of incompatible datasets during the assembly of the dataset
matrix for the PODI calculation with different levels of accuracy. Using the cube template grid, the solution fields from
the selected datasets are standardized by interpolating them at the template nodes. It is found that this operation is,
time-wise, costly as opposed to other subprocesses in the PODI calculation. In terms of accuracy, the results obtained
are found to be error-prone. That is especially noted when looking at the displacement field where the PODI results
show nonphysical behavior. This behavior is identified to be linked to the zero values assigned to nodes during the
standardization process in order to account for template nodes lying outside the dataset heart geometry. The second
DOFS method considered solves the problems encountered in the cube grid standardization procedure as it no longer
introduces the zero values in the dataset matrix. This is achieved by registering the dataset heart geometries to a tem-
plate heart geometry using the CPD method. As such, all nodes belonging to the dataset hearts are located inside or
along the surface of the template heart. The initial results obtained exhibit reduced errors in the displacement fields and
pressure-volume relationship curves that decrease by 39% or more. The nonphysical deformation is nevertheless still
present but less accentuated. It is subsequently shown that template grid refinement can provide a remedy for this
issue by optimizing the CPD registration and completely removing nonphysical deformations. For coarse mesh den-
sities, the cube template required less calculation time than the heart one. But for higher mesh densities, this difference
vanishes (see Tables 8.9 and 8.12). In terms of total time, the cube template takes 643 s and the heart template takes
597 s for the finest mesh density. Here, the time required for the template projection alone is 628 s for the cube and 564 s
for the heart template. One way of reducing the total calculation time would be to create a database of already stan-
dardized datasets instead of standardizing the data during the PODI calculation.
Based on these results, it can be deduced that the presented cardiac PODI framework is suitable to carry out PODI
calculations of the whole cardiac cycle using heart geometries having different sizes and mesh discretization, as is the
case for patient-specific heart models. Importantly, low calculation times are accomplished at a good solution
accuracy.
With those encouraging results, the next steps of this research will be to look at actual patient-specific hearts as
obtained from CMR scans and create a database that considers additional characteristics such as gender, age, state
of health, fitness, or any other relevant parameters so as to extend its range of applicability.

APPENDIX

A.1 Moving Least Square Approximation
As mentioned previously, the PODI method is based on the use of an interpolation technique. In this research, the
MLS approximation method [33] is chosen for the PODI calculation as well as the fiber distribution approximation, as it
can deal with problems of arbitrary dimensionality and different size of data point sets.
Let us consider a function f(θ) defined over the domain M, which is here not a geometrical domain because each
point in M is not associated with spatial coordinates but with a set of parameters θ representing the characterizing
properties of a problem’s solution, for example, stiffness, anisotropy, etc. A possible approximation for f(θ) is given
by a polynomial P(θ) and its nonconstant coefficients a(θ):
h
f ðθÞ¼ PðθÞ aðθÞ: (A.1)
p
1
Now, let domain M be discretized by a finite number of parameter sets {θ , …, θ }, the so-called particles scattered in
domain M. Each particle is associated with a so-called weight function Φ of compact support. The size of the support
I
can be individually defined for each particle θ , I ¼ 1, p by ϱ I , the so-called influence radius of Φ. The collection of

I. BIOMECHANICS