Page 144 - Advances in Biomechanics and Tissue Regeneration

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

of the liver. Berkley et al. [19] also employed a linear model for the suturing procedure of biological tissue where
they deemed the dynamic effects to be negligible. As such, a calculation frequency of 30 Hz was achieved. Lim and
De [20] investigated two different approaches to address the nonlinear material behavior encountered for the sur-
gical tool-tip penetration of a kidney. The key idea consisted of considering nonlinear mechanics in the region
where the surgery tool tip is in contact with the organ and treat the rest of the organ in a linear fashion. For
the solving procedure, they made use of a mesh-free point collocation-based method and a modified Newton-
Raphson scheme, achieving a calculation frequency of 1000 Hz. Lastly, Courtecuisse et al. [21] employed graphical
processing units (GPU)-based algorithms of a linear finite element model to emulate the cutting of a liver at a sim-
ulation frequency of 29 Hz. Even though linear elastic models provide fast computation times, the solution accu-
racy in the context of large deformation is usually poor with, for instance, an error rate of up to 30% as encountered
in the work of Lim and De [20].
In this chapter, it is proposed to use a reduced order method (ROM) to drastically reduce the computation time of
full-cycle heart simulations at high levels of accuracy. Some of the most popular ROM techniques are proper orthog-
onal decomposition (POD) [22], reduced order basis (ROB) [23], and the proper generalized decomposition (PGD) [24].
The POD and ROB are based on an a posteriori approach, where predefined sets of data are needed to start the com-
putation [22, 23]. Yet, between the two of them, POD has been found to be more computationally efficient [25]. By
contrast, the PGD method is based on an a priori procedure and as such does not require predefined sets of data. Even
though these ROM techniques have been used across a wide variety of different fields, their application to cardiac
modeling is limited. Only POD has been employed so far but targeted specifically toward the electrophysiology of
the heart [26, 27]. The applications of POD, ROB, and PGD are mostly geared toward reducing the system of equations,
which can consequently be easily solved. However, assembling these equation systems still remains computationally
expensive, especially for nonlinear problems.
The POD is well established in solid mechanics, for example, Refs. [28, 29]. For our purpose of achieving real-time
modeling, the direct application of POD to, for example, the FEM, does not result in a large enough reduction in cal-
culation time because only the solving of the discrete equation system is accelerated. One particular variant of POD
bypasses the setting up of the discrete equation system while exploiting the reduction capabilities of the POD; it is
called the proper orthogonal decomposition with interpolation (PODI) method, developed by Ly and Tran [30]. Recent
research carried out by Niroomandi et al. and Coelho et al. has found that the PODI approach allows for subsecond
calculation times to be achieved and therefore makes high-frequency computation feasible [31, 32]. The PODI
method makes use of a collection of datasets of the problem under consideration, describing its mechanics for a range
of variations in terms of geometry, material properties, loading conditions, etc. These datasets are then utilized to
interpolate the mechanics of a similar problem of the same category where its mechanical behavior is unknown or
has not been determined yet. In our case, the moving least square (MLS) approximation [33] has been chosen as the
interpolation technique due to its ability in scaling up smoothly to several dimensions when multiparametric sim-
ulations are needed. The collection of datasets is stored offline in a database-type format for ease of access and is
comprised of full-scale simulation results of the human heart obtained using either the element-free Galerkin method
(EFG) or FEM, considering variations in the cardiac tissue characteristics. In practice, it is expected that the used
characteristic cardiac parameters will be related to in vivo data. These can be heart anatomy, strain, or hemodynam-
ics data from magnetic resonance imaging (MRI), fiber orientation vectors from diffusion tensor magnetic resonance
imaging (DTMRI), or other categorizing parameters such as gender, pathologies, etc. In this sense, a clear advantage
of the proposed method is that not only patient-specific heart modeling can be achieved at drastically reduced com-
putation times, but also the accuracy of the computations will be continuously improved by the addition of new
datasets over time. A recent paper by Rama et al. [34] addressed this problem and provided an in-depth investigation
on the evolution of the computational cost when using PODI. Subsecond calculation times were achieved using a
standard desktop computer with only a 1 CPU core and limited memory resources while the calculation errors were
kept at minimal levels. Rama et al. applied PODI to heart modeling, which resulted in a calculation frequency of
about 550 Hz.
In this chapter, a first application will look at modeling a full heartbeat using PODI [35] and is an extension of the
work of Rama et al. [34], where only the diastole filling was investigated. To achieve that, a so-called time standard-
ization method is developed. This procedure is required because the time steps needed for each simulated dataset vary
according to the step-size increments or the numerical stability level of the corresponding full-scale simulation. Hence,
choosing the solution fields for the PODI calculation necessitates that they are suitably synchronized with each other.
This involves the creation of a time line with fixed standardized points applicable to each dataset and an additional
temporal PODI calculation to obtain the solution fields at those points.

I. BIOMECHANICS