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8.2 CARDIAC MECHANICS AND MODEL 141

For our second application, a PODI-based cardiac modeling for geometrical variations in heart anatomy is consid-
ered [36]. Every heart shares common anatomical features such as two ventricles and two atria as well as ventricular
and aortic valves. Furthermore, a single heartbeat comprises four phases, namely, diastole filling, isovolumetric con-
traction, ejection, and isovolumetric relaxation where the heart expands, contracts, and twists on itself. In this sense,
the heart behavior of different individuals is generally very similar and can be extracted using POD in terms of proper
orthogonal modes (POMs). However, despite the similarities, distinct anatomical, physiological, and biomechanical
differences remain. Each heart has a different size as well as atrial and ventricular cavity volumes, see, for example,
Refs. [37–42]. The elastic material properties of the heart tissue determined from biaxial and triaxial tests by Sommer
et al. [43] show clear variations in the stress-strain response, in particular for larger strains. Within the PODI frame-
work, the anatomical differences are of primal importance but difficult to integrate. The problem is linked to the extrac-
tion of the POMs. In this work, the PODI database is constructed from a collection of solution fields obtained from
simulations carried out for a range of elastic material parameters. For the purpose of projection and interpolation,
a subset of these datasets is assembled in a matrix. In order to build the matrix, the data size of all incorporated solution
fields and their respective discrete spatial locations must be the same. Due to the anatomical differences of patient-
specific hearts, the corresponding discretized computational models will always exhibit a significant variation in nodal
numbers and spatial distribution. Consequently, assembling the resulting solution fields into a single matrix will lead
to incompatible vectors in the data matrix, effectively preventing the extraction of POMs from it.
Most PODI implementations found in the literature made use of a fixed mesh configuration. Only very recently
have variations in geometry and mesh configurations been explored [44–46]. In Amsallem et al. [45], the POMs linked
to different discretization layouts were minimized with respect to the POMs of a reference discretization reassembling
the geometrical shape of the problem at hand. The method considered POMs extracted from different series of datasets,
each computed from different mesh configurations. As such, the resulting POMs were not consistent with each other
and an additional step was required to enforce consistency. In González et al. [46], another approach was investigated.
The authors embedded different liver geometries on a benchmark cube mesh grid and computed a so-called distance
field with respect to the boundary surface of the organ. Following that, they then employed a method called locally
linear embedding to find the weight of the different registered liver configurations to reproduce the anatomical model
of the liver at hand and to carry out the interpolation. It remains, however, somewhat unclear how the nodes inside
the liver geometry were treated. Iuliano and Quagliarella [44] stacked the nodal coordinates in the data matrix such
that the extracted POMs reflected the spatial layout of node distribution. This approach only considered an identical
number of nodes, mainly focusing on mesh optimization. Generally, however, different mesh discretizations will con-
sist of a unique number of nodes and degrees of freedom. Therefore, for this research to facilitate patient-specific heart
modeling, an alternative approach is proposed that will be referred to as the degrees of freedom standardization
(DOFS) method.
The DOFS method consists of establishing a set of nodes, the so-called template nodes, that will be common for
every dataset in the database. Making use of a three-dimensional (3D) MLS scheme, all solution fields of each involved
dataset will be projected onto the set of template nodes and will consequently share the same degrees of freedom and
spatial locations. It has to be noted that the examples considered in the DOFS study looked at solution fields that rep-
resent only the diastole filling. For the choice of the spatial locations of the template nodes, two different approaches
will be explored: a cube grid and a heart-shaped grid. In order to enforce a high degree of accuracy in the interpolation
and to ensure that the spatial distribution of the dataset’s solution fields is correctly captured by the template nodes
of the heart-shaped grid, a nonrigid registration algorithm, the coherent point drift (CPD) method introduced by
Myronenko et al. [47], will be used to morph the heart datasets onto the template. The great benefit of this method
is that the datasets do not have to possess a geometry of the same size to obtain a good registration [48].
The layout of the chapter is as follows: In Section 8.2, the cardiac mechanics equations along with the model used for
the simulation of the heart will be introduced. Section 8.3 deals with the ROM where the POD method will be revisited
in Section 8.3.1 and the PODI method will then be elaborated in Section 8.3.2 with a focus on the parametric PODI and
the temporal PODI. The time standardization scheme is subsequently outlined in Section 8.4 and the degree of freedom
standardization scheme is presented in Section 8.5. Finally, the conclusion of the chapter is given in Section 8.6.

8.2 CARDIAC MECHANICS AND MODEL

A full heartbeat cycle consists of four phases: diastolic filling, isovolumetric contraction, ejection, and isovolumetric
relaxation. These stages are defined by mathematical models that are introduced in the following.

I. BIOMECHANICS