Page 150 - Advances in Biomechanics and Tissue Regeneration

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

as the former represents the dominant modes of deformation. Interestingly, it was found that with very few modes, one
can easily conserve up to 99% of the energy in the system. In Ref. [80], Lin et al. noted that, with their set of data com-
prising 24 snapshots, only the first two POVs were enough to achieve an energy conservation of 99.9997%, leading us
to believe that a 99% energy threshold is enough to achieve a reasonable accuracy.

8.3.2 POD With Interpolation

8.3.2.1 Parametric PODI
For the purpose of real-time modeling, the direct application of POD to a system of equations is not suitable.
When applied to the FEM, only the solving part of the overall process is accelerated while the preprocessing time
and repetitive equation system assembly time prohibit a drastic drop in calculation time. Hence, in order to cir-
cumvent this problem, the PODI method, developed by Ly and Tran [30], hasbeenemployedinthiswork. Their
p
1
idea starts by first constructing the matrix U from an ensemble of displacement vectors, {u , …,u }. Each dataset,
p
a
1
a
a
a
1
p
1
p
u , where u 2{u , …,u }, corresponds to a set of heart-defining parameters, θ , where θ 2{θ , …, θ }. Here, {θ , …, θ }is
defined as a sequence of parameter sets, each containing characteristic values of the properties of the respective heart.
From there, the POD calculation is carried out and used to transfer the datasets to a low-dimensional space using the
POMs, where an interpolation technique is set up to find the adequate solution. Once this is done, the solution is then
projected back to the high-dimensional space. The derivation of the method is given by first expressing U in terms of
the basis matrix, Φ, and the coefficient matrix, Ψ:
U ΦΨ, (8.21)
a
where a single displacement vector, u , can be defined as:
a
a
2
a
r
1
a
u ψ Φ + ψ Φ + ⋯ + ψ Φ : (8.22)
1 2 r
p
1
Now, let us suppose that an unknown displacement field, ^ u (with ^ u62fu ,…,u g), which corresponds to the set of
^
^
^
1
1
p
p
parameters θ (where θ < θ θ and θ62fθ ,…,θ g), is to be found using the earlier POD basis. Therefore, Eq. (8.22)
can be rewritten as:
r
2
1
^ u ^ ψ Φ + ^ ψ Φ + ⋯ + ^ ψ Φ : (8.23)
r
1
2
^
^
1
p
In Eq. (8.23), ^ u cannot be found as Ψ ¼ð^ ψ , ^ ψ ,…, ^ ψ Þ is unknown and Ψ62fΨ ,…,Ψ g. In this case, Ly and Tran [30]
2
r
1
^
suggested interpolating Ψ from matrix Ψ through:
^
Ψ ¼ ΨN, (8.24)
where N are the MLS interpolants that can be derived from
^ (8.25)
θ ¼ ΘN,
^
p
1
2
as θ and Θ are both known. Θ ¼ (θ , θ , …, θ ) and is a matrix. Each row of that matrix is defined by specific parameters
of a particular set of parameters while the columns are the different sets of parameters. In this chapter, the MLS approx-
imation method [33] was chosen as the interpolation scheme because it can deal with problems of arbitrary dimen-
sionality and different sizes of data points.
8.3.2.2 Temporal PODI
A temporal PODI calculation is very similar to the parametric one. The difference lies in the definition of the ensem-
ble matrix and the setting up of the interpolants. As discussed in the previous section, the ensemble matrix represents a
set of displacement vectors spanning over a parametric space of heart-defining characteristics but corresponding to a
particular time step. As such, the interpolation scheme is built up using the parametric space. In this case, the roles of
the parametric and temporal space are inverted. Here, the ensemble matrix is defined by a set of displacement vectors
spanning over a temporal space and corresponds to a particular parametric point. Hence, the interpolation scheme is
constructed using the temporal space. An example of a parametric as compared with a temporal ensemble matrix is
given in Fig. 8.2.
I. BIOMECHANICS