**1. Introduction**

Manufactured deformable microcapsules are intended to be used as drug carriers within the human vascular network to deliver drugs at specific targets (tumors, etc.). In order to design reliable capsules, one can make help of numerical simulation and high performance computing. The transportation of such capsules into microchannels is a three-dimensional fluid-structure interaction (FSI) problem involving a fluid flow within a confined environment and the deformation of hyperelastic membranes [1, 2]. The behavior of the capsule depends on dimensionless parameters such as the capillary number denoted by *Ca* and the aspect ratio *a=ℓ* between the capsule radius *a* and the channel characteristic length ℓ. The parameter vector *μ* ¼ ð Þ *Ca*, *a=*ℓ , for which a capsule steady shape exists, lies in a bounded domain *D* ⊂ <sup>2</sup> . We look for the time evolution of the capsule shape under a Lagrangian description. From an initial shape *X*, we are interested in determining the capsule position *x X*ð Þ , *t*, *μ* in the microchannel domain Ω ∈ <sup>3</sup> at time *t* for a parameter vector *μ*∈ *D*. By denoting *u* the displacement vector from the initial position, we have

$$\mathbf{x}(\mathbf{X}, \mu, t) = \mathbf{X} + \mathfrak{u}(\mathbf{X}, \mu, t) \tag{1}$$

with *u X*ð Þ¼ , *μ*, 0 **0**. The governing equations of the FSI problem include both kinematics and motion equations. At the membrane, we have equilibrium of the mechanical forces (mechanical equilibrium of the membrane and viscous stresses from the fluid). By denoting *v* the vector field of velocity at the membrane, the system of differential algebraic equations in abstract form reads

$$
\dot{\mathfrak{u}} = \mathfrak{v},
\tag{2}
$$

$$
\sigma = \mathcal{F}\_{\mu}(\mathfrak{u}, \mathbf{X}).\tag{3}
$$

Practically, there are different candidate computational approaches to discretize this system of equations. First, the initial capsule membrane has to be discretized by using a finite element triangular mesh made of nodes f g *X<sup>i</sup> I <sup>i</sup>*¼<sup>1</sup>. Regarding time discretization, in [2], an explicit time scheme is used and the velocity field is determined by the use of a boundary integral method (BIM) coupled with a finite element method (FEM). A numerical stability condition imposes the use of small time steps. For a given parameter *μ*, the time evolution of capsule dynamics on the time intervals of interest generally requires hours of CPU time. To better understand the membrane behavior with respect to *μ*, a design of computer experiment (DoCE) is done: from a set of *J* parameter samples of *μ <sup>j</sup>* ∈ *D*, *j* ¼ 1, … , *J*, a spatio-temporal solution is computed for each *μ <sup>j</sup>* , leading to a database of shape solutions under the form of a third-order tensor

$$\mathcal{T}\_{\mathbf{x}} = \left( \mathfrak{x} \left( \mathbf{X}\_i, \mu\_j, t^n \right) \right)\_{i=1, \dots, I, j=1, \dots, J, n=0, \dots, N,} \in \mathbb{R}^{3I \times J \times (N+1)} \tag{4}$$

using a triangular finite element discretization of the membrane, a time discretization *t <sup>n</sup>* <sup>¼</sup> *<sup>n</sup>*Δ*<sup>t</sup>* (assuming that the time step is constant) and the parameter samples *μ <sup>j</sup>* . Typically, for practical computations, *I* ¼ *O*ð Þ 1000 , *N* ¼ *O*ð Þ 10000 and *J* ¼ *O*ð Þ 100 , so that the tensor database becomes rather huge (about *O*ð Þ 10 gigabytes). Of course, one can only store the solutions at coarser times steps and reduce *N* to *O*ð Þ 100 but the database remains rather big even in this case.

From this data tensor, one can imagine different use cases leading to different tools:


*Space-Time-Parameter PCA for Data-Driven Modeling with Application to Bioengineering DOI: http://dx.doi.org/10.5772/intechopen.103756*

First and second items can be achieved by means of data dimensionality reduction. This leads to a lower storage of data in memory as well as a lower numerical complexity of processing and manipulation. In this chapter, we will consider a Higher Order Singular Value Decomposition (HOSVD), which is a generalization of Principal Analysis Component (PCA) to tensors. The third item involves a model-order reduction (MOR) methodology. From computed data, we would like to derive a lightweight dynamical system that reproduces the data and, even more, that is able to accurately estimate solutions for different parameter values *μ*. Data-driven modelorder reduction first makes use of data-dimensionality reduction by a low-order tensor decomposition of the solutions according to some suitable spatial, temporal and parameter reduced bases (see [3]). In our application, this will give the truncated decomposition (where the solutions are here seen as functions):

$$\bar{\mathfrak{x}}(\mathbf{X},\boldsymbol{\mu},t) = \mathbf{X} + \sum\_{k=1}^{K} \sum\_{\ell=1}^{L} \sum\_{m=1}^{M} a\_{k\ell m} \boldsymbol{\mathfrak{o}}\_{k}(\mathbf{X}) \boldsymbol{\upchi}\_{\ell}(\boldsymbol{\mu}) \boldsymbol{\upalpha}\_{m}(t) \tag{5}$$

for some expansion coefficients *ak*<sup>ℓ</sup>*<sup>m</sup>* and some spatial functions *φk*, parameter functions *ψ*<sup>ℓ</sup> and temporal functions *ω<sup>m</sup>* with *ωm*ð Þ¼ 0 0 ensuring *x X*ð Þ¼ , *μ*, 0 *X*. The truncation ranks *K*, *L* and *M* are expected to be small enough (*K* ≪ *I*, *L* ≪ *J* and *M* ≪ *N*). Discretized shape solutions returned by the Full-Order computational Model (FOM) are stored into a third-order tensor of data T *<sup>x</sup>*. Let ~ T *<sup>x</sup>* denote the truncated tensor expansion related to (5). It reads:

$$\bar{\mathcal{T}}\_{\boldsymbol{x}} = \boldsymbol{A}^{0} \otimes \boldsymbol{\mathsf{e}}^{N} + \sum\_{k=1}^{K} \sum\_{\ell=1}^{L} \sum\_{m=1}^{M} a\_{k\ell m} \boldsymbol{\Phi}\_{k} \otimes \boldsymbol{\Psi}\_{\ell} \otimes \boldsymbol{\mathbf{w}}\_{m} \approx \boldsymbol{\mathsf{T}}\_{\boldsymbol{x}} \tag{6}$$

where *A*<sup>0</sup> ∈ *<sup>I</sup>*�*<sup>J</sup>* , *A*<sup>0</sup> � � *ij* <sup>¼</sup> *<sup>X</sup>i*, *<sup>e</sup><sup>N</sup>* <sup>¼</sup> ð Þ 1, … , 1 *<sup>T</sup>* <sup>∈</sup> *<sup>N</sup>*, **<sup>Φ</sup>***<sup>k</sup>* <sup>∈</sup> *<sup>I</sup>* , **Ψ**<sup>ℓ</sup> ∈ *<sup>J</sup>* and *w<sup>m</sup>* ∈ *<sup>N</sup>*. In this chapter, we will seamlessly use the functional representation or the tensor one.

From this reduced form, we will then apply a kernel-based Dynamic Mode Decomposition (*k*-DMD, see [4]) to derive a dynamical system able to predict the capsule shape evolution over time for any value of the parameter *μ*. This will be developed in the next sections.
