Numerical simulation of bioartificial capsules and cells in flow

A. Scientific challenges and objectives

In most applications, capsules are suspended in a carrying fluid (figure 1a). When the suspension is flowing, the particles deform in a complex manner under the hydrodynamic stresses. Modeling the motion and deformation of a capsule in flow is a difficult non-linear problem, as non-conventional fluid-structure interactions intervene: the fluid motion is governed by viscous effects, the structure is undergoing large deformation and inertial effects are negligible. The assumptions that are typically made are:

Assuming the capsule wall as an elastic surface without bending rigidity is only valid as long as the membrane is under traction. When the membrane experiences in plane compression in one direction, it tends to fold and buckle, these phenomena being governed by its bending rigidity. The existence of such phenomena has been demonstrated experimentally (figure 1b). The problem is that they tend to make unstable numerical codes based on a membrane model.

 
The objective has therefore been to develop a new coupling method (solving for the fluid-structure interactions) that remains numerically stable in zones of compression and that has a very precision.

a.   b.

Figure 1: Ovalbumin-membrane capsules. (a) Capsules in suspension in a liquid at rest. (b) Capsule flowing in a tube: folds form in the azimuthal direction. (UMR CNRS 6600)


B. New fluid-structure coupling strategy

We introduce a new numerical method to model the fluid–structure interaction between a microcapsule and an external flow. An explicit finite element method is used to model the large deformation of the capsule wall, which is treated as a bidimensional hyperelastic membrane. It is coupled with a boundary integral method to solve for the internal and external Stokes flows. Our results are compared with previous studies in two classical test cases: a capsule in a simple shear flow and in a planar hyperbolic flow. The method is found to be numerically stable, even when the membrane undergoes in-plane compression, which had been shown to be a destabilizing factor for other methods. The stiffness introduced by the numerical method contributes to the stability of the problem because it enriches the membrane model with some bending stiffness. It allows the numerical procedure to remain stable during transient phases when in-plane compression and high curvatures may render other methods unstable. It also has the advantage that steady states with negative tensions can be computed and studied. While it stabilizes the numerical procedure, the stiffness introduced by the elements is a byproduct of the numerical method and cannot be controlled or used to model the physical bending stiffness of a capsule.

The results are in very good agreement with the literature. When the capillary number, ratio of the fluid viscous forces to the membrane elastic forces, is increased, three regimes are found for both flow cases (Figure 2). Our method allows a precise characterization of the critical parameters governing the transitions.

a.   b.   c.

Figure 2: Deformed shape of an initially spherical capsule in a linear shear flow at steady state (neo-Hookean membrane law). Persistence of compressive normal forces at low (a) and high (c) values of the capillary number.


C. Influence of internal viscosity on the large deformation and buckling of a spherical capsule in a simple shear flow

We numerically studied the motion and deformation of a spherical elastic capsule freely suspended in a simple shear flow, focussing on the effect of the internal-to-external viscosity ratio. For low viscosity ratios, the internal viscosity no longer affects the capsule deformation. Inversely, for large viscosity ratios, the slowing effect of the internal motion lowers the overall capsule deformation; the deformation is asymptotically independent of the flow strength and membrane behaviour. An important result is that increasing the internal viscosity leads to membrane compression and possibly buckling. Above a critical value of the viscosity ratio, compression zones are found on the capsule membrane for all flow strengths. This shows that very viscous capsules tend to buckle easily.

 

D. Ellipsoidal capsules in simple shear flow: prolate versus oblate initial shapes

The large deformations of an initially-ellipsoidal capsule in a simple shear flow are studied by coupling a boundary integral method for the internal and external flows and a finite element method for the capsule wall motion. Oblate and prolate spheroids are considered (initial aspect ratios: 0.5 and 2) in the case where the internal and external fluids have the same viscosity and the revolution axis of the initial spheroid lies in the shear plane. The influence of the membrane mechanical properties (mechanical law, ratio of shear to area dilatation moduli) on the capsule behaviour is investigated.

Two regimes are found depending on the value of a capillary number comparing viscous and elastic forces. At low capillary number, the capsule tumbles, behaving mostly like a solid particle (figure 3). At higher capillary numbers, the capsule has a fluid-like behaviour and oscillates in the shear flow while its membrane continuously rotates around its deformed shape. During the tumbling-to-swinging transition, the capsule transits through an almost circular profile in the shear plane for which a long axis can no longer be defined. The critical transition capillary number is found to depend mainly on the initial shape of the capsule and on its shear modulus, and weakly on the area-dilatation modulus. Qualitatively, oblate and prolate capsules are found to behave similarly, particularly at large capillary numbers when the influence of the initial state fades out. However, the capillary number at which the transition occurs is significantly lower for oblate spheroids.

Figure 3: Evolution of the capsule shape in the shear plane over one half period. The initial shape is a prolate spheroid, a/b = 2, and the membrane follows the Sk law with C = 1. The grey scale corresponds to the normal component of the load. The dot shows the position of material point P, originally on the short axis. (a) Tumbling at low capillary number, (b) Transition, (c) Swinging at higher capillary number.

E. Flow of a spherical capsule in a pore with circular or square cross-section

The motion and deformation of a spherical elastic capsule freely flowing in a pore of comparable dimension is studied. The thin capsule membrane has a neo-Hookean shear
softening constitutive law. The three-dimensional fluid-structure interactions are modelled by coupling a boundary integral method (for the internal and external fluid motion)
with a finite element method (for the membrane deformation). In a cylindrical tube with a circular cross-section, the confinement effect of the channel walls leads to compression
of the capsule in the hoop direction. The membrane then tends to buckle and to fold as observed experimentally. The capsule deformation is three-dimensional but can be fairly
well approximated by an axisymmetric model that ignores the folds. In a microfluidic pore with a square cross-section, the capsule deformation is fully three-dimensional. For
the same size ratio and flow rate, a capsule is more deformed in a circular than in a square cross-section pore. We provide new graphs of the deformation parameters and
capsule velocity as a function of flow strength and size ratio in a square section pore. We show how these graphs can be used to analyze experimental data on the deformation of artificial capsules in such channels.

F. Comparison between spring network models and continuum constitutive laws: application to the large deformation of a capsule in shear flow

Recently, many researchers have used discrete spring network models to express the membrane mechanics of capsules and biological cells. However, it is unclear whether such modeling is sufficiently accurate to solve for capsule deformation. We examined the correlations between the mechanical properties of the discrete spring network model and continuum constitutive laws. We first compared uniaxial and isotropic deformations of a two-dimensional sheet, both analytically and numerically. The 2D sheet was discretized with four kinds of mesh to analyze the effect of the spring network configuration (Figure 4). We derived the relationships between the spring constant and continuum properties, such as the Young modulus, Poisson ratio, area dilation modulus and shear modulus. It was found that the mechanical properties of spring networks are strongly dependent on the mesh configuration. We then calculated the deformation of a capsule under inflation and in a simple shear flow in the Stokes flow regime, using various membrane models. To achieve high accuracy in the flow calculation, the boundary-element method was used. Comparing the results between the different membrane models, we found that it is hard to express the area incompressibility observed in biological membranes using a simple spring network model.


Figure 4: Configurations of spring network used for the discrete model: cross, cross center, regular triangle, unstructured


Collaborators

Prof. Dominique Barthès-Biesel, BMBI, UTC
Prof. Patrick Le Tallec, LMS, Ecole Polytechnique
Dr. Dominique Chapelle, INRIA Rocquencourt
Dr. Marina Vidrascu, INRIA Rocquencourt
Dr. Annie Viallat, INSERM Marseille
Dr. Marc Leonetti, IRPHE Marseille
Prof. Marc Jaeger, M2P2 Marseille
Prof. Takami Yamaguchi, Tohoku University, Sendai (Japan)
Prof. Takuji Ishikawa, Tohoku University, Sendai (Japan)
Prof. Yohsuke Imai, Tohoku University, Sendai (Japan)

Johann Walter, Couplage intégrales de frontière - éléments finis : application aux capsules sphériques et ellipsoïdales en écoulement, PhD Thesis, UTC, December 2009.
Xuqu Hu (PhD student), Simulation numérique des interactions fluide structure de capsules dans un tube tri-dimensionnel, UTC.
Toshi Omori, PhD Thesis, Tohoku University, Sendai (Japan), April 2012.
Etienne Foessel, Simulation numérique de capsules en cisaillement simple : effet du rapport de viscosité, Master Thesis, July 2010.
Claire Dupont, Mouvement d’une capsule ellipsoïdale en cisaillement, Master Thesis, September 2011.
Jean-Baptiste Bourjade, Modélisation des effets de flexion pour des capsules sous écoulement, Master Thesis, July 2011.
Claire Dupont (PhD student), Dynamique de capsules sous écoulement, UTC - Ecole Polytechnique.
Benjamin Sévénié (PhD student), Capsules en écoulement dans un réseau capillaire complexe , UTC.