Biomedical Engineering Reference
In-Depth Information
A number of numerical models have been developed recently including a contin-
uum description [1, 23, 24, 25] and a discrete approximation on the spectrin molec-
ular level [26, 27] as well as on the mesoscopic scale [28, 29, 30, 31]. Some of the
models suffer from the assumption of purely elastic membrane, and are able to cap-
ture only the RBC mechanical response, but cannot quantitatively represent realistic
RBC rheology and dynamics. Fully continuum (fluid and solid) models often suffer
from non-trivial coupling between nonlinear solid deformations and fluid flow with
consequential computational expense. Therefore, “semi-continuum” models [23, 25]
of deformable particles which use immersed boundary or front-tracking techniques
are developing rapidly. In these, a membrane is represented by a set of points which
are tracked in Lagrangian fashion and are coupled to an Eulerian discretization of
fluid domain. These models employ the same external and internal fluids and do
not take into account the existing viscosity contrast between them. In addition, con-
tinuum models omit some mesoscopic and microscopic scale phenomena such as
membrane thermal fluctuations which affect RBC rheology and dynamics [32]. On
the microscopic scale, detailed spectrin molecular models of RBCs are much lim-
ited by the demanding computational expense. Therefore, we will focus here on an
accurate mesoscopic modelling of red blood cells.
There exist several mesoscopic methods [28, 29, 30, 31] for modelling de-
formable particles such as RBCs. Dzwinel et al. [29] model RBCs as a volume of
elastic material having an inner skeleton. This model does not take into account the
main structural concept of red blood cell, namely a membrane filled with a fluid,
and therefore it cannot capture properly the dynamics of RBCs, for example, the
observed tumbling and tank-treading behaviour in shear flow [14, 33]. Three other
aforementioned methods [28, 30, 31] employ a very similar approach to the method
we will present here, where the RBC is represented by a network of springs in com-
bination with bending rigidity and constraints for surface-area and volume conser-
vation. Dupin et al. [28] couple the discrete RBC to a fluid described by the Lat-
tice Boltzmann method [34]. They obtained promising results, however the model
does not consider external and internal fluids separation, membrane viscosity, and
thermal fluctuations. Noguchi and Gompper [30] employed Multiparticle Collision
Dynamics [35] and present encouraging results on vesicles and RBCs, however they
do not use realistic RBC properties and probe only a single aspect of RBC dynam-
ics. Pivkin and Karniadakis [31] used Dissipative Particle Dynamics (DPD) [36]
for a multiscale RBC model which will be the basis of the general multiscale RBC
(MS-RBC) model we will present here. The MS-RBC model is able to successfully
capture RBC mechanics, rheology, and dynamics; this very accurate model was first
published in [37]. Potential membrane strain hardening or softening as well as the
effects of metabolic activity can also be incorporated into the model leading to pre-
dictive capabilities on the progression of diseases such as malaria. Theoretical analy-
sis of the hexagonal network yields its linear mechanical properties, and completely
eliminates adjustment of the model parameters. Such models can be used to repre-
sent seamlessly the RBC membrane, cytoskeleton, cytosol, the surrounding plasma
and even the parasite, e.g. in malaria-infected RBC, see Fig. 10.1. However, it is
quite expensive computationally, and to this end, we also present a low-dimensional
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