Biomedical Engineering Reference
In-Depth Information
red blood cell model (LD-RBC), also based on DPD, that is more appropriate for
blood flow simulations in large arterioles [38].
This chapter is organized as follows: in Sect. 10.2 we review the basic DPD the-
ory, the two RBC models, as well as aspects of the aggregation and adhesion mod-
els that are especially important in modelling hematologic disorders. In Sect. 10.3
we present some details on how we can use diverse single-cell static and dynamic
measurements to estimate key macroscopic parameters, which upon mapping to the
network (microscopic) parameters serve as input to the models. In Sect. 10.4 we first
present validation tests based on single-cell experiments. Subsequently, we present
validation tests for whole blood, demonstrating that both models can predict the
human blood viscosity in a wide range of shear rate values, including the low shear
rate regime, where aggregation and rouleaux formation are responsible for the strong
non-Newtonian blood behaviour. In Sect. 10.5, we apply to malaria the framework
we developed, i.e. from single-cell-measurements parameter estimation to predicting
the mechanical and rheological behaviour of infected blood in malaria. We conclude
in Sect. 10.6 with a brief summary and a discussion on the potential of multiscale
modelling to predicting the state and evolution of hematologic disorders.
10.2 Methods and models
We first review two formulations of the dissipative particle dynamics (DPD) method
that we employ in modelling RBCs and blood flow. We then provide specific details
on the multiscale RBC model (MS-RBC) and subsequently on the low-dimensional
RBC model (LD-RBC), including the aggregation and adhesion models. Finally, we
present details on the scaling from DPD units to physical units.
10.2.1 Dissipative particle dynamics: original method
Dissipative Particle Dynamics (DPD) [36, 39] is a mesoscopic particle method,
where each particle represents a
molecular cluster
rather than an individual atom,
and can be thought of as a soft lump of fluid. A first-principles derivation of the DPD
method from the Liouville equation is presented in [40]. The DPD system consists
of
N
point particles of mass
m
i
, position
r
i
and velocity
v
i
. DPD particles interact
through three forces: conservative (
F
ij
), dissipative (
F
ij
), and random (
F
ij
) forces
given by
F
ij
=
F
ij
(
)
r
ij
,
r
ij
F
ij
=
−
γω
D
(
r
ij
)(
v
ij
·
r
ij
)
r
ij
,
(10.1)
r
ij
)
ξ
ij
F
ij
=
σω
R
r
ij
,
(
√
dt
where
r
ij
=
/
=
−
r
ij
r
ij
,and
v
ij
v
i
v
j
. The coefficients
γ
and
σ
define the strength
D
and
R
are weight
of dissipative and random forces, respectively. In addition,
ω
ω
functions, and
ξ
ij
is a normally distributed random variable with zero mean, unit
variance, and
ξ
ij
=
ξ
ji
. All forces are truncated beyond the cutoff radius
r
c
, which
