Mathematical Models for Blood Coagulation - Fluid-Structure Interaction and Biomedical Applications - page 508

LangevinDynamics(LD) . The molecules of the solvent (in which the molecules or
proteins of interest are dissolved) are approximated using additional force terms
expressing drag and random collisions associated with the thermal motions of
the solvent molecules [ 19 , 81 ]. The modified molecular dynamics equation can
now be written as

m i d 2 r i
dt 2

D f i dr i

dt C R.t/

i D 1;:::;N p (7.3)

where the first additional term reflects the friction force while the second, time-
dependent term R.t/, approximates the stochastic collision force. The exclusion
of the solvent molecules from the simulated particle set reduces the number of
numerical degrees of freedom of the problem and saves a significant part of the
computational time.
BrownianDynamics(BD) . This method can be seen as a viscous limit of the
Langevin Dynamics. The viscosity is assumed to be large and the inertial effects
are suppressed [ 85 ].
NormalModeAnalysis(NMA) . This method is based on the harmonic analysis
of the system (e.g., protein) oscillations about its local minimum energy state.
Based on experimental observations it is assumed that only the slowest harmonic
modes of protein oscillations have some functional consequences. In the original
implementation of NMA, the force field is required as in MD simulations.
However, instead of solving the Newtonian equations of motion, an harmonic
analysis is performed to find the slow, most important oscillation modes. For
further details on this method, see, e.g., [ 84 , 145 , 156 , 218 ]. Some simplifications
were introduced using Coarse Grained NMA [ 18 ], ElasticNetwork Model [ 102 ],
or EssentialDynamics [ 5 , 236 ].
2. Microscale Models

In this case the coagulation models consider microscale objects, i.e., cellular scale
matter with dimensions of the order of 1-10m. The need for modeling blood flow
and coagulation at microscale has at least two different motivations.
Micromechanics of blood clotting . The importance of the role of RBCs, platelets,
and other microparticles in the blood coagulation process can be better understood
and captured by models that are actually resolving all these objects. The motion,
deformation, aggregation, and adhesion dynamics as well as other microscopic
behavior of blood cells can only be accurately described in this scale. So the first
motivation is to gain detailed and high-resolution simulation results.
Blood coagulation in microvessels . The blood flow and thus also blood coagula-
tion in microvessels, which are vessels with diameter comparable with the size of
RBCs, cannot be accurately described by any macroscopic (continuous or statistical)
model. The continuum hypothesis is no more valid at this scale and thus the use of
models explicitly taking into account the blood cells is inevitable [ 70 ]. The second
motivation is thus the necessity of adjusting the scale of the resolved object to the
spatial size of the domain of interest.