Mathematical Models for Blood Coagulation - Fluid-Structure Interaction and Biomedical Applications - page 544

important in evaluating the artificial mechanical devices coming into contact with
blood (e.g., blood pumps or mechanical heart valves).
Several models have been developed to quantify the blood damage and platelet
activation due to flow-induced stress. The most widely used approach defines the
Blood Damage Index

D using an empirical power law relation

D D C Ǜ t exp

(7.108)

where is a scalar measure of stress and t exp is the exposure time. The constants
C, Ǜ,andLJ have to be determined experimentally depending on the type of
solvent, blood cells (RBCs or platelets), and other typical flow parameters. For
an overview of available model parameters see the summary in [ 114 ] and related
references therein. This model was used to integrate the blood damage along particle
trajectories in computational analysis of a blood pump in [ 223 ], or for numerical
investigation of the effects of channel geometry on platelet activation and blood
damage in [ 259 ]. A similar model, but a bit simpler in its form is the blood damage
index proposed in [ 204 ]

C
2 t exp

D D (7.109)

On the other hand, a more complicated differential model for this quantity was
proposed in [ 200 ]

d dt D C

.1 D / ı 2

(7.110)

Here C and ı are case-dependent constants to be determined experimentally. A
model of this type 93 was used, e.g., in [ 4 ] to study the flow-induced platelet
activation and damage accumulation in a mechanical heart valve. This approach
was generalized in [ 271 ]:

d D .t/

dt D D 0 C F.

;/C F.P /

D (7.111)

D 0 is the constant activation/damage rate, F.

D

where ;/ is the stress-dependent

part, and F.P / is the stress rate-dependent contribution. A similar approach was
recently adopted in [ 222 ] to develop a mathematical model of activation and
sensitization of platelets subjected to dynamic stress histories.
These simple empirical laws are suitable for situations that can easily be
characterized by a single stress level and exposure time. This is, however, seldom the
case of realistic blood flow simulations. In order to be able to handle also complex
flows a more general Lagrangian approach is often adopted. The one introduced for
blood (erythrocyte) damage in [ 97 ]and[ 98 ] was extended for platelets activation in

93 In Lagrangian particle tracking form.