Kinetic Monte Carlo

The lattice Kinetic Monte Carlo (LKMC) module was used to simulate the dynamics of platelet transport and deposition under blood flow. The LKMC simulation was executed on a square grid with lattice spacing, h0.5m, while platelets were assumed to be circular with radius (1.5 μm). Platelet transport events were defined as (1)

diffusion with rate _D platelet₂

LKMC D h

  , where D_platelet was the effective diffusivity accounting

for both Brownian motion and the red blood cell (RBC) dispersion effect [41], and (2) advection with rate i

C

LKMC v h

  , where v_i was the fluid velocity component along the

lattice direction e_i. The total rate of motion for each platelet was the sum of the convective and diffusive rates. This approach was validated in our earlier work [39]. Red blood cells (RBC) also play a significant role in platelet motion. Besides augmenting the apparent diffusivity of platelets, flowing RBCs also generate an excess of platelet concentration in the plasma layer near the boundary, which was modelled as an additional radial drift velocity superimposed on the actual velocity that resulted in a 3-fold wall excess of platelets [40] (Figure 3-9). The drift velocity in the two dimensional channel is positive toward the wall within 3 μms, thereby causing extra momentum of hitting the thrombus.

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Figure 3-9: Inlet platelet concentration distribution and drift velocity.

The inlet platelet concentration distribution was biased near-wall due to the red blood cell effect. The platelet near wall has excess concentration (left) Platelet drift velocity due to RBC motions from Yeh, Calvez, and Eckstein [40]. A positive velocity points away from the wall, while the negative velocity within the center of the channel pushes platelets towards the wall (right).

In addition to the convection-dispersion-drift events for each platelet, we also considered the activation-dependent rates of attachment and detachment between 2 platelets, or between a platelet and the collagen boundary. Both cumulative and recent-history activation levels were considered for each platelet. The cumulative internal activation state, , of the i-th platelet at time t was defined as the accumulated integral calcium concentration above the basal level:

 

₀t

   



i t Cai Ca baseline d

 



  , (3-2)

where

 

100

baseline

Ca  nM [60]. The recent-history activation level was defined as the accumulated calcium level between the current time t and previous time t t ( t 30sec ):

   





, t t i _{t t} Cai Ca _baseline d _   

_

 . (3-3)

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The recent-history activation metric allows platelet integrins to return to a resting state (nonadhesive) if calcium has returned to baseline for 30 sec. One of the most widely used functional forms, the Hill functions were introduced to normalize, between _min = 0.001 and 1, the cumulative and 30-sec recent history activation states:

 

_min



_min



_, 50 1 , , n i i n n i i t i i F                , (3-4)

where represents the base level of activation, n represents the sharpness of the Hill function, and was the critical level for 50% activation. An overall rate of attachment of the i-th platelet to the collagen surface was defined by:

 

_{ }

,

collagen collagen

att katt F  F t i

  . (3-5)

The attachment rate between the i-th and j-th platelet, via fibrinogen depending on both the cumulative and transient calcium level of two particles was modelled as:

 

 

 

,

 

,

fibrinogen fibrinogen

att katt F i F j F t i F t j

  . (3-6)

The detachment rate between platelet and the reactive surface was modelled using the Bell exponential [61] to describe the shear-dependent breakage of receptor-ligand bonds,

 

1

 

1

det det , exp

collagen collagen i i t i c k F  F           _{ }  , (3-7)

where _i was the local shear rate around the i-th platelet and _c was the characteristic shear rate required to initiate bond breakage. The detachment rate between the i-th and j- th platelet was given similarly by

 

 

1

 

 

1

det det , , exp

2 i j fibrinogen fibrinogen i j t i t j c k F  F  F  F              _{ }   . (3-8)

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In the current model, platelets could only dissociate as singlets from the thrombus and fracture of large chunks of platelets was not considered due to the complexity of stress propagation within the random aggregated clot. Platelets displayed no bulk aggregation as they entered the concentration boundary layer due to short exposure times and rarity of collisions in the bulk flow. Therefore, aggregation for free-flowing platelets was not considered.

At each time step, a specific event k with rate _k was chosen from the rate catalog with probability  _k / _total. A specific time step   ln

 

u /_total was also chosen by drawing a random number u from the unit interval (0, 1]. To update the system clock, the next reaction method was used where the presumptive wait times were calculated for each time step, which include the following steps:

1. Initialize particle occupancy based on the radial platelet distribution and density. 2. Calculate all the rate events: motion events, binding/unbinding of fibrinogen,

collagen if applicable.

3. Generate random wait times based on the rate information.

4. An event with the shortest wait time is executed. Reallocate the particle or update the binding/unbinding kinetics for all particles under influence.

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In document Multiscale Modelling Of Platelet Aggregation (Page 63-67)