Supporting Information

1. Model Equations and Boundary Conditions

Below are the general equations used to describe the coagulation cascade in the presence of blood flow. Fluid flow is assumed to be at steady-state; whereas mass transport is spatially and temporally variable.

Fluid

Navier-Stokes

Where ρ is the fluid density, µ is the fluid viscosity, u is the fluid velocity vector, and P is pressure.

Inlet

Where uin is the inlet velocity, is the specified shear rate, H is the vessel height, and y is the position perpendicular to the direction of flow.

Outlet

Surface Boundary Condition

Fluid Species

Convection-Diffusion

Where ci is the concentration of species i, Ri are the volume reactions involving species i, D is the diffusivity of species i, and Ni is the volumetric flux of species i.

Inlet

Outlet

Surface Boundary Condition:

Where Ni,s is the surface flux of species i and n is the vector normal to the surface.

2. Reactions

Listed below are all the chemical reactions included in this model. They are grouped by reaction type: MichaelisMenten, association/dissociation, and adsorption/desorption. The rates used for each reaction are provided in tabular form in section 3, Parameter Values.

Activated coagulation factors are described as that factor number (in roman numerals) followed by a lowercase ‘a’. For instance that activated form of factor X is Xa. Chemical species that are bound to a surface as designated by a subscript ‘s’, such as TFs to indicate surface bound tissue factor. Furthermore, in order to differentiate between un-activated factors (factors which have yet to be activated) and inactivated factors (activated factors that have be de-activated), a superscript ‘in’ has been added to describe inactivated factors. These shorthand designations can be combined so that Xas,in represents surface-bound, activated factor X that has been de-activated.

Besides numbered coagulation factors, other abbreviations are: TF (tissue factor), Pro (prothrombinase), Ten (intrinsic tenase), PC (protein C), TM (thrombomodulin), APC (activated protein C), ATIII (antithrombin III), TFPI (tissue factor protein inhibitor), H (heparin of any length), and Pi(the density of free surface sites specific to species i).

MichaelisMenten Reactions

MM1:

MM2:

MM3:

MM4:

MM5:

MM6:

MM7:

MM8:

MM9:

MM10:

MM11:

Association/Dissociation Reactions

R1:

R2:

R3:

R4:

R5:

R6:

R7:

R8:

R9:

R10:

R11:

Adsorption/Desorption Reactions

A5:

A8:

A9:

A10:

3. Parameter Values

This section contains 3 tables with kinetic rate constants, material properties, and initial chemical concentrations used in the formation of the numerical model.

Table 1 contains kinetic rate constants for the reactions described in section 2 (Reactions). In the case of fast association reactions where forward rates are too high to be measured experimentally, the rate is assumed to be transport limited and the diffusion limited rate of 100e3 m3/(mol*s) is used. The reverse rate (dissociation) is calculated based on the published equilibrium constant.

Table 3 lists the chemical concentrations used to initialize the numerical model. For numbered coagulation factors the name is abbreviated with an ‘F’ first and then the Roman numeral. For example, factor VIIa is listed as FVIIa. Other abbreviations are: PC (protein C), ATIII (antithrombin III), and TFPI (tissue factor protein inhibitor). Additionally, the total number of species-specific surface-sites is given for factors V, VIII, IX and X.

Table 1. Kinetic Values

Reaction Label / Notes / F/Sa / kab / kdc / kcatd / KMe / Ref.
R1 / Transport limited
Ref gives KD between 0.6-1 nM / S / 100e3 / 0.06 / (1)
R2 / Transport limited / S / 100e3 / 0.01 / (2)
R3 / Transport limited / S / 100e3 / 0.07 / (3)
R4 / Transport limited
Assume as fast as Prothrombinase / S / 100e3 / 0.01 / (2)
R5 / F / 6.8 / --- / (4)
R6 / Assume inactivation of IXa is the same as for Xa / F/S / 2.6 / --- / (4)
R7 / F / 1.4e4 / 6.7e-4 / (5)
R8 / S / 1e4 / --- / (5)
R9 / Transport limited / S / 100e3 / 2.3/
0.5/
0.5f / (4)
R10 / S / 820/
6.3e3/
140e3f / --- / (4)
R11 / S / 11/
1.7e4/
1.2e5f / --- / (4)
MM1 / Taken from Table 1 (10.5mM)
Assume all VIIa is complexed with TF in ref / S / 1.15 / 4.5e-4 / (6)
MM2 / Data in ref was based on Xaconc, altered hereg / S / 26.4 / 1.06e-3 / (7)
MM3 / Assume all TM is complexed in ref / S / 5.58 / 0.7e-3 / (8)
MM4 / Volume to surface rate conversionh
Data taken in the presence of PS
Assume VIIIa rate is same as Va
Assume all Va is bound / S / 3.0e-4i / 1.8e-8j / (9)
MM5 / Several values of similar order are given in this ref / S / 0.34 / 2.43e-4 / (10)
MM6 / kcat come from ((11)) and kcat/KM from ((12)) / F / 0.9 / 1.8e-4 / (11),
(12)
MM7 / Based on moles of bound IXa / S / 39.85 / 1.6e-4 / (13)
MM8 / F / 0.233 / 7.17e-5 / (14)
MM9 / S / 15.2 / 1.2e-3 / (15)
MM10 / S / 0.043 / 1.04e-5 / (14)
MM11 / Ref states that MM11 proceeds at a rate equal to ~30% of MM9 / S / 4.56 / 1.2e-3 / (15)
A5 / Use rates found with PCPS vesicles / S / 5.7e4 / 0.17 / (16)
A8 / Use KD found with vesicles (2.9nM)(17) and match on rate with that of FV / S / 5.7e4 / 0.17 / (17)
A9 / Use KD found with BAOEC (4.9nM)(18) and match on rate with that of FX / S / 2.9e4 / 0.14 / (18)
A10 / Use KD found with BAOEC (21nM)(18) and on rate found with vesicles(16) / S / 2.9e4 / 0.61 / (16, 18)
  1. F for fluid phase, S for surface phase
  2. ka [=] m3/(mol*s)
  3. kd [=] 1/s
  4. kcat [=] 1/s
  5. KM [=] mol/m3
  6. Rates are given for surface bound heparin of lengths 5, 26, and 70 saccharides, respectively
  7. Ref states that 85% of Xa is complexed with Va. kcat used here is the original kcat divided by 0.85
  8. Ref uses PC/PS vesicles to determine volume reaction rate. Since substrate is a surface species kcat and Km need to be converted to surface units. This is done by dividing each constant by the total vesicle surface area (as an estimate) during the experiment. Reactions took place with 10uM 75:25 PCPS vesicles, where the concentration is based on the molecular units of the vesicle. 75:25 PCPS vesicles have an average radius of approximately 163 Å(19) giving a surface area of 3.34e-15 m2/vesicle. MW was determined using a (w/w) conversion factor of 25 (phospholipid/P)(20) =774.344g/mol. The number of vesicles can be estimated using the specific volume of pure PC vesicles (0.98400 ml/gm(21)) to find a vesicle concentration of 4.2e17 vesicles/m3 or 6.97e-10mol vesicles/L. The total surface area is then 1402 m2/m3. Ref provided constants are then divided by this number.
  9. Unit is m/s. Value determined by Vmax in ref and [APC]= 0.44nM
  10. Unit is mol/m2

Table 2. Material Parameters

Description / Notes / Value / Ref
D / Diffusivity / Estimate using blood viscosity and MW range ~34-330
Use for all species / 5e-11 m2/s / (22)
µ / Dynamic viscosity / Estimate around 50s-1 shear rate / 5e-3 kg/m/s / (23)
ρ / Density / 1050 kg/m3 / (24)

Table 3. Initial Concentrations

Description / Notes / Value / Ref
FII / 1400e-6 mol/m3 / (25)
FV / 20 e-6 mol/m3 / (25)
FVII / 10 e-6 mol/m3 / (25)
FVIIa / 0.1 e-6 mol/m3 / (25)
FVIII / 0.7 e-6 mol/m3 / (25)
FIX / 90 e-6 mol/m3 / (25)
FX / 170 e-6 mol/m3 / (25)
PC / 60 e-6 mol/m3 / (25)
ATIII / 2400 e-6 mol/m3 / (25)
TFPI / 2.5 e-6 mol/m3 / (25)
FV surface sites / a / 2.5e-10 mol/m2 / (26)
FVIII surface sites / a / 3.7e-11 mol/m2 / (17)
FIX surface sites / a / 4.2e-11 mol/m2 / (27)
FX surface sites / a / 2.2e-10 mol/m2 / (28)
  1. Literature sources give density in terms of sites/platelet; this is converted to a surface density assuming a platelet surface area of 20 m2 which corresponds to the surface of a 2.5 m sphere. Original data is 3000 sites/platelet (V), 450 (VIII), 500(IX), and 2700 (X).

References

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