Mathematical Modeling of Immune-Mediated Processes in Coagulation and Anticoagulation Therapy

Abstract

Thousands of people each year succumb to complications from deep vein thrombosis (DVT). DVT occurs when blood flow in the veins is blocked by a clot. Individuals with DVT are at increased risk of experiencing pulmonary embolism (PE), in which small pieces of the clot break off and travel to the lungs. PE can lead to lung damage and even death. Mechanisms implicated in DVT may comprise an imbalance in the body, poor circulation, inflammation, or an immune response. Given the complex interplay of mediating factors in DVT, it is important to understand the role that each plays in thrombus and embolus formation. In this work, we develop and implement a mathematical model of venous clot formation and dissolution by describing interactions between inflammation, blood cells, and various chemical factors. We then use the model to identify factors essential to embolus formation and discuss implications for clinical treatment.

Appendices

Appendix

Model Variables

The model consists of interactions between 86 state variables, many of them taken from [4]. A schematic of these interactions is given in Fig. 7. We gather here descriptions of all of the state variables for ease of reference; for more details, refer to [4].

The largest group of state variables consists of clotting factors (molecules) and complexes that are involved in the coagulation cascade. These factors are often denoted by roman numerals, e.g., “factor IX,” or “factor II.” Clotting factors can be either inactive, or active, typically denoted by an “a,” as in factor IX (inactive) and factor IXa (active). Here, we use Z _i to denote the inactive clotting factor i (zymogen) and E _i to denote the active factor (enzyme). Concentrations are shown with square brackets, or, to simplify the notation, concentrations of factors are denoted by lowercase letters. So [Z ₉] = z ₉ and [E ₉] = e ₉ represent the concentrations of factor IX and factor IXa, respectively. Factors that are bound to platelets are denoted with a superscript m, as in $e_9^m$ or $z_9^m$. Certain enzymes can bind to specific sites on platelets. These specifically bound enzymes are denoted with superscripts m ∗ or h, as in $e_{11}^{m*}$, $e_{11}^{h}$ or, when both binding sites are used, $e_{11}^{h,m*}$.

In addition to these clotting factors, we have the following variables, some of which have already been listed in Sect. 3.

F :

Fibrin

F _g :

Fibrinogen (fibrin precursor)

P :

Plasmin

P _g :

Plasminogen (plasmin precursor)

TFPI :

Tissue factor pathway inhibitor

APC :

Activated protein C

TM :

Thrombomodulin

W :

Vessel wall (endothelial) activation level, with values between 0 (unactivated) and 1 (fully activated)

[NET] :

Concentration of leukocyte-derived NETs

TF :

Average concentration of active (decrypted) tissue factor expressed per activated monocyte

$L_u^m$, $L_a^p$, $L_a^e$ :

Concentration of unactivated and mobile, activated and bound to platelets, and activated and endothelium-bound leukocytes, respectively

$P_a^b$, $P_a^l$, $P_a^e$,$P_a^m$ :

Concentration of activated platelets bound to other platelets, leukocytes, endothelium, or unbound (mobile), respectively

Complexes are shown as molecules joined by a colon, as in TM : E ₂ or TM : E ₂ : APC. Some complexes have special names:

TEN = VIII :IXa = platelet-bound tenase

PRO = Va:Xa = prothrombinase.

Model Equations

Here we give the complete list of model equations incorporating immune-mediated mechanisms to coagulation in the veins. Following [6], we eliminate the endothelium-dependent kinetic equations from the model in [4], as the current model does not distinguish between sub- and intact endothelium. As such, the following variables are omitted, with the necessary reactions appearing in the modified equations: $e_2^{ec},~APC^{ec},~e_9^{ec},~e_{10}^{ec}$. Specific changes from the previous model [4] which models arterial coagulation (in vivo framework) are highlighted in purple.

(1)

(2)

(3)

(4)

(5)

(6)

$$\displaystyle \begin{aligned} &{\genfrac{}{}{}{}{\mathrm{d}{P_u^m}}{\mathrm{d}{t}}} = k_{\mathrm{flow}}^p (P^{\mathrm{up}}-P_u^m) + k^-_{\text{adh}} P_a^e \end{aligned} $$

(7)

(8)

(9)

(10)

(11)

$$\displaystyle \begin{aligned} &\genfrac{}{}{}{}{\mathrm{d}{z_{9}}}{{\mathrm{d}}{t}} = -k_{9}^{\mathrm{on}}z_{9}p_{9}^{\mathrm{avail}} +k_{9}^{\mathrm{off}}z_{9}^{m} +k_{\mathrm{flow}}(z_9^{\mathrm{up}}-z_9) -k_{z_9:e_7^m}^+ z_9e_7^m + k_{z_9:e_7^m}^- [Z_9:E_7^m] \end{aligned} $$

(12)

(13)

(14)

(15)

(16)

(17)

(18)

(19)

$$\displaystyle \begin{aligned} &\genfrac{}{}{}{}{\mathrm{d}{}}{\mathrm{d}{t}}z_7 = -k_7^{\mathrm{on}} z_7 [TF]^{\mathrm{avail}} + k_7^{\mathrm{off}}z_7^m - k_{z_7:e_2}^+ z_7e_2 + k_{z_7:e_2}^- [Z_7:E_2] - k_{z_7:e_{10}}^+ z_7 e_{10} {} \end{aligned} $$

(20)

(21)

(22)

(23)

(24)

$$\displaystyle \begin{aligned} &\genfrac{}{}{}{}{\mathrm{d}{}}{\mathrm{d}{t}}e_{10} = -k_{10}^{\mathrm{on}}e_{10}p_{10}^{\mathrm{avail}} +k_{10}^{\mathrm{off}}e_{10}^{m} -k_{z_{10}:e_7^m}^{\mathrm{cat}}[Z_{10}:E_7^m] \end{aligned} $$

(25)

(26)

(27)

(28)

$$\displaystyle \begin{aligned} &\genfrac{}{}{}{}{\mathrm{d}{}}{\mathrm{d}{t}}e_5 = -k_{5}^{\mathrm{on}}e_{5}p_{5}^{\mathrm{avail}} +k_{5}^{\mathrm{off}}e_{5}^{m} +k_{z_5:e_2}^{\mathrm{cat}} [Z_5:E_2] + k_{e_5:APC}^- [APC:E_5] \end{aligned} $$

(29)

(30)

(31)

(32)

$$\displaystyle \begin{aligned} &\genfrac{}{}{}{}{\mathrm{d}{}}{\mathrm{d}{t}}e_8 = -k_{8}^{\mathrm{on}}e_{8}p_{8}^{\mathrm{avail}} +k_{8}^{\mathrm{off}}e_{8}^{m}+k_{\mathrm{flow}}(e_8^{\mathrm{up}}-e_8) +k_{z8:e2}^{\mathrm{cat}}[Z_8:E_2] - k_{8}^{deg} e_8 \end{aligned} $$

(33)

(34)

(35)

(36)

(37)

(38)

$$\displaystyle \begin{aligned} &\genfrac{}{}{}{}{\mathrm{d}{}}{\mathrm{d}{t}}z_2^m = k_2^{\mathrm{on}}p_2^{\mathrm{avail}}z_2-k_2^{\mathrm{off}}z_2^m -k_{z_2^m:PRO}^{+}[PRO]z_2^m+k_{z_2^m:PRO}^{-}[Z_2^m:PRO] \end{aligned} $$

(39)

$$\displaystyle \begin{aligned} &\genfrac{}{}{}{}{\mathrm{d}{}}{\mathrm{d}{t}}e_2^m = k_2^{\mathrm{on}}p_{2s}^{\mathrm{avail}}e_2-k_2^{\mathrm{off}}e_2^m \end{aligned} $$

(40)

(41)

(42)

(43)

(44)

(45)

(46)

$$\displaystyle \begin{aligned} &\genfrac{}{}{}{}{\mathrm{d}{}}{\mathrm{d}{t}}[Z_7:E_2] =k_{z_7:e_2}^+ z_7e_2 - (k_{z_7:e_2}^{\mathrm{cat}} + k_{z_7:e_2}^-) [Z_7:E_2]-k_{\mathrm{flow}}[Z_7:E_2] \end{aligned} $$

(47)

$$\displaystyle \begin{aligned} &\genfrac{}{}{}{}{\mathrm{d}{}}{\mathrm{d}{t}}[Z_7:E_{10}] = k_{z_7:e_{10}}^+ z_7e_{10} - (k_{z_7:e_{10}}^{\mathrm{cat}} + k_{z_7:e_{10}}^-) [Z_7:E_{10}]-k_{\mathrm{flow}}[Z_7:E_{10}] \end{aligned} $$

(48)

$$\displaystyle \begin{aligned} &\genfrac{}{}{}{}{\mathrm{d}{}}{\mathrm{d}{t}}[Z_7^m:E_{10}] = k_{z_7^m:e_{10}}^+ z_7^me_{10} - (k_{z_7^m:e_{10}}^{\mathrm{cat}} + k_{z_7^m:e_{10}}^-) [Z_7^m:E_{10}] \end{aligned} $$

(49)

$$\displaystyle \begin{aligned} &\genfrac{}{}{}{}{\mathrm{d}{}}{\mathrm{d}{t}}[Z_7^m:E_{2}] = k_{z_7:e_{2}}^+ z_7^me_{2} - (k_{z_7:e_{2}}^{\mathrm{cat}} + k_{z_7:e_{2}}^-) [Z_7^m:E_{2}] \end{aligned} $$

(50)

$$\displaystyle \begin{aligned} &\genfrac{}{}{}{}{\mathrm{d}{}}{\mathrm{d}{t}}[Z_{10}:E_{7}^m] = k_{z_{10}:e_{7}^m}^+ z_{10}e_7^m - (k_{z_{10}:e_{7}^m}^{\mathrm{cat}} + k_{z_{10}:e_{7}^m}^-) [Z_{10}:E_{7}^m] \end{aligned} $$

(51)

$$\displaystyle \begin{aligned} &\genfrac{}{}{}{}{\mathrm{d}{}}{\mathrm{d}{t}}[Z_{10}^m:TEN] =k_{z_{10}^m:TEN}^+[TEN]z_{10}^m-(k_{z_{10}^m:TEN}^{\mathrm{cat}}+k_{z_{10}^m:TEN}^-)[Z_{10}^m:TEN] \end{aligned} $$

(52)

$$\displaystyle \begin{aligned} &\genfrac{}{}{}{}{\mathrm{d}{}}{\mathrm{d}{t}}[Z_5:E_2] =k_{z_5:e_2}^+ z_5e_2 - (k_{z_5:e_2}^{\mathrm{cat}} + k_{z_5:e_2}^-)[Z_5:E_2]-k_{\mathrm{flow}}[Z_5:E_2] \end{aligned} $$

(53)

$$\displaystyle \begin{aligned} &\genfrac{}{}{}{}{\mathrm{d}{}}{\mathrm{d}{t}}[Z_5^m:E_{10}^m] =k_{z_5^m:e_{10}^m}^+ z_5^me_{10}^m - (k_{z_5^m:e_{10}^m}^{\mathrm{cat}} + k_{z_5^m:e_{10}^m}^-)[Z_5^m:E_{10}^m] \end{aligned} $$

(54)

$$\displaystyle \begin{aligned} &\genfrac{}{}{}{}{\mathrm{d}{}}{\mathrm{d}{t}}[Z_5^m:E_2^m] =k_{z_5:e_2^m}^+ z_5^me_2^m - (k_{z_5:e_2^m}^{\mathrm{cat}} + k_{z_5:e_2^m}^-)[Z_5^m:E_2^m] \end{aligned} $$

(55)

$$\displaystyle \begin{aligned} &\genfrac{}{}{}{}{\mathrm{d}{}}{\mathrm{d}{t}}[Z_8^m:E_{10}^m] =k_{z_8^m:e_{10}^m}^+ z_8^me_{10}^m - (k_{z_8^m:e_{10}^m}^{\mathrm{cat}} + k_{z_8^m:e_{10}^m}^-)[Z_8^m:E_{10}^m] \end{aligned} $$

(56)

$$\displaystyle \begin{aligned} &\genfrac{}{}{}{}{\mathrm{d}{}}{\mathrm{d}{t}}[Z_8^m:E_2^m] =k_{z_8^m:e_2^m}^+ z_8^me_2^m - (k_{z_8^m:e_2^m}^{\mathrm{cat}} + k_{z_8^m:e_2^m}^-)[Z_8^m:E_2^m] \end{aligned} $$

(57)

$$\displaystyle \begin{aligned} &\genfrac{}{}{}{}{\mathrm{d}{}}{\mathrm{d}{t}}[Z_8:E_2] =k_{z_8:e_2}^+ z_8e_2 - (k_{z_8:e_2}^{\mathrm{cat}} + k_{z_8:e_2}^-)[Z_8:E_2]-k_{\mathrm{flow}}[Z_8:E_2] \end{aligned} $$

(58)

$$\displaystyle \begin{aligned} &\genfrac{}{}{}{}{\mathrm{d}{}}{\mathrm{d}{t}}[APC:E_8^m] =k_{e_8^m:APC}^+[APC]e_8^m-(k_{e_8^m:APC}^{\mathrm{cat}}+k_{e_8^m:APC}^-)[APC:E_8^m] \end{aligned} $$

(59)

$$\displaystyle \begin{aligned} &\genfrac{}{}{}{}{\mathrm{d}{}}{\mathrm{d}{t}}[Z_9:E_7^m] = k_{z_9:e_7^m}^+z_9e_7^m - (k_{z_9:e_7^m}^{\mathrm{cat}} + k_{z_9:e_7^m}^-)[Z_9:E_7^m] \end{aligned} $$

(60)

$$\displaystyle \begin{aligned} &\genfrac{}{}{}{}{\mathrm{d}{}}{\mathrm{d}{t}}[Z_2^m:PRO] =k_{z_2^m:PRO}^+z_2^m[PRO]-(k_{z_2^m:PRO}^{\mathrm{cat}}+k_{z_2^m:PRO}^-)[Z_2^m:PRO] \end{aligned} $$

(61)

$$\displaystyle \begin{aligned} &\genfrac{}{}{}{}{\mathrm{d}{}}{\mathrm{d}{t}}[APC:E_5^m] =k_{e_5^m:APC}^+[APC]e_5^m-(k_{e_5^m:APC}^{\mathrm{cat}}+k_{e_5^m:APC}^-)[APC:E_5^m] \end{aligned} $$

(62)

$$\displaystyle \begin{aligned} &{\genfrac{}{}{}{}{\mathrm{d}{}}{\mathrm{d}{t}}[Z_7:E_9] } {=k_{z_7:e_9}^+z_7e_9 - (k_{z_7:e_9}^{\mathrm{cat}} + k_{z_7:e_9}^-)[Z_7:E_9] } \end{aligned} $$

(63)

$$\displaystyle \begin{aligned} &\genfrac{}{}{}{}{\mathrm{d}{}}{\mathrm{d}{t}}e_9^{m*} =k_9^{\mathrm{on}}p_{91}^{\mathrm{avail}}e_9 -k_9^{\mathrm{off}}e_9^{m*}+k_{e_8^m:e_9^m}^-[TEN^*]-k_{e_8^m:e_9^m}^+e_8^me_9^{m*} \end{aligned} $$

(64)

$$\displaystyle \begin{aligned} &\genfrac{}{}{}{}{\mathrm{d}{}}{\mathrm{d}{t}}[TEN^*] =-k_{e_8^m:e_9^m}^-[TEN^*]+k_{e_8^m:e_9^m}^+e_8^me_9^{m*} \end{aligned} $$

(65)

(66)

$$\displaystyle \begin{aligned} &\genfrac{}{}{}{}{\mathrm{d}{}}{\mathrm{d}{t}}[APC:E_5] = k_{e_5:APC}^+e_5[APC]-(k_{e_5:APC}^{\mathrm{cat}} + k_{e_5:APC}^-)[APC:E_5] \end{aligned} $$

(67)

$$\displaystyle \begin{aligned} &\genfrac{}{}{}{}{\mathrm{d}{}}{\mathrm{d}{t}}[APC:E_8] = k_{e_8:APC}^+e_8[APC]-(k_{e_8:APC}^{\mathrm{cat}} + k_{e_8:APC}^-)[APC:E_8] \end{aligned} $$

(68)

$$\displaystyle \begin{aligned} &\genfrac{}{}{}{}{\mathrm{d}{}}{\mathrm{d}{t}}z_{11} = k_{\mathrm{flow}}(z_{11}^{\mathrm{up}}-z_{11}) -k_{z_{11}}^{on}z_{11}p_{11}^{\mathrm{avail}}+k_{z_{11}}^{\mathrm{off}}z_{11}^{m} -k_{z_{11}:e_2}^+z_{11}e_2 + k_{z_{11}:e_2}^- [Z_{11}:E_2] \end{aligned} $$

(69)

$$\displaystyle \begin{aligned} &\genfrac{}{}{}{}{\mathrm{d}{}}{\mathrm{d}{t}}e_{11} = k_{\mathrm{flow}}(e_{11}^{\mathrm{up}}-e_{11}) -k_{e_{11}}^{\mathrm{on}}e_{11}p_{111}^{\mathrm{avail}}+k_{e_{11}}^{\mathrm{off}}e_{11}^{m*} \end{aligned} $$

(70)

(71)

$$\displaystyle \begin{aligned} &\genfrac{}{}{}{}{\mathrm{d}{}}{\mathrm{d}{t}}e_{11}^{m*} = k_{e_{11}}^{\mathrm{on}}e_{11}p_{111}^{\mathrm{avail}}-k_{e_{11}}^{\mathrm{off}}e_{11}^{m*} \end{aligned} $$

(72)

(73)

$$\displaystyle \begin{aligned} &\genfrac{}{}{}{}{\mathrm{d}{}}{\mathrm{d}{t}}[Z_{9}^m:E_{11}^{m*}] =k_{z_9:e_{11}}^+z_9^me_{11}^{m*}-(k_{z_9:e_{11}}^-+k_{z_9:e_{11}}^{\mathrm{cat}})[Z_9^m:E_{11}^{m*}] \end{aligned} $$

(74)

$$\displaystyle \begin{aligned} &\genfrac{}{}{}{}{\mathrm{d}{}}{\mathrm{d}{t}}[Z_{11}:E_2] = k_{\mathrm{flow}}([Z_{11}:E_2] ^{\mathrm{up}}-[Z_{11}:E_2]) + k_{z_{11}:e_2}^+ z_{11}e_2 - (k_{z_{11}:e_2}^- + k_{z_{11}:e_2}^{\mathrm{cat}})[Z_{11}:E_2] \end{aligned} $$

(75)

$$\displaystyle \begin{aligned} &\genfrac{}{}{}{}{\mathrm{d}{}}{\mathrm{d}{t}}[Z_9:E_{11}] = k_{\mathrm{flow}}([Z_9:E_{11}]^{\mathrm{up}}-[Z_9:E_{11}])+k_{z_9:e_{11}}^+ z_9e_{11} - (k_{z_9:e_{11}}^-+k_{z_9:e_{11}}^{\mathrm{cat}})[Z_9:E_{11}] \end{aligned} $$

(76)

$$\displaystyle \begin{aligned} &\genfrac{}{}{}{}{\mathrm{d}{}}{\mathrm{d}{t}}e_{11}^h = k_{\mathrm{flow}}(e_{11}^{h,up}-e_{11}^{h}) -k_{e_{11}}^{\mathrm{on}}e_{11}^hp_{111}^{\mathrm{avail}} +k_{e_{11}}^{\mathrm{off}}e_{11}^{h,m*} -k_{z_{11}}^{\mathrm{on}}e_{11}^hp_{11}^{\mathrm{avail}}+k_{z_{11}}^{\mathrm{off}}e_{11}^{h,m} \end{aligned} $$

(77)

(78)

(79)

$$\displaystyle \begin{aligned} &\genfrac{}{}{}{}{\mathrm{d}{}}{\mathrm{d}{t}}[Z_9:E_{11}^h] = k_{\mathrm{flow}}([Z_9:E_{11}^{h}]^{\mathrm{up}} - [Z_9:E_{11}^h] )+ k_{z_9:e_{11}}^+ z_9e_{11}^h - (k_{z_9:e_{11}}^- + k_{z_9:e_{11}}^{\mathrm{cat}})[Z_9:E_{11}^h] \end{aligned} $$

(80)

$$\displaystyle \begin{aligned} &\genfrac{}{}{}{}{\mathrm{d}{}}{\mathrm{d}{t}}[Z_9^m:E_{11}^{h,m}] = k_{z_9:e_{11}}^+z_9^me_{11}^{h,m} - (k_{z_9:e_{11}}^-+k_{z_9:e_{11}}^{\mathrm{cat}})[Z_9^m:E_{11}^{h,m}] \end{aligned} $$

(81)

$$\displaystyle \begin{aligned} &\genfrac{}{}{}{}{\mathrm{d}{}}{\mathrm{d}{t}}[E_{11}^h:E_2] =k_{\mathrm{flow}}([E_{11}^h:E_2]^{\mathrm{up}} - [E_{11}^h:E_2]) + k_{z_{11}:e_2}^+ e_{11}^h e_{2} - (k_{z_{11}:e_2}^- + k_{z_{11}:e_2}^{\mathrm{cat}})[E_{11}^h:E_{2}] \end{aligned} $$

(82)

$$\displaystyle \begin{aligned} &\genfrac{}{}{}{}{\mathrm{d}{}}{{\mathrm{d}}{t}}[E_{11}^{h,m*}:E_2^m] = k_{e_{11}^{h,m*}e_2^m}^+e_{11}^{h,m*}e_2^m - (k_{e_{11}^{h,m*}e_2^m}^-+k_{e_{11}^{h,m*}e_2^m}^{\mathrm{cat}})[E_{11}^{h,m*}:E_2^m] {} \end{aligned} $$

(83)

$$\displaystyle \begin{aligned} &\genfrac{}{}{}{}{\mathrm{d}{}}{\mathrm{d}{t}} [Fg] = -\frac{v_{14}e_2[Fg]}{k_{14a}+{[Fg]}} -\frac{v_{15}[P][Fg]}{k_{15a}+[P]}+d_{Fg}(1-[Fg]) {} \end{aligned} $$

(84)

(85)

$$\displaystyle \begin{aligned} &\genfrac{}{}{}{}{\mathrm{d}{}}{\mathrm{d}{t}} [Pg] = -\frac{v_{21}e_2[P_g]}{k_{21a}+e_2} -\frac{v_{23}{[APC][VKH_2]}[Pg]}{(k_{23a}([APC]+c_{37a})+[VKH_2][APC])} {} \end{aligned} $$

(86)

(87)

Platelet Variables

Activated platelet dynamics are calculated as in Fogelson et al. [10]. At the beginning of each step, we store the total active platelet concentration , and the total active leukocyte population: $L_a^i = L_a^p + L_a^e \cdot ls/leuk$. Then the change in platelet concentration during a given time step is calculated as

The current height of the platelet layer, nhc, is updated using

and, from this, the volume of the chemical boundary layer, v _ch, and the platelet volume, v _pl, are calculated:

These volumes are then used to adjust the platelet and chemical concentrations. Then platelet binding site availabilities are updated as follows:

$$\displaystyle \begin{aligned} p_2 &= p_2 + d_{PL}(p\cdot np_2/s_2) \end{aligned} $$

(88)

$$\displaystyle \begin{aligned} p_5 &= p_5 + d_{PL}(p\cdot np_5/s_5) \end{aligned} $$

(89)

$$\displaystyle \begin{aligned} p_8 &= p_8 + d_{PL}(p\cdot np_8/s_8) \end{aligned} $$

(90)

$$\displaystyle \begin{aligned} p_9 &= p_9 + d_{PL}(p\cdot np_9/s_9) \end{aligned} $$

(91)

$$\displaystyle \begin{aligned} p_{10} &= p_{10} + d_{PL}(p\cdot np_{10}/s_{10}) \end{aligned} $$

(92)

$$\displaystyle \begin{aligned} p_{11} &= p_{11} + d_{PL}(p\cdot np_{11}/s_{11}) \end{aligned} $$

(93)

$$\displaystyle \begin{aligned} p_{111} &= p_{111} + d_{PL}(p\cdot np_{111}/s_{111}). \end{aligned} $$

(94)

Additionally, the binding site availabilities for surface binding interactions are defined as follows (each step):

$$\displaystyle \begin{aligned} p^{\mathrm{avail}}_{PLAS}&= p_{PLAS} - [P_a^e] \end{aligned} $$

(95)

$$\displaystyle \begin{aligned} p_{10}^{\mathrm{avail}} &=p_{10}-z_{10}^m-e_{10}^m-[Z_{10}^m:TEN]-[Z_5^m:E_{10}^m] \end{aligned} $$

(96)

(97)

(98)

(99)

$$\displaystyle \begin{aligned} p_{2}^{\mathrm{avail}} &=p_2-z_2^m-[Z_2^m:PRO] -[Z_{11}^m:E_2^m] - [E_{11}^{h,m*}:E_2^m] \end{aligned} $$

(100)

$$\displaystyle \begin{aligned} p_{2s}^{\mathrm{avail}} &=p_{2s}-e_2^m-[Z_5^m:E_2^m]-[Z_8^m:E_2^m]-[Z_{11}^m:E_2^m] - [E_{11}^{h,m*}:E_2^m] \end{aligned} $$

(101)

$$\displaystyle \begin{aligned} p_{91}^{\mathrm{avail}} &=p_{91}-e_9^{m*}-([TEN^*]+[Z_{10}^m:TEN^*]) \end{aligned} $$

(102)

$$\displaystyle \begin{aligned} p_{11}^{\mathrm{avail}} &= p_{11}-z_{11}^m-e_{11}^{h,m*}-[Z_9^m:E_{11}^{h,m*}] -[Z_{11}^m:E_2^m] \end{aligned} $$

(103)

$$\displaystyle \begin{aligned} p_{111}^{\mathrm{avail}} &=p_{111}-e_{11}^{h,m*}-e_{11}^{m*}-[Z_9^m:E_{11}^{m*}]-[E_{11}^{h,m*}:E_2^m] \end{aligned} $$

(104)

$$\displaystyle \begin{aligned}{}[TM]^{\mathrm{avail}} &= [TM]-[TM:E_2]-[TM:E_2:APC]. \end{aligned} $$

(105)

Complete Parameter List

New Model Parameters

See Table 1.

Table 1 List of parameters used in the inflammation-based modification in the present work

Original Model Parameters

See Tables 2, 3, and 4.

Table 2 Modified list of reaction parameters for the clotting pathway except for vitamin K components

Table 3 Modified list of surface binding (on/off) parameters

Table 4 Modified list of platelet, surface and volume scalings, and flow-related parameters

Initial Conditions

See Table 5.

Table 5 Modified list of concentration scalings and nondimensional initial values used for all variables