A mathematical model for in vitro coagulation of blood: role of platelet count and inhibition

Abstract

A mechanistic model including the role of platelets is proposed for clot formation and growth in plasma in vitro. Initiation of clot formation is by the addition of tissue factor, and initiation via the intrinsic pathway is neglected. Activation of zymogens follows the extrinsic pathway cascade and reactions on platelet membranes are included. Platelet activation occurs due to thrombin and also due to other activated platelets. Inhibition of the active clotting factors is by ATIII and TFPI, whereas inhibition due to APC is not relevant in the conditions modeled. The model predictions matched existing data for thrombin production in synthetic plasma. The model predicts that inhibition of platelet-driven activation of platelets has a major effect on concentration of activated platelets in PRP, normal plasma and PPP. Inhibition of platelet activation by (other activated) platelets significantly delays thrombin production in PRP and normal plasma as compared to that by thrombin. Further, sensitivity analysis shows that the model is most sensitive to the activation of platelet membrane-bound factor X by the intrinsic tenase complex.

Appendices

Appendix 1: Model reactions

$$\begin{aligned} G_\mathrm{TF}=&-k^{+}_{T7}[\mathrm{TF}][\mathrm{VII}]+k^{-}_{T7}[\mathrm{TF:VII}] \nonumber \\&-\, k^{+}_{T7a}[\mathrm{TF}][\mathrm{VIIa}]+k^{-}_{T7a}[\mathrm{TF:VIIa}]. \end{aligned}$$

(1)

$$\begin{aligned} G_\mathrm{VII}=&-k^{+}_{T7}[\mathrm{TF}][\mathrm{VII}]+k^{-}_{T7}[\mathrm{TF:VII}]-k_{TF7}[\mathrm{TF:VIIa}][\mathrm{VII}] \nonumber \\&-\, k_{10,7}[\mathrm{Xa}][\mathrm{VII}]-k_{2,7}[\mathrm{IIa}][\mathrm{VII}]. \end{aligned}$$

(2)

$$\begin{aligned} G_\mathrm{TF:VII}=&k^{+}_{T7}[\mathrm{TF}][\mathrm{VII}]-k^{-}_{T7}[\mathrm{VII}^{m}]. \end{aligned}$$

(3)

$$\begin{aligned} G_\mathrm{VIIa}=\,&-k^{+}_{T7a}[\mathrm{TF}][\mathrm{VIIa}]+k^{-}_{T7a}[\mathrm{TF:VIIa}]+k_{TF7}[\mathrm{TF:VIIa}][\mathrm{VII}] \nonumber \\&+\,k_{10,7}[\mathrm{Xa}][\mathrm{VII}]+k_{2,7}[\mathrm{IIa}][\mathrm{VII}]. \end{aligned}$$

(4)

$$\begin{aligned} G_\mathrm{TF:VIIa}=&k^{+}_{T7a}[\mathrm{TF}][\mathrm{VIIa}]-k^{-}_{T7a}[\mathrm{TF:VIIa}]-h^{TP}_{7}[\mathrm{TFPI:Xa}][\mathrm{TF:VIIa}]\nonumber \\&-\,h^{AT}_{7}[\mathrm{ATIII}][\mathrm{TF:VIIa}]. \end{aligned}$$

(5)

$$\begin{aligned} G_\mathrm{IX}=&-\dfrac{k_{9}[\mathrm{TF:VIIa}][\mathrm{IX}]}{K_{9M}+[\mathrm{IX}]}-k^{+}_{9}N_{9}[\mathrm{AP}][\mathrm{IX}]+k^{-}_{9}[\mathrm{IX}^{m}].\end{aligned}$$

(6)

$$\begin{aligned} G_\mathrm{IXa}=&\dfrac{k_{9}[\mathrm{TF:VIIa}][\mathrm{IX}]}{K_{9M}+[\mathrm{IX}]}-k^{+}_{9}N_{9}[\mathrm{AP}][\mathrm{IXa}]\nonumber \\&+k^{-}_{9}[\mathrm{IXa}^{m}] -h_{9}[\mathrm{IXa}][\mathrm{ATIII}]. \end{aligned}$$

(7)

$$\begin{aligned} G_\mathrm{IX^{m}}=&k^{+}_{9}N_{9}[\mathrm{AP}][\mathrm{IX}]-k^{-}_{9}[\mathrm{IX}^{m}]. \end{aligned}$$

(8)

$$\begin{aligned} G_\mathrm{IXa^{m}}=&-k^{+}_{TEN}[\mathrm{VIIIa}^{m}][\mathrm{IXa}^{m}]+k^{-}_{TEN}[\mathrm{VIIIa}^{m}:\mathrm{IXa}^{m}] \nonumber \\&+k^{+}_{9}N_{9}[\mathrm{AP}][\mathrm{IXa}]-k^{-}_{9}[\mathrm{IXa}^{m}]. \end{aligned}$$

(9)

$$\begin{aligned} G_\mathrm{X}=&-\dfrac{k_{7,10}[\mathrm{TF:VIIa}][\mathrm{X}]}{K_{7,10M}+[\mathrm{X}]}-k^{+}_{10}N_{10}[\mathrm{AP}][\mathrm{X}]+k^{-}_{10}[\mathrm{X}^{m}]. \end{aligned}$$

(10)

$$\begin{aligned} G_\mathrm{Xa}=&\dfrac{k_{7,10}[\mathrm{TF:VIIa}][\mathrm{X}]}{K_{7,10M}+[\mathrm{X}]}-h^{TP+}_{10}[\mathrm{TFPI}][\mathrm{Xa}]+h^{TP-}_{10}[\mathrm{Xa:TFPI}] \nonumber \\&-\, h^{AT}_{10}[\mathrm{ATIII}][\mathrm{Xa}]-k^{+}_{10}N_{10}[\mathrm{AP}][\mathrm{Xa}]+k^{-}_{10}[\mathrm{Xa^{m}}]. \end{aligned}$$

(11)

$$\begin{aligned} G_{\mathrm{X}^{m}}=&-\dfrac{k_{10}[\mathrm{VIIIa}^{m}:\mathrm{IXa}^{m}][\mathrm{X}^{m}]}{K_{10M}+[\mathrm{X}^{m}]}\nonumber \\&+\,k^{+}_{10}N_{10}[\mathrm{AP}][\mathrm{X}]-k^{-}_{10}[\mathrm{X}^{m}]. \end{aligned}$$

(12)

$$\begin{aligned} G_{\mathrm{Xa}^{m}}=&\dfrac{k_{10}[\mathrm{VIIIa}^{m}:\mathrm{IXa}^{m}][\mathrm{X}^{m}]}{K_{10M}+[\mathrm{X}^{m}]}-k^{+}_\mathrm{PRO}[\mathrm{Va}^{m}][\mathrm{Xa}^{m}] \nonumber \\&+\,k^{-}_\mathrm{PRO}[\mathrm{Va}^{m}:\mathrm{Xa}^{m}]+k^{+}_{10}N_{10}[\mathrm{AP}][\mathrm{Xa}]-k^{-}_{10}[\mathrm{Xa^{m}}]. \end{aligned}$$

(13)

$$\begin{aligned} G_\mathrm{II}=&-k_{2t}[\mathrm{Xa}][\mathrm{II}]-k^{+}_{2}N_{2}[\mathrm{AP}][\mathrm{II}]+k^{-}_{2}[\mathrm{II}^{m}]. \end{aligned}$$

(14)

$$\begin{aligned} G_\mathrm{IIa}=\,&k_{2t}[\mathrm{Xa}][\mathrm{II}]-k^{+}_{2}N_{2}[\mathrm{AP}][\mathrm{IIa}]+k^{-}_{2}[\mathrm{IIa}^{m}]\nonumber \\&-\, h_{2}[\mathrm{ATIII}][\mathrm{IIa}]. \end{aligned}$$

(15)

$$\begin{aligned} G_{\mathrm{II}^{m}}=&-\dfrac{k_{2}[\mathrm{Va}^{m}:\mathrm{Xa}^{m}][\mathrm{II}^{m}]}{K_{2M}+[\mathrm{II}^{m}]}+k^{+}_{2}N_{2}[\mathrm{AP}][\mathrm{II}]-k^{-}_{2}[\mathrm{II}^{m}]. \end{aligned}$$

(16)

$$\begin{aligned} G_{\mathrm{IIa}^{m}}=&\dfrac{k_{2}[\mathrm{Va}^{m}:\mathrm{Xa}^{m}][\mathrm{II}^{m}]}{K_{2M}+[\mathrm{II}^{m}]}+k^{+}_{2}N_{2}[\mathrm{AP}][\mathrm{IIa}]-k^{-}_{2}[\mathrm{IIa}^{m}]. \end{aligned}$$

(17)

$$\begin{aligned} G_\mathrm{PL}=&-kpp[\mathrm{PL}][\mathrm{AP}]-\dfrac{kp2[\mathrm{PL}][\mathrm{IIa}]}{1+[\mathrm{IIa}]}. \end{aligned}$$

(18)

$$\begin{aligned} G_\mathrm{AP}=&kpp[\mathrm{PL}][\mathrm{AP}]+\dfrac{kp2[PL][\mathrm{IIa}]}{1+[\mathrm{IIa}]}. \end{aligned}$$

(19)

$$\begin{aligned} G_\mathrm{VIII}=&-\dfrac{k_{8}[\mathrm{IIa}][\mathrm{VIII}]}{K_{8M}+[\mathrm{VIII}]}-k^{+}_{8}N_{8}[\mathrm{AP}][\mathrm{VIII}]+k^{-}_{8}[\mathrm{VIII}^{m}]. \end{aligned}$$

(20)

$$\begin{aligned} G_\mathrm{VIIIa}=&\dfrac{k_{8}[\mathrm{IIa}][\mathrm{VIII}]}{K_{8M}+[\mathrm{VIII}]}-k^{+}_{8}N_{8}[\mathrm{AP}][\mathrm{VIIIa}]\nonumber \\&+\, k^{-}_{8}[\mathrm{VIIIa}^{m}]-h_{8}[\mathrm{VIIIa}]. \end{aligned}$$

(21)

$$\begin{aligned} G_{\mathrm{VIII}^{m}}=&-\dfrac{k^{m}_{8}[\mathrm{IIa}^{m}][\mathrm{VIII}^{m}]}{K^{m}_{8M}+[\mathrm{VIII}^{m}]}-\dfrac{k^{m}_{8t}[\mathrm{Xa}^{m}][\mathrm{VIII}^{m}]}{K^{m}_{8tM}+[\mathrm{VIII}^{m}]}\nonumber \\&+\, k^{+}_{8}N_{8}[\mathrm{AP}][\mathrm{VIII}]-k^{-}_{8}[\mathrm{VIII}^{m}]. \end{aligned}$$

(22)

$$\begin{aligned} G_{\mathrm{VIIIa}^{m}}=&\dfrac{k^{m}_{8}[\mathrm{IIa}^{m}][\mathrm{VIII}^{m}]}{K^{m}_{8M}+[\mathrm{VIII}^{m}]}+\dfrac{k^{m}_{8t}[\mathrm{Xa}^{m}][\mathrm{VIII}^{m}]}{K^{m}_{8tM}+[\mathrm{VIII}^{m}]}+k^{+}_{8}N_{8}[\mathrm{AP}][\mathrm{VIIIa}]\nonumber \\&-\,k^{-}_{8}[\mathrm{VIIIa}^{m}]-k^{+}_{TEN}[\mathrm{VIIIa}^{m}][\mathrm{IXa}^{m}] \nonumber \\&+\, k^{-}_{TEN}[\mathrm{VIIIa}^{m}:\mathrm{IXa}^{m}]. \end{aligned}$$

(23)

$$\begin{aligned} G_{\mathrm{VIIIa}^{m}:\mathrm{IXa}^{m}}=\,&k^{+}_{TEN}[\mathrm{VIIIa}^{m}][\mathrm{IXa}^{m}]-k^{-}_{TEN}[\mathrm{VIIIa}^{m}:\mathrm{IXa}^{m}]. \end{aligned}$$

(24)

$$\begin{aligned} G_\mathrm{V}=&-\dfrac{k_{5}[\mathrm{IIa}][\mathrm{V}]}{K_{5M}+[\mathrm{V}]}-k^{+}_{5}N_{5}[\mathrm{AP}][\mathrm{V}]+k^{-}_{5}[\mathrm{V}^{m}]. \end{aligned}$$

(25)

$$\begin{aligned} G_\mathrm{Va}=&\dfrac{k_{5}[\mathrm{IIa}][\mathrm{V}]}{K_{5M}+[\mathrm{V}]}-k^{+}_{5}N_{5}[\mathrm{AP}][\mathrm{Va}]+k^{-}_{5}[\mathrm{Va}^{m}]-h_{5}[\mathrm{Va}]. \end{aligned}$$

(26)

$$\begin{aligned} G_{\mathrm{V}^{m}}=&-\dfrac{k^{m}_{5}[\mathrm{IIa}^{m}][\mathrm{V}^{m}]}{K^{m}_{5M}+[\mathrm{V}^{m}]}-\dfrac{k^{m}_{5t}[\mathrm{Xa}^{m}][\mathrm{V}^{m}]}{K^{m}_{5tM}+[\mathrm{V}^{m}]}\nonumber \\&+\, k^{+}_{5}N_{5}[\mathrm{AP}][\mathrm{V}]-k^{-}_{5}[\mathrm{V}^{m}]. \end{aligned}$$

(27)

$$\begin{aligned} G_{\mathrm{Va}^{m}}=&\dfrac{k^{m}_{5}[\mathrm{IIa}^{m}][\mathrm{V}^{m}]}{K^{m}_{5M}+[\mathrm{V}^{m}]}+\dfrac{k^{m}_{5t}[\mathrm{Xa}^{m}][\mathrm{V}^{m}]}{K^{m}_{5tM}+[\mathrm{V}^{m}]}-k^{+}_\mathrm{PRO}[\mathrm{Xa}^{m}][\mathrm{Va}^{m}]\nonumber \\&+\, k^{-}_\mathrm{PRO}[\mathrm{Xa}^{m}:\mathrm{Va}^{m}]+k^{+}_{5}N_{5}[\mathrm{AP}][\mathrm{Va}]-k^{-}_{5}[\mathrm{Va}^{m}]. \end{aligned}$$

(28)

$$\begin{aligned} G_{\mathrm{Xa}^{m}:\mathrm{Va}^{m}}=\,&k^{+}_\mathrm{PRO}[\mathrm{Xa}^{m}][\mathrm{Va}^{m}]-k^{-}_\mathrm{PRO}[\mathrm{Xa}^{m}:\mathrm{Va}^{m}]. \end{aligned}$$

(29)

$$\begin{aligned} G_{I}=&-\dfrac{k_{f}([\mathrm{IIa}])[\mathrm{I}]}{K_{fM}+[\mathrm{I}]}. \end{aligned}$$

(30)

$$\begin{aligned} G_\mathrm{Ia}=&\dfrac{k_{f}([\mathrm{IIa}])[\mathrm{I}]}{K_{fM}+[\mathrm{I}]}. \end{aligned}$$

(31)

$$\begin{aligned} G_\mathrm{TFPI}=&-h^{TP+}_{10}[\mathrm{Xa}][\mathrm{TFPI}]+h^{TP-}_{10}[\mathrm{Xa:TFPI}]. \end{aligned}$$

(32)

$$\begin{aligned} G_\mathrm{Xa:TFPI}=\,&h^{TP+}_{10}[\mathrm{Xa}][\mathrm{TFPI}]-h^{TP-}_{10}[\mathrm{Xa:TFPI}] \nonumber \\&-h^{TP}_{7}[\mathrm{TF:VIIa}][\mathrm{Xa:TFPI}]. \end{aligned}$$

(33)

$$\begin{aligned} G_\mathrm{ATIII}=&-[\mathrm{ATIII}](h^{AT}_{10}[\mathrm{Xa}]+h_{9}[\mathrm{IXa}]+h_{2}[\mathrm{IIa}] \nonumber \\&+h_{T7}[\mathrm{TF:VIIa}]). \end{aligned}$$

(34)

Appendix 2: Description of reactions

1.1 Generation and depletion of factors VIIa, VII:

Tissue Factor (TF), an integral membrane protein gets exposed to blood upon blood vessel injury. It binds clotting factor VII/VIIa present in the plasma. Free, and tissue factor-bound factor VII is activated by factors IIa and Xa [51]. The TF–VIIa complex initiates the “extrinsic pathway” of blood coagulation. Clot formation proceeds by activation of zymogens factor IX and factor X [52]. TFPI exhibits a Xa-dependent inhibition of TF-VIIa [19].

1.2 Generation and depletion of factors IXa, IX:

While activation of factor IX by the tissue factor–factor VIIa complex is predominant and requires a membrane surface, that by factor XI is also important and is independent of a membrane surface [21, 53]. However, since we assume that the intrinsic pathway is inhibited, we neglect the role of factor XI in the present model. Inactivation of the activated factor IX occurs by the action of ATIII in the plasma [37, 54].

1.3 Generation and depletion of factors Xa, X:

Activation of factor X occurs by the tissue factor–factor VIIa complex, i.e., the extrinsic tenase complex (via extrinsic pathway), and by the factor VIIIa–factor IXa complex, i.e., the intrinsic tenase complex (via intrinsic pathway). Activated factor X then combines with activated factor V to form the platelet membrane-bound prothrombinase complex [12, 55]. Following the inactivation of factor Va (in the prothrombinase complex) by APC, dissociation of factor Xa from the membrane surface occurs. The dissociated factor Xa is then removed by flow and/or inactivated by ATIII and TFPI in plasma [56].

1.4 Generation and depletion of factors IIa, II:

Activation of the zymogen prothrombin to the enzyme thrombin occurs predominantly by the action of prothrombinase on activated membrane surface [20]. However, small amounts of prothrombin are also activated in plasma at a low rate by factor Xa [10]. Thrombin is primarily inhibited by ATIII [37].

1.5 Generation and depletion of activated and resting platelets:

The tissue factor and collagen exposed due to disruption of sub-endothelium lead to activation of platelets. In addition to converting fibrinogen to fibrin, thrombin plays a major role in activating platelets [12, 57]. Also, platelets are constitutively activated once they come in contact with other activated platelets in the plasma or on the sub-endothelial surface [57].

1.6 Generation and depletion of factors VIIIa, VIII:

Thrombin-activation of factor VIII does not necessarily require a membrane surface. However, activation of factor VIII by activated factor X, which is also an important reaction, requires activated platelets [58, 59]. On being activated, factor VIII shows cofactor activity in conjunction with activated factor IX [21]. Inhibition of activated factor VIII occurs due to the proteolytic attack by activated protein C (APC) on the membrane surface, apart from spontaneous decay [60].

1.7 Formation and dissociation of intrinsic tenase complex ($\hbox {VIIIa}^{m}$ : $\hbox {IXa}^{m}$):

On activation, activated factor IX combines with its active cofactor, factor VIIIa, and forms intrinsic tenase complex on the surface of activated platelets. This procoagulant complex then activates factor X via the intrinsic pathway [21, 25].

1.8 Generation and depletion of factor Va, V:

Activation of factor V to its cofactor state, factor Va, occurs by the action of thrombin [7], as well as by that of factor Xa (on the membrane surface) [61]. Competitive binding of APC to factor Va (which we have not included) followed by cleavage leads to inactivation of factor Va [62], apart from spontaneous decay just like factor VIIIa.

1.9 Formation and dissociation of prothrombinase complex ($\hbox {Xa}^{m}$ : $\hbox {Va}^{m}$):

Factor Va, the cofactor of factor Xa, binds it on the platelet membrane surface giving rise to the prothrombinase complex which converts prothrombin to thrombin [55, 62].

1.10 Generation and depletion of ATIII and TFPI:

While ATIII is a major physiologic inhibitor of almost all serine proteases produced during the process of blood coagulation process, TFPI mainly targets factor Xa and the TF–VIIa-factor Xa complex. Heparin-catalyzed ATIII-inhibition of factors IX, X and II is faster than the uncatalyzed reaction [54]. Since heparin is released from endothelial cells upon injury, in our case, we assume that the amount of heparin present is negligible as endothelial cells are not present in plasma in vitro. Inactivation by ATIII and TFPI occurs once the clotting enzymes escape into the plasma from the site of thrombus formation [20].