Analysis of the Activated Partial Thromboplastin Time Test Using Mathematical Modeling

Thrombosis Research 101 (2001) 299 ± 310

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Analysis of the Activated Partial Thromboplastin Time Test Using Mathematical Modeling Alexander E. Kogan1, Denis V. Kardakov2 and Mikhail A. Khanin2 1 Biological Faculty, Department of Biochemistry, Room 129, Moscow State University, Moscow 119899, Russia; 2 Laboratory of Theoretical Biology, Russian State University of Technology, Russia (Received 20 October 2000 by Editor I. Bokarew; revised/accepted 1 November 2000)

Abstract Activated partial thromboplastin time (APTT) is a laboratory test for the diagnosis of blood coagulation disorders. The test consists of two stages: The first one is the preincubation of a plasma sample with negatively charged materials (kaolin, ellagic acid etc.) to activate factors XII and XI; the second stage begins after the addition of calcium ions that triggers a chain of calciumdependent enzymatic reactions resulting in fibrinogen clotting. Mathematical modeling was used for the analysis of the APTT test. The process of coagulation was described by a set of coupled differential equations that were solved by the numerical method. It was found that as little as 2.3  10 ÿ 9 mM of factor XIIa (1/10 000 of its plasma concentration) is enough to cause the complete activation of factor XII and prekallikrein (PK) during the first 20 s of the preincubation phase. By the end of this phase, kallikrein (K) is completely inhibited, residual activity of factor XIIa is 54%, and factor XI is activated by 26%. Once a clot is formed, factor II is activated Abbreviations: APTT, activated partial thromboplastin time; PK, prekallikrein; K, kallikrein; C1-Inh, C1-inhibitor; AT-III, antithrombin III; a1-AT, a1-antitrypsin (a1-proteinase inhibitor); a2AP, a2-antiplasmin; a2M, a2-macroglobulin; PAI-1, plasminogen activator inhibitor type 1; TFPI, tissue factor pathway inhibitor; HMWK, high molecular weight kininogen. Corresponding author: Dr. Alexander E. Kogan, Biological Faculty, Department of Biochemistry, Room 129, Moscow State University, Moscow 119899, Russia. Fax: +7 (95) 9392788; E-mail: .

by 4%, factor X by 5%, factor IX by 90%, and factor XI by 39%. Calculated clotting time using protein concentrations found in the blood of healthy people was 40.5 s. The most pronounced prolongation of APTT is caused by a decrease in factor X concentration. D 2001 Elsevier Science Ltd. All rights reserved. Key Words: APTT test; Mathematical modeling; Contact activation

A

ctivated partial thromboplastin time (APTT) is one of the commonly used coagulation tests in clinical practice. It is applied to diagnose a wide range of acquired and congenital coagulation disorders in which proteins of the intrinsic coagulation pathway are involved. A prolongation of the APTT is observed in congenital deficiencies of factor VIII (hemophilia A), factor IX (hemophilia B), factors XI and XII. The APTT test is performed in two stages. The first stage is the preincubation of a plasma sample with phospholipids and negatively charged materials (kaolin, ellagic acid, dextran sulfate, and others) in the absence of calcium ions. During this stage, factor XII is autoactivated and converts prekallikrein (PK) to kallikrein (K) which, in turn, quickly activates factor XII to XIIa. Factor XIIa then partially activates factor XI. The second stage begins after the addition of calcium ions that in the presence of phospholipids triggers a coagulation cascade resulting in thrombin generation and conversion of fibrinogen to fibrin. APTT is a time

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A.E. Kogan et al./Thrombosis Research 101 (2001) 299±310

sic [16] pathways. Mathematical models of the prothrombin time test were also proposed [17,18]. The purpose of this work was to study the APTT test using mathematical modeling to determine the kinetics of clotting factor activation and the influence of clotting factor deficiencies on APTT.

1. Methods Fig. 1. A scheme of the contact stage of the APTT test.

when a fibrin clot becomes visible or detectable by instruments. Despite the long-term application of the APTT test in clinical practice, some details are still not clear. The most arguable question concerns the mechanism of the activation of factor XII. It is not clear whether it is activated by a trace amount of factor XIIa present in plasma [1] or factor XII being adsorbed on the negatively charged surface becoming active due to conformational changes [2]. Another question concerns the kinetics and degree of the activation of clotting factors during the test, because it is difficult to study the activity of a single enzyme in the enzyme mixture in gelating plasma by biochemical techniques. To solve these problems, the use of mathematical modeling seems to be very expedient. The first linear mathematical model of blood coagulation was proposed by Dr. Levin [3] in which the amplification effect in the enzymatic cascade was confirmed. Later, the nonlinear models made it possible to predict the threshold effects in the activation of extrinsic [4] and intrinsic [5] pathways. The threshold effects were further confirmed by biochemical study [6,7]. Most of the models describe the different parts of the coagulation system. The mathematical analysis of the contact phase of coagulation [8,9], the prothrombinase complex [10,11], thrombin generation [6,12], the inhibition of serine proteases in the tissue factor pathway [13], the interaction between blood coagulation system and platelets [14] had been published, as well as the model for the tissue factor [15] and intrin-

The present study was made by the method of mathematical modeling. The schemes of the contact activation phase and the intrinsic coagulation pathway are shown in Figs. 1 and 2. Chemical reactions and corresponding kinetic constants are summarized in Tables 1 and 2. All constants except for the reactions of the contact phase were taken in the presence of calcium ions. Second-order rate constants for the inhibition of enzymes by antithrombin III (AT-III) were taken in the absence of heparin. The constants of inhibition of prothrombinase and tenase complexes by AT-III were taken as one-third of the values for free factor Xa and IXa, respectively [44]. Activation of factor XI by thrombin was left out of account (see Discussion). Plasma concentrations of coagulation factors were taken as follows: PK, 0.58 mM [53]; factor XII, 0.3 mM [54]; factor XIIa, 2.3  10 ÿ 5 mM [55]; factor XI, 0.025 mM [56]; factor X, 0.133 mM [57];

Fig. 2. A scheme of the calcium-dependent stage of the APTT test.

A.E. Kogan et al./Thrombosis Research 101 (2001) 299±310

301

Table 1. Enzyme reactions and kinetic constants used in the model kcat (min ÿ 1) N

Reaction

1

XIIa

XII ! XIIa

2

PK ! K

3

PK ! K

4

XII ! XIIa

5 6

XIIa ! XIIf XIIa XI ! XIa

7 8

XII ! XIIa XIa IX ! IXa

XIIa

XIIf

K

Published data 1.98 216 2400 342

K

3.4  10 ÿ 2

XIa

10.4 25 39.6 144 462

IXa

9

X ! Xa

3.8  10 ÿ 2 9.5  10 ÿ 2

IXaÿVIIIaÿPL

X ÿÿÿÿÿ! Xa

10

Xa

11

II ! IIa

12

II ÿÿÿÿÿ! IIa

13

V ! Va

14

V ! Va

15

VIII ! VIIIa

16

I ! Ia

1344 2100

IIa

14

IIa

IIa

b c d e

2.25

XaÿVaÿPL

Xa

a

1740

2.6 7.2 5040

km (MM)

Used in the model

6.8  10 ÿ 1b 34c

225d

14 2.6 60

e

5040

37

0.51

342 0.34a

200

0.091

37

2400

2.25

11

0.091

216

1740

Used in the model

11

1.98

4.0  10

Published data

2 2 3.1  10 ÿ 1 3.7  10 ÿ 1 3  10 ÿ 1 4.9  10 ÿ 1 8.1  10

ÿ2

0.51 0.5

ÿ1

9.4  10 ÿ 1 1.4  10 ÿ 1 1.9  10 ÿ 1 5.8  10

ÿ2

1.06 1.0 7.17  10 ÿ 2 1.04  10 ÿ 2 2.0  10 ÿ 2 7.2

2 0.5

3.5  10 ÿ 1

2.0 1.9  10 ÿ 1 5.8  10 ÿ 2 1.0 7.17  10 ÿ 2 1.04  10 ÿ 2 2.0  10 ÿ 2 7.2

Ref. [19] [19] [19] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [26] [28] [29] [30] [31] [32] [32] [33] [34]

Estimated according to Dunn et al. [35]. kcat was increased according to the data that the rate of this reaction is accelerated in the presence of kaolin [36,37]. kcat was estimated according to Griffin [38]. kcat was increased since the rate of the activation increases in the presence of phospholipids [39]. kcat was increased because reported constants were measured at 25°C [33].

factor IX, 0.09 mM [58]; factor VIII, 0.7 nM [59]; factor V, 0.021 mM [60]; factor II, 1.4 mM [61]; factor I, 8.3 mM [62]; C1-inhibitor (C1-Inh), 1.7 mM [63]; AT-III, 3.4 mM [64]; a1AT, 24.5 mM [63]; a2AP, 0.9 mM [63]; a2M, 3.5 mM [63]; plasminogen activator inhibitor type 1 (PAI-1), 4.6  10 ÿ 4 mM [65]; tissue factor pathway inhibitor (TFPI), 2.5 nM [66].

The initial concentrations of activated factors were taken equal to zero, except for factor XIIa. The rates of enzymatic reactions were calculated using Michaelis±Menten kinetics. The rates of enzyme inhibition and complex formation were calculated using the equation for secondorder reactions. The balance of clotting factors

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Table 2. Second-order reactions and kinetic constants used in the model K (MM min) ÿ 1 N

Reaction

17 18 19

Va + Xa ! Va ± Xa VIIIa + IXa ! VIIIa ± IXa IIa + AT-III ! IIa ± AT-III

20

IIa + a1AT ! IIa ± a2AT

Published data

4.25  10 ÿ 1 3.36  10 ÿ 1 2.77  10 ÿ 1 ÿ3

6.51  10 2.88  10 ÿ 3 ÿ2

21

IIa + a2M ! IIa ± a2M

2.93  10

22

Xa + AT-III ! Xa ± AT-III

1.88  10 ÿ 1 3.4  10 ÿ 2 ÿ2

23

Xa + a1AT ! Xa ± a1AT

1.57  10

24

Xa + TFPI ! Xa ± TFPI

9.6  102

25

IXa + AT-III ! IXa ± AT-III

2.94  10 ÿ 2

26

XIa + C1-Inh ! XIa ± C1-Inh

4  10 ÿ 2

27

XIa + a1AT ! XIa ± a1AT

1.3  10 ÿ 2

28

XIa + AT-III ! XIa ± AT-III

3  10 ÿ 2

29

XIa + a2AP ! XIa ± a2AP

6  10 ÿ 2

30

XIa + PAI-1 ! XIa ± PAI-1

12.6

31

XIIa + C1-Inh ! XIIa ± C1-Inh

2.2  10 ÿ 1

32

XIIa + a2AP ! XIIa ± a2AP

1.1  10 ÿ 2

33

XIIa + a2M ! XII ± a2M

5  10 ÿ 3

34

XIIa + AT-III ! XIIa ± AT-III

1.3  10 ÿ 3

35

XIIa + PAI-1 ! XIIa ± PAI-1

9.6  10 ÿ 1

36

XIIf + C1-Inh ! XIIf ± C1-Inh

1.85  10 ÿ 1

37

XIIf + a2AP ! XIIf ± a2AP

0.91  10 ÿ 2

38

XIIf + AT-III ! XIIf ± AT-III

0.32  10 ÿ 2

39

K + C1-Inh ! K ± C1-Inh

1

40

K + a2M ! K ± a2M

2.9  10 ÿ 1

41

K + PAI-1 ! K ± PAI-1

3.6

42

K + AT-III ! K ± AT-III

9.6  10 ÿ 3

Used in the model 104a 104a

3.5  10 ÿ 1 4.7  10 ÿ 3 2.93  10 ÿ 2 1.1  10 ÿ 1 1.57  10 ÿ 2 9.6  102 2.94  10 ÿ 2 1  10 ÿ 3b 4  10 ÿ 3b 1  10 ÿ 2b 3  10 ÿ 2b 12.6 4.6  10 ÿ 2c 1.1  10 ÿ 2 5  10 ÿ 3 1.3  10 ÿ 3 9.6  10 ÿ 1 1.85  10 ÿ 1 0.91  10 ÿ 2 0.32  10 ÿ 2 1 2.9  10 ÿ 1 3.6 9.6  10 ÿ 3

Ref.

[40] [41] [42] [41] [43] [43] [40] [44] [44] [45] [40] [46] [46] [46] [46] [47] [48] [48] [48] [48] [47] [49] [49] [49] [49] [50] [47] [50]

A.E. Kogan et al./Thrombosis Research 101 (2001) 299±310

during the APTT test was described using ordinary differential equations. A set of dynamic equations [Eq. (1)] describing the preincubation stage of the APTT test is as follows: dK kcat2  XIIa  PK kcat3  XIIf  PK ˆ ‡ dt km2 ‡ PK km3 ‡ PK ÿk39  C1 ÿ Inh  K ÿ k40  a2 M  K ÿ k41 PAI ÿ 1  K ÿ k42  AT ÿ III  K; dPK kcat2  XIIa  PK kcat3  XIIf  PK ˆ ÿ ; dt km2 ‡ PK km3 ‡ PK dXIIa kcat4  K  XII kcat7  XIa  XII ˆ ‡ dt km4 ‡ XII km7 ‡ XII ‡

kcat1  XIIa  XII kcat5 K  XIIa ÿ ÿ k31 km1 ‡ XII km5 ‡ XIIa

C1 ÿ Inh  XIIa ÿ k32  a2 AP XIIa ÿ k33  a2 M  XIIa ÿ k34 AT ÿ III  XIIa ÿ k35  PAI ÿ 1  XIIa; dXIIf kcat5  K  XIIa ˆ ÿ k36  C1 ÿ Inh dt km5 ‡ XIIa XIIf ÿ k37  a2 AP  XIIf ÿ k38 AT ÿ III  XIIf; dXII kcat4  K  XII ˆÿ dt km4 ‡ XII ÿ

kcat7  XIa  XII kcat1  XIIa  XII ÿ ; km7 ‡ XII km1 ‡ XII

dXIa kcat6  XIIa  XI ˆ ÿ k26  C1 ÿ Inh dt km6 ‡ XI XIa ÿ k27  a1 AT  XIa ÿ k28 AT ÿ III  XIa ÿ k29  a2 AP XIa ÿ k30  PAI ÿ 1  XIa; dXI kcat6  XIIa  XI ˆÿ ; dt km6 ‡ XI

303

dC1 ÿ Inh ˆ C1 ÿ Inh…k39  K ‡ k31  XIIa dt ‡k36  XIIf ‡ k26  XIa†; da2 M ˆ ÿa2 M…k40 K ‡ k33  XIIa†; dt dPAI ÿ 1 ˆ dt ÿPAI ÿ 1…k41  K ‡ k35  XIIa ‡ k30  XIa†; dAT ÿ III ˆ ÿAT ÿ III…k42  K ‡ k34  XIIa dt ‡k38  XIIf ‡ k28  XIa†; da2 AP ˆ dt ÿa2 AP…k32  XIIa ÿ k37  XIIf ÿ k29  XIa†; da1 AT …1† ˆ ÿk27  a1 AT  XIa: dt A set of dynamic equations [Eq. (2)] describing the second stage of the APTT test is as follows: dXI kcat6  XIIa  XI ˆÿ ; dt km6 ‡ XI dXIa kcat6  XIIa  XI ˆ ÿ k26  C1 ÿ Inh dt km6 ‡ XI XIa ÿ k27  a1 AT  XIa ÿ k28 AT ÿ III  XIa ÿ k29  a2 AP XIa ÿ k30  PAI ÿ 1  XIa; dIX kcat8  XIa  IX ; ˆÿ dt km8 ‡ IX dIXa kcat8  XIa  IX ˆ ÿ k18 dt km8 ‡ IX VIIIa  IXa ÿ k25  AT ÿ III  IXa; dX kcat9  IXa  X kcat10  ‰VIIIaIXaŠ  X ˆÿ ÿ ; dt km9 ‡ X km10 ‡ X dXa kcat9  IXa  X kcat10  ‰VIIIaIXaŠ  X ˆ ‡ dt km9 ‡ X km10 ‡ X

Notes to Table 2 a Estimated according to Krishnaswamy et al. [51]. b Decreased about three times using data on the influence of HMWK on these reactions [46]. c Decreased 4.8 times according to the data on the influence of kaolin on the reaction rate [52].

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A.E. Kogan et al./Thrombosis Research 101 (2001) 299±310

ÿk17  Va  Xa ÿ k22  AT ÿIII  Xa ÿ k24  TFPI  Xa ÿk23  a1 AT  Xa; dII kcat11  Xa  II kcat12  ‰VaXaŠ  II ˆÿ ÿ ; dt km11 ‡ II km12 ‡ II dIIa kcat11  Xa  II kcat12  ‰VaXaŠ  II ˆ ‡ ÿ k19 dt km11 ‡ II km12 ‡ II AT ÿ III  IIa ÿ k20  a1 AT IIa ÿ k21  a2 M  IIa; dV kcat13  IIa  V kcat14  Xa  V ˆÿ ‡ ; dt km13 ‡ V km14 ‡ V dVa kcat13  IIa  V kcat14  Xa  V ˆ ‡ dt km13 ‡ V km14 ‡ V ÿk17  Va  Xa; dVIII kcat15  IIa  VIII ˆÿ ; dt km15 ‡ VIII dVIIIa kcat15  IIa  VIII ˆ ÿ k18  VIIIa  IXa; dt km15 ‡ VIII dI kcat16  IIa  I ˆÿ ; dt km16 ‡ I dIa kcat16  IIa  I ˆ ; dt km16 ‡ I d‰VaXaŠ ˆ k17 Va  Xa ÿ k22  AT ÿ III dt ‰VaXaŠ ÿ k23  a1 AT  ‰VaXaŠ;

Fig. 4. Kinetics of proenzyme consumption during the contact stage of the APTT test.

d‰VIIIaIXaŠ ˆ k18  VIIIa  IXa ÿ k25 dt AT ÿ III  ‰VIIIaIXaŠ; dC1 ÿ Inh ˆ ÿk26  C1 ÿ Inh  XIa; dt da1 AT ˆ ÿa1 AT…k27  XIa ‡ k20  IIa dt ‡k23  Xa ‡ k23  ‰XaVaŠ†; dATIII ˆ ÿAT ÿ III…k28  XIa ‡ k25  IXa ‡ k25 dt ‰IXaVIIIaŠ ‡ k22  Xa ‡ k22 ‰XaVaŠ ‡ k19  IIa†; da2 AP ˆ ÿk29  a2 AP  XIa; dt dPAI ÿ 1 ˆ ÿk30  PAI ÿ 1  XIa; dt dTFPI ˆ ÿk24  TFPI  Xa: dt

…2†

The systems of Eqs. (1) and (2) were solved by the numerical method. Enzyme competition for substrates was taken into account (not shown in order not to overcomplicate the equations).

Fig. 3. Kinetics of factor XII decay in plasma during storage.

Fig. 5. Kinetics of enzyme formation during the contact stage of the APTT test.

A.E. Kogan et al./Thrombosis Research 101 (2001) 299±310

Fig. 6. Kinetics of proenzyme consumption during the calcium-dependent stage of the APTT test.

Clotting time was assumed to be the time necessary to convert 30% of fibrinogen to fibrin monomer [67,68].

2. Results 2.1. The Decay of Factor XIIa in a Plasma Sample During Storage A decrease in the concentration of factor XIIa due to its inhibition by C1-Inh, AT-III, a2antiplasmin (a2AP), a2-macroglobulin (a2M), and plasminogen activator inhibitor was calculated (Fig. 3). As little as 1/10 000 of the initial factor XIIa concentration remains after 1.5 h. The concentration of 2.3  10 ÿ 9 mM was taken for further modeling to initiate the contact activation. 2.2. The Activation of Clotting Factors During the Preincubation Stage Time-courses of the activation of factor XII, factor XI, and PK during the preincubation stage of the APTT test are given in Figs. 4 and 5. It was found

Fig. 7. Kinetics of enzyme formation during the calcium-dependent stage of the APTT test.

305

Fig. 8. Kinetics of prothrombinase formation during the APTT test. Concentration of Va ± Xa complex is given as percentages of factor V.

that during the preincubation phase, factor XII and K were completely activated within 20 s. K then was entirely inhibited during 1 min. By the end of the preincubation phase, factor XI was activated by 26%, and the residual activity of factor XIIa was 54%. 2.3. The Activation of Clotting Factors During the Second Stage of the APTT Test Time-courses of the proenzyme consumption and enzyme formation during the second stage of the APTT test are shown in Figs. 6 and 7, respectively. Once a clot is formed, factor II is activated by 4%, factor X by 5%, factor IX by 90%, and factor XI by 39%. The proenzyme activation continues inside the clot. A two-peak curve for factor Xa concentration was observed. Inactivation of coagulation enzymes also continues after clot formation. As shown in Figs. 8 and 9, factors V and VIII, being activated rapidly, form complexes with factors Xa and IXa, respectively.

Fig. 9. Kinetics of tenase formation during the APTT test. Concentration of VIIIa ± IXa complex is given as percentages of factor VIII.

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Table 3. APTT at different concentrations of clotting factors (in seconds) Factor concentration (%) 100 50 25 10 5 1 0.5

Factor II

Factor VIII

Factor IX

Factor X

Factor XI

Factor XII

40.5 43.5 48 56.5 65 94 114

40.5 44 48 52 55 60.5 62

40.5 45.5 51.5 62 71.5 110.5 139

40.5 48 59 78 98.5 178 239

40.5 44.5 49.5 57 64 87 101

40.5 44.5 49.5 57 64 86.5 100

2.4. The Relationship Between Clotting Times and Concentrations of Coagulation Factors The correlation between clotting times in the APTT test and concentrations of coagulation factors was determined using mathematical modeling. As can be seen in Table 3, the APTT test is more sensitive to the decrease in factors X and IX concentrations. A decrease in factors II, XI, and XII concentrations results in an approximately equal prolongation of APTT. A decrease in the factor VIII concentration below 1% almost does not lead to the further prolongation of APTT.

3. Discussion What initiates the activation of factor XII? It is the main theoretical question concerning the contact activation of blood coagulation in the presence of the negatively charge surface. There are several hypotheses of the activation of factor XII. One of them explains the activation of factor XII by a trace amount of its activated form present in plasma [1,69]. According to another hypothesis, factor XII becomes active due to conformational changes after adsorption on the negatively charged materials (conformational changes in the factor XII molecule in the presence of dextran sulfate were reported [70]). The analysis of kinetics of factor XIIa decay in plasma due to its inactivation by inhibitors shows that about 1/10 000 of the initial concentration of factor XIIa remains in plasma after 1.5 h of storage. The mathematical modeling proves that this concentration of factor XIIa is enough to cause the complete activation of factor XII and PK during the first 20 s of the preincuba-

tion phase. A high rate of the activation is achieved by a reciprocal activation of PK and factor XII. Regardless of the possibility of factor XII activation due to the conformational changes after adsorption on the negatively charged surface, the activation of factor XII in the presence of PK by a trace amount of factor XIIa appears to be the predominant mechanism in the APTT test. The second question concerns the activation of factor XI by thrombin. This reaction was not included in the coagulation scheme of the APTT because, despite publications reporting the activation of purified factor XI by thrombin in the presence of dextran sulfate [20], it was shown that fibrinogen prevents thrombin-mediated activation of factor XI [71], and this reaction does not go in plasma surroundings [72]. In addition, unlike dextran sulfate, kaolin did not support the activation of factor XI by thrombin [73]. This reaction may occur in vivo in the presence of activated platelets [74]. The mathematical modeling reveals that during the preincubation stage, factor XII and K are completely activated. K then is entirely inhibited by plasma inhibitors, whereas factor XIIa is only partially inactivated. Factor XI is activated by 26% by the end of the preincubation stage. These data explain the dependence of APTT on the duration of the preincubation period. Only small amounts of factors II and X are activated before sample clotting, but this process continues after the clot formation. Unlike them, factor IX is almost completely activated due to its activation by factor XIa formed during the preincubation phase. A two-peak curve for factor Xa concentration (inside the clot) was observed. It can be explained by the binding of factor Xa formed into the Va±Xa complex.

A.E. Kogan et al./Thrombosis Research 101 (2001) 299±310

Mathematical modeling results show that the inactivation of coagulation enzymes continues after clot formation. The decrease in the initial concentrations of coagulation factors results in the prolongation of APTT. From all proteins studied in the model, the test appeared most sensitive to the decrease in factor X. The decrease in factors XI and XII concentrations results in an almost equal prolongation of APTT. The decrease in the factor VIII concentration below 1% does not lead to a further prolongation of APTT because, at very low concentration of factor VIII, factor X is activated chiefly by free factor IXa but not by the VIIIa± IXa complex. There is a discrepancy between some calculated clotting times in the APTT test and results obtained using APTT kits. The reason for that is the wide scatter in published kinetic constants (see Tables 1 and 2). In addition, kinetic constants for tenase and prothrombinase reactions are measured in the presence of phospholipid mixture consisting of phosphatidylcholine and posphatidylserine [26,28,30,31], whereas the extracts of brain and lung tissues, which are used as phospholipids in the APTT kits, contain many various phospholypids. The advantage of mathematical modeling is in the possibility to get generalized regularities and to obtain data that are difficult to measure in experiments.

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