Full text of DOI:10.1134/1.1342454

Doklady Physics, Vol. 45, No. 12, 2000, pp. 701–704. Translated from Doklady Akademii Nauk, Vol. 375, No. 6, 2000, pp. 774–777.
Original Russian Text Copyright © 2000 by Obraztsov, Barynin, Khanin.
MECHANICS
A Mechanism for Restriction of the Activation-Zone Propagation
for a Hemostasis System (A Mathematical Model)
Academician I. F. Obraztsov, Yu. A. Barynin, and M. A. Khanin
Received September 9, 1999
INTRODUCTION
In the mathematical model proposed, the following
processes are represented:
Using methods of mathematical simulation, a
hemostasis system was found to be bistable [1–4].
Later on, the concept of bistability and the existence of
threshold effects in the external-path activation were
(
i) fermentative reactions resulting in the activation
of coagulation factors;
(ii) chemical reactions of the second order resulting
supported by the results of biochemical studies. As is in the formation of activated-factor complexes;
well known, the bistability of the hemostasis system
protects against the formation of inadequate thrombo-
sis. On the other hand, this feature suggests the possi-
bility that there exists an autowave mechanism for the
propagation of an activation zone of a hemostasis sys-
tem. It was hypothesized in [5] that activation-zone
propagation is terminated owing to the propagation of
(
iii) diffusion of proenzymes, procofactors,
enzymes, cofactors, as well as complexes of activated
factors and cofactors;
(iv) activation of the external path by the tissue
factor;
(v) distinction between the rate of fermentative
an autowave of a certain inhibitor that catches up to the reactions on thrombocyte membranes and that in the
activation wave. However, no inhibitor satisfying the blood bulk;
necessary requirements is known. The goal of the
(vi) transfer of the coagulation factors by the blood
present study is to investigate the mechanism for the
processes terminating the propagation of the activation-
zone for a hemostasis system using a mathematical-
simulation method. The same mechanisms must also
insure that the thrombus size is adequate relative to the
extent of blood-vessel damage.
flow.
MATHEMATICAL MODEL
FOR PROPAGATING THE ACTIVATION ZONE
With the structure of the external path and the pro-
cesses under consideration taken into account, the
mathematical model being developed takes the follow-
ing form.
BASIC PROCESSES FOR CONSTRUCTING
A MATHEMATICAL MODEL
We consider only the external path of the hemoco-
agulation schematically shown in Fig. 1. As is seen
from this figure, the external path is composed of the
following components:
TF
VII
[
TF VII]
[TF VIIa]
(1) cascade of fermentative reactions;
(
2) positive feedback caused by the action of a
VII
VIIa
cofactor, i.e., the factor Va;
(
3) positive feedback appearing in the mutual acti-
Va
V
X
Xa
vation of factors VII and X;
(
4) negative feedback caused by the action of the C
protein.
[Xa Va]
II
IIa
PCa
PC
Moscow Institute of Aviation and Technology (MATI),
ul. Orshanskaya 3, Moscow, 121552 Russia
Fig. 1. Diagram for the hemocoagulation external path.
1
028-3358/00/4512-0701$20.00 © 2000 MAIK “Nauka/Interperiodica”

7
02
OBRAZTSOV et al.
The process proceeding on cell membranes:
[TF]
∂[TFVII] ∂[TFVIIa]
[
II][Xa]
α k2 _ 10a--------------------------------- + ---- -- ---------- --
+ [II] δ ∂x
x = 0
D
∂[II]
2
–
,
k
∂
m2 _ 10a
-
------------- = – ----------------------- – --------------------------,
∂
t
∂t
∂t
∂
[IIa]
[II][XaVa]
-------------------------------------
-
------------ -- = α k
∂
[TFVII]
2 _ 10a5a
∂
t
k
+ [II]
m2 _ 10a5a
-
---------------------- = ktf _ 7[TF][VII] – Etf 7[TFVII]
∂
t
[
II][Xa]
D
2a ∂[IIa]
[
TFVII][Xa]
+ α k2 _ 10a--------------------------------- + ------ -- ------------- --
,
–
ktf 7 _ 10a------------------------------------------------,
k
+ [II]
δ
∂x
m2 _ 10a
x = 0
k
+ [TFVII]
mtf 7 _ 10a
∂
[PC]
∂t
[PC][IIa]
+ [PC]
mPC _ 2a
∂
[TFVIIa]
[TFVII][Xa]
+ [TFVII]
mtf 7 _ 10a
--------------- = –k*
--------------------------------------
PC _ 2a
-
------------------------- = k
------------------------------------------------
k
tf 7 _ 10a
∂
t
k
D ∂[PC]
PC
–
ETF7a [TFVIIa],
------- -- ---------------
+
,
δ
∂x
x = 0
∂
[VII]
[VII][Xa]
-
------------- -- = –α k
-------------------------------------
_ 10a
7
∂
[PCa]
[PC][IIa]
--------------------------------------
∂
t
k
+ [VII]
m7 _ 10a
*
PC _ 2a
-
---------------- -- = k
∂
t
k
+ [PC]
mPC _ 2a
D ∂[VII]
7
–
ktf _ VII[TF][VII] + Etf 7[TFVII] + ---- -- -------------- --
,
D
PCa
∂[PCa]
δ
∂x
x = 0
+ ------------------------------
∂x
.
δ
x = 0
∂
[VIIa]
[VII][Xa]
----------------- -- = α k
-------------------------------------
_ 10a
7
Here, [i] and D are the i-factor concentration and diffu-
∂
t
k
+ [VII]
i
m7 _ 10a
sivity, respectively; ki, ma is the constant for the rate of
activation of the i-factor by the mth-ferment; kmi, ma is
the Michaelis constant in the fermentative activation
reaction for the i-factor by the mth-ferment; δ is the
thickness of the near-membrane layer; and E is the
constant of the i-factor inactivation.
D ∂[VIIa]
7
a
+
Etf 7a[TFVIIa] + ------ -- ----------------- --
,
δ
∂x
x = 0
∂
[Xa]
[X][VIIa]
i
-
------------- = α k
--------------------------------
0 _ 7a
1
∂
t
k
+ [X]
m10 _ 7a
The process proceeding in a blood bulk:
[
X][TFVII]
k10 _ TFVII---------------------------------------
+ [X]
+
k
m10 _ TFVII
∂
[VII]
∂t
[VII][Xa]
-
------------- -- = –k
-------------------------------------
_ 10a
7
[
X][TFVIIa]
k10 _ TFVIIa-----------------------------------------
+ [X]
k
+ [VII]
m7 _ 10a
+
k
m10 _ TFVIIa
2
x
L
x (VIIo – [VII])
∂ [VII]


+
H-- 2 – -- --------------------------------- -- + D ---------------- -- ,
D
∂[Xa]
7
2
1
0a


L
L
–
k10a _ 5a[Xa][Va] + -------- -- --------------
∂x
,
y
∂x
δ
x = 0
∂
[VIIa]
[VII][Xa]
k + [VII]
m7 _ 10a
-
---------------- -- = k
-------------------------------------
7 _ 10a
∂
[V]
[V][IIa]
∂
t
-
---------- = –α k
------------------------------
5
_ 2a
∂
t
k
+ [V]
m5 _ 2a
2
(1)
x
L
x [VIIa]
∂ [VIIa]


L
–
H-- 2 – -- --------------- + D ------------------- -- ,
[
V][Xa]
D
5
∂[V]
7a


2
–
α k5 _ 10a-------------------------------- + ---- -- -----------
,
L
y
∂x
k
+ [V] δ ∂x
m5 _ 10a
x = 0
∂
[X]
∂t
[X][VIIa]
∂
[Va]
[V][IIa]
+ [V]
m5 _ 2a
--------------------------------
0 _ 7a
-
---------- = –k
1
-
------------- = α k
------------------------------
_ 2a
5
k
+ [X]
m10 _ 7a
∂
t
k
(2)
2
x
L
x (Xo – [X])
∂ [X]
[
V][Xa]
α k5 _ 10a--------------------------------
+ [V]


L
+
H-- 2 – -- ------------------------ -- + D -------------,
+
10


2
L
k
y
∂x
m5 _ 10a
D ∂[Va]
δ
5
a
∂
[Xa]
∂t
[XVIIa]
--------------------------------
10 _ 7a
–
k10a _ 5a[Xa][Va] + ------ -- --------------
∂x
,
-
------------- = k
– E10 _ ATIII[Xa]
x = 0
k
+ [X]
m10 _ 7a
∂
[XaVa]
D
10a5a ∂[XaVa]
[XaVa][PCa]
+ k10a5a _ PCa--------------------------------------------------
-
------------------- -- = k
[Xa][Va] + ------------------------------------
0a _ 5a
,
1
∂
t
δ
∂x
km10a5a _ PCa + [PCa]
x = 0
2
x
L
x [Xa]
∂ [Xa]
∂
[II]
[II][XaVa]


k10a _ 5a[Xa][Va] – H-- 2 – -- ----------- + D ----------------,
-
--------- -- = –α k
-------------------------------------
–
10a
2
_ 10a5a
2


L
L
∂
t
k
+ [II]
y
∂x
m2 _ 10a5a
DOKLADY PHYSICS Vol. 45 No. 12 2000

A MECHANISM FOR RESTRICTION OF THE ACTIVATION-ZONE PROPAGATION
703
The boundary conditions. Equations (1) define the
boundary condition of the problem at x = 0, i.e., at the
boundary of a near-membrane layer.
Another boundary condition at x = L, where L is a
value on the order of the vessel radius, is of the form
∂
[V]
[V][IIa] [V][Xa]
------------------------------ – k --------------------------------
5 _ 10a
-
---------- = –k
5
_ 2a
∂
t
k
+ [V]
k
+ [V]
m5 _ 10a
m5 _ 2a
2
x
L
x (Vo – [V])
∂ [V]


+
H-- 2 – -- ------------------------- + D -------------,
5


L
2
L
y
∂x
∂[i]
---------
∂x
∂
[Va]
[V][IIa]
------------------------------
= 0.
-
------------- = k
5
_ 2a
x = L
∂
t
k
+ [V]
m5 _ 2a
For t = 0, [i] = [i ] and [ia] = 0.
0
[
V][Xa]
k5 _ 10a-------------------------------- + E10a5a _ ATIII[XaVa]
+ [V]
+
k
m5 _ 10a
RESULTS
[
Va][PCa]
k5a _ PCa--------------------------------------- – k10a _ 5a[Xa][Va]
+ [Va]
In Figs. 2 and 3, the concentration distributions for
the factors II, IIa (thrombin), V, and Va are shown along
–
k
m5a _ PCa
2
x
L
x [Va]
∂ [Va]


II, IIa, µm
1.6
–
H-- 2 – -- --------- -- + D ----------------,
5
a


L
L
y
2
∂x
1
.4
1
a
∂
[XaVa]
2
a 3a
4a
-
------------------- -- = k10a _ 5a[Xa][Va] – E10a5a _ ATIII[XaVa]
1.2
.0
0.8
.6
0.4
∂
t
1
[
XaVa][PCa]
–
k10a5a _ PCa-----------------------------------------------------
k
+ [XaVa]
m10a5a _ PCa
0
2
x
L
x [XaVa]
∂ [XaVa]


–
H-- 2 – -- ------------------ + D
10a5a
---------------------- -- ,


L
2
L
y
∂x
0
.2
4
b
2
b 3b
1
b
∂
[II]
[II][XaVa]
0
30 60 90 120 150 180 210 240 270 300
---------- -- = –k
-------------------------------------
_ 10a5a
2
∂
t
k
+ [II]
m2 _ 10a5a
X, µmol
[
II][Xa]
k2 _ 10a---------------------------------
+ [II]
Fig. 2. Distributions of the prothrombin and thrombin con-
centrations over the external-path activation zone. The
curves with indices a and b correspond to prothrombin and
thrombin, respectively. The indices 1, 2, 3, and 4 correspond
to the time moments of 1, 2, 3, and 4 min after the onset of
the process. After an elapsed time of 4 min, the activation-
zone front becomes stabilized.
–
k
m2 _ 10a
2
x
L
x (IIo – [II])
∂ [II]


L
+
H-- 2 – -- --------------------------- + D ------------ -- ,
2


2
L
y
∂x
∂
[IIa]
[II][XaVa]
+ [II]
m2 _ 10a5a
V, Va, µm
0.06
-
------------ -- = k
-------------------------------------
2
_ 10a5a
∂
t
k
[
II][Xa]
+
k2 _ 10a--------------------------------- – k2a _ ATIII[IIa]
0.05
k
+ [II]
1a
4a
m2 _ 10a
0
.04
0.03
.02
2a
3a
2
x
L
x [IIa]
∂ [IIa]


–
H-- 2 – -- ---------- -- + D -----------------,
2
a


L
L
y
2
∂x
0
1
b
∂
[PC]
[PC][IIa]
-
-------------- = –k
--------------------------------------
PC _ 2a
∂
t
k
+ [PC]
0.01
0
mPC _ 2a
2
b 3b 4b
2
x
L
x (PCo – [PC])
∂ [PC]


+
H-- 2 – -- ---------------------------------- + D -----------------,
40 80 120 160 200 240 280 320 360 400
PC


L
2
L
y
∂x
X, µmol
Fig. 3. Concentration distributions for the factors V and Va
over the external path activation zone. The curves with indi-
ces a and b correspond to the factors V and Va, respectively.
The indices 1, 2, 3, and 4 correspond to the time moments of
∂
[PCa]
[PC][IIa]
-
---------------- -- = k
--------------------------------------
PC _ 2a
∂
t
k
+ [PC]
mPC _ 2a
2
1
, 2, 3, and 4 min after the onset of the process. After an
x
L
x [PCa]
∂ [PCa]


L
–
H-- 2 – -- --------------- + D ------------------- -- .
elapsed time of 4 min, the activation-zone front becomes
stabilized.
PCa


2
L
y
∂x
DOKLADY PHYSICS Vol. 45 No. 12
2000

7
04
OBRAZTSOV et al.
the coordinate axis normal to the vessel-wall surface adequate for the extent of vessel damage. The mecha-
for various instants of time. As is seen from Figs. 2 and nism for regulation of the penetration depth of the acti-
3
, the activation-zone front moves deeply into the ves- vation zone associated with the hemodynamic transport
sel and attains a certain steady position. If the terms of activated factors is essential only in the case of exist-
corresponding to hemodynamic transport are discarded ence of a cascade scheme for fermentative reactions.
in Eqs. (1) and (2), the wave front propagates continu-
ously. The same effect is also observed in the absence
of a cascade in the hemocoagulation scheme. Thus, a
cascade of fermentative reactions enhances the effect of
the hemodynamic entrainment as a factor restricting
propagation of the activation zone for a hemostasis sys-
tem. The greater the number of fermentative reactions
involved in a cascade, the smaller the path traversed by
the activation zone. For example, an internal path
involving a “longer” cascade forms an activation zone
with a smaller penetration depth compared to that of the
external path. This corresponds to the physiological
functions of the external and internal paths, the former
of which is fitted at significantly heavier damage of the
vessels.
The results obtained are appropriate for the case of
the activation-zone propagation in a blood flow.
ACKNOWLEDGMENTS
This study was supported by the International Sci-
ence Foundation (grants NAK-000 and ITN-4200).
REFERENCES
1
2
3
. V. V. Semenov and M. A. Khanin, Biofizika 35, 139
(1990).
. M. A. Khanin, Izv. Vyssh. Uchebn. Zaved., Prikl.
Nelineœnaya Din., Nos. 3/4, 65 (1994).
. M. A. Khanin, V. L. Leytin, and A. P. Popov, Thromb.
Res. 64, 659 (1991).
CONCLUSIONS
4. M. A. Khanin and V. V. Semenov, J. Theor. Biol. 136,
27 (1989).
1
Employing the method of mathematical simulation,
the hemodynamic entrainment of activated factors is
established as the basic mechanism resulting in the ter-
mination of hemostasis activation-zone propagation.
The same mechanism insures that the thrombus size is
5
. F. I. Attaulakhanov, G. T. Guriya, and A. Yu. Safrosh-
kina, Biofizika 39, 89 (1994).
Translated by V. Tsarev
DOKLADY PHYSICS Vol. 45 No. 12
2000