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Control of basophil degranulation
Znt J Biomed Cornput, 32 (1993) 151-159 Elsevier Scientific Publishers Ireland Ltd.
151
CONTROL OF BASOPHIL
DEGRANULATION
B. HADJAJB, Y. CHERRUAULTa
and J. SAINT’E LAUDYb
“Universitk Paris 6, MEDZMAT, IS, rue de I’Ecole de MPdecine 75270 Paris Gdex 06 and bLaboratoire de la R&ublique. 40 rue a’ela RPpublique. 69002 Lyon (France) (Received September 14th 1992) (Accepted October Sth, 1992) We first present a linear compartmental model describing human basophiol degranulation by means of an antigen concentration. Then we study an optimal control problem associated to this linear compartmental model in presence of high dilutions. For solving this control problem we used the LAPLACE transform. Key wordr: Modelization; Optimal control; Optimization; LAPLACE transform
1. Introdnetion
The basophil activation is induced by action of antigens. The extra-cellular calcium goes through the cell membrane and neutralizes the negative charges of the granule and precedes the release of active mediators. The high dilution acts on cell for inhibing the activation of basophils. We obtain the Fig. 1. 2. The Human BaaophII Degranalation Test (I-IBDT) Our data JJ,.(Oi) were obtained by using the method described by Benveniste [ 1 ] . The HBDT method permits the count of number (NJ of non reacted basophils and the number (No) of basophils in a control. We note that our high dilution data vary from 1 to 20 Ch (equivalent to theoretical concentrations of low2 to 10V40mol/l) and the antigen dilution varies from 10s4 to 9.5 x 10-‘” mg/ml [2].
VjfDi)
=
CNO
-
Ni) * lOO/No
i= l,...,n j=
(1)
l,...,m
3. Linear ModeIIIog 3. I. Introduction Let us consider a linear compartmental model for human basophil degranulation [ 31 described in Fig. 2, where the compartments represent: (1) percentage of degranulated basophils (Xi); (2) percentage of histamine (X2). Correspondence to: B. Hadjaj, Universite Paris 6, MEDIMAT, 15, rue de 1’Ecole de Mtdecine 75270 Paris Cedex 06, France. 0020-7101/93/SO6.00 0 1993 Elsevier Scientific Publishers Ireland Ltd. Printed and Published in Ireland
B. Haajaj et al.
152
r
I
Dilution
Where
:
Ag
= antigen
*
=
4.
ca++
9
ca++=
basophil calcium
Granule
Fig. 1. Scheme of &granulation of basophil mechanism in presence of high dilution.
3.2. Mathematical modeling The linear differential equations corresponding to the previous system are
-
L
=
k,XI - k2X2
X1 = 0 and X2 = 0 at
(2) t= 0
Fig. 2. A scheme of linear compartmental model.
Control oj basophil &granulation
153
TABLE I
Time (min) 1
3
5
6
7
8
10
X:’
3
20
31
31
37
40
44
X,(q)
6.15
17.34
27.20
31.68
35.88
39.82
46.90
3.2. Numerical results
For solving this problem we used Rung-Kutta’s method [ 41 and we have to idenare as close as tify the kr, k2, kJ, parameters such that the differences (J&-&(t)) possible to zero. The identification problem consists of finding k,, k2, k3 minimizing the functional:
J(W&)
[&(tj) - x&)1 2 where X = (X,;XJ
= i
(3)
j=l
The functional (3) represents the sum of differences between the exact solutions Xc(t) and the experimental data X,. Therefore the direct method of so-called ‘local
variations’ [5] can be used. A variant of the method is the Vignes’s method [6] in which J is minimized in several directions, taking into account variables influences on each other. We obtain the following parameters: kl = 4.1665E-2;
k2 = l.O058E-3;
kl, = 2.2072E-2
(For the results and the comparison of experimental and calculated data see respectively Tables I, II and Fig. 3). In Tables I-IV we defined: Degranulated
basophils experimental data
Degranulated
basophils calculated data
TABLE II
Time (min) 2
4
8
16
32
X5
8
12
23
43
56
Wtj)
7.74
14.55
25.83
41.36
56.30
154
B. Hdjaj
et al.
e
r c au-e n t
40 --
a !J e
20 -,..i"
Od 0
5
10
15
20
25
30
Time(min)
Fig. 3. The solid curve represents a solution of system (2) (with out dilution) and (0) and (D) experimental percentage of basophil degranulation and release histamine respectively.
Histamine experimental data
4. Optimal control [6] 4.1. Introduction Our principal aim is to control the evolution of this system by acting with high dilutions in compartment 1 (XI), which leads a diminishing of degranulation of basophils. The previous compartmental model is transformed according to Fig. 4. The control problem consists of finding a function u(t) taking into account high dilutions concentration such that the basophils activation becomes optimal. Find u(t) such that the following differential system is satisfied:
Fig. 4. A scheme of linear compartmental model in presence of high dilution.
155
Control of basophil akgmnuiation
TABLE III
Time (min) 1
3
5
6
7
8
10
A?:’
2
16
25
25
29
32
35
xl(rj)
4.15
13.52
21.38
25.001
28.42
31.66
37.63
-= -(k,
-
+ k3)X, + k2X2 + u
klX, - k2X2
=
and the following functional is minimized
J(u) =
sTt-,F-
Ye)2dt
(51
0
where Y,: experimental data. For solving this problem, the principle consists of finding an explicit relation, as possible, between Xi and the control u. An expression may be found: X,(t) = F@(t), kg). In order to do this, we used the LAPLACE transform [ 61. 4.2. Numerical mathematical analysis Let us consider the general system: .? = A x(t) + u(t)
(6) such that X and u verified the system and minimizes the following functional:
J(u) = min u(t)
ks1 s O
a is a fixed constant
- a)2dt
defined from biological considerations.
(6’)
The LAPLACE
transform gives:
/f(s) = (sZ - A)-$2(s)
(7)
156
B. Haajaj et al.
We deduce:
where q&,kU) = det(sZ - A) (it’s a polynomial of the nth degree in s) P, _ ,(s,kv) = (it’s a polynomial of the (n - 1)th degree in S)
Therefore the original of Eqn. (8) is: 1 T,,(T) X,(t - T) dr s (in Eqn. 9 the sign * is Convolution product) u(t) = m(t) * X,(t)
=
(9)
where m
r,(t) = k
c =,
%(d cl - 1 (Sk)
eSkt
(10)
Eqn. 9 gives the explicit formula for u(t) XI = F(s,k)d So a precedent minimisation problem becomes:
Hence F(s,kv)d = ?-
s
In conclusion, we obtain the following convolution f(t,Z$) + u(t) = a
equation:
Cf= F)
Applying this to our case, we get:
zd.9= 60) 49
with B(s)=
s + k2 (s _ x,)(s _ x
)
2
where Ai are eigenvalue of matrix A. Therefore 40) = Ai e hlf
+
A2 eh2’
(11)
157
Controlof basophil&granulation TABLE IV
Time (min) 2
4
XS
6
xs(tj)
5.91
8
16
32
18
34
45
20.09
32.709
45.40
9 11.19
where K2 + XI
Al =
XI -
A2 =
x2
K2 + A2 A2 -
h
(12)
The X1(t) = q(t) * u(t) 5. Results and Conclusion
The optimization problem consists of finding u which minimizing the functional J. From Eqn. 12 it may be shown that the optimal solution is such
4(t) * 40 =
ye
(13)
Differentiating Eqn. 13 we obtain:
4’0) * u(t) f q(O)u(t) = q”(t)
ye
* u(t) + q(0) u(t) + q(0) u’(t)
=
Y”,
It may be shown that K(r) is a solution of a following differential equation q(t) + arq’(t) + flq”(t) = 0, that is to say: o!q(O) u(t) + @q’(O) u(t) + /34(O) u’(t)
=
Y, + Y’, + Y”,
(14)
u(t) is solution of a linear differential Eqn. 14 with a coefficients and second term. An explicit solution can be obtained.
TABLE V
Time (min)
U(tj) (10-q
2
4
8
16
20
26
28
0.0991
0.2611
0.4777
0.7419
1.0472
1.3883 1.7606
32 2.1598
B. Hadjaj et al.
158
C
e
40
0
5
10
15
20
25 Time (min)
30
Fig. 5. The solid curve represents a solution of system (4) (with high dilution) and (0) and (D) experimental percentage of basophil degranulation and release histamine respectively.
In Tables III-V the values of X,(t), X*(t) and u(t) are respectively given (the graphic comparison for differential system (4) see Fig. 5). In conclusion, the results given by this optimal control allow to justify the effect of high dilutions on the basophils activation. Indeed the comparison of the degranulation curves with diltuion (Fig. 6, curve 2) and without dilution (Fig. 6,
c w-e
_______--------
n t
40 --
: e
2o 0 0
5
10
15
20
25
Time (mtn)
~
Curve n*l ------
Curve no2
Fig. 6. Percentage of degranulation (curve 1) without dilution and with dilution (curve 2).
30
Control of basophil &granulation
159
curve l), proves an effect. We can see that the percentage of degranulation is decreasing and that the dilution acts on the system. But at the same time another effect arises on the histamine release. We do not know enough arguments for explaining the action of high dilutions on the histamine release. We can just formulate many hypothesis impossible to justify. References 1
2 3 4 5 6
Benveniste J: Human basophil degranulation as in vitro test for the diagnosis of allergy. C/in. Alfergy 11 (1981) I-11. Guellal S et al.: Human basophil receptivity. Comput Math Applic Vol. 20 (1990). Cherruauh Y: Biomath&natiques. P.U.F. Coil. Que sais-je? (1983). de calcul numbique. Edit du P.S.1, Tome 1 (1982). Nowakowski C: h4kthodes Vignes J: ‘Algorithmes numeriques. Analyse et mise en oeuvre.’ Ed Tech, 1980. Cherruault Y: Mathematical Modeling in Biomedicine. Optimal Control of Biomedical Systems, Reidel, New York, 1986.
151
CONTROL OF BASOPHIL
DEGRANULATION
B. HADJAJB, Y. CHERRUAULTa
and J. SAINT’E LAUDYb
“Universitk Paris 6, MEDZMAT, IS, rue de I’Ecole de MPdecine 75270 Paris Gdex 06 and bLaboratoire de la R&ublique. 40 rue a’ela RPpublique. 69002 Lyon (France) (Received September 14th 1992) (Accepted October Sth, 1992) We first present a linear compartmental model describing human basophiol degranulation by means of an antigen concentration. Then we study an optimal control problem associated to this linear compartmental model in presence of high dilutions. For solving this control problem we used the LAPLACE transform. Key wordr: Modelization; Optimal control; Optimization; LAPLACE transform
1. Introdnetion
The basophil activation is induced by action of antigens. The extra-cellular calcium goes through the cell membrane and neutralizes the negative charges of the granule and precedes the release of active mediators. The high dilution acts on cell for inhibing the activation of basophils. We obtain the Fig. 1. 2. The Human BaaophII Degranalation Test (I-IBDT) Our data JJ,.(Oi) were obtained by using the method described by Benveniste [ 1 ] . The HBDT method permits the count of number (NJ of non reacted basophils and the number (No) of basophils in a control. We note that our high dilution data vary from 1 to 20 Ch (equivalent to theoretical concentrations of low2 to 10V40mol/l) and the antigen dilution varies from 10s4 to 9.5 x 10-‘” mg/ml [2].
VjfDi)
=
CNO
-
Ni) * lOO/No
i= l,...,n j=
(1)
l,...,m
3. Linear ModeIIIog 3. I. Introduction Let us consider a linear compartmental model for human basophil degranulation [ 31 described in Fig. 2, where the compartments represent: (1) percentage of degranulated basophils (Xi); (2) percentage of histamine (X2). Correspondence to: B. Hadjaj, Universite Paris 6, MEDIMAT, 15, rue de 1’Ecole de Mtdecine 75270 Paris Cedex 06, France. 0020-7101/93/SO6.00 0 1993 Elsevier Scientific Publishers Ireland Ltd. Printed and Published in Ireland
B. Haajaj et al.
152
r
I
Dilution
Where
:
Ag
= antigen
*
=
4.
ca++
9
ca++=
basophil calcium
Granule
Fig. 1. Scheme of &granulation of basophil mechanism in presence of high dilution.
3.2. Mathematical modeling The linear differential equations corresponding to the previous system are
-
L
=
k,XI - k2X2
X1 = 0 and X2 = 0 at
(2) t= 0
Fig. 2. A scheme of linear compartmental model.
Control oj basophil &granulation
153
TABLE I
Time (min) 1
3
5
6
7
8
10
X:’
3
20
31
31
37
40
44
X,(q)
6.15
17.34
27.20
31.68
35.88
39.82
46.90
3.2. Numerical results
For solving this problem we used Rung-Kutta’s method [ 41 and we have to idenare as close as tify the kr, k2, kJ, parameters such that the differences (J&-&(t)) possible to zero. The identification problem consists of finding k,, k2, k3 minimizing the functional:
J(W&)
[&(tj) - x&)1 2 where X = (X,;XJ
= i
(3)
j=l
The functional (3) represents the sum of differences between the exact solutions Xc(t) and the experimental data X,. Therefore the direct method of so-called ‘local
variations’ [5] can be used. A variant of the method is the Vignes’s method [6] in which J is minimized in several directions, taking into account variables influences on each other. We obtain the following parameters: kl = 4.1665E-2;
k2 = l.O058E-3;
kl, = 2.2072E-2
(For the results and the comparison of experimental and calculated data see respectively Tables I, II and Fig. 3). In Tables I-IV we defined: Degranulated
basophils experimental data
Degranulated
basophils calculated data
TABLE II
Time (min) 2
4
8
16
32
X5
8
12
23
43
56
Wtj)
7.74
14.55
25.83
41.36
56.30
154
B. Hdjaj
et al.
e
r c au-e n t
40 --
a !J e
20 -,..i"
Od 0
5
10
15
20
25
30
Time(min)
Fig. 3. The solid curve represents a solution of system (2) (with out dilution) and (0) and (D) experimental percentage of basophil degranulation and release histamine respectively.
Histamine experimental data
4. Optimal control [6] 4.1. Introduction Our principal aim is to control the evolution of this system by acting with high dilutions in compartment 1 (XI), which leads a diminishing of degranulation of basophils. The previous compartmental model is transformed according to Fig. 4. The control problem consists of finding a function u(t) taking into account high dilutions concentration such that the basophils activation becomes optimal. Find u(t) such that the following differential system is satisfied:
Fig. 4. A scheme of linear compartmental model in presence of high dilution.
155
Control of basophil akgmnuiation
TABLE III
Time (min) 1
3
5
6
7
8
10
A?:’
2
16
25
25
29
32
35
xl(rj)
4.15
13.52
21.38
25.001
28.42
31.66
37.63
-= -(k,
-
+ k3)X, + k2X2 + u
klX, - k2X2
=
and the following functional is minimized
J(u) =
sTt-,F-
Ye)2dt
(51
0
where Y,: experimental data. For solving this problem, the principle consists of finding an explicit relation, as possible, between Xi and the control u. An expression may be found: X,(t) = F@(t), kg). In order to do this, we used the LAPLACE transform [ 61. 4.2. Numerical mathematical analysis Let us consider the general system: .? = A x(t) + u(t)
(6) such that X and u verified the system and minimizes the following functional:
J(u) = min u(t)
ks1 s O
a is a fixed constant
- a)2dt
defined from biological considerations.
(6’)
The LAPLACE
transform gives:
/f(s) = (sZ - A)-$2(s)
(7)
156
B. Haajaj et al.
We deduce:
where q&,kU) = det(sZ - A) (it’s a polynomial of the nth degree in s) P, _ ,(s,kv) = (it’s a polynomial of the (n - 1)th degree in S)
Therefore the original of Eqn. (8) is: 1 T,,(T) X,(t - T) dr s (in Eqn. 9 the sign * is Convolution product) u(t) = m(t) * X,(t)
=
(9)
where m
r,(t) = k
c =,
%(d cl - 1 (Sk)
eSkt
(10)
Eqn. 9 gives the explicit formula for u(t) XI = F(s,k)d So a precedent minimisation problem becomes:
Hence F(s,kv)d = ?-
s
In conclusion, we obtain the following convolution f(t,Z$) + u(t) = a
equation:
Cf= F)
Applying this to our case, we get:
zd.9= 60) 49
with B(s)=
s + k2 (s _ x,)(s _ x
)
2
where Ai are eigenvalue of matrix A. Therefore 40) = Ai e hlf
+
A2 eh2’
(11)
157
Controlof basophil&granulation TABLE IV
Time (min) 2
4
XS
6
xs(tj)
5.91
8
16
32
18
34
45
20.09
32.709
45.40
9 11.19
where K2 + XI
Al =
XI -
A2 =
x2
K2 + A2 A2 -
h
(12)
The X1(t) = q(t) * u(t) 5. Results and Conclusion
The optimization problem consists of finding u which minimizing the functional J. From Eqn. 12 it may be shown that the optimal solution is such
4(t) * 40 =
ye
(13)
Differentiating Eqn. 13 we obtain:
4’0) * u(t) f q(O)u(t) = q”(t)
ye
* u(t) + q(0) u(t) + q(0) u’(t)
=
Y”,
It may be shown that K(r) is a solution of a following differential equation q(t) + arq’(t) + flq”(t) = 0, that is to say: o!q(O) u(t) + @q’(O) u(t) + /34(O) u’(t)
=
Y, + Y’, + Y”,
(14)
u(t) is solution of a linear differential Eqn. 14 with a coefficients and second term. An explicit solution can be obtained.
TABLE V
Time (min)
U(tj) (10-q
2
4
8
16
20
26
28
0.0991
0.2611
0.4777
0.7419
1.0472
1.3883 1.7606
32 2.1598
B. Hadjaj et al.
158
C
e
40
0
5
10
15
20
25 Time (min)
30
Fig. 5. The solid curve represents a solution of system (4) (with high dilution) and (0) and (D) experimental percentage of basophil degranulation and release histamine respectively.
In Tables III-V the values of X,(t), X*(t) and u(t) are respectively given (the graphic comparison for differential system (4) see Fig. 5). In conclusion, the results given by this optimal control allow to justify the effect of high dilutions on the basophils activation. Indeed the comparison of the degranulation curves with diltuion (Fig. 6, curve 2) and without dilution (Fig. 6,
c w-e
_______--------
n t
40 --
: e
2o 0 0
5
10
15
20
25
Time (mtn)
~
Curve n*l ------
Curve no2
Fig. 6. Percentage of degranulation (curve 1) without dilution and with dilution (curve 2).
30
Control of basophil &granulation
159
curve l), proves an effect. We can see that the percentage of degranulation is decreasing and that the dilution acts on the system. But at the same time another effect arises on the histamine release. We do not know enough arguments for explaining the action of high dilutions on the histamine release. We can just formulate many hypothesis impossible to justify. References 1
2 3 4 5 6
Benveniste J: Human basophil degranulation as in vitro test for the diagnosis of allergy. C/in. Alfergy 11 (1981) I-11. Guellal S et al.: Human basophil receptivity. Comput Math Applic Vol. 20 (1990). Cherruauh Y: Biomath&natiques. P.U.F. Coil. Que sais-je? (1983). de calcul numbique. Edit du P.S.1, Tome 1 (1982). Nowakowski C: h4kthodes Vignes J: ‘Algorithmes numeriques. Analyse et mise en oeuvre.’ Ed Tech, 1980. Cherruault Y: Mathematical Modeling in Biomedicine. Optimal Control of Biomedical Systems, Reidel, New York, 1986.