BULLETIN OF XATHEMATICAL BIOLOQY VOLUME 39, 1977
IMMUNE SURVEILLANCE AND NEOPLASIA-II A TWO-STAGE MATHEMATICAL MODEL
q ALDO RESCI~NOand &LES DELISI Laboratory of Theoretical Biology, National Cancer Institute, Bethesda, MD 20014, U.S.A.
In a previous paper (DeLisi and Rescigno, 1977) a model for the interaction of tumor cells and killer lymphocytes was presented. Although that model was highly simplified, the qualitative behavior was in accord with intuitive expectations and a wide range of data, It could not however account for de nova tumor development. In this paper a slightly more realistic model is presented by introducing a delay in the formation of killer lymphocytea. This is done by requiring two stages in the production of a killer. We show that introduction of this second stage allows tumor development from even a single cell, thus removing an important limitation of two variable systems.
1. Introduction. In a previous paper (DeLisi and Rescigno, 1977) we described a mathematical model of carcinogenesis, in which the tumor cells elicit an immune response. While many inferences of that model were in agreement wifh the experimental evidence, one difficulty was the inability to predict de novo tumor development. Here we show that this difficulty vanishes when the model is made slightly more realistic by recluiring that lymphocytes go through two stages of development, and that only lymphocytes in the second stage (“activated” lymphocytes) are effective in killing tumor cells. Let L1 and Lz denote the number of lymphocytes in stages one and two respect,ivel>-; let C and C; denote the total number of tumor cells and the number of free cells (i.e., not bound by lymphocytes), respectively. We make now the following assumptions : the target, or stage 1, lymphocytes are produced at, a fixed rate (21Lo) (in the absence of tumor cells), plus a rate proport,ional to the product of the number of free tumor cells and killer (stage 487
488
AID0
RESCIGNO AND CHARLES DELIS1
2) lymphocytes. Thus the target population is reseeded by transformation of the immunologically active population (Bell, 1970, Cline, 1975). This last relative rate, decreases as the number of killer lymphocytes increases, and we introduce an exponential factor a; exp ( -L&C,), representing the saturation of the system with these lymphocytes. Stage 1 lymphocyte8 are transformed at a fkd rate (Al); and stage 2 die at a fixed rate (Q). Tumor cells reproduce with a rate constant 122only if they are free, and are killed with a rate co&ant a; by mature lymphocytes. In addition one expects, on biological grounds, some delay in the formation of L2 by tumor cell stimulation. As an approximation to this delay we require that the progeny of LZ musf first pass through stage LI before becoming mature lymphocytes. With these a8sumption8, the equation8 of the model are :
Ll = L2
=
-ML1
MI
-LoI
+a;CfL2
exp
(-L2/&),
-42,
Strictly speaking, the parameter Al in the first two equations should be replaced by two distinct parametera, since L1 is lost by death a8 well as by conversion to L2. However under the conditions of interest, loss due to the former should be relatively small and we therefore ignore it. If the relation between free tumor cells and stage 2 lymphocyks is equilibrium controlled, (Bell, 1973; DeLisi & Rescigno, 1977) then
K =
(fJ-CdlW2,
with K constant; thence
C, = C/(1 +KL2) and the differential equations become
x1 =
-Ll(Ll-Lo)+a;
1 $TfL exp ( -L2/-U, 2
22 = AILI -asL2, 0
12 - a:L,
= l+KL2
c.
With the substitution8
X = KLl, Y = KL2, Z = KC, a1 = a;/K, a2 = .;/K, x0 _ KLo,
yc
I
KL,,
IMMUNE
SURVEILLANCE
AND
NEOPLASIA
489
the equations become Xt=
YZ
- Izl(X -X0)
+ a1 -exp
(-Y/Ye),
1+Y
(1)
P = 11X-a3Y,
jJ=
12-a2Y
(2)
2.
1+Y
(3)
2. Qualitative Solution. In this section we analyze the qualitative behavior of the solution of (l), (2), (3) under different conditions of parameters. The analysis will be confined to the first octant of the X-Y-Z space, aa only nonnegative values of X, Y, 2 are of interest ; also we consider only positive values of the parameters 011,~12,tls, Al,&, Xc, Ye. The equations above have two singular points, A and B, with coordinates: Point A: XA = X0, YA = ’
X0, & = 0,
a3 a3l2
Point B: XB = -
yB
a221 '
=412,.& a2
= ~(l+~)($-X0)exp($~).
Point B is in the first octant only if X0
First case.
a&/a&l.
(5)
In this case A must be globally stable, i.e., for all possible initial conditions in the first octant, the system always moves toward A. In fact integrating (1) , -w
+ a1 exp [ therefore,
ts
= X(t0) exp [ -i.l(t-to)]
id]
to
+iZlXf~exp [ -Ilt]
exp [&I
X(t) >=SO Sexp [ -Al(t
t
s to
exp [AIT]dz
Y(Mr) eq [ - Y(z)/Ye] d7 I+ Y(t)
-to)][X(to)
thus, we can find a time tl such that for any t > tl s(t)-So > - E, where E > 0 is an arbitrary number.
-X0];
ALDO ~S~~NO
490
IdqM.ng
W) =
AND ClEAFiLES DELIS1
(2), Yftlf~xp[-as(t-tl)]f11exp[-a~t]
ts ts tt
thtXlCt3
Y(t) > Yfti) exp [-as@-h)] +A1 exp [ -a$]
expEwl X(4 dz,
expEwl
(XO
- e) dz,
t1
and
Y(t) -$ x0 > [Y(t+$ (X0-e)
exp 1
[-as(t-tl)l-$&.
Therefore given an arbitrary number 6 > 0, we can find a tg such that for anyt > $2,
Y(t) -: x0 >
-6.
Ifwe now choose
6
=-
A1
A2
x0--& aa
we can
find a time tp,suchthat, for any t > tz, Y(t)-$X0
>&$X0,
i.e., Y(t) > &/a2, and for (3), % < 0 ever after. &z& case. X0 < a&/a& Inthiscase
(Two singularitiesin the firsto&ant).
Therefore&small perturbationof Z from ZA J 0 makes & > 0, i.e., point A is unstable. We have seen in the previous section that in this case point B is ia the first o&ant ; the Jacobian of (1)) (2)) (3) at PointB is
-as
IMMUNE
SURVEILLANCE
AND
NEOPLASIA
491
and the characteristic equation 1
1
‘+_
y,-YB
l+YB
The conditions of Hurwitz (18%) for the stability of point B are (1) XB-XO
> 0,
The first condition is inequality (4). The second is weaker than the third. The third can be written x,“-“x,
&(2&2;
but the right hand side above is positive if and only if l+yB
a2 ->1+a3 +A1
’
yc
thus the conditions for the stability of point B can be summarized as follows : a2 -
0, g changes from + to - , and vice-versa.
Figure 1. The transitions where two or three derivatives change sign simultaneously are extremely improbable and can be neglected. When point A is stable, i.e., when inequality (5) holds, no trajectory can be in zone VI, because in that zone (l), (2), (3) imply
x > x0,
Y > (Il/cQ)X, Y < 1,2/cQ,
thence a3A2
Xo
IMMUNE SURVEILLANCE AND NEOPLASIA-II A TWO-STAGE MATHEMATICAL MODEL
q ALDO RESCI~NOand &LES DELISI Laboratory of Theoretical Biology, National Cancer Institute, Bethesda, MD 20014, U.S.A.
In a previous paper (DeLisi and Rescigno, 1977) a model for the interaction of tumor cells and killer lymphocytes was presented. Although that model was highly simplified, the qualitative behavior was in accord with intuitive expectations and a wide range of data, It could not however account for de nova tumor development. In this paper a slightly more realistic model is presented by introducing a delay in the formation of killer lymphocytea. This is done by requiring two stages in the production of a killer. We show that introduction of this second stage allows tumor development from even a single cell, thus removing an important limitation of two variable systems.
1. Introduction. In a previous paper (DeLisi and Rescigno, 1977) we described a mathematical model of carcinogenesis, in which the tumor cells elicit an immune response. While many inferences of that model were in agreement wifh the experimental evidence, one difficulty was the inability to predict de novo tumor development. Here we show that this difficulty vanishes when the model is made slightly more realistic by recluiring that lymphocytes go through two stages of development, and that only lymphocytes in the second stage (“activated” lymphocytes) are effective in killing tumor cells. Let L1 and Lz denote the number of lymphocytes in stages one and two respect,ivel>-; let C and C; denote the total number of tumor cells and the number of free cells (i.e., not bound by lymphocytes), respectively. We make now the following assumptions : the target, or stage 1, lymphocytes are produced at, a fixed rate (21Lo) (in the absence of tumor cells), plus a rate proport,ional to the product of the number of free tumor cells and killer (stage 487
488
AID0
RESCIGNO AND CHARLES DELIS1
2) lymphocytes. Thus the target population is reseeded by transformation of the immunologically active population (Bell, 1970, Cline, 1975). This last relative rate, decreases as the number of killer lymphocytes increases, and we introduce an exponential factor a; exp ( -L&C,), representing the saturation of the system with these lymphocytes. Stage 1 lymphocyte8 are transformed at a fkd rate (Al); and stage 2 die at a fixed rate (Q). Tumor cells reproduce with a rate constant 122only if they are free, and are killed with a rate co&ant a; by mature lymphocytes. In addition one expects, on biological grounds, some delay in the formation of L2 by tumor cell stimulation. As an approximation to this delay we require that the progeny of LZ musf first pass through stage LI before becoming mature lymphocytes. With these a8sumption8, the equation8 of the model are :
Ll = L2
=
-ML1
MI
-LoI
+a;CfL2
exp
(-L2/&),
-42,
Strictly speaking, the parameter Al in the first two equations should be replaced by two distinct parametera, since L1 is lost by death a8 well as by conversion to L2. However under the conditions of interest, loss due to the former should be relatively small and we therefore ignore it. If the relation between free tumor cells and stage 2 lymphocyks is equilibrium controlled, (Bell, 1973; DeLisi & Rescigno, 1977) then
K =
(fJ-CdlW2,
with K constant; thence
C, = C/(1 +KL2) and the differential equations become
x1 =
-Ll(Ll-Lo)+a;
1 $TfL exp ( -L2/-U, 2
22 = AILI -asL2, 0
12 - a:L,
= l+KL2
c.
With the substitution8
X = KLl, Y = KL2, Z = KC, a1 = a;/K, a2 = .;/K, x0 _ KLo,
yc
I
KL,,
IMMUNE
SURVEILLANCE
AND
NEOPLASIA
489
the equations become Xt=
YZ
- Izl(X -X0)
+ a1 -exp
(-Y/Ye),
1+Y
(1)
P = 11X-a3Y,
jJ=
12-a2Y
(2)
2.
1+Y
(3)
2. Qualitative Solution. In this section we analyze the qualitative behavior of the solution of (l), (2), (3) under different conditions of parameters. The analysis will be confined to the first octant of the X-Y-Z space, aa only nonnegative values of X, Y, 2 are of interest ; also we consider only positive values of the parameters 011,~12,tls, Al,&, Xc, Ye. The equations above have two singular points, A and B, with coordinates: Point A: XA = X0, YA = ’
X0, & = 0,
a3 a3l2
Point B: XB = -
yB
a221 '
=412,.& a2
= ~(l+~)($-X0)exp($~).
Point B is in the first octant only if X0
First case.
a&/a&l.
(5)
In this case A must be globally stable, i.e., for all possible initial conditions in the first octant, the system always moves toward A. In fact integrating (1) , -w
+ a1 exp [ therefore,
ts
= X(t0) exp [ -i.l(t-to)]
id]
to
+iZlXf~exp [ -Ilt]
exp [&I
X(t) >=SO Sexp [ -Al(t
t
s to
exp [AIT]dz
Y(Mr) eq [ - Y(z)/Ye] d7 I+ Y(t)
-to)][X(to)
thus, we can find a time tl such that for any t > tl s(t)-So > - E, where E > 0 is an arbitrary number.
-X0];
ALDO ~S~~NO
490
IdqM.ng
W) =
AND ClEAFiLES DELIS1
(2), Yftlf~xp[-as(t-tl)]f11exp[-a~t]
ts ts tt
thtXlCt3
Y(t) > Yfti) exp [-as@-h)] +A1 exp [ -a$]
expEwl X(4 dz,
expEwl
(XO
- e) dz,
t1
and
Y(t) -$ x0 > [Y(t+$ (X0-e)
exp 1
[-as(t-tl)l-$&.
Therefore given an arbitrary number 6 > 0, we can find a tg such that for anyt > $2,
Y(t) -: x0 >
-6.
Ifwe now choose
6
=-
A1
A2
x0--& aa
we can
find a time tp,suchthat, for any t > tz, Y(t)-$X0
>&$X0,
i.e., Y(t) > &/a2, and for (3), % < 0 ever after. &z& case. X0 < a&/a& Inthiscase
(Two singularitiesin the firsto&ant).
Therefore&small perturbationof Z from ZA J 0 makes & > 0, i.e., point A is unstable. We have seen in the previous section that in this case point B is ia the first o&ant ; the Jacobian of (1)) (2)) (3) at PointB is
-as
IMMUNE
SURVEILLANCE
AND
NEOPLASIA
491
and the characteristic equation 1
1
‘+_
y,-YB
l+YB
The conditions of Hurwitz (18%) for the stability of point B are (1) XB-XO
> 0,
The first condition is inequality (4). The second is weaker than the third. The third can be written x,“-“x,
&(2&2;
but the right hand side above is positive if and only if l+yB
a2 ->1+a3 +A1
’
yc
thus the conditions for the stability of point B can be summarized as follows : a2 -
0, g changes from + to - , and vice-versa.
Figure 1. The transitions where two or three derivatives change sign simultaneously are extremely improbable and can be neglected. When point A is stable, i.e., when inequality (5) holds, no trajectory can be in zone VI, because in that zone (l), (2), (3) imply
x > x0,
Y > (Il/cQ)X, Y < 1,2/cQ,
thence a3A2
Xo
