dournal o[
J. Math. Biology (1981) 12:363-373
Mathematical
Biology
by Springer-Verlag1981
A Model of the Role of Natural Killer Cells in Immune
Surveillance - I*
Stephen J. Merrill
Division of Applied Mathematics, Lefschetz Center for Dynamical Systems, Brown University,
Providence, RI 02912, and Department of Mathematics, Statistics and Computer Science, Marquette
University, Milwaukee, WI 53233, USA
Summary. The theory of immune surveillance of Thomas and Burnet stated in
part that antigenic differences between neoplastic and normal cells provide the
stimulus for their destruction by cells of the immune system. Burnet pointed to
the T lymphocyte as the cell which mediated this surveillance. The existence of
some form of surveillance in cases of no Tlymphocyte functioning presents the
possibility that surveillance, if present at all, is mediated by non T cells.
Cells identified as naturally cytotoxic killer (NK) cells appear to have
properties required of a surveillance effector population. This paper utilizes
properties of N K cells and the effects of interferon on this population to
construct a mathematical model of the characteristics that an NK cell
surveillance would have. A two level theory of immune surveillance is proposed.
Key words: N K cells - Immune surveillance
O. Introduction
The theory of immune surveillance of cancer suggested by Thomas 1"35]and Burnet
12] and developed by Burnet 1,3], 14] proposed that changes which take place on the
surface of a neoplastic cell are utilized by cells of the immune response to eliminate
those neoplastic colonies in much the same way as transplants are rejected.
Although Tlymphocytes (T-cells), which are primarily responsible for the rejection
of most transplants, have been shown to be cytotoxic to tumors 1'131 resistance to
tumors whose surface antigens (TAA) are weakly antigenic (e.g., spontaneous
tumors 1191) would not readily be recognized and eliminated by T-cells. In nude
mice, where T-cells are naturally absent due to lack of a thymus, spontaneous
tumors are exceedingly rare 1"30] and occur no more frequently than spontaneous
tumors in mice with functioning T-cells. Taken together, this suggests that at least
for spontaneous tumors and others whose TAA are weak antigens, other
mechanisms are responsible for antitumor defense.
* This research has been supported in part by the National Science Foundation under grant # NSFEng. 7904852
0303 -- 6812/81/0012/0363/$02.20
364
S.J. Merrill
The discovery of the existence of naturally cytotoxic cells [14], [26] pointed to
the possibility of a new mechanism for resistance against tumors. Natural
cytotoxicity is defined as cytotoxicity by a population of cells from peripheral blood
and lymphoid organs against various tumor ceils which is independent of any
antigenic priming. The recognition of strong natural cytotoxicity in nude mice and
the susceptibility of most spontaneous tumors to N K lysis gives a reasonable
explanation for in vivo data concerning tumor development and growth in these
immunologicaUy compromised individuals.
In this paper, a model of surveillance as it would be mediated by N K ceils is
proposed and analyzed. As a result, a modification of the theory of immune
surveillance is suggested which recognizes the (nearly) independent contributions to
tumor resistance by both T and N K populations.
Other mathematical models which describe interactions of tumor cells with cells
of the immune system include Lefever and Garay [21], [22], Garay and Lefever
[10], Grossman and Berke [11], Albert, Freedman and Perelson [1], DeLisi and
Rescigno [7], Reseigno and DeLisi [28] and Merrill [24]. Several of these efforts
were directed at the description of surveillance and the failure of surveillance
("sneaking through").
Mathematics used ~n cancer research has been surveyed by Swan [33], Eisen [9]
and Thames [34].
1. Properties of N K CellsNatural killer (N/Q cells are described as a collection of cells primarily found in the
spleen and peripheral blood with the ability to lyse some tumor cells without
antigen priming. The kinetic difference between NKcells and T~cells in their action
against in vitro tumor cells illustrates the differences. If N K cells are incubated with
susceptible tumor cells, lysis of tumor cells will begin almost immediately and
continue at a fairly constant rate. T-cells incubated with tumor will however not
immediately lyse tumor. Lysis will begin only following a delay (the priming) after
which the rate of lysis will rise quickly, reach a peak and then decline. The parallels
between this and the mechanism involved in the rejection of transplants gives one of
the arguments for the existence of some kind of T-cell mediated surveillance in vivo.
Because N K cells are functionally defined, their description morphologically has
been difficult. They are now known to be mononuclear cells originating in the bone
marrow [29]~ Surface markers which would indicate the hemopoietic origin as
lymphoid are lacking or deficient, although one theory of their origin is as prethymic T-cells [15]. Other evidence suggests that N K cells may be promonocytes
[23]~ Recent evidence from the mouse N K cell [20] suggests that N K cells are a
heterogeneous class of cells which need not have a single origin.
The identification of "large granular lymphocytes" (LGL) as the primary
human N K cell by Timonen and Saksela and the description of their response to
contact with tumor cells [36] gives the biological basis for the construction of this
model.
Timonen et al. [36] found that LGL from human peripheral blood produced
interferon when in contact with tumor cells. One effect of this interferon is to
augment the cytotoxic activity of inactive "pre-NK" cells. The amplification of the
The Role of Natural Killer Cells in Immune Surveillance
365
cytotoxic response and the reduced tumor growth rate due to this endogenous
interferon [8] is assumed in this model to be the main mechanism of N K mediated
tumor surveillance. Interferon may also reduce the NK killing of tumor cells,
especially evident at concentrations which may be present in clinical trials. It has
been suggested [37] that this may be due to difficulty in NK-tumor binding caused
by the interferon.
Although in vitro anti-tumor action by N K cells is not questioned, the in vivo
role of these ceils in tumor defense is not established~ The question has been
primarily examined by correlating resistence to tumor challenge to in vitro NK
activity. High in vitro N K activity correlates strongly with tumor resistance in vivo
if the tumor is small (103 - 104 cells in the mouse) [ 18]. Tumor types which escape in
vivo in general have not been susceptible to N K lysis in vitro [15]. These findings
suggest that in vivo, NK cells are at least participating in the activity against tumors
recogn~ed by these cells.
N K ~ctivity in vitro can be suppressed by suppessor T-cells and macrophagelike cells [6]. As the dynamics of the suppressor cell role in the sneaking through
phenomenon are still cloudy [25], the model constructed here assumes very general
terms for the dynamics of tumor growth and elimination as well as the dynamics of
interferon production due to tumor-NK cell contact so that most types of
interaction will be included.
N K cells seem to include K cells, those cells which mediate antibody dependent
cytotoxicity (ADCC) [15]. This model will assume there is no anti-tumor antibody
present in the system (or equivalently, if present, that its concentration is constant
in time).
2. Construction of the Model
The assumptions used in the construction of this model are:
1) The source of pre-NK ceils is the bone marrow. In the absence of any
stimulation or suppression of the marrow, pre-NKcells appear at constant rate $1.
2) The rate of maturation of pre-NK to NK cells is an increasing function of
interferon concentration. In the absence of interferon, this rate is proportional to
pre-NK cell concentration.
3) N K cells produce interferon in contact with tumor. This interferon acts on
pre-NK cells and the tumor. Interferon concentration is nearly homogeneous
spatially (well-mixed body).
4) Tumor growth is affected by interferon present and the rate of cytotoxicity
by N K cells. Other mechanisms which affect tumor growth are assumed to be
approximately unchanging in relative magnitude during the lifetime of the model.
By this is meant that all other effects are approximately in the form ~Tfor cceither a
positive or negative constant.
Assuming the law of mass action, the equation governing the pre-NK
population, NKv, is
dNKp. = $I - k I N K p l F - kzNK v.
dt
(1)
In (1), IF is the concentration of interferon, St, kl and k2 are positive constants.
366
S~ J. Merrill
k2NK v is the rate of pre-NK death plus baseline pre-NK maturation (not interferon
dependent). N K f l F is proportional to the rate of pre-NK contact with interferon,
which is assumed proportional to the increase in pre-NK maturation.
The "mature" N K cell population, NKm, similarly has dynamics governed by
dNKm
dt = k t N K p l F - k3NK~ + k'2NKp
(2)
where k~ and k3 are positive constants, k3NKm gives the rate of exit from the mature
NK class (through death or further differentiation) and k'2NKp is the entrance rate
into the mature class by natural maturation (not interferon dependent).
The equation governing interferon concentration is
dlF
- - ~ --- Sz - k j F + ks(NK., T) - k'INKplF - kv(IF, T)
(3)
where $2 and k] are non-negative constants and k4 is a positive constant. Sz is the
rate of (external) supply of interferon, k~ the removal or decay rate, k'~NKgF gives
the rate of interferon removal due to interaction with pre-NK cells, ks(NK~, T) is a
non-negative function which is proportional to the rate at which interferon is
produced when NKm mature NKcells and T tumor cells are present, k7 is a general
function which gives the rate at which interferon is removed by tumor-interferon
contact. A basic assumption of this model is that the arena of the interactions is a wellmixed body (by blood and lymph circulation) and local variations in interferon
concentration is assumed to be of short duration and not essential in the process. The
functions ks and k~ in the analysis will be assumed to satisfy
0ks
aks
ks(O, T) = O,
ks(NK,~, O) = O,
~
> O,
> O,
ONI~
aT
kT(o, 73 = o,
kT(ZF, O) = O.
(4)
The equation governing the population of tumor cells, T, is assumed to be
dT
- ~ m r(IF, T) -- k6(NK., IF, T)
(5)
where 7 is a function of interferon concentration and tumor population expressing
the growth rate and ks is a non-negative function of NK,., IF and T expressing the
removal rate by mature N K cells. The functions y and k6 are assumed to satisfy
y( IF, O) = k6(O, IF, 73 = ks(NK., IF, O) = O,
Ok6
ONK,,, > 0
and
Ok6
0---T> 0.
3. Analysis
The model developed in the previous section is
dNKp
- ' ~ = $1 - kxNKvlF - k2NKp,
dlF <~O,
(6)
The Role of Natural Killer Cells in ImmuneSurveillance
367
dNKm
dt = k l N K p l F - k3NK,, + k'2NKp,
dlF
= $2 - k, I F + ks(NKm, ~ - k'INKplF- kT(IF, 7~,
,it
dT
- - = ~(IF, T) - k6(NK., IF, T)
dt
(7)
with constants St, $2, kt, k't, k2, k~, k3, k,/> 0. Functions k~, k6, k7 and ~,satisfying
(4) and (6) and initial conditions NKp(O), NK.(O), IF(O), T(O) >10.
The analysis of (7) will follow the natural functioning o f the system. As the
natural state has no tumor present, the analysis begins by examining the three
equations derived from (7) when T - - 0 with the same assumptions on the
parameters"
dNKp
d--~- = $1 - k I N K p I F - kzNKp,
dNK~
dt = k l N K p l F - k3NKm + k'2NKp,
dlF
- ~ = $2 - k , I F - k]NKpIF.
(8)
Define a box, B, in NKp, NK,,, IF space by
B= {(NK,,NK.,,IF)IO<~NK, <<~,O<< NK. < ktStS______22 k'2St
k2kak,+k-~'O'lF<'~,}"
Theorem 1. Solutions to (8) whose initial conditions (NK~(O), NK,(O), IF(O)) ~ B exist,
are unique and for all t > O, (NKp(t), NK.,(t), IF(t))eB.
Proof Follows from the standard existence and uniqueness theorems [5], ['12] and
from the vector field defined by (8) always pointing inward on surface of the
boundary of B.
As the solutions beginning in the positively invariant region B are constrained to
that compact set for all t > 0, as t--* oo, each solution must approach some
connected, compact set, its co-limit. The simplest possible is an equilibrium point,
and there is a unique equilibrium point to (8) lying in the box, call it (Xo,Yo, Zo).
Linearizing (8) about this point we find that
(
NKp- xo,'
( - ( k l z o + k2)
NK., - Yo] ,~
klzo + k'2
i F - Zo /
- k' Zo
0
- k3
0
-ktxo.
,/NK~- xo,
k,xo
l [ N K . - Yo]
- (k, + V, x o ) l \ X F /
when (NKp, NK., IF) is close to (xo, Yo, Zo). The eigenvalues of the Jacobian matrix
are the roots of
(2 + k3)(~,2 + (k~zo + k2 + k'xxo + k,)2 + klk,zo + k'lk2x o + kzk,) = O.
368
S.J. Merrill
By the assumption on the parameters, all roots have negative real parts. The critical
point (Xo, Yo, Zo) is thus locally asymptotically stable. In fact, the following holds:
Theorem 2. I f (NKp(t), NKm(t), IF(t)) is a solution o f (8) with (NKp(O),
NKm(O), 1F(O)) >>,O, then
(i) ( N K p ( t l ) , N K ~ ( t l ) , I F ( t l ) ) ~ B for some tl >i 0 and for all t > tl, and
(ii) lira,_ ~(NKp(t), NKm(t), IF(t)) = (xo, Yo, Zo).
Theorem 2 implies that (xo,Yo, Zo) attracts everything in the positive octant.
Proof Let
2 = N K v - Xo,
= NKm
-
Yo,
~,=IF-zo
~.
and define
v(2,37, z') =
where
kl gz
k•
b=
2 z + b2~ + k'~
'
2k~xo + 2klzo
k l g o + k 2 + k,, -b k'lX o "
In coordinates 2,)~, L (8) becomes
= - k12~. - klzo2 - klxo~. - k22,
= k12~. + k : o 2 + k~xo3. - k3P + k'22,
(9)
= - k , 2 - kt2~. - k'lzo2 - k'~xo2
and (0, 0, 0) is the unique critical point of the system in the region of interest
(3 >/ - Xo, 37 t> - Yo, z >t - zo). As the derivative of the positive semi-definite
function V along a solution of (9) [5] is
dV
dt
dV(2(t),fi(t),~.(t)
dt
(2)2
2ki(z o + ~
bki(zo + ~
+ (z')2[ - 2 k l ( x o + 2 ) - b k l ( x
2
k2
o +2)-2~k,~]
~<0
in the region of interest, the lim,_~ V(t) exists and is 0. Let (2(t),y(t), 3.(0) be a
solution of (9) in this region, then lim,_~(2, 37,z') = {(0, 37,0)137>t - Yo} = E. As the
a~-limit of (2(t),y(t),3.(t)) must itself be invariant and (m-limit) = E, the w-limit
must be an invariant set inside E. By examining the second equation of (9) when
2 = 5 = 0, we find that 37 = 0 is the only invariant set contained in Eo We have
shown that limt_~(2(t),5(t))= (0,0). Thus 2 and 5 must stay bounded. This
implies in (9) that )7 also stays bounded, and )7(t) ~ 0 as t --, oo. In coordinates
The Role of Natural Killer Cells in Immune Surveillance
369
(NKp, NK~,, IF), as (Xo, Yo, Zo) corresponds to (0, 0, 0) and lies inside B, the theorem
is proven.
We
now
examine
(7).
Critical
points
of this
system
must
satisfy
= 7(IF, T) - k6(NK,,, IF, T) --- 0. Expanding this difference in a Taylor series
about T = 0 (using (6))
v(IF, T) - k6(NK--,n,IF, T) = v(IF, O) - k6(NKm, IF, O)
~k6
T ( dV (IF, O) --~(NKm, IF, O))
+ kOT
/
a2y
Tz
a2k6
+
I
(IF, r - -ff-fT (NXm, IF, r
for some r e (0, T),
{ a~'(/F, O)
= r {.dr
Ok~
T
az~,
c32k6
---~(NK,,,,IF, O)+-.~.(-~--s162162
Thus T = 0 is always a solution of T = 0 and as a result, (Xo, Yo, Zo, 0) is always one
critical point of (7). Linearizing about this point the Jacobian matrix is
r-(k'z~
-k30
-k~xo
ktxo
Ok.~
-ktzo +k'2
3k~
~
/Ok5
]
Ok~
"~
From (4) and (6),
ak5
Ok6
Ok7
..-3V (zo, 0) = ak6
O---~(y o, O) = O---~=(yo, Zo, O) = -~(Zo, O) = ale
-b-~(yo, Zo, O) = O.
As a result, the eigenvalues of A consist of the eigenvalues from the linearization of
(8) about (Xo, Yo, zo) with the addition of
aV .
Ok6
,to = b--f tzo, 0) - - ~ (yo, zo, 0).
We have
Theorem 3. The critical point (xo, Yo, Zo, 0) of(7) is locally asymptotically stable if
2o < 0 and unstable if;to > 0. Moreover the stable manifold of (xo, Yo, zo, O) always
contains the manifoM T = O.
Biologically Theorem 3 says that if (xo, Yo, Zo) lies in a region where 2o < 0 (20
being determined by the growth characteristics of the tumor and its susceptibility to
NK lysis), small tumors will be eliminated with the result that the system returns to
(Xo, Yo, Zo, 0).
Description of the curve 2o = 0. Set
20 = h(IF, NK,,,) =
0
Ok6
(IF, O) - --~ (NKm, IF, 0).
3~
S.J. Memll
IF
~o < 0
).o > 0
|
~r
|
NKm
Fig. 1. A typical curve ;to = 0 separating the positive octant into regidns
where (NK,,,,IF) -- (Yo,Zo) would determine the stability of (Xo,Yo, Zo, 0)
according to Theorem 3
h is determined by the response of a particular tumor to interferon and its
susceptibility to NK lysis. These parameter functions should be able to be
approximated in vitro and this model would then be able to predict the probability
that a tumor of that type would escape the N K surveillance once an individual's
status (Xo,Yo, Zo) is determined. Fix IF > 0o As ~h/ONK,, < 0, either there exists a
unique number ~ ( I F ) > 0 such that h(IF, ~ ( I F ) ) = 0 or no such S~ exists. If
h([F, 0) > 0 and no ~ ( I F ) exists for any IF > 0, (Xo,Yo, zo) is always unstable and
the tumor will evade extinction~ If h(IF, 0) < 0, for all IF and no ~ ( I F ) exists,
(xo, Yo, zo, 0) is the only critical point and Tis always decreasing. On the other hand,
if for some IF = IFo, ,9'(IFo) exists, h(IF, ~ ) = 0 can be solved for ~ as a function
of IF near (IFo, ~(IFo)) since c~h/O~ < O.
Figure 1 displays the relationship between NK,, and IF along such a curve
;to = 0 if h(IF, 0) > 0 for some IFo > 0 and c~h/alF (IFo, SP(IFo)) < 0~
4. Discussion
According to the model presented here, tumors satisfying NK susceptibility
parameters of Theorem 3, ensuring ;to < 0, would be eliminated by this natural
cytotoxic mechanism. A second part of that theorem (that T = 0 comprises the
stable manifold in the case ;to > 0) predicts that a tumor which has ;to > 0 will never
be totally eliminated by this mechanism naturally.
It is expected that the parameters Yo and Zo can be determined by standard
methods outlined in the cited literature while Yo + xo should be the total number of
NK active cells after treatment withsuitable levels of interferon~
The computation of ;to for a particular tumor in a particular individual may be
accomplished by estimating the relative growth rate for small tumors in the
presence of interferon at concentration Zo but no N K cells as (gy/aT)(zo, 0)
y(Zo, s)/a for small tumor population s. The assumption of exponential growth for
small tumor size is most likely appropriate. (ak6/~T)(yo, Zo, 0) involves estimating
The Role of Natural Killer Cells in Immune Surveillance
371
the mean survival time of a tumor in the presence of the appropriate concentration
of N K cells and interferon as (Ok6/OT)(yo, Zo, 0) ~ k6(Yo, Zo, ~)/e.
This model may be tested experimentally in vivo by artificially altering
parameters xo,Yo,.Zo to attempt to modulate tumor viability in a nude mouse.
N K Role in Immune Surveillance
In the model developed here, N K cells would effectively eliminate tumors with
growth and N K susceptibility parameters satisfying the conditions given in
Theorem 3. Any tumor which escapes this mechanism of defense would still have to
deal with the immune system side, mediated primarily by T cells. Tumors that lack
sufficient antigenicity, however, would most likely encounter only the N K cells and
conversely, tumors resistant to N K cells would primarily encounter the T cell
mediated mechanism.
In this light, "immune surveillance" can best be seen as the result of two
essentially independent anti-tumor mechanisms, each recognizing and responding
to different structures on the surface of a neoplastic cell.
Given an individual whose N K status and immune status are known, the types
of tumors which could escape both N K and T surveillance can be commented on.
Each tumor has a susceptibility to N K lysis and a susceptibility to lysis by T-ceU
mediated mechanisms (roughly the antigenicity if the tumor has not been
previously encountered and blocking effects ['24] are not considered). The tumor
also has growth kinetics determined by the characteristics of the tumor in the
environment in which it is placed (including interferon present, natural antibody
present and other parameters). In order to survive, the tumor must have the N K
status of the individual in the ,lo > 0 region of (NK~, IF) plane and must be able to
grow to a critical mass before the priming of T-cells would enable the elimination by
that mechanism. How the antigenicity affects the time that the tumor has to reach
critical mass is a fundamental question in this area. Also, the stochastic nature of
this event most likely also plays a critical role.
An analogy of Prehn [27] given in 1970 can be easily extended to explain the
interaction between the two systems.
The analogy can be made to a very large warehouse with multitudinous roomsfilled
with combustible materials. Spontaneous combustion is frequent and, in addition,
arsonists (represented by viruses, radiation and chemical oncogens) lurk in the halls
and passageways.
In this analogy, N K defense can be represented as smoke detectors whose
triggering immediately starts an automatic sprinkling system. The T-cell defense is
represented by heat detectors, which are connected to alarms in the fire department
some blocks away. In each case, there are certain fires that are detected sufficiently
early by one but not the other. The independent nature of the systems enhances the
reliability and for most challenges the systems will cooperate.
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Received November 18, 1980/Revised February 5, 1981