Analysis of a lumped model for tissue inflammation dynamics

Analysis of a lumped DOUGLAS Department

A. LAUFFENBURGER. AND CLINTON of Chemical and Biochemical Engineering,

University of Penmyluania, Receiwd

Model for Tissue Inflammation

Philadelphia,

Penn.yloania

Dynamics

R. KENNEDY+ 19104

6 Mqv I980

ABSTRACT The inflammatory

response

to bacterial

invasion

of tissue is a complex

combination

of

chemical and physical events, and the outcome of a particular challenge is dependent upon the interaction of these. Thus a wide variety of behavior can be observed for the operation of this response. In this paper we develop and analyze a mathematical model for a generalized inflammatory response to bacterial invasion of a tissue region assumed to be homogeneous on a macroscopic scale, in order to study the dynamical system. Our analysis allows interpretation of the outcome of a challenge

behavior of the in terms of key

parameters representing the rate processes involved. It demonstrates how abnormalities these processes can lead to pathological behavior. Numerical values for the parameters estimated from experimental literature sources, and used in example illustrate the predictions of the model under physiological conditions.

1.

in are

computations

to

INTRODUCTION

Inflammation has been a well-known phenomenon from early times, although its purpose and function have been a subject of debate ever since (it is instructive to read the turn-of-the-century arguments of Met&n&off [22]). It has been established that inflammation is primarily a defensive mechanism against tissue infection (see, for example, [31]), but sometimes becomes a pathological condition in itself [28]. The central role of the process in response to tissue invasion by bacteria or other antigens appears to be provision for rapid accumulation of leukocytes in the affected lesion [31]. Here they attempt to repulse the bacterial population, by phagocytosis and intracellular killing or by release of cytotoxic enzymes, before the invaders can cause extensive disruption of local or systemic host functions by proliferation or toxicity. In some cases, such as periodontal inflamma*To whom correspondence +Present address: Mobil Paulsboro,

should be addressed. Research and Development

Corp.,

Paulsboro

Laboratory,

N.J.

MATHEMATICAL

BIOSCIENCES

53:189-221

0Elsevier North Holland, Inc., 1981 52 Vanderbilt Ave., New York, NY 10017

189

(1981)

0025-5564/81/020189+33$02.50

190

DOUGLAS A. LAUFFENBURGER

AND CLINTON R. KENNEDY

tion or rheumatoid arthritis, most tissue damage appears to be caused by degradative enzymes released by the accumulated leukocytes, rather than by invading antigens [21]. The ability

of the inflammatory

response

to provide effective

defense

against infection seems to depend critically upon promptness of leukocyte accumulation at the lesion, efficiency of leukocyte localization within the lesion, and strength of leukocyte bactericidal and phagocytic activity. Because inflammation involves a quite complex combination of interacting vascular, biochemical, and cellular events (see, for example, [ 1 l]), there are many possible ways in which the system may break down. There is, in fact, a rapidly

growing body

of literature

enumerating

cases of deficient

in-

flammatory functions leading to recurrent or severe infection (for recent surveys, see [26] and [35]). However, there is a major difficulty in proving definitive cause-and-effect connection between specific defects in the response mechanisms and overall defense failure [36]. Because it is so difficult to experimentally control the interacting elements constituting the response, it is difficult to show that a certain defect will lead to pathological

results.

In such situations, mathematical modeling unraveling the threads. Through mathematical

tool in can be

varied at will, allowing

elucidation

of possible

can be a helpful models parameters cause-and-effect

relation-

ships. Success in so doing could be of potentially enormous practical importance in light of recent efforts to modify some of the component inflammatory mechanisms (for example, [8, 12, 241). In particular, an ability to relate in vitro measurements of the key response parameters to in uivo system performance could provide a much better informed basis for diagnosis and therapy of inflammatory conditions. At present, key response parameters need to be defined. Initial efforts in that direction have been recently presented [ 17, 181, using a simple model of local tissue inflammatory response to bacterial invasion. Considering this situation as a predator-prey-type system, as suggested so long ago by Metchnikoff [22], spatially distributed conservation equations for local tissue bacterial and leukocyte cell population densities in the neighborhood of a microcirculatory blood vessel were developed. Relevant kinetic and transport processes were described by phenomenological rate expressions with physiologically meaningful parameters. Effective defense was represented by elimination of the bacterial population, while unchecked bacterial proliferation indicated establishment of infection. The analysis demonstrated that defense effectiveness depends chiefly upon the values of the following parameters: bacterial growth and leukocyte phagocytosis rate constants, leukocyte random motility and chemotaxis coefficients, and leukocyte vessel emigration coefficients. In particular, low values of the leukocyte chemotactic coefficient can result in poor accumulation and localization of leukocytes in the lesion, leading to enhanced bacterial proliferation. This is consistent with clinical observations that deficient

INFLAMMATION

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191

leukocyte chemotaxis is correlated with increased susceptibility to and severity of infection (see, for example [23, 26, 291). These results have encouraged us to believe that in vitro measurements of leukocyte motility properties may be of value in understanding and treating inflammatory conditions. Analysis of that model required simplifying assumptions, e.g., neglecting certain nonlinearities in the kinetic rate expressions. The purpose of the present work is to present a different model that includes the desired nonlinear rate expressions. This is accomplished by viewing the tissue from a greater perspective, so that blood vessels do not appear as discrete leukocyte sources. Rather, we focus on a larger, macroscopic tissue region in which the leukocyte sources are assumed to be continuously distributed. We then provide an analysis of this model, with the aim of discovering the dynamical behavior that can occur. Our purpose is to show how the variable dynamical behavior observed in the inflammatory response can be accounted for by the underlying mechanisms, and to suggest what the key parameter relationships may be. 2.

MATHEMATICAL

MODEL

The general situation for inflammatory response to bacterial invasion of tissue is pictured in Fig. 1. Bacteria are growing in the tissue, and leukocytes are emigrating from the local microcirculatory bloodstream. The leukocytes squeeze through the walls of the postcapillary venules, and can phagocytose and kill bacteria after contact in the tissue. It is known that leukocytes are chemotactically attracted by a number of diffusible chemical substances, many of which are produced in inflammatory lesions either directly by the bacteria or by reaction of bacterial products with plasma proteins [36]. This attraction may serve to promote leukocyte-bacterial contact, which otherwise would result solely from random movement of the cells. Detailed discussions of the events of inflammation are available (for example [ 11, 361) and a summary presentation from a modeling perspective is given by Lauffenburger [ 161. The model to be considered here focuses on a macroscopic tissue region, homogeneously perfused by the microcirculation. Thus we assume that the postcapillary venules, the source of the emigrating leukocytes, are also homogeneously distributed throughout this tissue region. This contrasts with our previous model [17, 181, which analyzed a microscopic tissue region near an individual venule. If we consider the cell population distributions to be uniform in the local tissue region, we may write the following conservation equations for the bacterial density b and the leukocyte density c:

db=G

b’

(la)

dc _=

Gc’

(lb)

dt

dt

192

DOUGLAS

A. LAUFFENBURGER

AND CLINTON

R. KENNEDY

FIG. 1. Illustration of tissue inflammatory response to bacterial infection. Top: Microscopic environment surrounding a venule. Middle: Microcirculation. Bottom: Macroscopic tissue region. The model presented here focuses on the macroscopic scale. The model presented in Refs. [ 171 and [ 181 focuses on the microscopic scale.

INFLAMMATION

DYNAMICS

193

where Gb and G, are the net generation rates of viable cell biomass per unit volume for bacteria and leukocytes, respectively. This is therefore a lumpedparameter model, rather than a distributed-parameter model, since spatial gradients are neglected. We write the bacterial density generation rate as a combination of two components: bacterial growth, and bacterial death by leukocyte phagocytosis. Bacterial growth should commonly be exponential, because essential nutrients are generally available in the body tissues. At large bacterial densities, however, growth-inhibiting effects may show up, perhaps due to substrate depletion or waste production, causing growth to become linear. We thus write the growth rate as follows: bacterial

kszb

growth rate = -.

k, is the exponential growth rate constant, and Ki is the inhibition constant. When b= Ki, growth proceeds at half the maximal rate. Experiments by Stossel [32] and Leijh et al. [19] suggest that the rate of leukocyte phagocytosis of bacteria might be described by a saturation expression, such as phagocytosis

rate = $$

.

b

kd is the maximum killing rate per leukocyte, achieved at high bacterial densities at which the mechanism by which leukocytes ingest bacteria becomes saturated. The parameter K, characterizes this saturation; when b= Kb, the phagocytosis rate per leukocyte will be half the maximal rate. The leukocyte generation rate also consists of two components: the rate at which leukocytes enter the tissue from the venules, and the rate at which they die in the tissue. The death rate is most simply described as an exponential decay of viable leukocytes, since they have a fairly constant lifespan and emigration into tissue is a random process [3, 51: leukocyte

death rate = gc.

(4)

g is the death rate constant. Some evidence suggests that death may be hastened by ingestion of bacteria [9], and it is also possible that leukocytes are killed by bacterial toxins. In spite of this, we will assume g to be constant in this work. The rate of emigration from the venules is proportional to the leukocyte density in the bloodstream [3, 41. Under normal local tissue conditions there is a small, approximately constant leukocyte emigration rate, but it increases dramatically during inflammation [36]. This increase is probably

DOUGLAS

194

A. LAUFFENBURGER

AND CLINTON

R. KENNEDY

due to the combined effects of two phenomena. First, the local blood flow velocity decreases during inflammation, due to the production of vasoactive substances such as histamine [ 111. Second, there is production of chemical factors which enhance adherence of circulating leukocytes to the vessel wall [7]. Both factors should tend to increase leukocyte emigration through the venule walls, by causing adhesive forces between cell and wall to become stronger than the shearing force of the blood flow that would tend to pull cells away from the wall. Furthermore, the effect seems to be related to the local bacterial density, since invasion of the tissue causes production or release of chemical factors involved in both phenomena [27]. These observations can be summarized in simplest form by writing the leukocyte emigration rate expression in the following way: leukocyte emigration

rate=(h,

+h,b)

(

g

1

c~.

cb is the circulating leukocyte density in the local bloodstream, and A/V is the ratio of venule wall surface area to volume of tissue. ho is the emigration coefficient under normal conditions, and h, is the increase per unit local bacterial density. We will refer to h, as the normal-emigration coefficient, and h, as the enhanced-emigration coefficient. Whether the emigration rate increase is actually linearly proportional to the bacterial density is not known, but it is plausible as a working hypothesis since the bacteria cause production of the chemical mediators involved. h, and hi include the processes of margination of leukocytes, adherence to the venule wall, diapedesis through the wall, and (in this lumped model) migration into the tissue interior. Thus, h, might be related to the chemotactic response of the leukocytes, and both h, and h, to the random motility. Combining Eqs. (l)-(5), we obtain the following mathematical model equations for the bacterial and leukocyte densities in the local tissue region:

dbdt g

kgb 1 +b/K,

=h,( +

k,bc -K,’

(64

+q

PI

;)c,b-gc.

We will explore the dynamical behavior of these equations. Knowing the time course of the bacterial and leukocyte densities will be quite important, since tissue damage may be caused by chemical substances produced by these cells. 3.

ANALYSIS

In order to obtain an idea of the relative importance of the parameters in the problem, and to reduce the number of parameters to a minimum, we

INFLAMMATION

introduce

195

DYNAMICS

the following b

v=Ki9 h,Ki

u=ho’ Y=z’

kgKi

dimensionless

quantities:

k,cot

u=c

T=Kiy

co

Kb

gKi

a==9

K=Ki9 co=

hO(~/Wb g .

co is the background steady-state leukocyte density under normal tions, from Eq. (6b) with dc/dt = 0 and b = 0. Then the dimensionless equations are

dy = d7

YV

--1+v

condimodel

uv K+V

du - =(Y(l +av-u). d7

’

(3

(7b)

v and u are normalized bacterial and leukocyte densities. When v- 1, the bacterial density is such that growth inhibition is a significant factor. When u= 1, leukocyte death is exactly balanced by the background migration into the tissue (i.e. in the absence of an inflammatory stimulus). The parameter y represents the ratio of maximum bacterial growth rate to maximum phagocytosis rate, so when y is large the bacteria have great potential for rapid proliferation. cx represents the ratio of leukocyte death rate to maximum phagocytosis rate, so when cx is small the leukocytes will persist for long times. K is a paramater describing the effect of bacterial density on the ratio of bacterial growth rate to phagocytosis rate. This latter ratio can be expressed, for any given value of c, as a function of b: p(b)---.

k,Ki Kb +b k,c

Ki+b

For K < 1, we have Kb < K, and p increases as b increases. For K > 1, we have Kb > Ki and p decreases as b increases. So when K is small, increasing bacterial population density will tend to diminish the phagocytosis rate more than the growth rate. Finally, (I, the dimensionless enhanced-emigration coefficient, is a most significant quantity. It represents the sensitivity of the leukocyte infiltration rate to the local bacterial density. When u is large, the leukocyte infiltration rate increases dramatically when bacteria are present. It is to be noted that the steady-state solutions of equations (7a, b) depend only on the parameters y, u, and K, while the stability behavior of those states will depend also upon (Y.

DOUGLAS

196 A.

STEADY-STATE

A. LAUFFENBURGER

EXISTENCE

AND

AND CLINTON

MULTIPLICITY

There are two separate classes of steady-state (u,, u,), to be considered. They are:

1. “elimination”: 2. “compromise”:

R. KENNEDY

solutions,

denoted

by

US’l,

US=o, u, > 0,

u, = 1 + ou, .

The first class is termed the elimination state, since the bacterial population is completely eliminated and the leukocyte density is at its normal, “background” level. The second is called a compromise state, since a nonzero bacterial population exists and the leukocyte density is increased. Whether or not this compromise state corresponds to an acute infection depends on the actual magnitudes of b, and cS, where b,=K,u,

Any compromise

and

c,=cr,w,

state must satisfy the algebraic Y(K+U,)=(l

+us)(l

relation

+‘%),

(8)

which has two solutions: u,=$-((y-l-o)?

(1+o-y)2+4~(yK-l)

).

(9)

These are physically acceptable solutions only if they are real and positive. To examine these possibilities, Eq. (8) is rewritten as y(u)=

(l+vw+~u)

>o

(‘0)

K+U

Then y, as a function

of u, has the properties:

y(o)= l/K, UU*

+hJKU+K(l+U)-

v’(v)=

(K+U)’

1

K(l+U)-

u’(O)=

>

1

K2

9

where the prime denotes the derivative of y with respect to v. The derivative becomes zero at two points, and the one which might be positive is

fi=

K(l+U)-

1 (I

(‘1)

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DYNAMICS

197

FIG. 2. Plot of curve in (u, K) plane which divides parameter space into regions of qualitatively different steady-state properties.

This gives rise to two distinct

1. If 2. If

K> K <

possibilities:

l/(1 +a), then y’(O)>0 and there exists no 6>0. l/( 1+ a), then y’(0) < 0 and there does exist a t? < 0.

These two cases divide the positive quadrant of the (a, K) plane along the curve K = l/(1 + a) as shown in Fig. 2. The qualitative structure of the steady states is different in these two regions, as shown in Figs. 3(a) and (b). In both situations, the elimination state exists for all values of y. For K> (1 + 6) -i [Fig. 3(a)] there exists a compromise state for y> l/~ in addition to the elimination state. For K < (1 + u) - 1 [Fig. 3(b)] there is always a compromise state for y > 1 /K, but in addition there are two compromise states for T< y< l/~, where ?=~(a). Notice that u, -y/u as y+cc in both cases, so that even though a compromise state must exist, its magnitude can be arbitrarily small if the enhanced leukocyte emigration rate is large enough. The steady-state multiplicity behavior is summarized in Table 1. The qualitative nature of the dynamics and the biological implications of each of these two cases are quite different, as we shall see.

198

DOUGLAS

A. LAUFFENBURGER

AND

CLINTON

R. KENNEDY

la1 K>Il.el-’

FIG. 3. Example plots of leukocyte steady-state population det J>O; -----, detJ(l+a)-‘, assumed

density versus (b) ~<(l+o)-‘.

here that trace J < 0.

TABLE Steady-State Relationship between o and K K>(l+lJ)-’

K<(l+a)-’

1

Multiplicity Value of y O
Behavior Number of steady-states 1 2

o
1

Y
3 2

lc-’
y. -, It is

INFLAMMATION 8.

DYNAMICS

STEADY-STATE

199

STABILITY

The local stability and character of the steady states found in the last section is determined by examining the eigenvalues of the matrix obtained by linearizing Eq. (7a, b) around a steady state (us, 0,):

au a1 KU

J-

*-A

These eigenvalues,

denoted

K+Vs

@+*A2

i

*S

--

.

(‘4

by A, are found from the equation

The local behavior about the steady states gives important clues to constructing the global behavior of solutions to the equations (7a, b). When (II,, v,) corresponds to the elimination state, J reduces to: y-i I The eigenvalues

0

CYU

are just the diagonal h,=U-l/K,

Since A, is always negative, A, CO, i.e.

(13)

--a 1. elements A,=-a.

the elimination

state is stable if and only if

y
Thus, for y>~-’ the elimination state is unstable, as indicated dashed lines in Fig. 3. For the compromise state J becomes, in terms of v, only, 1+a*, (K+lQ2 i

1+av ---.-2 -I au

(

-_K

by the

*S

--

K+V,

(‘4)

-a

Both eigenvalues will have negative real parts, and the steady state will be locally stable, if and only if the determinant and trace of J satisfy the conditions det J>O trace J < 0

DOUGLAS

200

Examining

A. LAUFFENBURGER

the determinant

condition

AND CLINTON

R. KENNEDY

first, it can be shown that

det J=

(‘5)

If ~‘(0,) < 0, the compromise state is unstable; but the reverse statement cannot be made without further consideration of the trace condition. Referring to Fig. 3(b), this means that the smaller compromise state in the interval 7 $ y Q l/~ is always unstable. This situation is analogous to the well-known fact from chemical-reactor analysis that the “middle” steady state is always unstable (cf. [2]). Indeed, we could have anticipated this result from intuitive arguments. Since y represents a normalized maximum growth rate, then y’( 0,) < 0 implies that the steady-state bacterial population decreases as the maximum growth rate increases-a physically unrealistic result. In Fig. 3(b), this unstable compromise state is also indicated by a dashed line. Figure 3 displays the result that as y increases, the elimination state becomes unstable through the formation of a compromise state. This means that if y is large enough, the leukocytes cannot completely eliminate the bacterial population (although a large enhanced-emigration coefficient can make it very small). The model tells us, however, that there are two ways for this to happen. In the first, as in Fig. 3(a), a small-amplitude solution grows out of the elimination state in a gradual, continuous manner at y = 1/K. In the second, as in Fig. 3(b), the transition from the elimination to the compromise state occurs in a saltatory fashion as y is increased beyond 1/K. This jump might be though of physically as the sudden onset of inflammation when, perhaps, some characteristic of the system changes slightly-e.g., if k, (kd) becomes large (small) enough to make y > l/K. On the other hand, it is not even necessary that the elimination state become unstable for a stable compromise state to exist. For y< y< l/~, it is possible that both the elimination and the upper compromise state may be stable. Although we have not yet examined the trace condition for the compromise state, we will find later that it is often satisfied. Thus, when ~<(l +a) -’ there can simultaneously be two locally stable steady states for 7
(K+d

I+OU

----K Y

I

-cU>o.

INFLAMMATION

DYNAMICS

201

First notice that v, is independent (16) to hold is

of LY,so that a necessary

condition

1 +au, >yfC. Using Eq. (8), this is equivalent

for

(‘7)

to K<

(‘8)

1,

so that the trace condition can be violated saturates before the growth term does.

only if the bacterial

death rate

When trace J goes through zero with det J > 0 (this is possible only for the upper compromise state), the compromise state becomes unstable through a pair of complex-conjugate eigenvalues, leading to periodic solutions of the equations (7a, b) through a Hopf bifurcation. We therefore go on to examine the trace condition, assuming that (18) is satisfied. Towards this end, define v(l+au) (1 +v)(lc+v)2 Examining

(I-K)>o.

(‘9)

F, we find F(0) = 0,

(i) (ii)

F(v)---

(iii)

dF _= dv

U(l

-K) V

as

V--fCO,

-(l-K) G(v) ’ (K+v)3(l +v)2

where G(v)=uv’+(2-UK)V2+(1-2UK)V-K.

F(v) has its extremum at G(v)=O, and by use of the Routh test we can show that there is one and only one positive solution of G(v) = 0. Call it v*. Thus there is only one positive maximum of F, and it is at u=u*. Now define a*=F(v*).

(20)

Then, since F(v) is independent of a, Fig. 4 makes clear the truth of the following statement: If K< 1, there exists an a* such that there is a range of u (and hence y) where the inequality (16) is satisfied if a < (Y*.If we call this

202

DOUGLAS

A. LAUFFENBURGER

AND

CLINTON

R. KENNEDY

FIG. 4. Plot of F(o) [Eq. (19)] versus o, illustrating effect of a on the sign of the trace condition [Eq. (16)]. For a
0, =u* =u*

at

(~=a*,

(ii)

u,
for

(~
(iii)

limu,(a)=O a-0

and

limq(a)=

+cc.

U-SO

These properties are obvious from an examination

of Fig. 4. From Eq. (10)

we can define YI =Y(ul)>

Y2

=y(u2),

and depending on the shape of the y(u) curve, there may be a range of y’s over which violation of the trace condition produces instability. Information concerning the existence, multiplicity, and stability of the steady states is summarized in figures similar to Fig. 3. Figure 5(a)-(d) shows a number of possible configurations. Any state is stable only if both trace and determinant conditions are satisfied. Figure 5(a) and (b) show cases where violation of the trace condition produces instability over the range (y,, y2). Figure 5(c) shows a case in which violation of the trace condition produces instability over the range (y, y2). Finally, in (d), u, < u2
INFLAMMATION

L

“a

“2

203

DYNAMICS

(al

.. ..*.

/

..‘. *.a*

”

“I

__L_______L__-

f/K

Y,

Y2

Y

ILL&_____,_____ 5

FIG. 5.

Example

plots of leukocyte

detJ
(a) ~>(l

+o)-‘;

(b), (c), (d), ~<(l

)‘K

population

violation of trace condition [Eq. (16)]. detJ>O, traceJ>O; -.-.-.-.-. traceJ
values of a allowing ----

steady-state

v,

--

Y,

density

V

versus y, for

det J >O, trace J O.

+a)-‘.

As previously mentioned, if the trace condition is violated at a point where a steady state was stable, a periodic solution will appear. We will not pursue sustained oscihatory behavior further in this paper, but just remark that it is similar to behavior found for a model for the specific immune response [25]. In some cases, a periodic solution may represent a successful defense against a challenge, if in the first cycle the minimum value of the

204

DOUGLAS

A. LAUFFENBURGER

AND

CLINTON

R. KENNEDY

bacterial density is so small as to imply that there are practically no bacteria in the tissue. Even though the solution predicts renewed increase of the bacteria, the deterministic nature of our model would not be appropriate at such low densities so that calculation of the chances of reproliferation requires a stochastic model. In general, oscillatory solutions to this model have little meaning, since the specific immune response may come into action within some hours or days after the challenge, and the system will no longer be governed solely by the nonspecific inflammatory response. The main purpose of studying the trace condition is to discover whether the system will approach the compromise state or instead must evolve around a limit cycle surrounding the compromise state. C.

PHASE

PORTRAITS

Analysis of the stability of the steady states has indicated the conditions for which these points are locally attracting or repelling. Information on the global behavior of the solutions to the initial-value problem [i.e., Eqs. (7a, b) with appropriate values for u( T = 0) and u( r = O)] can be obtained by studying the structure of the (u, U) phase plane. First we note that zi >0 for u = 0 and v > 0, while 1_5 = 0 when v = 0 (the dots denote the derivatives with respect to r). Thus the first quadrant of the (u, u) plane is invariant, and hence any positive initial condition generates a solution which remains nonnegative, i.e., physically realistic. It is also clear that for any u > 0, all solutions in the first quadrant are bounded. Most of the structure of the phase plane can be obtained from knowledge of the zero-slope isoclines. There are two parts to the d=O isocline: v=O

and

II==,

(2hb)

and the li=O isocline is the straight line v=-.

u-1 0

Steady states exist at the intersection of the zi = 0 and 6= 0 isoclines. The trajectories in the phase plane can be sketched by analyzing the phase plane equation: y(K+V)-U(l+V)

G

(l+V)(K+V)(l+UV-U)

1 *

Along the d = 0 isocline dv/du = 0, while dv/du = co along the zi-0 isocline. There are several cases to be considered, but we can already see how increasing u can eliminate or diminish a compromise state by decreasing the

INFLAMMATION

DYNAMICS

205

slope of the zi= 0 isocline, Eq. (22). Also, transient bacterial populations are lowered by increasing u in the denominator of Eq. (23). then the elimination state is the only steady state, If ~>l and y<~-‘, and it is globally stable. This is the situation shown in Fig. 6(a). We have deliberately not tried to show too much local detail about the steady state, since we are primarily interested in the regions of attraction in this section. In this case, the entire positive quadrant is the region of attraction for the elimination state. Now, as y increases, the second part of the d=O isocline [Eq. (2 1b)] intersects the ti = 0 isocline along u = 0 at u = 1 when y = K -I, and thus creates a new steady state. This new point moves out along the line u=( u- 1)/a as y becomes larger. The elimination state becomes unstable, and the compromise state is stable, since the trace condition cannot be violated for K> 1 [Fig. 6(b)]. For K< 1, Eq. (21b) has a slope with sign opposite to that for K> 1. But for K > (1 + a) -I the type of intersection and the phase-plane trajectories are similar to those just discussed as long as the trace condition is not violated. When K< (1 +a) -I, however, the behavior is different. For y < y, the elimination state is the only steady state, and it is globally stable [Fig. 7(a)]. As y increases, the d=O isocline intersects u=(u1)/u tangentially when y-y, and thus creates two new steady states. One of these points moves down the line u=(u1)/u as y increases, and at Y=K -* it coalesces with the elimination state. For y > K -’ the elimination state becomes unstable and the single remaining compromise state moves out along the line u=( u- 1)/u as y increases. The stability of the compromise state now depends upon the trace condition [Fig. 7(b)]. The situation where there are three steady states, i.e. when ~<(l +u) -’ is the most interesting [Fig. 7(c)]. As we have already and f
and the discriminant

A defined by A=(trace

J)‘-4det

J

is always positive. If X is the negative eigenvalue, then the single stable trajectory approaching the middle steady state comes in along the eigenvector

206

DOUGLAS

A. LAUFFENBURGER

AND CLINTON

R. KENNEDY

FIG. 6. (a) Phase portrait for K> 1 and y 1 and y > K - ‘. The compromise state is globally stable.

INFLAMMATION

DYNAMICS

207

FIG. 7. (a) Phase portrait for K< (1 +a) -’ and U-C+. The elimination steady state is globally stable, (b) Phase portrait for ~<(l +a) -’ and Y>K -‘, with a such that the compromise steady state is globally stable.

DOUGLAS

OKY

A. LAUFFENBURGER

1

AND

CLINTON

R. KENNEDY

”

Cc) FIG. 7 (c). Phase portrait

for K<: (1 +(I) -

compromise and elimination steady states reached from points within region B.

’ and u < y < I( - ‘, with LYsuch that both the are stable.

The

compromise

steady

state

is

/’ =o ,/’

i

I’

/’

I’

,’

,’

,I’

,

,’ ,,‘i;OI’

_,’

”

which divides FIG. 8. Plot of phase plane separatrix promise and elimination steady states for a<(1 +e)-’ dividing line between regions A and B of Fig. 7(c).

regions and

of attraction <
for comThis

is the

INFLAMMATION

DYNAMICS

209

and thus has a positive slope in the (u, u) plane (see Fig. 8). This stable trajectory forms a separatrix which delineates the regions of attraction for the compromise and elimination states as shown in Fig. 7(c). If the initial conditions in the phase plane are near the elimination state, the small bacterial invasions, within the region of attraction of the elimination state (region A), are fought off successfully. Larger initial bacterial densities, within region B, move the system toward a compromise state, and the invading population establishes itself. Interestingly, for still larger initial bacterial densities, in region C, the system returns to the elimination state, but not before passing through relatively large values of u and u, perhaps causing extensive tissue damage in the process. This is due to the enhanced leukocyte emigration rate coefficient, h,. A large invading bacterial population causes rapid infiltration of leukocytes in the affected lesion, which eventually quashes the threat. A more moderate invasion is tolerated by the defensive mechanism. A very small challenge is easily defeated by the background “surveillant” leukocytes. Time courses of the bacterial populations for these three cases are illustrated in Fig. 9. In the above discussion we have always assumed that the trace condition is not violated, and the sketches have always shown the upper compromise state as a node. This state may also be a focus, depending on the sign of A.

to elimination

FIG. 9. Illustration of possible transient profiles for bacterial density for the situation of Fig. 7(c). The initial conditions A, B, and C correspond to the regions of attraction in the phase portrait of Fig. 7(c).

210

DOUGLAS A. LAUFFENBURGER

AND CLINTON R. KENNEDY

But even in the case of periodic solutions the regions of attraction will remain the same in a structural sense, and this has been our main concern. 4.

DISCUSSION

We have developed a mathematical model for the interaction of bacteria and leukocytes in a host-tissue inflammatory response. This model has assumed the cell density distributions in a macroscopic tissue region to be uniform, in order to focus this study on the behavior resulting from rate expressions for the kinetic processes. Results of analysis of the model equations indicate that, in general, two different classes of steady states can be attained by the inflammatory response. One possibility is that the bacteria will be completely eliminated from the local tissue region; we call this the elimination steady state, and it obviously represents efficient functioning of the defense response. The second possibility is that a finite bacterial population density will be established in the local tissue region; we call this the compromise steady state. If the bacterial density is large for this state, this might indicate an acute infection, with high levels of infectious agent present. If the bacterial density is rather small, it might indicate a low-level, perhaps chronic or tolerated, infected state. There is also the possibility of oscillations around the compromise steady state, which might represent recurrent infection, unless the minimum bacterial density in a cycle was so low as to effectively mean elimination in reality, or the maximum density was so high as to damage the tissue irreversibly. Which of these states is attained from a particular initial state and the time course of the transient approach will depend chiefly upon the values of three key dimensionless quantities: y, o, and K. These are defined in the analysis section as ratios of dimensional parameters used in modeling the relevant rate processes, and can be thought of in the following way: y=ratio of maximum bacterial growth rate to maximum leukocyte phagocytosis rate, (I = ratio of enhanced leukocyte tissue infiltration rate during inflammation to the normal “background” infiltration rate, K= ratio of the inhibition effect of increasing bacterial density on bacterial growth to the inhibition effect on phagocytosis. Large values of y and small values of (I and K may be expected to increase the potential for bacterial infection. There is a fourth dimensionless quantity, a, also defined in the analysis section, which can influence the transient approach to a steady state, but in a way we consider less important. a is the ratio of leukocyte death rate to phagocytosis rate; when it is small, oscillations around the compromise steady state can occur. Such long-time transient behavior is probably not of practical interest, because the model

INFLAMMATION

DYNAMICS

211

developed here is intended to represent the initial stage of the inflammatory response. At later times, other more specific immune processes not accounted for in our model may be called into action. The key predictions of this model can be summarized as follows: (1) For large enough values of K and u, so that K(U+ l)> 1, there is a parameter inequality which decides which of the two steady states (elimination or compromise) is reached: if y > K -I, then the compromise state is attained from all initial states, as in Fig. 6(b). In terms of the original dimensional parameters, this requires

(24) If y K -I, then the compromise state will be attained from all initial states, as in Fig. 7(b). But now, when but y>T [where f, which depends upon K and a, is determined y
In terms of the original dimensional

parameters,

this is

K&,g-kAo(A/Vh b(o)<

k,h,(A/V)c,

-K,k,g

Ki’

(26)

DOUGLAS

212

A. LAUFFENBURGER

AND CLINTON

R. KENNEDY

Thus, it is clear that the prediction concerning whether or not infection results from a bacterial invasion of tissue depends upon the size of the bacterial inoculum as well as upon the quantities y, u, and K. Such a dependence of outcome of bacterial invasion has been experimentally observed, for instance, in lung infection [34].

It may be instructive to enumerate the ways in which parameter changes can cause infection, rather than elimination, to result from tissue invasion. (1) If K( O+ 1) > 1, then an increase in y from less than to greater than will always result in infection. (2) If K(U+ l)< 1, again an increase in y from less than to greater than K -’ will always result in infection. Also, an increase in y from less than to greater than 7 can result in infection for large enough bacterial inoculum densities. Alternatively, a decrease in u can cause 7 to become smaller than y for a fixed value of y, again leading to infection for large enough initial bacterial densities. then a decrease in K or u could cause (3) If ~(u+l)> 1 and y<~Kl, K( u + 1) < 1 with 7 < y < K - ‘, leading to infection for large enough inoculum densities. (4) If ~(u+l)> 1 and y<~-’ or~(u+l)
In general, then, we see that parameter changes that increase y or decrease u can often change the predicted outcome from elimination to infection. The effect of parameter changes affecting K may vary. We will illustrate some of these results with numerical examples. 5.

NUMERICAL

EXAMPLES

In this section we illustrate our discussion by considering behavior for reasonable estimates of the parameters. A.

BACTERIAL

the model

GROWTH

The data of Hau et al. [lo] for E. coli indicate an in uiuo growth rate constant of 0.75 hr -‘. Since many species grow less rapidly, a reasonable range for k, may be 0.1-0.75 hr -‘. Since bacterial populations rarely grow without self-inhibitory effects, we to populations above 10 lo bacteria/cm3 will assume that Ki might be in the range of 10’ to lo9 bacteria/cm3. B.

PHAGOCYTOSIS

From Stossel’s data for phagocytosis of albumin-paraffin leukocytes [32] we estimate kd- lOa mgparticle/leukocytehr

particles by and K,--15

INFLAMMATION

DYNAMICS

213

mg/cm3. For bacteria of about 10 -s mg mass, these imply kd--100 bacteria/leukocyte hr and Kb - 1.5 X IO9 bacteria/cm3. From experiments by Leijh et al. [ 191 on phagocytosis of Staph. aureus and E. coli we estimate values of k d -60-80 bacteria/leukocyte hr. These experiments also showed that uptake was saturated when the ratio of bacteria to granulocytes was greater than 1, indicating that Kb is approximately the order of c in the tissue, or K,- 10s- IO’ bacteria/cm3. These estimates are all for fully opsonized particles. In both of these works it is shown that inadequate opsonization can reduce the value of kd by factors of about 10-25. It should also be noted that these experiments were carried out in agitated flasks. We expect the values of k, in vivo might be reduced, if limited by the leukocyte-bacterium collision rate. C.

LEUKOCYTE

DEATH

Normally leukocytes have an average lifetime in tissue of about a day [5], so that we can estimate g-5X lo-* hr-‘. If death is accelerated by bacterial toxins so that the lifetime averages only an hour, this could make g as large as about 1 hr -‘. D.

LEUKOCYTE

Eh4IGRA TION

The circulating leukocyte density normally ranges between 2X lo6 and 8 x lo6 leukocytes/cm3 [9, 151. The risk of infection seems to increase when C~ is less than lo6 leukocytes/cm3, and there is a striking rise in infection, morbidity, and mortality for cb less than 5 X 10’ leukocytes/cm [ 151. Normal values for the background tissue leukocyte density, c,, are about 10’ leukocytes/cm3 [IO, 30). Using the definition for u with the background value u- 1, we find

With the average value c, =105 leukocytes/cm3, g=5X 10m2 hr-‘, and c,=5X lo6 leukocyte/cm3, this requires that h,(A/V)=lO-’ hr-‘. An estimate for (A / V) can be made from that by Johnson [ 141that (A/V) = 600 (cm2 capillary surface area)/& tissue) for highly vascular tissue. With a tissue density of about 1 g/cm’, this gives (A / V) = 600 cm - ’ for capillaries. Lightfoot [20] presents values for the relative numbers of capillaries to venules in tissue, about 15 to 1. Using values for capillary diameter and length of 8 pm and 1 mm, and venule diameter and length of 30 pm and 2 mm, also from Lightfoot [20], we calculate that the rate of venule surface

214

DOUGLAS A. LAUFFENBURGER

AND CLINTON R. KENNEDY

area to tissue volume is about (600 cm-‘)X

(1)(3x10-3cm)(2x10-‘cm) (15)(8x

10e4 cm)(l x 10-l

=300cm_, ’

cm)

This yields an estimate for h, of about 3 X 10 -6 cm/hr. The data of Hau et al. [IO] for the dynamics of peritoneal infection by E. coli show that c, increases from about 2x IO5 leukocytes/cm3 to about 2 x lo6 leukocytes/cm3 within 3 hours of bacterial inoculation of about lo8 bacteria/cm3. Thus, an estimate for h, follows from 1.8X IO6 leukocytes/cm3 3 hr With values of (A/V)=300 cm-‘, cb =5X lo6 leukocyte/cm3, and b= lo8 bacteria/cm3, we get h, =4X lo-‘* cm/hr(bacteria/cm3). The product h,b would be 4x 10 -’ cm/hr. An upper bound on both h, and h, b should be the leukocyte linear rate of movement, approximately 20 pm/mm [l], or about 10 - ’ cm/hr. This would be an upper bound for at least two reasons: (1) migration is basically a biased random walk, and only approaches persistent linear movement with chemotaxis; (2) movement into the tissue interior will be slowed by the processes of margination, adherence to the venule wall, and diapadesis through the wall. For illustration of some of the results presented in the Discussion section, we propose the following scenarios (see Table 2 for listing of TABLE 2 Parameter Values for Illustrative Numerical Computations

% Ki k, K, he (A/P) cb h’ g Y K 0 a u

(hr-‘) @act/cm3) @act/leukhr) @act/cm3) (cm/hr) (cm-‘) (leuk/cm3) (cm/hr[hact/cm3]) (hr-‘)

Case I

Case II

Case III

Case IV

Normal host

Defective phagocytosis

Neutropenia

Defective enhanced emigration

1x10-’ 10s 40 10’ 5x10-6 2x 102 5x 106 4x lo-‘* 5x 10-a 25 1x10-2 800 125 -

1x10-’ IO9 2 10’ 5x10-6 2x 102 5x106 4x lo-‘* 5x 10-a 500 1x10-2 800 250 -

1x10-’ 109 40 10’ 5x10-6 2x 102 25x105 4x lo-‘* 5x10-2 500 1x10-a 800 250 -

1x10-’ IO9 40 10’ 5x 10-a 2x lo* 5x106 2x10-14 5x10-2 25 1x10-2 4 125 8.8

215

INFLAMMATION DYNAMICS

parameter values used) for numerical integration of the differential equations (6a, b). Case I. These parameter values might represent a normal healthy host challenged by a relatively nonvirulent bacterial species. For these values, K > (I+ a) - ’ and y < K - I. Thus, elimination should always result, although initial proliferation should occur for initial bacterial densities greater than about 3 x IO’ bacteria/cm’.

b,= 10' bacteria

lo2

I

0

\I 6

16

I

/cm3

I

24

32 TIME

LO

_

bacteria

____.

leukocyies I

I

I

16

56

61

i

Ihours)

FIG. 10. Transient cell densities for initial bacterial density of lo4 cells/cm”, a very small challenge. Curves labeled with Roman numerals correspond to cases enumerated in Table 2.

216

DOUGLAS A. LAUFFENBURGER

AND CLINTON R. KENNEDY

Case II. We have decreased the value of k, by a factor of 20, indicating an abnormally low rate of phagocytosis. This could be due perhaps to defective opsonization, or to a resistant bacterial strain, Now y > K-I, so a compromise steady state should be reached for all initial bacterial densities. Case III. We have decreased the value of cb by a factor of 20, representing a severe neutropenia. Again y > K -‘, so a compromise steady state should be reached for all initial bacterial densities. Case IV. We have decreased the value of h, by a factor of 200, representing an abnormally low enhanced emigration response. One possible cause for this might be a defective chemotactic response by the leukocytes. Now K < (1 + a) - ‘, so that although y < K - ‘, in fact y > 7, allowing a compromise steady state to be attained for some initial bacterial densities.

Results of numerical computations for these cases are shown in Figs. 10, 11, and 12. Figure 10 shows the transient cell densities for an initial bacterial density of lo4 bacteria/cm 3. In both cases I and IV the bacteria are rapidly eliminated, while in cases II and III the bacteria proliferate gradually, and, after some oscillations, will reach a steady state of 1.25 X 10’ bacteria/cm3, although there is almost no noticeable early leukocyte density increase. Figure 11 shows the transient cell densities for an initial bacterial density of lo6 bacteria/cm 3. Again the bacteria are rapidly eliminated in cases I and IV, but in cases II and III they will attain a compromise steady state of 1.25 x 10’ bacteria/cm3 after some oscillations. Here, however, there is a noticeable increase in leukocyte density within 24 hours for cases II and III. Finally, Fig. 12 shows the transient cell densities for an initial bacterial density of 10’ bacteria/cm3. Again the bacteria are eliminated rapidly in case I, despite a very short initial period of proliferation. In cases II and III, the bacteria proliferate briefly, and then are eliminated by the great numbers of leukocytes that rapidly accumulate. The model, of course, predicts again a compromise steady state of 1.25 x 10’ bacteria/cm3 for cases II and III, but during the first oscillation the density of the bacteria gets so small that they may be effectively eliminated. This result should serve to emphasize the importance of the initial transient period, so that sometimes the steady-state properties can be misleading. Case IV is now quite interesting. For this large initial bacterial density the deficient enhanced leukocyte infiltration response is costly, and the bacteria population can increase. A compromise steady state of 5 x lo9 bacteria/cm3 may now be attained, along with a large leukocyte density. Thus, an important result has been obtained: The rapid leukocyte infiltration in cases I, II, and III allow efficient bacterial elimination, while the slow leukocyte infiltration in case IV leads to bacterial growth. Notice that at 8 hours, there are larger leukocyte densities present for cases I-III than for case IV, but at times greater than 72 hours, the leukocyte density will be much larger for

INFLAMMATION DYNAMICS

217 -108

b,=106

bacteria/cm'

16

2L

32 TIME

40

-

bacteria

____

leukocytes

56

6L

72

Ihours)

FIG. 11. Transient ceII densities for initial bacterial density of lo6 cells/cm’, a moderate challenge.

case IV than for the other three cases. This is an especially important result for situations in which the tissue damage is primarily caused by lytic enzymes released by the leukocytes. As one application, we propose that the results obtained here may help explain the etiology of inflammatory diseases such as periodontitis. Patients with this disease are sometimes found to have deficient leukocyte chemotaxis, although the tissue damage is caused primarily by large numbers of

DOUGLAS A. LAUFFENBURGER

AND CLINTON R. KENNEDY -lo8

bo=108

bacteria/cm"

,_*'

,YS

____

_______--------____

_-

,/' II / I' 3' /' 1' /I

I'

,,,' ,' I'

____.--------_,<___

-IO4

_

bacteria

____ leukocytes

8

I 16

2L

32 TIME,

FIG. 12. challenge.

LO

I

I

40

56

6L

72

10'

(hours)

Transientcell profiles for initialbacterialdensity of lo* ce3s/cm3, a serious

leukocytes in the lesion [33]. Although seemingly paradoxical, this situation is consistent with the effects of decreasing the value of h, in this model, as in case IV above. The value of h, may be diminished if leukocyte chemotaxis is defective; this can result in a shift from rapid elimination of the stimulus to a compromise state in which many leukocytes are present in the tissue.

INFLAMMATION

6.

DYNAMICS

219

CONCLUSIONS

We have presented a simple model describing interactions between a bacterial invader of tissue and the local host inflammatory defense response. This model has focused on kinetic rate processes, in order to provide a foundation for further exploration of the effects of transport and motility processes. Analysis of this model has shown that quite interesting dynamical behavior of the bacteria and leukocyte tissue densities can result. In particular, we have outlined specific relationships that govern the outcome of the inflammatory defense against bacterial challenge-that is, whether the defense is effective and rapidly eliminates the bacteria, or whether the defense is inadequate and allows establishment of infection. The use of a lumped model can permit fairly extensive analytical study, and is quite useful if interpreted properly, A distributed model is necessary to properly assess the role of leukocyte motility and chemotaxis within the tissue [ 17, IS], although the lumped model yields a great deal of insight into the role of motility and chemotaxis for infiltration into the tissue. Motility and chemotaxis might be considered to be directly related to the values of ho and h,, respectively, in the lumped model, since these phenomena aid in emigration. To infer something about the roles of movement behavior within the tissue from the lumped model, one might guess that increasing motility and chemotaxis would effectively provide a larger value of k,, since contact with the bacteria should be enhanced. In this manner, we can learn about the relationship between the parameters of both the host defense and the bacterial invader, and about the dynamic behavior that can be observed for the inflammatory response. It is encouraging to note that methods for measurement of the dynamics of bacterial growth and leukocyte response in experimental infections are becoming zivailable [6, 131. The authors would like to thank Mr. Eric Worff for performing the numerical computations. D. Luuffenburger would also like to gratefully acknowledge the financial support of the National Science Foundation Chemical Processes Program, Grant CPELYO-06701. REFERENCES 1 2 3

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AND CLINTON

R. KENNEDY

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Reo.