Analysis of the effects of immune cell motility and chemotaxis on target elimination dynamics

Analysis of the Effects of Immune Cell Motlllty and Chemotaxis on Target Elimination Dynamics E. S. FISHER AND D. A. LAUFFENBURGER Department of Chemical Engineering, University of Pennsylvania, Philadelphia, Pennsylvania 191046393 Received 21 Februav

1989; revised 13 Jury 1989

ABSTRACT White blood cells of the immune system must encounter specific targets such as bacteria, malignant cells, virus-infected cells or other cells of the immune response in order to carry out their function of protecting the host from infectious and malignant disease. To analyze the dynamics of this process, a mathematical model has been developed for elimination of proliferating targets by a constant population of motile immune system cells in two dimensions. Encounter is assumed to be the rate-limiting step for elimination. This model makes use of a previously derived analysis of single cell-target encounter times, which yields an encounter rate constant that is incorporated into a kinetic conservation equation for target number density. This paper focuses on the influence of directed cell movement, or chemotaxis, as well as other cell motility properties, such as cell speed and persistence, on target elimination dynamics. A particularly significant result is that a given relative decrease in chemotactic responsiveness leads to much more severe deficiencies in target clearance rates for low levels of baseline chemotactic responsiveness than for high levels of baseline responsiveness. The general model results are then applied to the particular example of bacterial clearance from the lung surface by alveolar macrophages. It is shown that moderate levels of macrophage chemotactic responsiveness, similar to those measured in vitro, can account for the experimentally observed rates of bacterial elimination from the lung for typical values of bacterial specific growth rate and alveolar macrophage number density.

INTRODUCTION Encounter between white blood cells of the immune system (lymphocytes, macrophages, neutrophils, natural loller cells) and their targets is an essential first step in many aspects of the immune and inflammatory responses to infectious and malignant challenges. Neutrophils and macrophages must encounter bacteria and antibody-antigen complexes in order to destroy them by phagocytosis; cytotoxic lymphocytes and natural killer cells must encounter cancer cells and virus-infected cells in order to destroy them by MATHEMATICAL

BIOSCIENCES

OElsevier Science Publishing 655 Avenue of the Americas,

98:73-102

13

(1990)

Co., Inc., 1990 New York, NY 10010

0025-5564/90/$03.50

74

E. S. FISHER AND D. A. LAUFFENBURGER

cell-mediated cytotoxicity, and lymphocytes must encounter macrophages in order to be activated by antigen presentation. Encounter is important because it is the initial event in these phenomena, and it may often be the rate-limiting step for the overall process [18]. Factors that govern the rate of encounter may therefore critically influence the dynamics of the overall inflammatory and immune response. One significant factor that can affect the rate of immune cell-target encounter is the migration behavior of the immune cells. These diverse white blood cells share the capability for active locomotion, or motility, over two-dimensional surfaces and through three-dimensional matrices, which is clearly necessary for them to locate and encounter targets that are distributed on these surfaces and within these matrices. A key element of cell motility is chemotaxis, in which locomotion is biased in the direction of concentration gradients of chemical stimuli [36]. Such stimuli include bacterial peptide products, activated complement factor 5a, and various lymphokines released by the white cells themselves. Hence, the potential exists for immune cell-target encounter to be guided by these chemical signals, as local stimulus concentration gradients can form due to diffusion of the stimulus from a target source. The purpose of this work is to present a mathematical model for the general case of two-dimensional cell-target encounter, with the goal of elucidating the effects of cell motility properties-particularly chemotaxis-on the dynamics of target elimination by the cells. We will consider the situation in which a constant population of actively migrating immune cells encounters an immobile population of targets distributed homogeneously on a surface. Our model will consider that the targets are “eliminated” when encounter takes place; that is, they are removed from the target population. Thus, we can determine the effects of key system parameters on the transient behavior of the target population. We will consider cases in which the target population can proliferate exponentially as well as cases in which the targets cannot proliferate, allowing application to the wide range of physiological situations-bacterial or immune complex phagocytosis, cancer cell or virusinfected cell cytotoxic killing, and immune cell activation-mentioned above. In the cases of phagocytosis and cytotoxic killing, target elimination is equivalent to destruction, while in the case of activation, target elimination represents transfer of the target immune cell from the nonactivated population to the activated population. Our results may be helpful in interpretation of in vitro assays of phagocytosis, cytotoxicity, and activation [e.g., 16, 20, 21, 291, as well as the dynamics of these functions in vivo [e.g., 12-15,19,27, 28, 331. A specific example application we can consider with this type of model is that of phagocytic clearance of inhaled particles and bacteria from the lung surface by alveolar macrophages. These cells reside on the lung surface and

IMMUNE

CELL

75

MOTILITY

provide the primary line of defense against inhaled pathogens [ll]. The process by which macrophages encounter their targets is a very important one, since only a small fraction of the total lung surface is covered by these cells and by inhaled material [lo, 301, In vitro motility studies have shown that alveolar macrophages respond chemotactically to a variety of chemical attractants [4, 6, 301 and it is likely that some of these factors are present in the challenged lung [13, 23, 24, 351. Elsewhere we have published specific comparisons of our model results with literature data for the alveolar macrophage/lung surface clearance system [9]. The purpose of the present paper is to provide a more detailed development of the model along with a general analysis of the model predictions. MODEL GENERAL

DEVELOPMENT FORMULA

TION

The most fundamental assumption in this target elimination model is that the elimination rate is equal to the encounter rate, or that encounter is rate-limiting. Previous theoretical work [8] has shown that the time required for a phagocytic alveolar macrophage to encounter its target at typical target densities on the lung surface is on the order of tens of minutes. This is substantially greater than reported times for phagocytic killing by macrophages, roughly a few minutes [15, 161. Neutrophil migration speeds, and thus target .encounter rates, are somewhat faster [36]; it is still possible that encounter is also rate-limiting for these cells since phagocytosis rates are also slightly faster [22]. Delivery of lethal hits by cytotoxic lymphocytes similarly takes approximately a few minutes [26], so given their slower movement speeds [36] it is plausible that encounter is rate-limiting here as well. There are few available data on rates of lymphocyte activation by antigen-presenting cells, so this must be left uncertain for now. (In all these systems, of course, conclusions regarding the rate-limiting step will depend heavily on the cell and target number densities; the smaller the densities, the more likely encounter will be rate-limiting.) Based on the assumption of encounter rate limitation, the net rate of elimination of exponentially proliferating targets by a constant population of immune cells is

$=k,b-R(b

,c,w,

(1)

where b and c represent target and immune cell surface densities, respectively, k, is the target proliferation rate constant, and R is the encounter rate. We will show that R can be expressed as a function of the target and immune cell densities and W, the mean encounter time for a single cell and target. When encounter is not rate-limiting the elimination rate will need to

76

E. S. FISHER AND D. A. LAUFFENBURGER

include a model expression for the phagocytosis, cytotoxic killing, or antigen presentation step upon encounter. As an example, a model previously offered by Lauffenburger and Kennedy [17] considers the situation in which a phagocytosis step is rate-limiting. We next assume that the number of immune cells remains constant throughout the elimination process. With small challenges by most types of invading bacteria, this assumption is a reasonable one for resident tissue macrophages, such as on the lung surface [14, 331. For large challenges, substantial numbers of neutrophils can accumulate in local tissue foci from the bloodstream within a few hours [5, 251. Therefore, the assumption that their number remains constant will be strictly correct only for low challenge levels or later stages of elimination. Situations involving lymphocytes, whether for antigen presentation or elimination, can be approximated fairly well in this manner because of the slower kinetics of lymphocyte generation. For in vitro phagocytosis and cytotoxicity assays this assumption is certainly valid. However, a comprehensive dynamic model would in general include transient variations in immune cell number. Again, the model by Lauffenburger and Kennedy [17] considers this possibility. The encounter rate R is a function that depends, as indicated above, on the time required for encounter between a single cell and target. The single cell- target mean encounter time can be viewed as a reciprocal of the rate constant for the process of encounter between multiple cells and targets [32], and one of the parameters on which it depends is the maximum distance between a cell and its nearest target, R. That is, W = W(R). R, in turn, depends on the surface densities of targets or cells. When targets outnumber cells (c < b), R is equal to one-half the distance between targets. That is, we can consider the smaller number of cells to be moving within a larger number of two-dimensional unit “disks” surrounding the targets (see Figure 1). If the targets are distributed uniformly on the surface at an area density b [targets/area], then b = l/vR’, or

c< b.

(2)

The assumption of uniform distribution of cells and targets simplifies the calculation of the maximum distance between cell and target. A more realistic assumption might be of random target distribution, but computer simulations of a fairly similar situation (binding of solution ligand molecules to cell surface receptors) have demonstrated that only small differences are obtained for these two assumptions [2]. In any event, the mean encounter time here depends implicitly on the target density, that is, W = W( R( b)). Assuming that the cells act independently and do not compete for targets, the encounter rate can be written as the product of the cell density and the

77

IMMUNE CELL MOTILITY 0

0

0

l

0 0

0

0

0

a

0 0

0

0

0 0

if b > c. encounter rate = & W 63

0

0

0

0

0.0 0 0

if c > b. encounter rate = _b w 6) FIG. 1. Illustration of the two different situations dealt with by the dynamic clearance rate model for multiple cell-target systems. For target density greater than cell density (top), the single cell-target mean encounter time [8] depends on the average distance between targets; for target density smaller than cell density (bottom), the single cell-target mean encounter time depends on the average distance between cells.

inverse

encounter

time:

R=c/W(b),

cc b.

(3)

Targets will not always outnumber cells, however. Fortunately, the clearance model can easily account for this (see Figure 1). When there are more cells

78

E. S. FISHER

AND D. A. LAUFFENBURGER

than targets (c > b), the maximum distance between a target and the nearest cell will be approximately equal to half the average distance between cells. Now the mean encounter time depends implicitly on the cell density, so the encounter rate is equal to the product of the target density and the inverse encounter time. (This now assumes that the targets do not “compete” for cells.) The corresponding equations are R=l/&,

c>h,

(44

and R=b/W(c),

SINGLE

CELL-TARGET

MEAN

c> h.

ENCOUNTER

(4b)

TIMES

The mean encounter time for a single cell and target in two dimensions as a function of the maximum distance between them, W(R), has been calculated previously by Fisher and Lauffenburger [8]. This calculation was accomplished by solving a differential equation for the variation of encounter time with instantaneous distance between cell and target and then integrating the solution over all possible distances up to the maximum separation distance R. The resulting mean encounter time depends on R, on the distance A between cell and target centroids upon encounter, and on three parameters characterizing immune cell motility properties. These parameters are (1) cell speed s, (2) cell directional persistence time (i.e., average time between direction changes) T, and (3) the chemotactic index CI, which combines the chemotactic sensitivity of cells to an attractant gradient and the magnitude of the gradient. A simplistic calculation of CI is illustrated in Figure 2. We use this quantity to avoid specifying gradients explicitly for particular situations, allowing the model greater generality.

Random Motility

Chemotaxis

Attractant

FIG. 2. Illustration of net distance moved

of chemotactic index CI. The simplest definition of CI is the ratio toward the target to the total cell path distance: CI = x/L.

IMMUNE CELL MOTILITY

79

Based on this previous work, expressions dimensionless mean encounter time o,

have been obtained

for the

w= Ws/R,

(9

for the three general ranges of chemotactic index: CI = 0 (purely random motion); intermediate CI, 0 < CI < 1; and CI = 1 (perfectly directed motion). The expressions for CI = 0 and CI = 1 are exact, taken directly from the paper by Fisher and Lauffenburger [8]: o=

t[(A/R)‘-3][1-(A/R)‘] -ln(A/R)’ ’

(sT/R)[~-(A/R)~]

cl=o’

(6)

and w=

(UR)2-3(-W)2+2 ’

3[l-(A/R)2]

‘I=‘.

(7)

For intermediate values of the chemotactic index, no simple exact solution could be obtained. Rather, a closed-form integral solution must be evaluated numerically; this was done by Fisher and Lauffenburger [8]. This is less than desirable for our present purposes, since we would like to substitute an analytical expression for W ( = wR/s) into Equation (1). We can accomplish this is an approximate manner, however, by finding simple analytical expressions that represent the numerical computations satisfactorily. We have found that for 0.1~ CI < 1, the following form yields values for w within 3% of the true values: ~=q,-‘(A/R),

(8)

where 0.66 w” = CI - 0.023

(94

and 0.87 ’ = CI-0.049

’

t9b)

Detailed comparisons between these approximate expressions and the exact results can be found elsewhere [7]. One minor complication is that Equation (8) permits mean encounter times to be negative, while the true expression obviously cannot. With the approximate expression, there is a limiting value of b or c at which encounter time reaches zero, b = c -c [(~~/v)~]/nA~. This

80

E. S. FISHER AND D. A. LAUFFENBURGER

artifact does not affect the applicability of the clearance model, since the model cannot be applied in situations where encounter is no longer ratelimiting, that is, when encounter time approaches zero. INDIVIDUAL

CLEARANCE

EQUATIONS:

k, = 0

In this case, the following dimensionless B = brrA2,

variables

C=crA2,

T= a/A,

so that B and C represent fractions of monolayer with 0 < B, C < 1. The initial condition is B(T=T,) The resulting follows.

dimensionless

are used:

coverage of the surface,

= B,,.

equations

for the three different

- a,B(lB) (B-3)(1-B)-2lnB’

where a0 = F

CI cases are as

CI=O: BBC dB -= dT

dB -=-u,B dT

(10a)

%(IC) where ‘O= (C-3)(1-C)-2lnC‘

’

(lob)

0.1 gCI
BBC dB -= dT

- a,@ we-v@

(114

where a, = C.

C>B dB dT = - a,B,

where a, = c

( meal yE)

.

(lib)

CI=l:

B>C dB dT=

-3a,B(l-

B)

B2-3B+2@

’

where ai = a,.

(124

C>B dB _=_ dT

o,B,

where ui =

3cy,(l-

c)

C2 -3C+2c.

(12b)

IMMUNE CELL MOTILITY

81

The key parameter for the C > B situations, (I (representing a,, u,, and a,), is inversely proportional to the mean encounter time based on C. That is. A

u=sW(C;CI)

(13)

.

It can be shown for all three cases that u > 0. For CI = 0 and CI = 1, its asymptotic behavior is

C=O-u-+O(W-+co),

C=1*u~w(W+0).

When C = 0, u approaches 0 because the encounter time is infinite when no cells are present. u goes to infinity when C = 1 because then there is a monolayer of cells present and the encounter time is zero. For intermediate CI, the asymptotic behavior of u is 2 c=o*u;+O(w-+~),

c=

? (

-u,~oo(w+o). 1

When C = 0, a, approaches zero because, again, the encounter time is infinite when no cells are present. The quantity a, goes to infinity when encounter time is zero, and this occurs at the limiting value of C, that is, c < (w&J)*. INDIVIDUAL

CLEARANCE

In this case, changed to

EQUATIONS:

the definition

of dimensionless

T’= k,t = T/y,

Now, the dynamic

equations

k, > 0

time can profitably

be

where y = s/Ak,.

are as follows.

CI=O:

B>C “oy(l-

B)

(B-3)(1-B)-21nB

(144

C>B g

= B(l-

uay).

(14b)

82

E. S. FISHER AND D. A. LAUFFENBURGER

0.1 gCIC

C>B

(15b)

$=B(l-CTJ) CI=I: BBC

(164 C>B dB do’ = B(l-

(16b)

qy).

The key parameter for the C > B situation, ay (representing u,y, u,y, and a,~), can be written in another form to illustrate its relationship to the mean encounter time based on cell density C, as previously: 1 “’ = k,W( C; CI) As before, CI =l:

the following

asymptotic

behavior

c=0=,0y+0(w+co);

(17) is observed

C=l-uy+ca

for CI = 0 and

(W-O).

When C = 0, uy once again approaches 0 because the encounter time is infinite when no cells are present. uy goes to infinity when C = 1 because then there is a monolayer of cells present and the encounter time is zero. For intermediate CI, the asymptotic behavior of uy is different because of the artifact introduced by the use of the fitted expression:

c=o~u,y-+o(w+co);

C=

2

( 1 :

*u,y+cc

(W-O).

When C = 0, u,y approaches zero because the encounter time is infinite when no cells are present. The quantity u,y goes to infinity when encounter time is zero, and this again occurs at the limiting value of C.

83

IMMUNE CELL MOTILITY

RESULTS For a given value of CI, the two cases of B > C and C > B can be considered separately, and the relevant stable and unstable steady states can be determined for each equation individually. This information can then be combined to construct overall trajectory diagrams for B as a function of time for a variety of initial conditions and parameter values; we will refer to this as the “combined model,” for it combines the B > C and C > B cases. From these trajectories, a bifurcation diagram corresponding to the entire combined model for that choice of CI is constructed. This is quite straightforward to do, since there is only a single dependent variable to follow. Hence, it is easy to examine a full range of possible initial conditions. In the following sections, this bifurcation diagram is discussed for CI = 0, intermediate CI, and CI =l. We use u (for k, = 0) and ay (for k, > 0) as our bifurcation parameters. We propose that this is a highly appropriate choice, since Equations (13) and (17) demonstrate that these parameters essentially characterize dimensionless encounter rate constants. We will shortly show that the model steady states depend on the value of these quantities relative to unity for the k, > 0 cases. Some further intuition can be gained by rearranging the definition of u,y for the k, = 0, CI = 0 situation to the form

a,y=

[ 4sz;y’]C[

(I-C)/(‘VR)2 (C-3)(1-C)-lnC2

]

The first term represents the ratio of the rate of surface area coverage by a random searcher possessing speed s and directional persistence time T in a two-dimensional region of radius R, to the target proliferation rate. The second term represents the number of searchers. The third term represents the effective search area relative to the ratio of target size and search region size; for C - 1, this term goes to zero, while for C - 0 it approaches infinity. Similar arguments can be offered for the other situations. One might also be interested in an explicit view of the effects of the value of C on elimination behavior, but, as will be shown shortly, this can be elucidated easily from our analysis. CLEARANCE

RESULTS:

k, = 0

In the case where targets do not proliferate, the only possible steady state for the clearance system is complete elimination of targets, or B = 0, and this steady state is stable. This behavior is obvious from inspection of Equations (10) through (12) with the realization that a! and u are always positive. The rate of elimination of targets to this steady state will be of interest, however.

84

E. S. FISHER AND D. A. LAUFFENBURGER

CLEARANCE

RESULTS:

k, z 0

CI = 0. For the combined model, the relevant bifurcation parameter is u,y, which is related to (~a and C as shown in Equation (lob). For the B > C model, the following relationship holds at steady state: “0y=BS,-3-A Through

Equation

00-Y=

21nB 1- B,,

(18)

(18), u,y is related to B,, and C as follows:

(B,,-3)(1-B,,)-21% l - BSS

1-c (C-3)(1-C)-21nC

From Equation (19) it can be determined that u,y =l only when B,, = C. It can be further shown that Equation (19) yields only one solution for B,,. When uoy < 1, or when C < B,,, the only possible solution is B,,(C) > 0, which is stable. When u,y > 1, or C > B,,, the only steady state is B,, = 0, which is stable. When u,y = 1, if B is initially grdater than C, the number of targets decreases until the steady state B = C is reached, but if B is initially less than or equal to C this initial density is maintained as a steady state. The bifurcation diagrams for the combined model for two choices of C are shown in Figure 3. In Figure 4a, uoy is plotted against C to provide a different view of the steady-state behavior of the model. For uoy > 1 all targets are eliminated, while for uoy < 1 controlled proliferation to the steady-state value occurs. It should be noted that not all combinations of C and u,y are possible, since y is an intrinsic property of the cells and targets while u. depends on C. For any given values of the cell and target properties, however, the minimum value of C necessary for target elimination can be easily determined from the criterion comparing u,y to unity; this can be rewritten in the form (l-c)c (C-3)(1-C)-21nC The value of C making this an equality is plotted in Figure 4b as a function of the quantity k,A/s, using the particular value of 2sr/A = 3 (based on a set of values appropriate to alveolar macrophages: s = 3 pm/mm, T= 5 min, and A = 10 pm [8, 91). These values of C are shown compared to a typical experimental value for the quantity k,A/s = 0.03 (based on k, = 0.5 h-t, A =lO pm, s = 3 pm/n@ again relevant to lung clearance of bacteria by alveolar macrophages). Notice that in such a situation, for CI = 0, the immune cell (alveolar macrophage) number density must be equivalent to a fractional surface coverage of about 0.1 in order to achieve

IMMUNE

CELL

85

MOTILITY A 0.8

0.6

0.8

0.6 B ss 0.4

0.2

0.0

FIG. 3. Example bifurcation diagram for the CI = 0 combined q,y, with target proliferation (kg > 0). (a) C = 0.1; (b) C = 0.5.

clearance

model,

B,, vs.

elimination of the bacterial targets. This is an order of magnitude greater than typical surface coverages by alveolar macrophages on the lung surface, which we can estimate as roughly 0.001 to 0.01 [7]. (This estimate is obtained from a typical value of about one resident macrophage per alveolus, with alveolus radii approximately 50 to 150 pm.) Thus, it appears that purely random migration of alveolar macrophages could not allow them to eliminate bacteria from the lung surface when the bacteria are proliferating with a doubling time of about 1.5 h. We define this critical value of C required to provide for elimination when k,A/s = 0.03 as C*. In Figure 4c we plot C* as a function of CI; so

E. S. FISHER

1

1

o

1

AND D. A. LAUFFENBURGER

elimination

.

0

~~prol~

(

.

1

2

3

C (a)

0

-2

log

i

[kgA/s] -3

Cl=1 Cl = 0.6

9

Cl = 0.3

-A-

CI=O.l

-b-

Cl=0

-

(kgNs)eip

I -3

-2 log

-1

c

(b) FIG. 4. (a) steady-state behavior of the CI = 0 clearance model, as a function of dimensionless cell density and bifurcation parameter e,,y. (b) Minimum value of C required to allow elimination, from cry > 1 (with e = u,, for CI = 0, a, for intermediate CI, and et for CI = l), for a range of CI values. The horizontal line represents k,A/s = 0.03, a typical experimental value. The value of C for which a given curve intersects this hue is defined as C*. For CI = 0, we specify 2sr/A = 3. (c) Plot of C*, the value of C at which a curve in Figure 4b intersects the horizontal line k,A/s = 0.03, as a function of CI.

IMMUNE

87

CELL MOTILITY

-2

1ogc*

-

caxp

Q

c'(a)

-3 -

-4! 0.0

.,.,.,.,.I 0.2

0.4

0.6

0.6

1.0

CI

Cc) FIG. 4. (Continued)

far we have explained how the value for CI = 0 is determined; we now turn to presentation of the results for CI > 0. CI = 1. The bifurcation parameter for the combined model is a,y, which is related to C and a, as shown in Equation (12b). For the B > C model, the following relation holds at steady state:

B,‘,- 3% +-2/K alY =

3(1-

4,)

’

(20)

The function aiy( B,,) passes through a maximum in ~~iy at aiy* = 0.1395 and BsT = 0.2167. Through Equation (20), u,y is also related to B,,, as shown here:

DlY =

B: - 34, + 2JBss

1-

es

(21)

Clearly, a,y = 1 when B,, = C, but because the individual functions of B,, and C in the above equation are not monotonic in their argument, there is another point where u,y = 1. Figure 5 presents B,, as a function of C. Note that for each choice of C, there are two values of B,, where u,y ‘1, except at

E. S. FISHER

88

AND D. A. LAUFFENBURGER

1.0 4

0.0

0.4

0. t I

0.6

0.8

1.0

C

C = B,,* FIG. 5. Curves for the bifurcation parameter o,y as a function of B,, and C, for the CI = 1 combined clearance model. Shaded areas correspond to qy < 1; open areas corre-

spond to a,y > 1.

the point where B,, = C = BsT. This graph also delineates the ranges of B,, and C where a,y > 1 (open areas) and a,y < 1 (shaded areas). This graph can be used to construct plots of B as a function of time for the three ranges of the bifurcation parameter relative to unity. Recall that B,, > Bsz is the stable steady state for the B > C equation, so combinations of C and B,, in this range, as shown in Figure 5, pertain to trajectories relative to the stable steady state. The resulting trajectory information is summarized in the bifurcation diagrams in Figure 6. Figure 6a shows an example case in which C > @. Here for a,y < 1, B,, is the only steady state and is stable, while for uiy > 1, B = 0 is the only steady state and is stable. When u,y =l, trajectories beginning at B > C decrease to reach B = C, and trajectories beginning at B < C remain at their initial value. In Figure 6b, C < B,T, and when u,y < 1 the only steady state is still B,, and it is stable. At u,y > 1, however, the upper branch of the B,, curve, B > BsT, remains stable, and the lower branch of that curve is now an unstable steady state. B = 0 is also stable in this parameter range. When u,y =l, B,, is stable, as are the initial values of trajectories beginning at B < C. Trajectories beginning at values of B > C increase or decrease until they reach B,, (u,y = 1). From these bifurcation diagrams, it can be seen that there is a maximum value of uiy, uiy = uiy*, for which a nonzero steady-state level of B is permitted. This value depends on C as shown in Figure 7. The curve in this

IMMUNE

CELL

89

MOTILITY

B,,=C 0.6 -

B

0.2 -

\r

Bss=

I’

Bs1

B,,=C 0.0

0.5

1.0

q Y = =1

1.5

2.0

r*

FIG. 6. Example bifurcation diagrams, B,, vs. qy, for the CI = 1 clearance target proliferation (k, > 0). (a) C = 0.75; (b) C = 0.05.

model with

figure corresponds to ury = uty*. In each of the three regions indicated on this graph, the available steady states are known from the bifurcation diagram for the entire system. Only elimination is possible for ury > ary*, and only proliferation occurs when a, y < 1. The remaining region corresponds to ury* > uly > 1, and in this area, C < B$. Here, the equilibrium value of B depends on its initial value, and either proliferation or elimination can occur, as in Figure 6b. The key criterion for uty, though, can still be most simply viewed as its value relative to unity, since this is the minimum

E. S. FISHER

AND D. A. LAUFFENBURGER

4

OIY

‘I

3 Y= =ly*

elimination

elimination or limited proliferation

C = B,;

FIG. 7. Steady-state behavior of the CI =l clearance sionless cell density and bifurcation parameter qy.

model,

as a function

of dimen-

value required for elimination to be possible (a necessary but not sufficient condition for elimination). This criterion can be rewritten as 3(1-C)C C2-3C+2@

$A ‘7’

and it is plotted in Figure 4b. Notice that, in contrast to the CI = 0 case, for CI = 1 the critical value C* of the immune cell number density (necessary to allow target elimination for our typical value of k,A/s = 0.03) corresponds to a fractional surface coverage greater than about 0.0003. This is, in fact, much lower than the alveolar macrophage/lung surface coverage of roughly 0.01. Hence, we can conclude that perfectly directed alveolar macrophage migration would be more than sufficient to accomplish bacterial elimination from the lung surface under typical conditions. 0.1 < CI ( 1. For the combined model, the appropriate bifurcation parameter is u,y, which is related to (Y; and C as shown in Equation (llb). For the B > C case, the following relation is true at steady state:

BSS=

wa* i’,w(J- 4a,yu 2v

(

L

(22)

IMMUNE

CELL

91

MOTILITY

where the positive square root corresponds to the stable steady state. On a plot of B,, as a function of the parameter aiy, these steady states form a continuous curve with an extremum at a,y* and Bsz, where Bz = (0~/2v)~

(23a)

and (23b) The relationship rearranged:

between u,y and B,, can be seen clearly if Equation

(22) is

(24) so tha,

(25) Since the function appearing on the right-hand side of Equation (24) and twice on the right-hand side of Equation (25) goes through a maximum as its argument increases, there are two points where u,y = 1: one where B,, = C and a second where B,,=(q,/u-@)2.

(26)

These equations yield figures similar to Figure 5, for the dependence of the bifurcation parameter on B,, and C. The resulting bifurcation diagrams for the combined model are also quite similar to those produced by the combined model for CI = 1. The exact diagram that results depends on the value of C, as illustrated in Figure 8. When C > Bsz, as shown in Figure 8a, the upper branch of the B,, curve is the only steady state for u,y < 1 and it is stable, and B = 0 is the only steady state for a, y > 1 and it is stable. At a, y = 1, trajectories that begin above B = C will decrease until they reach this point and remain there, and trajectories that start at B Q C will remain at their initial levels. Figure 8b shows model behavior for C < Bsz; here the upper branch of the B,, curve is always stable, and for u,y < 1 it is the only steady state. The lower branch of the curve is an unstable steady state for uiy > 1, and B = 0 is a stable steady state in this region. When u,y = 1, trajectories that begin at B > C will eventually reach B,, on the upper branch of the curve, and trajectories that start at B < C will remain at their initial values.

92

E. S. FISHER

AND D. A. LAUFFENBURGER

0.5

A

1

B ss

B,,=C 0.2 -

B ss

B,,=C 0

1

2 qY

c,Y=

q*

FIG. 8. Example bifurcation diagrams for the 0.1~ CI < 1, B,, vs. 0, y, clearance with target proliferation (kn > 0) (a) C = 0.25; (b) C = 0.05.

model

Just as in the CI = 1 case, these bifurcation diagrams indicate that there is a maximum value of qy, qy = a,~*, above which the only possible steady state is B = 0.Thisvalue of a,y depends on C as shown in Figure 7, where qy can be substituted for qy. As before, there are three regions indicated on this graph, and the available steady states for each region are known from the bifurcation diagram for the entire system. For u, y > a,y*, the only possible steady state is elimination, and only proliferation occurs when a, y < 1. The remaining region corresponds to ui y* > a, y > 1, where C is less

IMMUNE CELL MOTILITY

93

than &T. Here, the steady-state value of B depends on its initial value, and either proliferation or elimination can occur. Again, though, we use the simple necessary criterion based on a, y > 1:

having substituted in the expressions for w0 and v. This relationship is plotted in Figure 4b for CI = 0.1,0.3, and 0.6. The critical values required to allow elimination for our typical value k,A/s = 0.03 are, as before, plotted in Figure 4c. This latter figure reveals that, given typical experimental values for C of about 0.01, the chemotactic index must be at least 0.2 or so in order for elimination of proliferating bacteria to be accomplished by the resident alveolar macrophages. This minimum value is similar to that found experimentally in vitro [7, 341. DISCUSSION The central goal of this modeling work is to consider the effect of immune cell chemotaxis, relative to the effect of other parameters, on the rate of elimination of targets by immune cells. The graphs presented in the previous section lay the foundation for this by showing the effects of the model parameters on the steady-state outcome of a challenge: elimination or controlled proliferation. In this section, system parameter values are selected that correspond to a representative class of immune system cells, alveolar macrophages, which are responsible for eliminating bacteria and other inhaled particles from the lung surface, since they are involved in a particular application for which there are good quantitative data on elimination dynamics (see [8], and [9]). Model predictions of elimination rates are examined over reasonable ranges of cell speed s, persistence time T, and chemotactic index CI, as well as bacterial specific proliferation rate constant k,. We typically use s = 3 pm/mm, T = 5 min as a base case (see [7], and [34]) and vary CI between 0 and 1. Experimental observations currently give CI values of approximately 0.2 to 0.3 [7, 341 for this cell type. Cell and target densities, c and b, respectively, are chosen from realistic experimental lung surface values (see [9]), yielding fractional surface area coverages on the order of 0.0001 to 0.01; these are also consistent with our fundamental assumption that encounter is rate-limiting within our chosen parameter range. Specific proliferation rate constants used are in the range 0.2 to 0.8 h-‘, a common range for bacterial growth [31]. The first set of results particularly relevant to this macrophage/lung surface system are illustrated in Figures 4b and 4c and have been discussed

94

E. S. FISHER AND D. A. LAUFFENBURGER

earlier. Figure 4b gives a plot of the minimum value of C (the dimensionless cell density in terms of fractional surface coverage) necessary to allow targets proliferating with a dimensionless specific growth rate constant k, A /s, for a range of CI. This is derived from the criterion ay > 1, using a,, a,, and (I~ appropriate for the different values of CI [Equations (lob), (llb), and (12b), respectively]. As k,A/s increases, so does this required value of C, with a roughly linear proportionality between the respective logarithms. The required value of C is greater for a given k,A/s for smaller CI, and the effect of C is correspondingly greater. For comparison, a typical experimental value of k,A/s is shown on this plot, equal to 3 x lo-* (based on k, = 0.5 h-‘, A = 10 pm, s = 3 pm/mm). Figure 4c gives a plot of C*, the value of C required to allow elimination of targets with this dimensionless specific growth rate constant, as a function of CL Given that typical experimental estimates of C for the alveolar macrophage/lung surface system are on the order 0.001 to 0.01 (see [8], and [9]), this figure indicates that a minimum value for CI of at least 0.2 is necessary for alveolar macrophages to successfully eliminate a typical challenge of proliferating bacteria from the lung surface. It is interesting to note that maximum values of CI found experimentally in vitro for alveolar macrophages responding to a bacteriarelated peptide [7, 341 and activated complement C5a [B. Farrell, R. Daniele, and D. Lauffenburger, unpublished observations] are in the range 0.2 to 0.3. Figures 9a and 9b illustrate the influence of chemotactic index on target elimination rates for nonproliferating and slowly proliferating targets, respectively. In both cases, a small amount of chemotactic response above purely random motion, say CI about 0.1 to 0.2, dramatically increases the elimination rate. Elimination rates for CI values greater than about 0.6 become very nearly independent of CI. The effect of cell speed on elimination rates is presented in Figures 10a and lob, in the form of the influence of speed on the time required to clear half an initial number of targets (i.e., clearance half-time). Increasing the cell speed decreases the clearance half-time. More interesting, notice that speed has a greater influence when CI is small than when it is larger. This effect is observed both when no proliferation of targets occurs (Figure 1Oa) and when moderate proliferation takes place (Figure lob). An especially meaningful result is seen in Figure 11, which shows the elimination half-time versus target proliferation rate constant for a range of values of CI. Not surprisingly, as the target proliferation rate constant increases, clearance half-time increases also. In addition, there is a critical value of the target proliferation rate constant above which elimination cannot occur; this critical value increases as CI increases. The effect of target proliferation rate constant is similar to the effect of cell speed, in that both parameters have a much stronger influence on half-times for cells with small CI than those with larger CI. This behavior has particularly significant

95

IMMUNE CELL MOTILITY

0

1

2

4

3

t (hours) B

=O.l

B 1.0 BO

= 0.2

0.0 0

1

2

3

4

t (hours) FIG. 9. Effect of CI on clearance dynamics, for C= 0.004, B,,= 0.001, and s = 3 pm/min. (a) k, = 0; (b) k, = 0.2 h-l.

implications for attempts to relate measurements from in vitro experimental assays for cell motility properties to in vivo immune cell function. Figure 11 illustrates clearly that for cells with normally low CI values, a given relative decrease in CI should lead to much more severe problems than would an equal relative decrease in CI for cells with normally high CI values. Evidently, determining merely relative deficiencies in immune cell chemotactic responsiveness is not sufficient to understand whether physiological function is likely to be impaired; rather, absolute measurements of responsiveness are also necessary.

96

E. S. FISHER

%

AND D. A. LAUFFENBURGER

2

(hours)

3

4

5

6

7

8

9

10

8

9

10

s (microns/minute)

(hours)

3

4

5

6

7

s (microns/minute) FIG. 10. Effect of cell speed on clearance half-time, for C = B, = 0.0004. (a) k, = 0, and a range of CI; (b) plain curves correspond to k, = 0 for CI = 0.2 and 1, while bold curves correspond to k, = 0.2 h-’ for CI = 0.2 and 1.

Figures 12a and 12b present the effect of immune cell density on elimination rates, for nonproliferating and slowly proliferating targets, respectively, for CI = 0 and CI = 1. The three values of C used represent 4B,, B,, and B, /4, with B,, = 10P3. Obviously, increasing C provides for more rapid target elimination or slower target proliferation. In contrast to our results for other system parameters, the number of immune cells has a greater effect on clearance rates for cells with more directed movement. This appears to be simply due to the fact that any given cell finds a target more

IMMUNE

CELL

97

MOTILITY

% (hours)

0.0

0.2

FIG. 11. Effect of target proliferation rate constant = 0.004 and s = 3 pm/min, for a range of CL

0.4

on clearance

0.6

half-time,

for C = B,,

quickly when it has more directed movement, so the addition of more cells has a greater effect on elimination. Notice that the transient curves in this figure are consistent with the conservative predictions of Figures 4b and 4c. Our model predictions can be compared to experimental data for the situation mentioned in the introduction-that of lung clearance by alveolar macrophages. Clearance data are generally reported in the literature as numbers of bacteria remaining viable in the lung as a function of time, and initial numbers of bacteria and cells. The relevant bacterial growth rate constants are typically not provided, however. Therefore, we assume that k, lies somewhere within the range of 0.2 to 0.8 h-’ [12, 311, and for each k, in this range the value of CI required for the model to best agree with the reported data can be determined [7]. This has been done for ten different sets of experimental bacterial clearance data, as shown in Figure 13 [9]. For each set of experimental data (see figure legend for experimental references), the best-fit value of CI is plotted for a given value of k,. This value of CI represents the level of macrophage chemotactic response to a target-derived attractant necessary to give the experimentally observed rate of bacterial elimination. As expected, this level of CI increases as k, increases. It is quite interesting, though, that over this entire range of expected bacterial growth rate constants and a wide spectrum of experimental reports the apparently relevant values of CI group together surprisingly well. In no case is CI = 0, or purely random macrophage motility, adequate for clearance of the bacteria at observed rates. At the same time, in almost no case is CI = 1, or

98

E. S. FISHER

0

1

2

AND D. A. LAUFFENBURGER

3

4

t (hours) B

2.5 1

B

0

1

2

3

4

t (hours) FIG. 12. Effect of dimensionless cell density on clearance dynamics. Bold curves correspond to CI = 0; plain curves correspond to CI = 1. B, = 0.001 and s = 3 pm/min. (a) k, = 0; (b) k, = 0.2 h-l.

perfectly directed macrophage motility, required to account for these clearance rates. These results suggest that a moderate macrophage chemotactic response to attractants generated by targets is critical to proper lung defense against challenge by proliferating bacteria. A more detailed discussion of the physiological significance of our model is presented elsewhere [9]. Again, it is useful to note that in vitro measured values of CI for this system are roughly 0.2 to 0.3 [7, 34; Farrell et al., unpublished observations]. Figure 13 suggests that these estimates could account for the experimental clearance data for

IMMUNE

CELL

MOTILITY

Cl

0.01 0.2

0.6

0.4 Kg

FIG. 13. Comparison of model predictions to challenges in the lung, from [9]. This plot shows experimental clearance data as a function of the different experimental reports: (X) Heidbrink Pesanti and Nugent [27], (m) Rehm et al. [28], (0) [12], (A) Jakob

and Green [15]. (0)

0.8

(/hour)

experimental clearance data for bacterial the values of CI required to account for value of k, assumed. Symbols represent et al. (141, (+) Gross et al. [13], (m,r) Green and Kass [ll], (m) Green and Kass

Kim et al. [16], (m) Laurenzi

et al. [19].

bacterial growth rate constants in the low end of the expected range. It is known that nonspecific physical mechanisms, neglected in this model, can clear a portion of inhaled particles and bacteria [lo-121; these mechanisms may be responsible for helping to eliminate some faster-proliferating bacteria despite these low estimates of CI. When the resident macrophage population is inadequate to deal with a challenge, they can release chemotactic attractants leading to influx of neutrophils [5], apparently raising the level of C to an effective value. As mentioned in the Model Development section, we have chosen to quantify immune cell chemotactic responsiveness in terms of the parameter CI, which combines both cell sensitivity per unit attractant gradient and the magnitude of the attractant gradient. Thus, a change in CI could be due to either a change in the cell sensitivity per se or to a change in the magnitude of the attractant gradient present. The latter represents a system defect (in

100

E. S. FISHER

AND D. A. LAUFFENBURGER

the production of the appropriate attractant by the targets) rather than a cellular defect. One could break this parameter down into its two component parts for more rigor. Given the current paucity of information concerning attractant gradients that might be expected in vivo, however, we have chosen to use the simpler approach for now. If encounter is not rate-limiting for a particular situation, this model can be modified to consider the overall elimination process as a series of steps (rather than effectively one step, the encounter step). The model could then account for different rates of ingestion, activation, or killing, if these were occurring as slowly as encounter, or it could account for the possibility that a cell pauses for a significant period of time after consuming or killing a target, although this is infrequently observed (1, 31. The model could also be made more widely applicable if recruitment of additional immune cells were included (along with immune cell death), with a conservation equation describing the rate of change of the immune cell density included in the model. An earlier effort directed toward analysis of these effects, though without a rigorous analysis of cell motility effects, has been published by Lauffenburger and Kennedy [17]. The central point we would like to make is that, in order to understand the implications of any abnormalities in immune cell motility properties as measured in in vitro assays, for in vivo function of the immune response a dynamical model including the cell motility phenomena must be analyzed. As demonstrated in this present work, this sort of model can be quite informative. For instance, we have seen here (e.g., see Figure 11) that sometimes even seemingly minor defects in chemotaxis may lead to dramatic deficiencies in target elimination rate, while sometimes apparently large defects in chemotaxis may have rather negligible consequences. Clearly, such results argue for the need to include rigorous quantitation of cell motility properties in dynamic models.

This work was partialb supported by U.S. Department of Energy grant ER60564 and a grant from the University of Pennsylvania Research Foundation, along with an AA UW Dissertation Fellowship Award to E.S.F.

REFERENCES 1 2 3

4

R. B. Allan and P. C. Wilkinson, A visual analysis of chemotactic and chemokinetic locomotion of human neutrophil leukocytes, Exp. Cell Res. 111:191-203 (1978). H. C. Berg and E. M. Purcell, Physics of chemoreception. Biophys. J. 20:193-219 (1977). C. Dahlgren, J. Hed, K.-E. Magnusson, and T. Sundqvist, Characteristics of individual polymorphonuclear leukocyte motility obtained with a new opto-electronic method, &and. J. Immunol. 9:537-545 (1979). J. H. Dauber and R. P. Daniele, Chemotactic activity of guinea pig alveolar macrophages, Am. Reo. Respir. Dis. 117:673-684 (1978).

IMMUNE 5

6 I 8 9

10 11 12 13 14

15 16

17 18

19

20

21

22

23

CELL

J. M. Davis

101

MOTILITY and J. I. Gallin,

The neutrophil,

in Cellular Functions in Immunity

and

~n/ummarion, J. Oppenheim, D. Rosenstreich, and M. Potter, eds., Elsevier, New York, 1981. R. Fine, J. 0. Shaw, and W. R. Rogers, Effects of C5a on baboon alveolar macrophage migration, Am. Rev. Respir. Dis. 123:110-114 (1981). E. S. Fisher, Effects of chemotaxis on cell-target encounter rates and particle clearance, PhD thesis, University of Pennsylvania, 1988. E. S. Fisher and D. A. Lauffenburger, Mathematical analysis of cell-target encounter rates in two dimensions, Biophys. J. 51:705-716 (1987). E. S. Fisher, R. P. Daniele, and D. A. Lauffenburger, The effect of alveolar macrophage chemotaxis on bacterial clearance from the lung surface, Am. Reu. Respir. Dis. 137:1129-1134 (1988). E. Goldstein, W. Lippert, and D. Warshauer, Pulmonary alveolar macrophage: defender against bacterial infection of the lung. J. Clin. Invest. 54:519-528 (1974). G. M. Green and E. H. Kass, The role of the alveolar macrophage in the clearance of bacteria from the lung, J. Exp. Med. 119:167-179 (1964). G. M. Green and E. H. Kass, Factors influencing the clearance of bacteria by the lung, J. Clin. tnvest. 43~769-776 (1964). G. N. Gross, S. R. Rehm, and A. K. Pierce, The effect of complement depletion on lung clearance of bacteria, J. C/in. Invest. 62:373-378 (1978). P. J. Heidbrink, G. B. Toews, G. N. Gross, and A. K. Pierce, Mechanisms of complement-mediated clearance of bacteria from the murine lung, Am. Reu. Respir. Dis. 125:517-520 (1982). G. J. Jakob and G. M. Green, The effect of Sendai virus on bactericidal and transport mechanisms of the murine lung, .I. C/in. Inuest. 51:1989-1998 (1972). M. Kim, E. Goldstein, J. P. Lewis, W. Lippert, and D. Warshauer, Murine pulmonary alveolar macrophages: rates of bacterial ingestion, inactivation, and destruction. J. lnfec. Dis. 133:310-320 (1976). D. Lauffenburger and C. R. Kennedy, Analysis of a lumped model for tissue inflammation dynamics, Math. Biosci. 53:189-221 (1981). D. Lauffenburger, D. Hammer, R. Tranquillo, H. Buettner, and E. Fisher, How immune cells find their targets: quantitative studies of cell adhesion, migration, and chemotaxis, in Theoretical Immunology, A. Perelson, ed., Santa Fe Institute, AddisonWesley, Reading, Mass., Part 2, pp. 3-18, 1988. G. A. Lauremi, L. Berman, M. First, and E. H. Kass, A quantitative study of the deposition and clearance of bacteria in the murine lung, J. C/in. Inuest. 43:759-768 (1964). B. E. Lehnert and P. E. Morrow, The initial lag in phagocytic rate by macrophages in monolayer is related to particle encounters and binding, fmmunol. Invest. 14:515-521 (1985). B. E. Lehnert and C. Tech, Quantitative evaluation of opsonin-independent phagocytosis by alveolar macrophages in monolayer using polystyrene microspheres, J. Immunol. Methods 78:337-344 (1985). P. Leijh, M. van den Barselaar, T. van Zwet, I. Dubbeldeman-Rempt, and R. van Furth, Kinetics of phagocytosis of Staphylococcus aurew and Escherichia coli by human granulocytes, Immunology 37:453-465 (1979). W. A. Marasco, S. H. Phan, H. Krutsch, H. J. Showell, D. E. Feltner, R. Naim, E. L. Becker, and P. A. Ward, Purification and identification of formyl-methionyl-leucylphenylalanine as the major neutrophil chemotactic factor produced by Escherichia co/i, J. Biol. Chem. 259:5430-5439 (1984).

102 24 25 26

21 28 29

30 31 32 33

E. S. FISHER

AND D. A. LAUFFENBURGER

K. Miller, A. Calverley, and E. Kagan, Evidence of a quartz-induced chemotactic factor for guinea pig alveolar macrophages, Environ. Res. 22:31-39 (1980). H. Z. Movat, Cellular emigration, in The Inflnmmarory Reaction, Elsevier, New York, Chapter F 1985. A. Perelsoo and G. Bell, Delivery of lethal hits by cytotoxic T-lymphocytes in multicellular 129:2796-2801 E. L. Pesanti antioxidants,

conjugates occurs sequentially but at random times, J. Immunol. (1982). and K. M. Nugeot, Modulation of pulmonary clearance of bacteria by Infect. Immunol. 48:57-61 (1985).

S. R. Rehm, G. N. Gross, and A. K. Pierce, Early bacterial clearance from murioe lungs: species-dependent phagocyte response, J. Clin. Invest. 66:194-199 (1980). T. L. Rothstein, M. Mage, G. Jones, and L. McHugh, Cytotoxic T lymphocyte sequential killing of immobilized allogeoeic tumor target cells measured by time-lapse microcinematography, J. Immunol. 121:1652-1655 (1978). L. W. Schwartz and C. A. Christman, Alveolar macrophage migration: influence of lung lining material and acute lung insult, Am. Rev. Respir. Dis. 120:429-439 (1979). R. Y. Stanier, E. A. Adelberg, and J. L. Ingraham, The Microbial World, 4th ed., Prentice Hall, Eoglewood Cliffs, NJ., 1976. A. Szabo, K. Schulteo, and Z. Schulteo, First passage time approach to diffusion controlled reactions, J. Chem. Phys. 72:4350-4357 (1980). G. B. Toews, G. N. Gross, and A. K. Pierce, The relationship bacterial clearance and phagocytic cell response in mice,

of iooculum size to lung Am. Reu. Respir. Dis.

35

120:559-566 (1979). R. Tranquillo, E. S. Fisher, B. Farrell, and D. A. Lauffeoburger, A stochastic model for chemosensory cell movement: application to oeutrophil and macrophage persistence and orientation, Murh. Biosci. 90:287-303 (1988). D. B. Warheit, L. H. Hill, G. George, And A. R. Brody, Time course of chemotactic

36

factor generation and the corresponding macrophage response to asbestos inhalation, Am. Rev. Respir. Dis. 134:128-133 (1986). P. C. Wilkinson, Chemotoxis and Inflammation, 2nd ed., Churchill Livingston, Edio-

34

burgh,

1982.