Leukocytes respond to a bacterial invasion by moving up a gradient of some chemical attractant produced by the bacteria and then ingesting the bacterium when it is encountered. Here we present a onedimensional model (Alt and Lauffenberger, 1987) to show if and when the leukocytes successfully defend against a bacterial invasion.
There are three concentrations that must be determined. These are the bacterial, attractant, and leukocyte concentrations, denoted by b, a, and c, respectively. The governing equations for these concentrations follow from the following assumptions concerning their behavior:
1. Bacteria diffuse, reproduce, and are destroyed when they come in contact with leukocytes:
dt dy2
2. The chemoattractant is produced by bacterial metabolism and diffuses:
3. The leukocytes are chemotactically attracted to the attractant, and they die as they digest the bacteria, so that dc dJr
dt dy
For this model we assume that the leukocyte flux is given by (16.61), although more general descriptions are readily incorporated.
To specify boundary conditions we assume that y = 0 is the skin surface and that a bloodtransporting capillary or venule lies at distance y = L from the skin surface. We assume that the bacteria cannot leave the tissue domain, although the attractant may diffuse into the bloodstream. Leukocytes enter the tissue from the bloodstream at a rate proportional to the circulating leukocyte density rb. When chemotactic attractant is present, the emigration rate increases, because leukocytes that would normally flow in the bloodstream tend to adhere to the vessel wall (margination) and then migrate into the interstitium. These considerations lead to the boundary conditions db
The governing equations are made dimensionless by setting x = y/L, t = kgt, u = r/cb, v = b/b0, and w = a/a0. We find that dv d2v
du ( d2U d ( dw\ . _ . t= P« hrr  a—[«—)  Y0(1 + v)u, (16.70)
dT \ dx2 dx \ dx where a0 = L2kvb{0ÎD,b{0 = g0/g1,a = xa0/^,Pv = , P« = kjp,Pw = kL, % = kdrb/kg, Y0 = g0/kg.
In nondimensional form the boundary conditions become dv
\dx dx J y Y0(j0 + j1w)(1 — u) atx = 1, where a = haL/D, = = hL.
There is at least one steadystate solution for this system of equations. It is the elimination state, in which v = w = 0 and u(x) = — cosh ( /Y^x, (16.74)
a \y pu j where A = cosh ^.Jp^ + sinh ^Jy). In this state, all bacteria are eliminated, and the leukocyte density is independent of any bacterial properties. This should represent the normal state for healthy tissue. If Y0/pu is small, then this steady distribution of leukocytes is nearly constant, at level (1 + 1)—1.
Bacterial diffusion is generally much smaller than the diffusion of leukocytes or of chemoattractant. Typical numbers are D = 10—6 cm2/s, \x = 10—7 cm2/s, < 10—8 cm2/s, kg = 0.5 h—1, and L = 100 ^m. With these numbers, pu and pw are relatively large, while pv is small. This leads us to consider an approximation in which bacterial diffusion is ignored, while attractant and leukocyte diffusion are viewed as fast. In this approximation, airborne bacteria can attach to the surface, but they do not move much on the time scale of leukocyte and chemoattractant motion.
Our first approximation is to ignore bacterial diffusion (take pv = 0) and then to assume that a bacterial invasion occurs at the skin surface x = 0. This is a reasonable assumption for periodontal, peritoneal, and epidermal infections, which are highly localized, slowly moving infections. Then, since we neglect bacterial diffusion, we specify the bacterial distribution by v(x,t) = V (t )S(x), (16.75)
where S(x) is the Dirac delta function. The governing equation for V(t) is dV
Since v = 0 for x > 0, the equations for w and u simplify slightly to dw d2w
while the effect of the bacterial concentration at the origin is reflected in the boundary conditions at x = 0 (found by integrating (16.69) and (16.70) "across" the origin), dw
An identity that will be important below is found by integrating (16.78) with respect to x to obtain dU
Y0 1 = U  Vu(0, t) + (j0 + jiw(1, t))(1  u(1, t)), (16.81)
where U(t) = /J u(x, t)Xx is the total leukocyte population within the tissue.
Our second approximation is to assume that the chemoattractant diffusion is sufficiently large, so that the chemoattractant is in quasisteady state,
This implies that w(x) is a linear function of x with gradient dw
Finally, we assume that pu is large (taking pu ^ to), so that the leukocyte density is also in quasisteady state with Jc = 0, that is, du
We can solve this equation and find the leukocyte spatial distribution to be u(x, t) = U(T)F(aV)eaVx, (16.85)
where F(z) = is determined by requiring U(t) = /J u(x, t)Xx.
Now we are able to determine u(0, t),u(1, t) from (16.85) and w(1, t) from (16.72) and (16.83), which we substitute into the equation for total leukocyte mass (16.81) to obtain dU ( )
Y0 1 — = (j0 + jV) (1  UF(aV)eaV)  (VF(aV) + 1)U, (16.86)
where j = j\/a. Similarly, from (16.76) and (16.85), we find the equation governing V to be dV
Phaseplane analysis
The system of equations (16.86)(16.87) is a twovariable system of ordinary differential equations that can be studied using standard phaseplane methods. In this analysis we focus on the influence of two parameters: ¡3, which characterizes the enhanced leukocyte emigration from the bloodstream, and a, which measures the chemotactic response of the leukocytes to the attractant.
One steadystate solution that always exists is U = (1 + V = 0. This represents the elimination state in which there are no bacteria present. Any other steady solutions that exist with V > 0 are compromised states in which the bacteria are allowed to persist in the tissue.
We assume that the system is at steady state at time t = 0 with U(0) = U0 = (1 + ¡r)1 when a bacterial challenge with V(0) = V0 > 0 is presented. We begin the analysis with simple cases for which a = 0.
In this case the system reduces to du
There are three nullclines: —V = 0 on the vertical line U = 1 and on the horizontal line V = 0, and U = 0 on the hyperbola V = ¡0(U+1)U.
Two types of behavior are possible. If $U0 < 1, there are no steady states in the positive first quadrant. The only steady state is at U = U0, V = 0. For U < U0, —f> 0, so that U decreases and V increases without bound. The bacterial challenge cannot be met. This situation is depicted in Fig. 16.8. In this and all the following phase portraits, the nullcline for V = 0 is shown as a short dashed curve, and the nullcline for —U = 0 is shown as a long dashed curve. The solid curve shows a typical trajectory starting from initial data U = U0,V = V0.
If %U0 > 1, there is a nontrivial steady state in the first quadrant, which is a saddle point. This means that there is a value V* for which a trajectory starting at U = U0, V =
Figure 16.8 Phase portrait for the system (16.86M16.87) with "small" $ = 1.6, "small" 3 = 0.1, a = 0. Other parameters are ¡30 = 1.0, Yo = 0.2, so that U0 = 0.5.
V* is on the stable manifold of this steady state and divides the line U = U0 into two types of behavior. If V <V * initially, the trajectory evolves toward the elimination state, while if V > V* initially, the trajectory is unbounded. Thus, for large enough f and small enough initial bacterial population, the challenge can be withstood, but for a larger initial bacterial challenge, the bacterial population wins the competition. The number V* is a monotone increasing function of f, and limf^TO V* = to. This follows because to the right of U = 1 the stable manifold is an increasing curve as a function of U, so that V* lies above the the value of V at the saddle point. However, as a function of f, the steadystate value of V is monotone increasing as f increases, approaching to in the limit f ^to, so V* ^to as well.
The phase portrait for this situation is depicted in Fig. 16.9. In this situation the bacterial challenge is met only if f is large enough and V0 is small enough, so that the leukocytes are effective killers, although with a = j = 0 they are not good hunters. Note that f = kdcb/kg, where kd is the rate at which leukocytes kill bacteria, kg is the growth rate of the bacteria, and cb is the leukocyte density in the blood. Hence, large f means that leukocytes are effective killers, since they kill bacteria at a rate exceeding the growth rate of the bacteria.
Here, the leukocytes can respond to the bacterial challenge by enhanced emigration from the bloodstream, but they cannot localize preferentially within the tissue. The system of equations becomes du
The nullclines for V are unchanged from above. The nullcline —U = 0 is the hyperbola V = fr+j+ff. For small j, with j+j < U0, the behavior of the system changes only slightly from Case I. These phase portraits are as depicted in Figs. 16.8 and 16.9.
\ \  
\ \  
dV/dt= 0  
TA   
i ^ \  
\ \  
dU/dt = 0 \\  
\\  
\\ 
^—  
VS. 
_______  
V». 
^__—  
0.30 0.40 0.45 0.50 0.20 0.25 0.30 0.40 0.45 0.50 Figure 16.9 Phase portrait for the system (16.86M16.87) with "large" £ = 3.0, "small" ft = 0.1, a = 0. Other parameters are ¡30 = 1.0, Yo = 0.2, so that U0 = 0.5. If fU0 < 1, the bacterial population grows without bound, whereas if fU0 > 1, the bacterial population can be eliminated if V <V* initially. The value V* is a monotone increasing function of ft. Thus, with ft small, the leukocytes have an enhanced ability to eliminate a bacterial population. In fact, if fft > ft + 1 (phase portrait not shown), then V* = to, so that a bacterial invasion of any size can be eliminated. Notice that in this case, the bacterial invasion is controlled because the leukocytes are effective killers and they effectively deploy troops to withstand the invasion. There is still no mechanism making them effective hunters. In all of the above cases, the leukocyte population decreases initially, and if the bacterial population is controllable, the leukocyte population eventually rebounds back to normal. If ft is large enough, with > U0, then the response to a bacterial invasion is with an initial increase in leukocyte population. If fft < ft + 1, then the bacterial population is unbounded; the invasion cannot be withstood. If fft > ft + 1 and fU0 < 1, there is a nontrivial steady state in the positive first quadrant that is a stable attractor. All trajectories starting at U = U0 go to this stable steadystate solution with U > U0. Since V > 0 for this steady solution, the bacterial population is controlled but not eliminated. This situation is depicted in Fig. 16.10. Finally, if fU0 > 1, the leukocyte population initially increases and then decreases back to normal as the bacterial population is eliminated. This situation is depicted in Fig. 16.11. The above information is summarized in Fig. 16.12, where four regions with differing behaviors are shown, plotted in the (1/ft, f) parameter space. The four regions are bounded by the curves f = 1/U0 and f = 1 + 1/ft and are identified by the asymptotic state for V, limT^TO V(t). For f > 1/U0 and f > 1 + 1/ft, the bacteria are always eliminated. For f > 1/U0 and f < 1 + 1/ft, there are two possibilities, either elimination or unbounded bacterial growth, depending on the initial size of the bacterial population. For f < 1/U0 and f > 1 + 1/ft, the bacteria survive but are controlled at population size Vp, and finally, for f < 1/U0 and f < 1 + 1/ft, the bacterial population cannot be controlled but becomes infinite. 0.70 0.70 Figure 16.10 Phase portrait for the system (16.86M16.87) with "small" f = 1.6, "large" ft = 3.0, a = 0. Other parameters are ft0 = 1.0, y0 = 0.2, so that U0 = 0.5. Figure 16.11 Phase portrait for the system (16.86M16.87) with "large" f = 3.0, "large" 0 = 3.0, a = 0. Other parameters are 00 = 1.0, y0 = 0.2, so that U0 = 0.5.

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