Page 39 - Biomedical Engineering and Design Handbook Volume 1, Fundamentals

16 BIOMEDICAL SYSTEMS ANALYSIS

into the blood stream via the thoracic duct. Lymphocytes recirculate from the blood stream through
the tissue and return to the blood stream via the lymphatic system. When foreign cells are introduced
into the tissue, blood lymphocytes migrate into the tissue at an increased rate and bring about the
destruction of the target cells. Lymphocytes have memory and they exhibit an increased secondary
response, e.g., if after the rejection of the first graft, a second graft is introduced into the host, the
second graft is rejected much faster. A similar situation occurs in delayed hypersensitivity, which is
another cell-mediated reaction. In this analysis, let us assume that blood and tissue are well-stirred
compartments and that the newly produced lymphocytes are introduced into the blood compartment
(Reddy and Krouskop, 1978).
For sensitization to occur, a lymphocyte has to come in contact with a target cell. The number of
lymphocytes becoming sensitized at any given time (L (t)) is a function of the number of lympho-
s
cytes in the tissue (L (t)) and the number of target (foreign) cells (g(t))
T
L (t) = C L (t)g(t) (1.40)
s 1 T
Certain lymphocytes, upon encountering target cells, are transformed into memory cells. The
memory cell formation depends upon the number of lymphocytes in the tissue and the number of
target cells. The number of memory cells formed at any time (t) may thus be expressed as
L (t) = C L (t)g(t) (1.41)
ms 1 T
Sensitized lymphocytes stimulate the production of immunocompetent lymphocytes and the
effect of each sensitized cell lasts for a given period of time. For the purpose of the present analysis,
it is assumed that the effect of each sensitized lymphocyte decays exponentially over a period of
time. The production rate of blood lymphocytes at any time (t) due to the primary response
(dL /dt) prim would then be equal to the sum of the residual effect of all the lymphocytes sensitized
B
between time 0 and time t − Φ , where Φ is the time lag between sensitization and production of
1
1
the lymphocytes.
The number of lymphocytes produced due to primary response between time t and time (t − Φ )
1
would be
L (t) − L (t − Δt ) = C {L (t − Φ )Δt + L (t − Φ − Δt )Δt + L (t − Φ − 2Δt)e −K1Δt Δt
B B 3 S 1 S 1 S 1
Due to lymphocytes Due to lymphocytes Due to lymphocytes
sensitized at t − Φ sensitized at t − Φ − Δt sensitized at t − 2Φ − Δt
1 1 1−
+ L (t − Φ − rΔt)e −K1 rΔt Δt + … } (1.42)
S 1
Due to lymphocytes
sensitized at t − Φ 1− = rΔt
= C 3L (t − Φ − rΔt)e −K1 rΔt Δt (1.43)
3 S 1
Dividing by Δt and taking the limits as Δt Π 0, the left-hand side becomes a derivative and the right-
hand side can be represented as an integral in terms of the hereditary function

(dL (t)/dt) = C ∫ t − Φ 1L (τ)e −K1(t − Φ − τ) dτ (1.44)
1
B primary 3 0 S
Substituting for L in terms of L ,
S T
(dL (t)/dt) = k ∫ t − Φ 1L (τ)e − K1(t − Φ − τ) dτ (1.45)
1
B primary 2 0 T
For the secondary response to appear, a memory cell must encounter a target cell, and therefore the
secondary response depends upon the number of memory cells and the number of target cells.
Similar to Eq. 1.45, Reddy and Krouskop (1978) expressed the secondary response in terms of a
hereditary function
(dL (t)/dt) = k ∫ t − Φ 2 L (τ)g(τ)e −K4(t − Φ − τ) g(t − Φ )dτ (1.46)
2
B secondary 3 0 T 3