Lymph node model

The single-compartmental model of the lymph node was associated with the forcing function described in section 4.4.1 and fitted the data obtained from some experiments. However, the results from this model could not clearly explain lymphocyte migration within the node as this single compartmental model assumes that all lymphocytes behave identically, ie one homogenous kinetic, which is unlikely to be the case in reality. Some previous studies have reported that different lymphocyte subtypes have different abilities to migrate in lymphoid tissue (Washington et al., 1988; Abernethy et al., 1990; Abernethy et al., 1991), ie more than one kinetic system is required to explain lymphocyte kinetics in lymphoid tissue in its entirety. Thus a more complicated model would be more appropriate to explain this.

The multi-compartmental model in Figure 4.4(a) was initially constructed according to the physiology of lymphocyte migration as previously outlined in section 4.3. A full description of this model was given in Figure 4.4(a). However, to fit with the available experimental data, there was a need to modify this physiologically based model to another model, namely the modified multi- compartmental model (Figure 4.4(b)). There are n pairs (ie loops) of

Compartment M and B in this model. The first Compartment M (M,) receive lymphocytes (ie an input) from the blood whilst the last (Mn) deliver lymphocytes to the external system (ie as monitored in the efferent lymph). Compartment M and B represent, respectively, mobile (ie fast dynamics) and bound stages (ie slow dynamics) of migrating lymphocytes within the lymphoid tissue.

In this model, each Compartment B acts as a lymphocyte reservoir by receiving lymphocytes from the corresponding Compartment M, and storing and releasing them at a later stage. Lymphocytes in Compartment M could either bind to the reticular mesh (ie enter to corresponding Compartment B via kbm) or migrate further to the next adjacent Compartment M via km/n.33 However, lymphocytes that reside in Compartment B can only leave the lymph node by passing through the corresponding Compartment M via k^, ie there is no real connection from Compartment B to the external system. This is relevant to the physiological reality since bound lymphocytes must detach themselves from the reticular mesh and become mobile lymphocytes before proceeding with migration.

33 Lrtm is defined as the transfer coefficient for lymphocytes transferred from Compartment M(n-i) to Mn and for lymphocytes transferred from the final M-Compartment to the external system. Where n is equal or greater than 2. However, this was defined will be used only in this thesis as it does not agree with the standard definition of compartmental model analysis.

The modified multi-compartmental model in Figure 4.4(b) was constructed by lumping together some parameters of the physiologically based model (Figure 4.4(a)). To achieve this, a few assumptions were made. First, all of the

parameters that may be relevant to delivery (D), adhesion (A) and emigration (E) in physiologically based (ie true) models were aggregated into the entire modified multi-compartmental model. This assumption was dependent upon the

knowledge that the time scale of those events, which is of the order of seconds for delivery and adhesion and of minutes for emigration (Bjerknes et al., 1986), would have a minor impact upon the lymphocyte migration model as a whole (ie the entire migration process would take up to several hours). Second, among a pair (ie loop) of M and B compartments, there are three transfer coefficients to be identified (ie ^ , kbm , and k j) . Subsequently, there will be 3 x n transfer

coefficients for a model with n M-B loops. However, to make the fitting procedure simpler, all of those three transfer coefficients were assumed to be identical for all individual loops in the model (ie each individual loop represents identical lymphocyte kinetics). This may be relevant to the existence of anatomical segmentation of lymphoid tissue in that lymphocytes within each individual segment would probably have similar kinetic properties.

Brief descriptions of mathematical equations (ie mass balance equations) representing the first loop (ie Compartment Mn and B^ of the modified multi- compartmental model of the lymph node can be expressed as a set of differential equations as follows

where qm{t) and qb(t) , represent respectively the number of lymphocytes in Compartment Mt and Bv i(t) is the external input of lymphocytes to

Compartment M,. In this case, this input is a fraction of the forcing function (previously defined in blood model, section 4.4.1) via kmX. Given i(t) is a constant value of / and an initial condition that qm(0) = qb{0) = 0 (ie there are no labelled lymphocytes in Compartment IV^ and B, at time zero) a set of solutions written in exponential equations can be solved from (4.4) and (4.5) as

= i ( f ) + (0 - Km * (0 - ( 0 (4.4)

and qb(t) = where Ä ( A + k mm + k bm X^2 +

KbiA~A)

_A

Theoretically, the solution of the remainder of the model could be achieved by means of the same approach following (4.4) to (4.8). It is noted that taking time at infinity, the solution of equations gives a steady stage distribution of cells in

dqb (°°)

the entire compartment. Subsequently, from (4.5), this leads to — -— = 0 and

at

k bm£l m ( 0 0 ) = k m b ^ b ( ° ° ) (4.9)

gm(°°) _ k mb

< ^ (° ° )

Km

(4.10) Equation (4.10) describes the relationship between two coefficients , kbm and mass in Compartment M and B at the steady stage (ie time infinity).

Figure 4.4: Diagram of lymph node models

a) Physiologically based model

Fast Dynamics Slow Dynamics

Figure 4.4(a) demonstrates a multi-compartmental model of the different stages of lymphocyte migration through the lymph node. D, A and E denote the delivery, adhesion, and emigration stages. There are serial compartments of the mobile and the bound stage. (M1,..,Mn and B1,..,Bn respectively). Each compartment of the mobile and bound stages is connected to each other. The fast dynamics

compartment comprises all stages of D, A, E and M Meanwhile, the slow dynamics compartment comprises all stages of B.

b) Modified multi-compartmental model

Figure 4.4(b): The modified multi-compartmental model represents lymphocyte kinetics within the lymph node. Compartment M and B represent, respectively, mobile (ie fast kinetics) and bound (ie slow kinetics) compartments in lymphoid tissue. The transfer coefficients kbm, and k ^ were assumed to be of the same value for all individual loops of the entire model. More detail of the model was given in section 4.4.2 of this chapter.