Mathematical modeling reveals kinetics of lymphocyte recirculation in the whole organism

Abstract

The kinetics of recirculation of naive lymphocytes in the body has important implications for the speed at which local infections are detected and controlled by immune responses. With a help of a novel mathematical model, we analyze experimental data on migration of 51Cr-labeled thoracic duct lymphocytes (TDLs) via major lymphoid and nonlymphoid tissues of rats in the absence of systemic antigenic stimulation. We show that at any point of time, 95% of lymphocytes in the blood travel via capillaries in the lung or sinusoids of the liver and only 5% migrate to secondary lymphoid tissues such as lymph nodes, Peyer's patches, or the spleen. Interestingly, our analysis suggests that lymphocytes travel via lung capillaries and liver sinusoids at an extremely rapid rate with the average residence time in these tissues being less than 1 minute. The model also predicts a relatively short average residence time of TDLs in the spleen (2.5 hours) and a longer average residence time of TDLs in major lymph nodes and Peyer's patches (10 hours). Surprisingly, we find that the average residence time of lymphocytes is similar in lymph nodes draining the skin (subcutaneous LNs) or the gut (mesenteric LNs) or in Peyer's patches. Applying our model to an additional dataset on lymphocyte migration via resting and antigen-stimulated lymph nodes we find that enlargement of antigen-stimulated lymph nodes occurs mainly due to increased entrance rate of TDLs into the nodes and not due to decreased exit rate as has been suggested in some studies. Taken together, our analysis for the first time provides a comprehensive, systems view of recirculation kinetics of thoracic duct lymphocytes in the whole organism.

Figure 1. Two types of experiments performed in rats by Smith & Ford .

In the first set of experiments (migration data, panel A), thoracic duct lymphocytes (TDLs) were collected overnight from male donor rats by cannulation and labeled with sodium-[Cr] chromate. Labeled lymphocytes were passaged from blood to lymph in vivo by thoracic duct cannulation in an intermediate male. Collected lymphocytes were injected into female recipient rats and the percent of injected donor lymphocytes was measured in major lymphoid and nonlymphoid organs of the recipient rats. In the second set of experiments (cannulation data, panel B), TDLs were passaged via an intermediate host and then injected into the final recipients. Donor lymphocytes were counted during the thoracic duct cannulation of the recipient rats over the period of 45 hours.

Figure 2. Cartoon representing migration of thoracic duct lymphocytes (TDLs) in rats.

Blood is the main compartment that connects all tissues, and the rate of lymphocyte migration from the blood to other tissues is denoted as where and . Cells leaving a particular organ return to the blood at a rate with the exception of lymphocytes in the Peyer's patches from which lymphocytes migrate to the mesenteric LNs at a rate [67, p. 470]. In these experiments the total number of labeled cells declined over time (Figure 2 in Text S1), and therefore we allow for a constant removal rate of TDLs from the blood occurring due to death and/or migration of lymphocytes to other tissues that were not sampled. The percent of transferred lymphocytes was measured in the blood (), lung (), liver (), spleen (), subcutaneous lymph nodes (SCLNs, ), mesenteric lymph nodes (MLNs, ), and Peyer's patches (PPs, ). Lymphocytes exiting lymph nodes return to the blood via right lymphatic and left lymphactic (thoracic) ducts (see also Figure 1 in Text S1). The thoracic duct collects lymph from all mesenteric lymph nodes and from approximately half () of subcutaneous lymph nodes . Lymph from other subcutaneous lymph nodes () enters the blood via the right lymphatic duct , .

Figure 3. Mathematical model accurately predicts the hierarchy of recirculation of thoracic duct lymphocytes (TDLs) between major murine organs.

Cr-labeled TDLs were passaged via an intermediate host and then were transferred into syngenic rats (Figure 1A). The percent of transferred cells was measured at different times after cell transfer in major lymphoid and nonlymphoid murine organs and is shown by markers. We fit the mathematical model of lymphocyte recirculation (eqn. (1) – (6)) to these experimental data using nonlinear least squares; model fits are shown as lines. Plots are for the first 30 minutes of the experiment (A) or for the whole experiment (B, abscissa values are plotted on the log-scale). Parameter estimates of the model are given in Table 1. Different y-scales in panels A and B were used for clarity.

Figure 4. Increase in the average residence time of lymphocytes in lymph nodes with time since cannulation is needed to explain the kinetics of labeled lymphocyte exit during thoracic duct cannulation.

Cr-labeled TDLs were passaged via an intermediate host and then transferred into final recipient rats. Recipients were cannulated via the thoracic duct (Figure 1B) and the rate of exit of labeled TDLs into the thoracic duct per hour was measured . The data are shown by markers (points). In panel A we show that for the parameter estimates from migration experiments (Table 1) the models with different number of subcompartments in LNs () fail to describe experimental data when of lymphocytes exiting SCLNs migrate to the blood via the thoracic duct. To explain the data, we let the rate of lymphocyte exit from the LNs to decline exponentially with the time since cannulation, (panel B). We fit the data on the output rate of labeled cells into the thoracic duct using the mathematical model (eqn. (1) – (6)). We fix all model parameters to values shown in parameters for and fit only parameters and . The best description of the data was found when 1) the fraction of lymphocytes in SCLNs enter the blood via the thoracic (left lymphatic) duct, and 2) the rate of lymphocyte migration via lymph nodes and Peyer's patches declines with time since cannulation at a rate min (solid line in panel B). The model fails to predict thoracic duct output data if residence times of lymphocytes in LNs is unaffected by cannulation (, large dashing lines in panels A and B).

Figure 5. Antigen-stimulated popliteal lymph nodes (pLN) accumulate higher numbers of Cr-labeled TDLs because of increased entrance rate of TDLs into the LN.

We plot the percent of labeled TDLs found in the antigen-stimulated (dots) and resting (triangles) popliteal LNs. Model fits of the data are shown by lines. We found that the entrance rate into stimulated and unstimulated pLNs is min and min, respectively. Exit rate of lymphocytes from antigen-stimulated and unstimulated pLNs are min and min (two sub-compartments in the LNs, ). To describe the dynamics of TDLs in the blood and other organs we used parameters given in Table 1. Our results suggest that antigenic stimulation of the lymph node with sheep erythrocytes increases the rate of entrance of TDLs into the LN almost 4 fold without a significant change in the lymphocyte exit rate (, ).

Figure 6. The mathematical model accurately describes the kinetics of lymphocyte migration via major lymphoid organs (panel A) and kinetics of labeled lymphocytes exit during the thoracic duct cannulation (panel B).

We fit the basic mathematical model (eqn. (1) – (6)) simultaneously to the data on lymphocyte migration (Figure 3) and lymphocyte output via the thoracic duct (Figure 4) using generalized likelihood method assuming subcompartments in LNs and PPs (see Materials and Methods). In panel A, symbol and line labeling is similar to that of Figure 3. The model predicts that the of lymphocytes exiting SCLNs migrate to the blood via the right lymphatic duct and thus are not sampled during the thoracic duct cannulation. Numbers in parentheses in panel B indicate the percent of cells exiting into blood from different LNs in 45 hours as predicted by the model (lines) or as observed in the data (dots). In the data, 55% of transferred TDLs were collected during 45 hours of the thoracic duct cannulation. The model also predicts the contribution of lymphocytes exiting SCLNs (short dashed line in panel B) and MLNs (long dashed line in panel B) via the thoracic duct. To explain the data, the rate of egress of lymphocytes from LNs and PPs declines exponentially with time during cannulation at an estimated rate min. Other parameters for the migration kinetics of TDLs are nearly identical to those given in Table 1. We fixed in fits of these data. Estimated standard errors are and (see eqn. (8)).