The Spleen Is a Major Source of Effectors During the Primary

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Modeling of Influenza-Specific CD8+ T Cells during the Primary Response Indicates that the Spleen Is a Major Source of Effectors

This information is current as Hulin Wu, Arun Kumar, Hongyu Miao, Jeanne of October 1, 2021. Holden-Wiltse, Timothy R. Mosmann, Alexandra M. Livingstone, Gabrielle T. Belz, Alan S. Perelson, Martin S. Zand and David J. Topham J Immunol 2011; 187:4474-4482; Prepublished online 23 September 2011; doi: 10.4049/jimmunol.1101443 Downloaded from http://www.jimmunol.org/content/187/9/4474

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The Journal of Immunology is published twice each month by The American Association of Immunologists, Inc., 1451 Rockville Pike, Suite 650, Rockville, MD 20852 Copyright © 2011 by The American Association of Immunologists, Inc. All rights reserved. Print ISSN: 0022-1767 Online ISSN: 1550-6606. The Journal of Immunology

Modeling of Inﬂuenza-Speciﬁc CD8+ T Cells during the Primary Response Indicates that the Spleen Is a Major Source of Effectors

Hulin Wu,*,1 Arun Kumar,*,1 Hongyu Miao,*,1 Jeanne Holden-Wiltse,* Timothy R. Mosmann,†,‡ Alexandra M. Livingstone,†,‡ Gabrielle T. Belz,x Alan S. Perelson,{ Martin S. Zand,{ and David J. Topham†,‡

The biological parameters that determine the distribution of virus-specific CD8+ T cells during influenza infection are not all directly measurable by experimental techniques but can be inferred through mathematical modeling. Mechanistic and semimechanistic ordinary differential equations were developed to describe the expansion, trafficking, and disappearance of activated virus-specific CD8+ T cells in lymph nodes, spleens, and lungs of mice during primary influenza A infection. An intensive sampling Downloaded from of virus-specific CD8+ T cells from these three compartments was used to inform the models. Rigorous statistical fitting of the models to the experimental data allowed estimation of important biological parameters. Although the draining lymph node is the first tissue in which Ag-specific CD8+ T cells are detected, it was found that the spleen contributes the greatest number of effector CD8+ T cells to the lung, with rates of expansion and migration that exceeded those of the draining lymph node. In addition, models that were based on the number and kinetics of professional APCs fit the data better than those based on viral load, suggesting that the immune response is limited by Ag presentation rather than the amount of virus. Modeling also suggests that http://www.jimmunol.org/ loss of effector T cells from the lung is significant and time dependent, increasing toward the end of the acute response. Together, these efforts provide a better understanding of the primary CD8+ T cell response to influenza infection, changing the view that the spleen plays a minor role in the primary immune response. The Journal of Immunology, 2011, 187: 4474–4482.

urrent strategies for preventing or decreasing the sever- served antigenic regions in the viruses are perhaps the most im- ity of influenza infection focus on increasing virus- portant immune component against emerging novel strains of the C neutralizing Ab titers through vaccination, as experi- virus. Understanding the factors that regulate the T cell response is ence indicates that this is the best way to prevent morbidity and key to the development of new immunization strategies designed mortality. It is generally accepted that T cells provide a substantial to promote cross-reactive immunity. by guest on October 1, 2021 degree of protection from disease, and cytotoxic CD8+ T cells With the exception of certain highly pathogenic strains of in- control the infection by eliminating infected cells when neutral- fluenza (4–6), the virus infection is usually restricted to the air- izing Abs are absent (1–3). Because influenza viruses are unique ways and lung tissues. This phenomenon is believed to be due to in their ability to undergo antigenic shift, resulting in a wholesale the limited expression of host enzymes required to cleave the viral evasion of neutralizing Abs, T cells that cross-react with con- hemagglutinin protein into its active conformation (7, 8). Because of this, it takes time for Ag (and APCs) to reach the draining + *Department of Biostatistics and Computational Biology, University of Rochester lymph node (9, 10) and also for virus-specific effector CD8 Medical Center, Rochester, NY 14642; †David H. Smith Center for Vaccine Biology T cells to migrate through the bloodstream back to the site of and Immunology, University of Rochester Medical Center, Rochester, NY 14642; ‡Department of Microbiology and Immunology, University of Rochester Medical infection. The dynamic nature of this process makes it very dif- Center, Rochester, NY 14642; xDivision of Molecular Immunology, Walter and Eliza ficult to study directly. Conventional approaches rely on tissue { Hall Institute of Medical Research, Melbourne, Victoria 3052, Australia; Theoret- sampling at defined time points to assemble a set of “snapshots” ical Biology and Biophysics Group, Los Alamos National Laboratory, Los Alamos, NM 87545; and ‖Division of Nephrology, Department of Medicine, University of that can be linked together to understand the system as a whole. Rochester Medical Center, Rochester, NY 14642 Unfortunately, this understanding still depends on a number of 1H.W., A.K., and H.M. contributed equally to this work. assumptions. For example, a given tissue sampling tells us where Received for publication May 16, 2011. Accepted for publication August 23, 2011. the cells are at that point in time but does not reveal where they This work was supported by National Institute of Allergy and Infectious Diseases came from. In addition, the rates of cell proliferation and traf- Contract HHSN272201000055C and Grant R01 AI069351 (to M.S.Z.). Portions of ficking into and out of a given tissue can, at best, be inferred. In this work were performed under the auspices of the U.S. Department of Energy under Contract DE-AC52-679 06NA25396 (to A.S.P.). the end, the number of cells at any single tissue site will depend on Address correspondence and reprint requests to Dr. Martin S. Zand or Dr. David the combined, simultaneous processes of proliferation, trafficking, J. Topham, University of Rochester Medical Center, 601 Elmwood Avenue, Box 675, retention, and cell loss due to death, phagocytosis, or sloughing. Rochester, NY 14642 (M.Z.) or University of Rochester Medical Center, 601 Elmwood Mathematical modeling and computational approaches to the Avenue, Box 609, Rochester, NY 14642 (D.J.T.). E-mail addresses: martin_zand@ urmc.rochester.edu (M.Z.) and [email protected] (D.J.T.) problem offer one of the best means to estimate the key biological Abbreviations used in this article: AIC, Akaike Information Criterion; cMEM, com- parameters that cannot be directly and unambiguously measured. plete minimum essential medium; DC, dendritic cell; IAV, influenza A virus; MLN, Mathematical models have long been used to investigate viral dy- mediastinal lymph node; NP, nucleoprotein; ODE, ordinary differential equation; PA, namics and immune responses to viral infections, including influenza acid polymerase; RSS, residual sum of squares. A virus (IAV) (11–18). In particular, we recently developed complex Copyright Ó 2011 by The American Association of Immunologists, Inc. 0022-1767/11/$16.00 differential equation models to quantify key kinetic parameters and www.jimmunol.org/cgi/doi/10.4049/jimmunol.1101443 The Journal of Immunology 4475 predict early and adaptive immune responses against IAV infection draining lymph nodes were digested for 20 min at room temperature with (19, 20). However, limited by available experimental information, collagenase/DNase and then treated for 5 min with EDTA to disrupt our previous studies mainly focused on the lung. Better under- T cell–dendritic cell (DC) complexes. Previously, it has been shown that depletion of cells expressing CD11c removed Ag-specific stimulatory standing of immune cell kinetics among multiple compartments is capacity indicating that Ag presentation was limited to CD11c+ DCs thus a natural and necessary next step. Toward this goal, in this (9, 22). The LacZ-inducible hybridomas specific for Db nucleoprotein b article we propose new mathematical models of multicompartment (NP)366–374 (BWZ-IFA.NP4) (9) and D acid polymerase (PA)224–233 cell trafficking and fit them to experimental data from mice infected (BWZ-IFA.PA1) were maintained in DMEM containing 10% FCS, 50 mM 2-mercaptoethanol, 2 mM L-glutamine, 100 U/ml penicillin, and 100 mg/ with the H3N2 influenza A/X31 strain. Based on model fitting re- ml streptomycin (hybridoma medium). These clones were used to analyze sults, we compare and verify several mechanistic hypotheses that Ag presentation by lymph node cells, as previously described (23). cannot be directly inferred without mathematical models. Mathematical models and assumptions Materials and Methods First, we consider a mechanistic model involving lymph node, spleen, and Murine kinetic experiments lung to describe CD8+ T cell response to primary IAV infection. For the mediastinal lymph node (MLN), we assume that there exist proliferation, Female C57BL/6 mice (The Jackson Laboratory, Bar Harbor, ME) from 6 to migration to spleen, migration to lung, migration to other compartments, 13 wk of age (n = 78) were anesthetized with 2,2,2-tribromoethanol and and apoptosis (death) or loss due to other mechanisms, and all these dy- intranasally inoculated with 0.03 ml of 1 3 105 EID H3N2 A/Hong 50 namic rates are proportional to the number of activated, effector CD8+ Kong/X31 IAV. Tissues for determination of CD8+ T cell counts were T cells, T m, in the MLN. Furthermore, we assume that the influx of CD8+ collected at days 0, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, and 28 postinfection. At E T cells from other compartments to the MLN can be ignored compared each time point, data were collected from six mice (for most of the time with the other kinetic components mentioned above. For spleen, the points) and from 12 mice on day 12 for flow cytometry. Data from days 5 + s number of CD8 T cells, T , is assumed to be mainly affected by pro- Downloaded from to 14 were used in modeling fitting to understand the cell trafficking during E liferation in place, influx from MLN, migration to lung, and loss due to the adaptive phase. All experiments involving animals were reviewed and death or other mechanisms. For CD8+ T cells in lung, T l , direct influx approved by the University Committee for Animal Resources (institutional E from both lymph node and spleen is considered, and the loss is again due animal care and use committee). to death or other reasons. We further make the initial assumptions that the Tissue collection activation of CD8+ T cells that leads to expansion of the virus-specific population in both MLN and spleen is stimulated mainly by Ag-bearing On the day of organ harvest, mice were euthanized using a lethal dose of 2,2,2- DCs, with negligible expansion due to proliferation in lung. Based on these tribromoethanol. Whole lung, mediastinal lymph nodes, and spleen tissues tenets, we formulated the following mechanistic model: http://www.jimmunol.org/ were isolated from individual mice. Single-cell suspensions were prepared by Dounce homogenization or mechanical disruption with a fine mesh and dT m E ¼½ Dmðt 2 Þ 2 T m 2 ð þ ÞT m suspended in MEM with 5% FBS. RBCs were lysed using buffered ammonium dt rm t dm E gms gml E chloride solution (Gey’s solution), and cells were resuspended in complete minimum essential medium (cMEM) for analysis. Total lymphocyte numbers dT s E ¼½ Dsðt 2 Þ 2 T s 2 T s þ T m ð Þ were obtained by manual counting using a hemocytometer and confirmed in dt rs t ds E gsl E gms E 1 many samples by flow cytometry measurements. + dT l Determination of virus-specific CD8 T cell immune responses E ¼ T m þ T s 2 T l ; dt gml E gsl E dl E

Cell phenotypes were determined by flow cytometry after surface and m s by guest on October 1, 2021 intracellular staining. To determine the number of virus-specific CD8+ where D denotes the number of mature Ag-bearing DCs in MLN; D is T cells, spleen cells from naive B6.SJL (CD45.1+) mice were used as APCs the number of mature DCs in spleen; t is the time delay of the effects of DCs + + and infected with influenza (multiplicity of infection = 1) in 1 ml serum- on CD8 T cell activation; rm and rs are the proliferation rates of CD8 free media for 60 min. Infected cells were then washed and resuspended in T cells stimulated by DCs in MLN and spleen, respectively; dm, ds,anddl cMEM. APCs (1 3 106) were added to 1 3 106 responders for a total are the loss rates in MLN, spleen, and lung, respectively; gms is the migration rate from MLN to spleen, gml the migration rate from MLN to lung, volume of 100 ml. GolgiPlug (BD Pharmingen) was then diluted to 1 ml/ml s in cMEM and 100 ml added to each well. Cells were incubated for 5∼6hat and gsl the migration rate from spleen to lung. Finally, data for D are not available in this study or the literature to the best knowledge of the authors. 37˚C. Samples were then surface stained for CD3, CD4, and CD8 and then m s washed and resuspended in Cytofix/Cytoperm (BD Pharmingen) 100 ml/ D is thus used to replace D in all calculations, with the assumptions that well for 15 min. After one Perm/Wash (BD Pharmingen), anti–IFN-g, the APCs in the MLN and spleen share similar temporal patterns but can anti–TNF-a, and anti–IL-2 Abs were added in Perm/Wash and cells in- differ in total number proportional to the size of the two organs. cubated for 30 min on ice in the dark. Samples were resuspended in PBS/ We make explicit and strong assumptions in the model above. However, + BSA for FACS. Samples were analyzed on a BD Biosciences LSRII cy- we recognize that numerous factors may affect CD8 T cell trafficking after tometer, and results were processed using FlowJo software (Tree Star). influenza virus infection, and their mechanisms are not fully understood; The number of Ag-specific CD8+ cells per lung was calculated as follows: therefore, the mechanistic model above may be oversimplified and thus not able accurately to interpret the experimental observations. We hence ex- Cytokineþ CD8þ T cell counts ðflowÞ plore three more hypotheses: first, that the effects of other undefined + Lymphocyte counts ðflowÞ factors and mechanisms that affect CD8 T cell growth and disappearance are not negligible; second, that the rise and fall of CD8+ T cell numbers in Total number of cells in pooled lungs 3 : the infection is determined by clonal expansion followed by increased Number of pooled lungs programmed cell death when the pathogen is cleared, versus a constant rate of loss over the acute phase; third, that an influx of T cells from other For this analysis, the total number of CD3+ CD8+ cells expressing one or 2 + + tissue compartments such as the liver (24) could explain late increases in more of the three cytokines TNF-a, IFN-g, and IL-2 (TNF IFN IL-2 , + 2 2 2 2 2 2 splenic CD8 T cell numbers not due to proliferation (Fig. 1). TNF IFN+IL-2 , TNF+IFN IL-2+, TNF+IFN IL-2 , TNF-IFN IL-2+, 2 To study the first hypothesis, we consider the following model: TNF+IFN+IL-2+, and TNF+IFN+IL-2 ) was used as the number of virus- + specific cells. See Fig. 1 for the observed CD8 T cell counts in lymph dT m E ¼ ðtÞT m 2 ð þ ÞT m node, spleen, and lung. dt km E gms gml E Detection of Ag presentation by lymph node dendritic cells dT s E ¼ ðtÞT s 2 T s þ T m ð Þ APCs were isolated essentially as described (10, 21). C57BL/6 mice for dt ks E gsl E gms E 2 influenza experiments were obtained from The Walter and Eliza Hall In- stitute of Medical Research. Mice were maintained under specific patho- l dTE m s l gen-free conditions. Experiments with mice began when they were be- ¼ gmlTE þ gslTE 2 dlTE ; tween 6 and 10 wk of age. Mice were anesthetized with methoxyflurane dt and then intranasally infected with a nonlethal challenge of H3N2 A/Hong where two unknown time-varying parameters kmðtÞ and ksðtÞ are in- m s Kong/X31 IAV diluted in 25 ml sterile PBS. The regional (mediastinal) troduced to replace ½rmD ðt 2 tÞ 2 dm and ½rsD ðt 2 tÞ 2 ds and 4476 CD8+ T CELL GROWTH AND MIGRATION should be interpreted as the net growth rates of CD8+ T cells in MLN and Some of our parameter estimations and simulations for ODE models spleen, respectively, not distinguishing proliferation from disappearance/ were performed using DEDiscover, a publicly available tool developed by death. To study the second hypothesis, we introduce a time-varying loss the University of Rochester Center for Biodefense Immune Modeling + rate of CD8 T cell dlðtÞin lung, removing the two previous unknown time- (https://cbim.urmc.rochester.edu/software). DEDiscover is a user-friendly varying parameters introduced in model 2, so the third model is modeling environment for both mathematical modelers and immunologists that provides a workflow-oriented interface and a comprehensive set of m dTE m m m computing algorithms. In particular, to help modelers and immunologists ¼½r D ðt 2 tÞ 2 dmT 2 ðg þ g ÞT dt m E ms ml E to compare and select models built upon alternative assumptions, DED- iscover also reports the AIC score after model fitting. dT s E ¼½ Dsðt 2 Þ 2 T s 2 T s þ T m ð Þ dt rs t ds E gsl E gms E 3 Results dT l E ¼ T m þ T s 2 ðtÞT l : Model fitting to determine key cellular parameters dt gml E gsl E dl E Our objective using the modeling approach is to estimate bi- + To study the third hypothesis, we consider a time-varying flux of CD8 ological parameters that are difficult or impossible to measure T cells from other additional compartments to spleen at a non-parametric + directly. These include such processes as cell trafficking from rate hsðtÞ, and we also keep the time-varying loss rate of CD8 T cell dlðtÞ in lung. This alternative model can be written as one anatomical compartment to another and growth of the virus- specific CD8+ T cell population. Initially, we fitted model 1 to m dT + E ¼ ½ Dmðt 2 Þ 2 T m 2 ð þ ÞT m the experimental data from mice for virus-specific CD8 T cells dt rm t dm E gms gml E in the MLN, spleen, and lung. The virus-specific T cells were

dT s identiﬁed and measured using an intracellular cytokine assay with Downloaded from E ¼½ Dsðt 2 Þ 2 T s 2 T s þ T m þ ðtÞðÞ dt rs t ds E gsl E gms E hs 4 live-virus restimulation and staining for IL-2, IFN-g, and TNF-a as previously described in our original global inﬂuenza model l + dTE m s l (19). CD8 T cells that stained positive for any combination of one ¼ g T þ g T 2 dlðtÞT : dt ml E sl E E to three cytokines were counted as positive. Prior to day 5, the In models 2 to 4, because the time-varying parameters do not depend on difference between virus-restimulated and control cultures was +

explicit assumptions about how CD8 T cells are stimulated or disappear in close to zero and inconsistent, making it impossible accurately to http://www.jimmunol.org/ spleen, we call such models “semimechanistic.” fit the model to the data. In contrast, virus-specific CD8+ T cells All these alternative hypotheses are explored and tested from our ex- become easily detectable from day 5 onward by a variety of perimental data using the statistical methods introduced below. We sum- marize our mathematical notations and parameter definitions in Table I. assays, including the intracellular cytokine approach (19, 24, 36– Also, the schematic illustration of the CD8+ T cell migration pathways is 39). Therefore, data from days 5 to 14 were used for the model depicted in Fig. 2. fitting. The experimental data and the fitted curves are plotted in Figs. 1 and 3, with the estimated parameters reported in Table II. Statistical methods Growth/loss rates from mechanistic model (model 1)

We performed structural and practical identifiability analysis for the four In the mechanistic model (model 1; see Materials and Methods), by guest on October 1, 2021 ordinary differential equation (ODE) models presented earlier before we we make the assumption that the expansion of CD8+ T cells in applied statistical methods to estimate the parameters in these ODE models (25, 26). From the structural identifiability analysis, all the parameters both MLN and spleen is mainly a consequence of Ag presenta- were found theoretically identifiable. Further, we performed practical tion analogous to mature Ag-bearing DCs that can activate naive identifiability analysis to confirm whether all the parameters are estimable T cells. We also allow a time delay for stimulation of T cell ex- from experimental data with noise using Monte Carlo simulations (25, 27, pansion, the length of which is unknown but can be estimated 28). Our results demonstrate that almost all the parameters are practically from data. The concentrations of DCs in MLN and spleen are identifiable except for dm, ds, and gml. We estimated all the kinetic parameters including both constant and time- based on the two different parameters rm and rs in our model. varying parameters as well as the delay time and the initial conditions for the In more explicit terms, our assumptions are that: 1) the stimula- ODE models from the time course data of activated CD8+ T cell numbers in tion strength for activation and growth of the virus-specific CD8+ MLN, spleen, and lung using the nonlinear least squares method (29–32). We used a hybrid global optimization algorithm DESQP in Ref. 30 to minimize the residual sum of squares. The log-transformation and stan- dardization of the data were used to stabilize the estimation algorithm. The time-varying parameters in models 2 to 4 were approximated by a sum of 3 third-order B-splines of df 3 (33), for example, kmðtÞ¼+j¼ amjbj;3ðtÞ ðtÞ¼+3 a b ðtÞ 1 and ks j¼1 sj j;3 , so that the time-varying parameter ODE models can be transformed into standard ODE models with constant parameters (32). The number of knots and basis functions can be determined using the model selection criterion such as the Akaike In- formation Criterion (AIC) (34, 35). We used Fisher Information Matrix to calculate 95% confidence interval of the parameters. We explore and evaluate various alternative models and model assumptions using the AIC score defined as (34, 35): ¼ n RSS þ k; AIC ln n 2 where RSS is the residual sum of squares, n is the number of observations, and k is the number of unknown parameters. The AIC is a score that can be FIGURE 1. Experimental observations and smoothed curves for effector used to evaluate different models by compromising the fitting residuals and + model complexity (measured by the number of parameters in the model). CD8 T cells identified by intracellular cytokine staining for IL-2, IFN-g, A smaller AIC score indicates a better model. In this study, the number of and TNF-a in spleen, lung, and lymph node after IAV infection. Experi- observations (n) is 180 in total (from day 5 to day 14), and the number of mental data indicated by *, spleen; :, lung; s, MLN. Smoothed curves parameters (k) is 9 in model 1, 12 in model 2, 11 in model 3, and 14 in indicated by solid line for MLN, dotted line for spleen, and dashed line for model 4. lung. The Journal of Immunology 4477

largest compared with the other three models, which suggests that this mechanistic model does not fit the data as well, possibly due to an oversimplification of the model assumptions or omission of key biological parameters included in the alternative models. The estimated DC/APC-dependent effector CD8+ T cell pop- m ulation expansion rates for MLN and spleen, rmD ðt 2 tÞ and s 21 21 rsD ðt 2 tÞ, respectively have peak values of 1.3 d and 3.5 d at around day 7, corresponding with a doubling time of 12 h and 4.8 h. This suggests that the growth of the virus-specific CD8+ T cell population in spleen is ∼2-fold greater than that in the MLN. Similar conclusions were drawn from the results derived from models 3 and 4 described later (see Table II and Fig. 4). The time delay for stimulation of T cell expansion by DCs is

+ calculated as 3.08 d, matching well with direct measurements in the FIGURE 2. Schematic diagram depicting pathways of CD8 T cell literature (40–43). The estimates of the migration rates of effector migration among three compartments. + CD8 T cells from MLN to spleen and from spleen to lung, gms 21 21 and gsl, are 0.16 d and 0.5 d , respectively, which suggests that migration from spleen to lung is substantial. The constant loss rate + T cell population is proportional to the number of DCs; and 2) the of effector CD8 T cells from the lung, dl, could be reliably (p , 2 Downloaded from temporal patterns of stimulation in MLN and spleen are similar, 0.001) estimated at ∼4d 1, corresponding with a half-life of 4.2 h. but the numbers can be very different based on the size of the However, it was not possible reliably to determine the values for + organs. Based on these assumptions, we ﬁt this model, and the the constant loss rates (dm and ds) of CD8 effectors from the results are summarized in Table II. Both the residual sum of MLN and spleen or the migration rate of activated CD8+ T cells squares (15.77) and AIC score (2420.29) of this model are the from MLN to lung (gml). This is due to the fact that the estimated http://www.jimmunol.org/ Table I. Variable/parameter deﬁnitions in models 1 to 4

Variable/ Parameter Deﬁnition Units Assay Description

m + TE Effector CD8 T cells in Cells per MLN ICS CD8 T cells secreting any MLN combination of TNF, IFN, IL-2

s + + TE Effector CD8 T cells in Cells per spleen ICS CD8 T cells secreting any

spleen combination of TNF, IFN, IL-2 by guest on October 1, 2021

l + + TE Effector CD8 T cells in Cells per lung ICS CD8 T cells secreting any lung combination of TNF, IFN, IL-2

Dm Matured DCs in MLN Cells per MLN PNAS LacZ-inducible Number of DbNP366–374 speciﬁc hybridomas for DbNP LacZ+ cells per MLN and DbPA (61) Ds Matured DCs in spleen Cells per spleen No data Replaced by Dm in calculation 21 21 rm Proliferation rate of effector d (cells per MLN) Estimated In models 1, 3, and 4 only CD8+ T cells in MLN stimulated by DCs 21 21 rs Proliferation rate of effector d (cells per spleen) Estimated In models 1, 3, and 4 only CD8+ T cells in spleen stimulated by DCs t Lag time for DCs to stimulate d Estimated In models 1, 3, and 4 only CD8+ T cell proliferation 21 dm Disappearance rate of effector d Estimated In models 1, 3, and 4 only CD8+ T cells in MLN 21 ds Disappearance rate of effector d Estimated In models 1, 3, and 4 only CD8+ T cells in spleen + 21 gms Migration rate of effector CD8 d Estimated In all models T cells from MLN to spleen + 21 gml Migration rate of effector CD8 d Estimated In all models T cells from MLN to lung + 21 gsl Migration rate of effector CD8 d Estimated In all models T cells from spleen to lung 21 dl Disappearance rate of effector d Estimated In all models, time-varying in CD8+ T cells from lung models 3 and 4 + hs(t) Net ﬂux rate of effector CD8 Cells per day Estimated In model 4 only, time-varying T cells into spleen from compartments other than MLN, spleen, and lung + 21 km(t) Net growth rate of effector CD8 d Estimated In model 2 only, time-varying T cells in MLN + 21 ks(t) Net growth rate of effector CD8 d Estimated In model 2 only, time-varying T cells in spleen 4478 CD8+ T CELL GROWTH AND MIGRATION

However, the estimated migration rates from MLN to spleen and from spleen to lung as well as the disappearance rate in lung (1.1 d21, 4.2 d21, and 32 d21, respectively) are all an order of magnitude higher in model 2, raising questions about the accuracy of these estimates. The high values could be due to the strong correlations between the proliferation parameters and the migration and disappearance rates. Estimates of effector CD8+ T cell migration and disappearance Next, the parameters for cell trafficking and constant (models 1 and 2) or time-varying death or loss (model 3) were estimated. These constant loss rates represent the disappearance of cells from the system due to death or physical removal (sloughing or phagocytosis), which cannot be easily resolved from one another and are treated as a single term. As mentioned earlier, in comparison with other parameters, the estimates for the constant pure loss rates for MLN and spleen, dm and ds, were essentially zero and could not be reliably determined. Similarly, the migration rate of virus-specific + CD8 T cells from MLN to lung, gml, was also estimated as close to zero. A low AIC score for models in which these parameters are Downloaded from excluded, indicating a good fit, reinforces the conclusion that these rates have a minimal impact on the fidelity of the model and the FIGURE 3. Fitting results from model 4 using parameter estimates from behavior of the system. Table II along with the experimental data plotted (indicated by the in- Notably, when we change the constant disappearance rate dl in dividual symbols in each plot) against day of infection. A–C, Solid lines lung in model 1 to time-varying dlðtÞ as in model 3, the fit of the indicate curves predicted by the model for effector CD8+ T cells in MLN model to the data improved further, and AIC score decreased from http://www.jimmunol.org/ (A), spleen (B), and lung (C). D, Actual (individual symbols) and predicted 2420.29 to 2473.40, which suggests the pattern of cell loss in numbers of Ag-bearing DCs are shown. lung cannot be simplified as a constant over time. Instead, the estimated time trajectory of dlðtÞis plotted in Fig. 4B, which shows + that the disappearance rate dlðtÞ of Ag-specific CD8 T cells in parameter values are low and close to zero and because of this lung monotonically increases from 0 at day 5 to 9.7 d21 at day 14. may be (statistically) obscured by noise in the data set. The shortest half-life of CD8+ T cells therefore occurs at day 14 ∼ + with a value 2 h, demonstrating an increase in the rate of effector Estimation of the growth rate of effector CD8 T cells cell loss at the end of the immune response. by guest on October 1, 2021 Because it is difficult directly to measure or estimate explicit values Collectively, these results suggest increased effector cell clear- for cell loss other than migration in the MLN and spleen, in model 2 ance toward the end of the acute response, particularly in the lung. we considered an alternative model in which we introduce com- In addition, there is a strong indication that the spleen plays a much bined terms for both cell proliferation and loss as the time-varying more substantial role in the overall CD8+ T cell response to the pure growth rates, kmðtÞ and ksðtÞ. The estimates of these two infection than may have previously been appreciated. parameters are plotted in Fig. 4. Model 2 makes no assumptions on the patterns and magnitudes of the two time-varying parame- Modeling the contribution of other tissues + ters kmðtÞ and ksðtÞ. That is, we did not make an explicit as- In contrast to the growth of the virus-specific CD8 T cell pop- sumption of what factors regulate the expansion of the CD8 ulation, T cell trafficking into peripheral tissues is an Ag- effector population. We believe this approach has its advantages independent process (44, 45). Activated T cells that enter the especially when some mechanisms are not completely clear. The circulation have the ability to traffic to many tissues other than the patterns of kmðtÞ and ksðtÞ are similar to one another but, as with site of infection in the lung (46, 47). Because T cell migration the population growth rates estimated in the first model, the value through peripheral tissues is dynamic and continuous, and as- of ksðtÞ is more than 2-fold greater than kmðtÞ, which suggests suming some T cells pass through the tissues and re-enter the a noticeably faster net generation of effector CD8+ T cells in the circulation via the lymphatics, we investigated the contribution of 21 + spleen than in MLN. For MLN, kmðtÞ decreases from 2.1 d at influx of activated CD8 T cells from compartments other than the 21 day 5 to 0.9 d at around day 10; for spleen, ksðtÞ decreases from three we considered. To accomplish this, model 4 was developed, 2 2 6.0 d 1 at day 5 to 3.6 d 1 at around day 12. The corresponding which included an additional compartment corresponding with all maximum doubling times of the virus-specific CD8+ T cell pop- other tissue sites combined (see model 4 in Materials and Meth- ulations in MLN and spleen are 8 h and 2.8 h, respectively, which ods). Also, as suggested by model 3, we retain the time-varying are more than 30% more rapid than the estimated doubling times disappearing parameter dlðtÞ in lung. The fitting results suggest in model 1 (at 12 h and 4.8 h in MLN and spleen, respectively). that a time-varying influx hs(t) from other tissues to spleen is Therefore, it is likely that factors important to the expansion of the significant, contributing a maximum of 8.2 3 104 cells per day. effector CD8+ T cell population in MLN and spleen are missing Also, the estimates of all other constant parameters are very close from model 1. The smaller AIC score for model 2 than that for to the estimates from models 1 and 3, suggesting that unlike model model 1 (2428.27 versus 2420.29) reinforced this interpretation 2, the introduction of the two time-varying parameters in model 4 indicating that model 2 is a better fit to the data. Also note that the do not affect the estimates of other constant parameters, and the short doubling time in the spleen suggests additional factors at results are therefore reliable. The smallest AIC score (2477.59) work, such as the contribution of other compartments to the also suggests that model 4 is superior to all three other models. growth rate. This is considered explicitly later. Suchresultssuggestthatitisvery likely that there exists a The Journal of Immunology 4479

Table II. Point estimates and 95% conﬁdence intervals for parameters in four models

Estimates (Conﬁdence Intervals)

Parameters and Units Model 1 Model 2 Model 3 Model 4 m TE (5) 3.96E+3 3.24E+3 4.43E+3 4.63E+3 Cells per MLN (3.65E+3, 4.27E+3) (2.91E+3, 3.58E+3) (3.34E+3, 5.52E+3) (3.35E+3, 5.92E+3) s TE (5) 3.64E+4 2.19E+4 3.49E+4 3.07E+4 Cells per spleen (3.16E+4, 4.12E+4) (1.96E+4, 2.43E+4) (2.33E+4, 4.64E+4) (2.11E+4, 4.04E+4) l TE (5) 1.31E+3 1.31E+3 1.33E+3 1.30E+3 Cells per lung (1.05E+3, 1.58E+3) (1.09E+3, 1.53E+3) (7.34E+2, 1.93E+3) (7.45E+2, 1.85E+3) rm 1.66E25 NA 1.58E25 1.55E25 d21 (cells per MLN)-1 (1.40E25, 1.92E25) (1.08E25, 2.08E25) (4.23E26, 2.68E25) rs 4.48E25 NA 3.59E25 4.57E25 d21 (cells per spleen)-1 (4.22E25, 4.74E25) (3.40E25, 3.78E25) (2.02E25, 7.12E25) t 3.08E+0 NA 3.53E+0 3.72E+0 d (2.80E+0, 3.35E+0) (2.42E+0, 4.64E+0) (3.34E+0, 4.09E+0) dm 0NA 0 0 d21 (dropped) (dropped) (dropped) ds 0NA 0 0 d21 (dropped) (dropped) (dropped) gms 1.57E21 1.09E+0 1.77E21 1.84E21 2 d 1 (1.42E21, 1.72E21) (1.05E+0, 1.14E+0) (1.12E22, 2.42E21) (1.32E21, 2.36E21) Downloaded from gml 0000 d21 (dropped) (dropped) (dropped) (dropped) gsl 4.95E21 4.18E+0 3.52E21 5.99E21 d21 (4.67E21, 5.23E21) (3.91E+0, 4.45E+0) (2.75E21, 4.28E21) (3.64E21, 8.34E21) dl 3.96E+0 3.15E+1 Varies with time; see Fig. 4 Varies with time; see Fig. 4 d21 (3.52E+0, 4.40E+0) (3.07E+1, 3.23E+1) hs(t) NA NA NA Varies with time; see Fig. 4 http://www.jimmunol.org/ Cells per day km(t) NA Varies with time; see Fig. 4 NA NA d21 ks(t) NA Varies with time; see Fig. 4 NA NA d21 RSS 15.77 14.59 11.48 10.85 AIC 2420.29 2428.27 2473.40 2477.59 NA, not available.

+ l significant, time-varying influx of CD8 T cells from other tis- For convenience, we plotted the product of dlðtÞand TE in Fig. 5. by guest on October 1, 2021 + l sues to spleen; also, the rate of disappearance of effector CD8 We see that the number of cells lost per day dlðtÞ×TE increases T cells in lung should be monotonically increasing (see Fig. 4). from 0 at day 5 to 2.7 3 105 cells per day at day 9 and then decreases down to 8 3 104 cells per day at day 14. These results suggest that sites other than the MLN and spleen are making substantial contributions to the effector CD8+ T cell pool and moderate the obvious contributions of the spleen as a sole organ. Instead, these results are consistent with the view that, in addition to in situ expansion, the spleen may collect effector T cells in the circulation that have come from other tissues. These findings do

FIGURE 5. Net population growth/loss rates at different sites. Estimated FIGURE 4. Plots of estimated time-varying parameters for growth of the number of cells gained or lost per day in MLN, spleen, or lung using model effector CD8+ T cell populations in spleen and MLN in model 2 (A); dis- 4 and the parameters from Table II. Results for MLN are also plotted on appearance of effectors in the lung in model 3 (B) and model 4 (C); and the a two log lower (103 versus 105) scale in the inset to better visualize the influx of effector CD8+ T cells from other compartments in model 4 (D). change in growth rate over time. 4480 CD8+ T CELL GROWTH AND MIGRATION not diminish the conclusion that cells from the spleen substantially dependent upon a combination of factors that include in situ contribute to the response in the lung. proliferation and an influx of T cells from other tissue compartments. These other compartments would include cells returning to the circulation from peripheral nonlymphoid tissues not infected Discussion with the virus, but may also include lymphoid organs such as the We have developed ODEs and time-delay ODEs to model the cervical lymph nodes and bone marrow that have the capacity to population growth, trafficking, and loss of virus-specific CD8+ contribute to the virus-specific T cells response (53, 56–58) but T cells in MLN, spleen, and lung in mice with primary influenza were not explicitly considered in our modeling or the experi- A infection. These models allowed us to challenge various as- mental data set. Alternatively, the growth of the CD8+ T cell sumptions and test hypotheses about how the system works. In numbers in the spleen could also include those cells that were addition, the model fitting process generated estimates of several activated to proliferate outside of the spleen but continued a pro- key biological parameters that are technically difficult or impos- gram of cell division (59). Other factors such as the local cytokine sible to measure empirically. Several assumptions that were held environment to promote T cell expansion were also not explicitly prior to the start of this work proved difficult to justify in the modeled, though such effects should still be proportional to the models. In particular, the role of the spleen in the primary CD8+ DC kinetics as suggested though our model fitting. T cell response to influenza infection may be more significant than In addition to a time-varying term for the contribution of other previously thought. sources of T cells to the spleen, the best fit to the data was provided There is a widely held view that during a primary influenza by the models with a time-varying loss rate for the effector T cells, infection, the development of the virus-specific CD8+ T cell re- with values increasing toward the end of the acute response and sponse occurs in the following sequential fashion: DCs bearing highest in the lung. Models with constant rates of cell loss did not Downloaded from viral Ags migrate from the lung to the draining MLN where they reliably reflect the experimental data. Regardless, effector CD8+ activate naive virus-specific T cells. The T cells proliferate in the T cell loss rates from the MLN and spleen were estimated to be MLN and then leave the lymph node to enter the blood, appear- very low and close to zero compared with other parameters. These ing next in the lung after extravasation (24, 48–51). This whole results suggest that T cell survival factors that include Ag or process takes 4–5 d before virus-specific CD8+ T cells can be cytokines must wane toward the end of the response. It is also detected at any one site. Very careful measurements made with possible that the features unique to influenza infection of the lung, http://www.jimmunol.org/ sensitive techniques such as MHC class I tetramer staining (24, such as the epithelial site of infection and effector cell trafficking 36) during the very early phase of the acute response (days 0–6) into the mucosa, drive some of these effects. support this general view (24, 51–53). However, these approaches A notable advantage of time-varying parameters is that uncer- only tell us where the T cells first become detectable, not where tainties in mechanisms can be well accounted for. Although this they came from. Nor do they consider the relative sizes of the approach usually requires frequent time-course data, it could be the T cell populations in each site, or the population growth and only right choice for complex dynamic problems. For example, trafficking rates that would be required for the MLN to be the although the introduction of one time-varying parameter will re- major source of effectors. Nevertheless, when looking at such sult in three more constant parameters in our models, the overall by guest on October 1, 2021 data, the earliest tetramer+ CD8+ T cells can be detected in the performance measured by AIC scores of time-varying parame- MLN at day 4–5, then appear simultaneously in the lung, spleen, ter models still outperformed model 1, which has only constant and organs such as the liver at 5–6 d (24, 52, 53). After this point, parameters. In more than one of the models, we found that the the numbers of cells at each site rises very rapidly, presumably growth rate of the virus-specific, effector CD8+ T cell population in driven by a combination of proliferation and migration, with little spleen is twice as large as that in MLN, and both the experimental initial loss. Unfortunately, these major contributing biological data and fitted-model simulations indicate that the MLN precedes parameters and sites become hard experimentally to deconvolute the spleen by about 1 d. Furthermore, the numbers of cells and through empirical observations. The various rates of activation, rate of accumulation in the spleen and the lung cannot be solely proliferation, migration, and death are difficult to measure directly explained by proliferation in and/or migration from the MLN, and are often confounded by the difficulty to distinguish experi- suggesting that the MLN is not the sole source of antiviral ef- mentally whether, for example, a T cell that has upregulated Ki67, fectors. Direct migration from the MLN to the lung and death diluted its CFSE label, or incorporated BrdU did so in the com- rates in the MLN and spleen were all estimated to be very small partment that was sampled or in another site the T cell just came relative to other contributors. Combined, these estimates support from (54, 55). However, by comparing alternative models through that the system operates differently than we originally assumed. a process of fitting, we can resolve some of the factors. The finding of a minimal contribution of effector CD8+ T cell We developed four models and compared them using mainly the migration from MLN to the growth of this population in the lung AIC score. It is possible for two models to have close scores (e.g., deserves some qualification. These results do not imply that CD8+ model 3 versus model 4); however, as suggested by Burnham and T cells do not traffic from the MLN to the lung, nor do they Anderson (35), the difference instead of the absolute value of AIC change the view that the adaptive response is initiated in the score matters. Even a small decrease in AIC scores will still draining lymph node. Instead, these estimates should be viewed in suggest superiority in model structures. In this study, the model consideration of the entire systemic response, the relative sizes of that produced both the best fit to the experimental observations the virus-specific T cell populations at each site over time, and in and the best AIC score is model 4. In this model, the expansion of comparison with the rates of cell migration and accumulation in the CD8+ effector T cell population is dependent on the number sites other than the lung. For example, the numbers of virus- + m s and kinetics of the professional, Ag-bearing APCs rather than the specific effector CD8 T cells at day 5, TE ð5Þ, TEð5Þ, and + l 3 4 viral load. The effector CD8 T cell populations expand in both TEð5Þ in model 3, are estimated as 4.4 3 10 , 3.5 3 10 , and the draining MLN and the spleen, but the spleen is estimated to 1.3 3 103 cells per MLN, spleen, and lung, respectively; the make a much larger contribution to the pool of cells that migrate spleen being an order of magnitude above the other sites. Shortly to the lung. Because cell loss in these sites is estimated to be very after virus-specific CD8+ T cells become detectable in the system, low, the size of the T cell population in the spleen, in turn, is likely the spleen becomes the major reservoir of these cells, and the The Journal of Immunology 4481 distribution cannot be mathematically explained by growth and in lung cannot be determined separately based on current models trafficking of the virus-specific effector population from the MLN and data. Therefore, we did not explicitly include the term for + alone. In contrast, the migration rates of activated CD8 T cells proliferation in the lung in our model. Instead, the parameter dl is, from MLN to spleen and from spleen to lung in models 1, 3, and 4 in effect, the net of the proliferation and disappearance rates for are all larger than from MLN to lung and close to each other at CD8+ cells in lung. The possibility of cell proliferation in the lung around 0.2 d21 and 0.5 d21, respectively. does not negate the conclusion that the spleen and other sites are This means that although common sense suggests that some of a major contributor to the overall CD8+ T cell response. Migration the cells must come directly from the MLN, in fact the contribution of T cells that have proliferated in the lung back to the MLN or of this source is very small and primarily observable very early in spleen is likely to be very small or negligible. Furthermore, al- the response when cell numbers are low. Later in the response, the though evidence of CD8+ T cell proliferation in the lung exists, only way to compensate would be to impose extraordinarily high it makes a substantial but not complete contribution to the total cell growth and migration rates for the MLN. number of CD8+ T cells in the lung and seems to contribute the The picture that emerges fits with observations made by most at early time points in the infection (54). Masopust, Reinhardt, and Marshall in 2001 showing that activated Notably, the lung was the only site in which cell loss, presumably effector T cells traffic all over the body and not just to the sites of due to death and other factors, was estimated to be time varying infection (46, 47, 53). Experiments in which the MLN is carefully with a maximum value of 9.7 per day, or a half-life of 1.7 h. The digested with enzymes to liberate T cells complexed to APCs (60) number of cells lost per day reached its maximum around day 9 do not change the values enough to alter this conclusion. This (2.7 3 105 cells per day, see Fig. 5). This is not out of line with view is less controversial when considering that the spleen can the notion that continued encounter with Ag-bearing target cells function as a site of primary immunity under conditions that and migration out into the infected airways (from which there is Downloaded from disrupt trafficking of naive T cells into the lymph nodes (56–58, no return) account for a significant disappearance of the effector 61). The original experiments, performed more than a decade ago, cells. If the cells in the lung were also proliferating as has been demonstrated a more than 10-fold decrease in virus-specific CTLs postulated, then the actual loss rate would have to be higher still. (and Th precursors) in the lung in mice that had been splenec- Keep in mind, however, that we treated cell loss as a time-varying tomized (56). The conclusions about the role of the spleen stand in parameter instead of a constant one. We believe our modeling contrast to early reports of experiments with splenectomized mice methods and techniques can provide a more powerful platform for http://www.jimmunol.org/ that concluded the spleen made little contribution to the infection immunologists to understand viral and immune system dynamics. (62). A more likely interpretation is that splenectomy does not Collectively, these efforts change the way the immune response affect control of mild influenza infection because there are more to influenza infection is viewed. The spleen can no longer be than enough CTLs generated to control the virus. This is consis- considered as a location used merely to contain the “spillover” of tent with the observation that the absence of a spleen reduces the excess virus-specific CD8+ T cells in the blood that were gener- total number of CD8+ T cells generated. ated in the MLN and not taken up by the lung or other tissues. One of the valuable features of modeling and simulation is Instead, the spleen is a major contributor of effector CD8+ T cells the ability explicitly to explore alternative hypotheses and model against the virus infection. This view also implies that the spleen by guest on October 1, 2021 structures. For example, we compared two models in which the contains Ag-bearing APCs that can drive T cell proliferation and T cell response was driven by either the amount of virus in the is an environment where T cell expansion can occur. As such, it system (results not shown) or the number of professional, Ag- may also be an important source of the memory T cells that re- bearing APCs (DCs, or D as a variable in this article). The DC- main at the end of the acute immune response. driven model provided a substantially, and statistically better, fit to the actual experimental data. This suggests that the kinetics and Acknowledgments magnitude of the CD8+ T cell response is limited more by the We acknowledge the excellent and dedicated technical support of Tina Pel- number of APCs available than the amount of Ag. legrin and Joe Hollenbaugh during the murine experiments. Several other key parameters were estimated from the models. These include the growth and disappearance of T cells in different Disclosures m compartments. 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