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SM, JLF, and DEK conceived and designed the experiments and wrote the paper. SM and PLL performed the experiments. SM, HP, PLL, JC, JLF, and DEK analyzed the data. SM, DS, HP, and PLL contributed reagents/materials/analysis tools.

Our collaborator on this project, Joanne Flynn from the University of Pittsburgh, has received funding in the form of a Research Grant from Amgen, the company that makes Etanercept. Although this does not directly support the research in this manuscript, the Research Grant does fund studies on TNF and TB.

The immune response to

Control of

TNF action increases the phagocytosis by macrophages and enhances mycobacterial killing in concert with IFN-γ [

TNF is initially a transmembrane (tmTNF) protein that undergoes cleavage by the specific metalloproteinase TNF-converting enzyme (TACE) to form a soluble trimer [soluble tumor necrosis factor (sTNF)] [

sTNF and tmTNF Effects on Lymphocytes and Monocytes/Macrophages

Several biologic inhibitors (antibodies and receptor fusion molecules) have been developed that interfere with TNF activity and are used to control inflammation in diseases such as rheumatoid arthritis [

The present study takes a theoretical approach toward characterizing the role of TNF in protection against the tubercle bacillus in both active and latent infection. We extend our previous models [

We describe results in these different areas of TNF study: mathematical modeling of typical infection progressions in humans, mechanisms driving infection outcomes, and anti-TNF therapies. Deletion and depletion experiments are discussed in

Negative control simulations have been performed on the model [

Shown are intracellular and extracellular bacterial loads (A), CD4+ and CD8+ T cells (B,C) (linear scale), cytokines (D,E) (linear scale), and macrophages (F). The volumetric unit for cell and bacteria populations is number per cm^{3} of a granulomatous tissue. The unit of measure for cytokine concentrations is pg/mL of granuloma homogenate.

BE, extracellular bacteria; BI, intracellular bacteria; MA, activated Mφ; MI, infected Mφ; MR, resident Mφ.

Shown are intracellular and extracellular bacterial loads (A), CD4+ and CD8+ T cells (B,C) (linear scale), cytokines (D) (linear scale) and (E) (linear-log scale), and macrophages (F) (linear scale). See _{52}), and increased extracellular bacteria growth rate (α_{20}).

Simulations predict that with an inoculum of 25 mycobateria [^{3} of granulomatous tissue), and all intracellular bacteria (

The roles played by different cellular sources of TNF involved in protective immunity remain unclear. During latency we evaluate and compare production of TNF by macrophages and lymphocytes (

As discussed in the ^{8} per cm^{3} of granulomatous tissue approximately at day 300 (^{5} cells per cm^{3} of granulomatous tissue). IL-12, IL-10 (

We investigate the importance of specific TNF-dependent mechanisms that allow for infection control via sensitivity and uncertainty analyses (see

Uncertainty and Sensitivity Analyses of the Model for TNF-Related Parameters

Our sensitivity analysis indicates a critical role for TNF production by both MIs (α_{30}) and Th1 cells (α_{32}) throughout the infection (negative correlation values −0.6 to −0.2, _{33}) is only significant in the first 250 days post-infection, suggesting that it is important for establishing latency but not maintaining it (see

The model predicts that enhanced recruitment of lymphocytes (Th1, T8, and TC) is a desirable strategy toward establishing latency, as suggested by the strong and very significant negative correlation of TNF-dependent recruitment parameters (_{3b}_{3a}_{3}_{3}

Among all TNF-related mechanisms, the uncertainty and sensitivity analyses indicate that lymphocyte recruitment and macrophage activation are the most influential toward controlling bacterial levels when compared with TNF-induced apoptosis (which is not significant,

Little or no data are available to indicate the fraction of TNF (_{α}_{α}_{α}_{α}_{α}

(A) Mathematical model simulations of total bacterial load corresponding to different proportions

(B) Simulated depletion of variable levels of sTNF. Until day 500, the system is in latency and

We use the mathematical model to simulate three virtual clinical trials (VCT) of anti-TNF treatments (protocols are described in detail in

Two classes of biological inhibitors were tested in the VCT: anti-TNF antibody and TNF receptor fusion (TNFR2Fc). We define each drug as having a specific ability to neutralize TNF at the granuloma site; these data are not currently known (i.e., the drug neutralizing power). We define

As shown above (

A series of VCT were simulated assuming different TNF bioavailability ranges induced by the two different treatments and a natural biological variation of

Virtual Clinical Trial 1 (VCT1)

If TNF bioavailability at the granuloma site is <20% of baseline latency level (at the initiation of therapy at 500 days), both treatments induce 100% reactivation. In the range 20%–30%, anti-TNF antibody always causes reactivation, while TNFR2Fc reactivates 83/100 virtual patients. At higher bioavailability ranges (30%–50%), the differential risk of reactivation goes down to 34 or 4 per 100 virtual patients for anti-TNF antibody versus TNFR2Fc, respectively. No reactivation occurs for both treatments if more than 50% of TNF is bioavailable. The prediction that anti-TNF antibody treatment has a stronger impact on reactivation risk than TNFR2Fc in the bioavailability range of 20%–50% suggests that other factors may be playing a role in reactivation in addition to bioavailability. To explore this, we simulated another clinical trial.

In addition to bioavailability, the percentage of total TNF cleaved (

Virtual Clinical Trial 2 (VCT2)

To determine other factors that contribute to reactivation differences between the two therapies, we now fix both TNF bioavailability and

Our sensitivity analysis (_{tmTNF-MA} and μ_{tmTNF-T8}).

The cell loss rate of MI negatively correlated with total bacterial load (μ_{tmTNF-MI}). Thus, anti-TNF treatment reduces the number of MAs and T8 cells and increases bacterial levels, increasing risk of reactivation. Although anti-TNF antibody also reduces the number of MIs, this is not sufficient to maintain latency. This may explain why a higher percentage of tmTNF has a negative impact on infection containment during anti-TNF antibody treatment: with more tmTNF, more MAs and T8 cells are lost from the granuloma.

Duration of treatment also affects risk of reactivation for both therapies.

Duration of Treatment study for VCT2

If treatment starts before infection with Mtb occurs, we assume that drug penetration is not relevant because the granuloma has not yet formed. We assume instead that a certain concentration of anti-TNF molecules are present in the lung where granulomas would begin to form in response to infection. Average serum concentrations of anti-TNF molecules are published [

The major findings from this study are that bioavailability of TNF following anti-TNF therapy is the primary factor for causing reactivation of latent infection, that anti-TNF therapy will likely lead to numerous incidents of primary TB if used in areas where exposure is likely, and that sTNF—even at very low levels—is essential for control of infection.

Our model predictions (see

We use the model to analyze the effects of anti-TNF therapy by simulating anti-TNF antibody and TNFR2Fc. The reported measure unit for a steady state or average concentration of anti-TNF drugs in serum is on the order of μg/ml. Data on soluble TNFRs concentration in serum are on the order of ng/ml [

The VCT simulations suggest that TNF bioavailability is the main factor leading to reactivation by anti-TNF treatments in latently infected patients. Reactivation always occurs if both drugs penetrate the granuloma equally well (TNF bioavailability less than 20%). High bacterial load at treatment initiation increases the likelihood of reactivation. This suggests that a complete regimen of antibiotic treatment for Mtb infection prior to anti-TNF treatment could reduce the risk of reactivation. If TNF bioavailability is equally affected by the two treatments, differential cell level losses induced by anti-TNF antibody therapy accounts for higher reactivation rates: activated CD8+ T cells and MA loss are not compensated by the apparently beneficial effect of MI loss. We speculate that the intracellular bacteria released after MI death induced by antibody binding to tmTNF (whether dependent on tmTNF reverse signaling or complement cascade) can only facilitate bacterial clearance by the host and does not enhance dissemination. Further, our results show that the longer patients are exposed to anti-TNF drugs through longer duration treatment protocols, the risk of reactivation increases. If infection with Mtb occurs after treatment is initiated, chances of developing active infection are very high if we assume reasonable levels of drug penetration into lungs (TNF bioavailability <50%). This is particularly important if anti-TNF treatments are implemented in regions of the world where infection risk is elevated. Bacteria grow uncontrolled when both sTNF and tmTNF are depleted (anti-TNF antibody therapy).

These data suggest that tmTNF plays a key role in controlling active infection, where tmTNF preserves a subset of the beneficial mechanisms of TNF while lacking detrimental effects. Our predictions and recent experimental data [

Current studies in both murine and NHP animal models by our group are now following up on these predictions. Our recent data from a mouse model showed that treatment with anti-TNF Ab in the chronic phase rapidly resulted in fulminant TB, while treatment with an etanercept-like molecule (receptor fusion) allowed mice to maintain

To better understand underlying dynamics of TNF production and function, we build on our mathematical model of Mtb in humans using 16 nonlinear ordinary differential equations. The updated model tracks three macrophage populations (resting, activated, and infected) and multiple T cell (Th0, Th1, Th2, and CD8+ T cell subsets) populations [

The nonlinear ODE model is based on [

Descriptive diagram of macrophage dynamics implemented in the mathematical model in

Descriptive diagram of lymphocyte dynamics implemented in the mathematical model in

Descriptive diagram of bacteria dynamics implemented in the mathematical model in

The equations describing dynamics for the macrophage subpopulations are given by:

Rate of change of resting macrophages (_{M}) and a natural death term (-μ_{MRM}_{R}). In the course of infection, additional resting macrophages are recruited in a TNF-dependent fashion at a rate Sr_{4B}, and this process is downregulated by IL-10. We also account for TNF-independent recruitment mechanisms (for both macrophages and lymphocytes) with a term that indirectly represent chemokines secreted primarily by MAs and MIs (_{A} + w M_{I}), 0 < w < 1): the magnitude of recruitment (α) varies from macrophages to lymphocytes. Resting macrophages at the site of infection can become chronically infected at a maximum rate k_{2} (dependent on the extracellular bacterial load) and activated at rate k_{3} (dependent on two signals from IFN-γ and either bacteria or TNF). Note that due to differences in measurement units, TNF is scaled by a factor β. IFN-γ induction is downregulated by IL-4.

MIs (_{I}. This mechanism has a maximal rate k_{17}, and is described by a Hill process. Immune responses also contribute to MI killing by several mechanisms. Both CD8+ and CD4+ T cells can use the Fas-FasL apoptotic pathway to induce apoptosis in these cells at a maximum rate k_{14a}. The half-saturation constant c_{4} describes the effector-target ratio (T_{t}:M_{I}) at which this process is half maximal. TNF can also induce apoptosis by binding to the TNFR1 receptor. This process is downregulated by IL-10 and occurs at a rate k_{14b}. Finally, CTL killing by CD8+ and CD4+ T cells happens at a rate of k_{52}. Specifically, CD4+ T cells have a limited contribution and this is accounted for by scaling the CD4+ T cell numbers (0 < w_{1} <1). CD8+ T cell numbers are scaled by a Michaelis-Menten term accounting for the indirect dependence on CD4+ T cells for their killing capability. MAs are generated from the term in _{MAM}_{A}). MAs can be deactivated by IL-10 at a rate k_{4.}

Similar to resting macrophages, recruitment of T cells occurs in both a TNF-independent and a TNF-dependent manner. The terms are similar, using different rates for the different T cell subsets (α_{1A}, Sr_{1B} for Th0 and T80 cells; α_{3A}, Sr_{3B} for Th1 and Th2 cells; α_{3Ac}, Sr_{3Bc} for CD8+ T cells, respectively). We assume that CD4+ T cells can arrive at the site of infection either as Th0 (majority), or a small fraction may arrive already differentiated intoTh1 or Th2 cells (see Wigginton et al. [

Upon arriving at the site of infection, Th0 cells (_{2}. Th0 cells can also differentiate into Th1 (_{T0}T_{0}). Th1 cells can be killed due to IFN-γ induced apoptosis in the presence of MAs at a rate μ_{Tγ}. Both Th1 and Th2 cells die naturally at rates μ_{T1} and μ_{T2}, respectively. As is the case for CD4+ T cells, we assume that CD8+ T cells can arrive at the site of infection as T80 (majority) (

CD8+ T cells also undergo IFN-γ induced apoptosis at a peak rate μ_{Tcγ}, and die at a rate μ_{Tc}. Since the T8s (

TNF (_{30}. MAs make TNF at a rate α_{31} in response to IFN-γ or bacteria and this process is inhibited by IL-10 and IL-4. Other sources of TNF are Th1 cells (rate α_{32}) and CD8+ T cells (rate α_{33}) in response to antigen, and TNF has a given half-life.

Th0, Th1, and CD8+ T cells produce IFN-γ (_{5A} and α_{5B}, respectively. Production by Th0 and T80 cells is further enhanced by IL-12, and inhibited by IL-10. Other sources of IFN-γ, such as NK cells, are also believed to play a role in TB infection. Since these are not accounted for in the model, we include an extra source term (s_{g}) dependent on extent of infection and IL-12 level.

Resting macrophages produce IL-12 (_{23}. MAs also produce IL-12, and this process is downregulated by IL-10. Dendritic cells are the primary source of IL-12 upon Mtb infection and are accounted for by an infection-dependent source term, s_{12}. Finally, there is a natural half-life for IL-12.

IL-10 (_{6}. Other sources such as Th1 cells, Th2 cells, and CD8+ T cells produce IL-10 at rates α_{16}, α_{17}, and α_{18}, respectively. IL-4 is produced by Th0, and Th2 cells produce (_{11} and α_{12}, respectively. IL-4 has a given half-life of μ_{i4}.

Intracellular bacteria (_{19} with logistic Hill kinetics accounting for a maximal carrying capacity of a macrophage. Extracellular bacteria (_{17}) adds to the extracellular subpopulation. To account for loss of intracellular bacteria due to various killing mechanisms, we assume each killed MI to hold an “average” number of bacteria, given by N_{AVG} (<=N). The corresponding gain in extracellular bacteria depends on the mechanism of killing: while Fas-FasL–induced apoptosis (k_{14a}) releases all intracellular bacteria, TNF-induced apoptosis (k_{14b}) eliminates approximately 50% of the bacteria within the macrophage, and this is shown by the N_{fraca} multiplier in the BE (extracellular bacteria) equation (_{52}) kills virtually all the intracellular bacteria, and does not add on to the BE (extracellular bacteria) population. Lastly, we assume that natural death of MIs also releases all intracellular bacteria, and this is modeled as a constant turnover of the bacteria (μ_{I}B_{I}) from intracellular to extracellular. Extracellular bacteria grow at a maximum rate α_{20}. They are taken up and killed by activated and resting macrophages at rates k_{15} and k_{18}, respectively.

sTNF is produced predominantly by cells of the macrophage lineage upon infection or exposure to bacteria or bacterial products [

Our previously published models of Mtb infection simulated cell recruitment as a function of MAs and MIs, the main producers of chemokine and sTNF. In our most recent model [_{α}) represents the dynamics of total sTNF and tmTNF in the system. Using the model, we investigate how different percentages of total TNF cleaved (i.e., sTNF) affect infection progression. We updated the model equations to address tmTNF effects on cell activation and apoptosis, based on

The strength of tmTNF effect on T cell activation through TNFR1 and TNFR2 is represented in the model by the coefficient

We also add new terms representing apoptosis or cell loss induced by anti-TNF antibody binding to tmTNF on macrophages [

These terms are present in the mathematical model only during anti-TNF antibody treatment simulations. The fraction of intracellular bacteria released in the extracellular domain due to tmTNF-induced apoptosis of MIs is likely very small [

Under pathological conditions (chronic inflammatory states), the presence of anti-TNF antibodies (and not TNF receptor fusion molecules) and subsequent binding to tmTNF can induce activation of the complement cascade (due to high concentration of Abs) [_{3}) as follows (bold term):

We do not directly include LT_{α} in the model, but we indirectly account for LT_{α}-dependent recruitment of macrophages and lymphocytes during anti-TNF therapy (namely TNF receptor fusion protein), since TNF receptor fusion protein binds LT_{α} while anti-TNF antibody does not (see

Once we derive the model and estimate parameters, we solve the system of 16 nonlinear ordinary differential equations to obtain temporal dynamics for each variable. To this end, we used Matlab version 7.1.0.183 (R14) Service Pack 3 (The Mathworks) platform and its numerical methods together with a computer code using a different solver written by our group.

As discussed previously [

Before simulations can be performed, parameters must be estimated from literature sources or by mathematical means. Values for most model parameters are estimated from published experimental data or data generated from our group. Data from human studies and Mtb experiments are favored over mice and other mycobacterial species, respectively. Where no appropriate data is available for a given parameter, we conduct uncertainty analysis to obtain a range within orders of magnitude. A detailed description of techniques used to evaluate model parameters, as well as a listing of parameters already estimated can be found in work previously published by our group [

Parameter values represent mechanisms in the host–pathogen system, and these were estimated from many different experimental sources. There is great variation that likely exists among them. In previous work [

There is an intrinsic biological and experimental variability in rates measured from in vivo or in vitro studies. Further, some interactions in the Mtb–host system are not currently measurable, particularly at the level of the lung granuloma. This complicates accurate estimation of model parameters (baseline values are unknown).

We quantify the importance of each TNF-related mechanism involved directly and indirectly in the infection dynamics using both uncertainty and sensitivity analyses as described previously [

The PRC method allows us to correlate the variability observed using the LHS method and to determine which parameters are responsible for the variation in outcomes. PRC coefficients (PRCCs) are between −1 and 1 and have a standard _{32}) is always very significant and negative (see

As a way to validate the mathematical model, we recapitulate experimental approaches such as TNF gene knockouts and TNF neutralization studies. These can be simulated with our mathematical model as virtual deletion and depletion simulations, respectively. Virtual deletions remove an element from the system at day zero while virtual depletions mimic experimental conditions where an element can be depleted or neutralized via antibody treatment at any time during the infection. We can selectively delete or deplete sTNF or transmembrane TNF (tmTNF) by varying the parameter

When all the TNF is deleted from the system on the same day that infection is initiated, the system goes to active disease (see ^{-/-}). This occurs with low-level cellularity, i.e., macrophage numbers are almost an order of magnitude lower (mainly infected and activated) than when disease occurs in the wild-type scenario (see

Mathematical model simulations of bacterial loads during TNF deletion (TNF^{−/-}) and depletion (TNF depl). The

The US Federal Drug Administration monitors the safety of TNF inhibitors through its Adverse Event Reporting System (AERS), a surveillance system to which drug manufacturers are required to submit reports of adverse events and to which health care professionals and consumers voluntarily send adverse event reports. Wallis et al. [_{3}. We capture the action of these two TNF neutralizing drugs by including an additional loss term in the TNF equation. This term accounts for concentration-dependent loss of TNF as a function of a drug's half-life, dissociation rate, bioavailability, and treatment regiment.

Infliximab is a human–mouse chimeric monoclonal TNF antibody that binds potently and essentially irreversibly to monomeric and trimeric TNF, both soluble and membrane-bound, but does not bind to soluble LTα_{3} [

Another anti-TNF drug, etanercept, is a TNF receptor p75-IgG fusion protein rather than an antibody. It binds selectively to human trimeric sTNF and LTα_{3}, with a 4-fold lower affinity for tmTNF with respect to infliximab [

Receptor fusion has a fast dissociation rate: it sheds 50% of sTNF and 90% of tmTNF in only 10 minutes, but can bind TNF again immediately [_{3} are engaged for very short time intervals, allowing for a redistribution of TNF throughout the system. However, if receptor fusion concentration is relatively high, those released molecules will quickly be reassociated with the receptor fusion and are therefore not free in the system. In contrast, under situations where TNF is released from the receptor fusion molecule and there are high numbers of cell associated TNF receptors present (such as in a granuloma) and possibly a lower level of receptor fusion (due to poor penetration), TNF might bind to the cell-associated TNFR1 or TNFR2 instead of back onto the receptor fusion. This contributes even more to lowering levels of bioavailable TNF in granulomas during receptor fusion treatment.

We indirectly test LTα-neutralization in the model by lowering all the TNF-independent recruitment parameters using the bioavailability coefficient (

We perform three VCTs to investigate what factors contribute most to reactivation during anti-TNF treatments if patients are latently infected or if exposure/infection occurs after anti-TNF treatment is initiated.

Several factors and mechanisms hypothesized to be involved in TB reactivation by anti-TNF drugs can be tested. These include the differential power of the drugs to neutralize TNF bioavailability [^{5} (latency level) during or after the end of the treatment. See

We define a

Each VCT comprises 100 simulations, where TNF bioavailability is varied in a specified interval. Each run is classified based on the bacterial load level, and reactivation is defined when bacterial loads grows uncontrolled. We define the reactivation subset of the 100 runs as the collection of all the reactivation cases with their bioavailabilities (from the uncertainty analysis). We obtain our RT as the average TNF bioavailability calculated on the reactivation subset. We statistically compare RTs between different trials by a standard

Contradictory data exist regarding levels of sTNF and sTNF receptors in lung epithelial lining fluid obtained by bronchoalveolar lavage [

Average steady-state concentrations of anti-TNF antibodies [^{3} of granulomatous tissue and we describe cytokine concentrations in pg/mL of granuloma homogenate.

We perform PRC (partial Spearman correlation on rank-transformed data) and

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New Parameter Estimates in addition to those estimated previously [

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We are grateful to the members of the Kirschner, Flynn, and Chan laboratories for helpful discussions.

Adverse Event Reporting System

latin hypercube sampling

activated macrophage

infected macrophage

partial rank correlation

soluble tumor necrosis factor

tuberculosis

transmembrane TNF

tumor necrosis factor, VCT, virtual clinical trial