# Mathematical model of sarcoidosis

^{a}Mathematical Biosciences Institute, The Ohio State University, Columbus, OH;^{b}Division of Pulmonary, Allergy, Critical Care, and Sleep Medicine, The Ohio State University Medical Center, Columbus, OH; and^{c}Mathematical Biosciences Institute & Department of Mathematics, The Ohio State University, Columbus, OH 43210

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Contributed by Avner Friedman, September 17, 2014 (sent for review August 8, 2014; reviewed by Marc Judson and Reinhard C. Laubenbacher)

## Significance

Sarcoidosis is a disease involving abnormal collection of granulomas that develop in the lung and other organs. The origin of the disease is unknown, clinical data are very limited, and there is no current effective treatment. This paper develops a mathematical model with simulations that are validated by the available clinical data. The model is then used to explore potential treatments of the disease, to suggest therapeutic targets that may reduce the disease activity, and thus to predict treatment responses in preclinical settings.

## Abstract

Sarcoidosis is a disease involving abnormal collection of inflammatory cells forming nodules, called granulomas. Such granulomas occur in the lung and the mediastinal lymph nodes, in the heart, and in other vital and nonvital organs. The origin of the disease is unknown, and there are only limited clinical data on lung tissue of patients. No current model of sarcoidosis exists. In this paper we develop a mathematical model on the dynamics of the disease in the lung and use patients’ lung tissue data to validate the model. The model is used to explore potential treatments.

Sarcoidosis is a disease involving an abnormal collection of inflammatory cells that can interact to form nodules, called granulomas, which are capable of altering the functions of affected tissues and organ systems. The granulomas contain macrophages, T lymphocytes whose functions are regulated by inflammatory mediators such as TNF-α, IFN-γ, IL-2, IL-10, IL-12 and TGF-β. The sarcoidosis granulomas are most commonly detected in the lungs and mediastinal lymph nodes; however, recent technological advances have improved disease detection in other organs, such as the heart, which is now recognized to be involved in one-third of cases, and other vital and nonvital organs (1⇓⇓–4). Sarcoidosis of the lungs and heart contributes to disability and increased mortality in these patients.

The primary cause of sarcoidosis remains a mystery, and progress has been limited by the lack of relevant disease models. It is unknown to what extent genetic predisposition or specific environmental exposures (e.g., antigens derived from infectious organisms) trigger the inflammatory immune response. It is reasonable, however, to assume that, in the lung, inflammation is initiated following inhalation of an environmental antigen, which leads to a typical Th1 immune response that is initiated by macrophages. Activated macrophages secrete proinflammatory cytokines such as IL-12 (5) and TNF-α (6, 7) and anti-inflammatory cytokine IL-10 (8) and IL-13 (9); they and Th17 cells secrete chemokine (C-C motif) ligand 20 (CCL20) CCL20 (10, 11). The CD4^{+} T cells in sarcoidosis are primarily Th1, Th17, and Treg. Th1 is activated by IL-12, and activated Th1 cells produce IFN-γ, which further activates macrophages; these processes are inhibited by IL-10 (12, 13). Cytokine CCL20 chemoattracts both Treg and Th17 cells (14) into the granuloma. Treg and Th17 are both activated by TGF-β (15). IL-2 secreted by Th1 (16) increases the proliferation of Th1 cells (16), blocks the proliferation of Th17 cells (17), and enhances the activation of Treg by TGF-β (18, 19); TGF-β is secreted by activated macrophages and Treg (20, 21).

A detailed diagram of the network involving the interactions among all these immune cells and cytokines, including GM-CSF production by macrophages, is shown schematically in Fig. 1.

No current models of sarcoidosis exist. In this paper we develop a mathematical model of sarcoidosis based on the diagram shown in Fig. 1. The model is represented by a system of partial differential equations. Parameters are determined by using the clinical data on cytokine levels in healthy lung tissue as reported in Crouser et al. (22). The model is then validated by data on cytokine levels in lung tissue of patients (22).

We use the model to explore the effect of anti–TNF-α (currently in use) and potential drugs, anti–IL-12, anti–IFN-γ, and TGF-β enhancement, in decreasing the size of sarcoid granulomas.

## Mathematical Model

The variables of the model are listed in Table 1. Here we assume that the granuloma occupies a region that varies in time and that macrophages and T cells are in movement with velocity **u** within the granuloma. The need to use a spatial model arises from the facts that granulomas are regions that evolve in time, and chemotaxis by chemokine CCL20 plays an important role in attracting both Th17 and Treg cells.

We assume that all species are dispersing or diffusing, in the granuloma, with appropriate diffusion coefficients. The equation for each species *m* is the production rate of

### Equation for Activated Macrophages ( M A ) .

Alveolar macrophages are M2 macrophages (23). They are activated by IFN-γ (24), GM-CSF (25), and TNF-α (26).

The density of the activated macrophages follows the equation*f* indicates the inflammation. As mentioned above, the use of the Michaelis–Menten law, for instance,

### Equation for Th1 Cells ( T 1 ) .

The density of Th1 cells satisfies the equation

In the first term on the right-hand side, Th1 cells are activated by IL-12 and direct contact with MHCII of activated macrophages and inhibited by IL-10 (12, 13). IL-2 increases the proliferation of Th1 cells (16). These processes are inhibited by Treg (27). Internalization of IL-12 is much smaller than the inhibition by IL-10, and therefore the term *SI Text*).

### Equation for Treg cells ( T r ) .

The density of Treg cells satisfies the equation

### Equation for Th17 Cells ( T 17 ) .

Th17, in direct contact with activated macrophages, is activated by TGF-β (15) and other cytokines including IL-6, IL-21, and IL-23 (29). For simplicity we include only TGF-β in our model and accordingly adjust its activation rate

### Equation for IFN-γ ( I γ ) .

The concentration of IFN-γ is modeled by

### Equation for TGF-β ( T β ) .

TGF-β is secreted by Treg (21), activated macrophages (20), and Th1 lymphocytes (21). Hence the TGF-β concentration satisfies the equation

### Equation for IL-12 ( I 12 ) .

IL-12 comes in two forms, IL-12 p40 and IL-12 p70 (5). Both forms are produced by activated macrophages and inhibited by IL-10 (31). This process is enhanced by IFN-γ, but the production of IL-12 p70 is negligible without the participation of IFN-γ (5). Hence the equations for the concentration of IL-12 are given by

### Equation for TNF-α ( T α ) .

The concentration of TNF-α evolves according to the equation

### Equation for IL-2 ( I 2 ) .

IL-2 is produced by Th1 cells (16):

### Equations for Other Cytokines: GM-CSF (*G*), IL-13 ( I 13 ) , IL-10 ( I 10 ) , and CCL20 (*C*).

Activated alveolar macrophages produce GM-CSF (34), IL-10 (8), IL-13 (9), and CCL20 (10). Hence,

The absorption term in Eq. **12** is based on the fact that IL-10 enters macrophages to block production of IL-12 at the transcription level (31). The rate of absorption depends on the level of IL-12 in the microenvironment and is taken to be

Eq. **14** includes a loss due to the chemotaxis by CCL20, which is bound and internalized by Treg and Th17 that are chemoattracted by CCL20.

### Equations for the Velocity u.

We assume that the cells are distributed uniformly throughout the granuloma, and their total density is 0.1 g/mL (35), so that**2**–**5**, we get

## Results

The parameter values of the system of Eqs. **1**–**14** are given *SI Text*. In this section we simulate the mathematical model developed in the previous section. For computational simplicity, we assume that the granuloma is a sphere with radius *u* is the component of the velocity **u** in the radial direction. We take the initial state to be that of a healthy individual. Some of the unknown parameters of the model were chosen so that the cytokine levels in healthy lung tissue coincide with the levels reported in ref. 22.

To validate the model, we assume that the initial stage of the disease triggers macrophage inflammatory reaction. We then simulated the development of a granuloma by taking a spherical tissue with initial radius **2**. Fig. 2 shows the profiles of all of the cells and cytokines for the first 100 d, at which time the disease reached steady state; note that the granuloma radius increased from *n* = 11) of cytokine concentration in lung tissue are reported in ref. 22. We can compare our results (at day 100) with the data in ref. 22. Fig. 3 shows a good fit of our simulation results with the patient data.

The slight discrepancy in the level of CCL20 can be attributed to the fact that in ref. 22 it was chemokine MIP-1α that was measured rather than chemokine CCL20.

### Treatment.

The most commonly used agents in the treatment of pulmonary sarcoidosis are corticosteroids; taken orally they provide relief of symptoms and control potentially disabling respiratory impairments (36). However, the exact mechanism of the action of the drugs is unknown, and they do not cure the disease (37). Infliximab, an anti–TNF-α drug, is used for chronic resistance sarcoidosis, but it has serious side effects and its effectiveness is uncertain (36).

From Fig. 2 we see that, starting from heathy state, the sarcoid granuloma radius will increase in 100 d from radius

Here we use our model to explore the efficacy of several drugs in terms of how they reduce the radius of the granuloma.

In the simulation of sarcoidosis in Fig. 3, we have taken the inflammation (in Eq. **1**) to be **8** by a factor

We proceed to use the model to explore other potential drugs: anti IL-12, anti IFN-γ, and injection of TGF-β.

We represented the effect of anti–IL-12 by reducing by half the production rates **8**. Fig. 5 shows a decrease in the granuloma radius. A steady state is reached after ∼20 wk.

Next we consider anti–IFN-γ and represent its effect by reducing the production rates **6**. Fig. 6 shows how radius

Finally we consider injection of TGF-β and represent its effect by introducing a source term (^{−1}⋅d^{−1}) in Eq. **7**. Fig. 7 shows the reduction of

We note that the steady states of

### Discussion.

Sarcoidosis is a disease whose origin remains a mystery. Pulmonary chronic sarcoidosis is currently treated by drugs that are generally known to reduce inflammation, but not curative. Among these drugs, infliximab is perhaps the most specific, an anti–TNF-α drug. In an attempt to explore the progression of the disease, we developed in this paper a mathematical model based on patient data (22). The model is represented by a system of partial differential equations within a granuloma of varying radius

Our model of sarcoid granuloma was based on Fig. 1. A similar network can be used to describe granuloma in tuberculosis (TB), although in that case one has to consider both classically activated and alternatively activated macrophages (43). However, the source of inflammation in TB arises from the TB antigen (i.e., the *Mycobacterium tuberculosis*); hence some of the model parameters will have to be changed in the TB case, leading to different conclusions. Indeed, BAL measurements in pulmonary sarcoidosis and pulmonary TB show differences in the expression of cytokines (44, 45). Big differences may occur when some of the cytokines are overexpressed or underexpressed. For instance, a genetic variant associated with excessive IFN-γ production in response to TB antigens may predispose the lung to sustained granuloma formation (sarcoidosis) while protecting against TB (because it will kill the bacteria and hence limit the inflammation). On the other hand, if the genetic defect were associated with impaired IFN-γ production to the same antigenic challenge, this condition would favor the development of latent TB (inability to kill/clear the organism) but would protect against sarcoidosis (inflammation would be self-limited and would readily resolve).

The present work is a step toward a more comprehensive study of sarcoidosis and its treatment. As more data become available, the model could be further refined. It would be important to include in this refined model adverse side effects of drugs.

## Methods

All of the computations used to solve the PDE system apply second-order finite difference discretization on the radial direction and a forward Euler method on the time direction.

To support the robustness of the simulation results, we ran sensitivity analysis on parameters that appear in the differential equations and in the boundary conditions; details are in *SI Text*.

The experimental results displayed in Fig. 3 were obtained by the following procedure: Gene expression analysis was performed on tissues obtained from patients with sarcoidosis at the time of diagnosis compared with normal lung tissue. Expression of select genes was further confirmed in lung tissue from a second series of patients with sarcoidosis and disease-free control subjects by semiquantitative RT-PCR. The expression of proteins corresponding to selected overexpressed genes was determined using fluorokine multiplex analysis, and immunohistochemistry.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: afriedman{at}mbi.osu.edu.

Author contributions: W.H., E.D.C., and A.F. designed research, performed research, analyzed data, and wrote the paper.

Reviewers: M.J., Albany Medical College; and R.C.L., University of Connecticut Health Center.

The authors declare no conflict of interest.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1417789111/-/DCSupplemental.

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