Combination therapy for melanoma with BRAF/MEK inhibitor and immune checkpoint inhibitor: a mathematical model

Background The B-raf gene is mutated in up to 66% of human malignant melanomas, and its protein product, BRAF kinase, is a key part of RAS-RAF-MEK-ERK (MAPK) pathway of cancer cell proliferation. BRAF-targeted therapy induces significant responses in the majority of patients, and the combination BRAF/MEK inhibitor enhances clinical efficacy, but the response to BRAF inhibitor and to BRAF/MEK inhibitor is short lived. On the other hand, treatment of melanoma with an immune checkpoint inhibitor, such as anti-PD-1, has lower response rate but the response is much more durable, lasting for years. For this reason, it was suggested that combination of BRAF/MEK and PD-1 inhibitors will significantly improve overall survival time. Results This paper develops a mathematical model to address the question of the correlation between BRAF/MEK inhibitor and PD-1 inhibitor in melanoma therapy. The model includes dendritic and cancer cells, CD 4+ and CD 8+ T cells, MDSC cells, interleukins IL-12, IL-2, IL-6, IL-10 and TGF- β, PD-1 and PD-L1, and the two drugs: BRAF/MEK inhibitor (with concentration γ B) and PD-1 inhibitor (with concentration γ A). The model is represented by a system of partial differential equations, and is used to develop an efficacy map for the combined concentrations (γ B,γ A). It is shown that the two drugs are positively correlated if γ B and γ A are at low doses, that is, the growth of the tumor volume decreases if either γ B or γ A is increased. On the other hand, the two drugs are antagonistic at some high doses, that is, there are zones of (γ B,γ A) where an increase in one of the two drugs will increase the tumor volume growth, rather than decrease it. Conclusions It will be important to identify, by animal experiments or by early clinical trials, the zones of (γ B,γ A) where antagonism occurs, in order to avoid these zones in more advanced clinical trials.


Background
PD-1 is an immunoinhibitory receptor predominantly expressed on activated T cells [1,2]. Its ligand PD-L1 is upregulated on the same activated T cells, and is also expressed by myeloid-derived suppressor cells (MDSCs) [2][3][4][5] and in some human cancer cells, including melanoma, lung cancer, colon cancer, and leukemia [2,3]. The complex PD-1-PD-L1 is known to inhibit T cell function [1]. Immune checkpoints are regulatory pathways in the immune system that inhibit its active response against *Correspondence: xiulanlai@ruc.edu.cn 1 Institute for Mathematical Sciences, Renmin University of China, 100872 Beijing, People's Republic of China Full list of author information is available at the end of the article specific targets. In the case of cancer, the complex PD-1-PD-L1 functions as an immune checkpoint for anti-tumor T cells. There has been much progress in recent years in developing checkpoint inhibitors, primarily anti-PD-1 and anti-PD-L1 [6].
The B-raf gene is mutated in up to 66% of human malignant melanomas, and its protein product, BRAF kinase, is a key part of the RAS-RAF-MEK-ERK (MAPK) pathway of cancer cell proliferation [7]. BRAF-targeted therapy induces significant response in the majority of patients but the response is short lived (about 6 months) [7][8][9]. Initial clinical trials indicate that concurrent inhibition of BRAF with MEK decreases MAPK-driven acquired resistance, resulting in enhanced clinical efficacy and decreased toxicity [10,11]. This provides a rational for using combined BRAF/MEK inhibition instead of BRAF inhibition alone [11]. Treatment of melanoma with immune checkpoint inhibitors has a lower response rate compared to treatment with BRAF/MEK inhibitors, but the response tends to be more durable, lasting for years [11][12][13]. It was therefore suggested that BRAF/MEK-targeted therapy may synergize with the PD-1 pathway blockade to enhance anti-tumor immunity [4,11,14,15]. Meta-Analysis of randomized clinical trials show that compared with other treatments of advanced BRAF-mutated melanoma, combined BRAF/MEK and PD-1 inhibitions significantly improved overall survival time [16].
In this paper we develop a mathematical model to address the efficacy of the combination of BRAF/MEK inhibitor (BRAF/MEKi) and anti-PD-1 (e.g. nivolumab). The model includes several types of T cells, MDSCs, and dendritic cells, as well as signaling molecules involved in the crosstalks among these cells.
Melanoma-derived factors alter the maturation and activation of tissue-resident dendritic cells, thus favoring tumor immune escape [17]. In BRAF mutant melanoma, BRAF inhibitor restores the compromised dendritic cells function, and, in particular, the production of IL-12 by dendritic cells [18]. Although MEK inhibitor (e.g. trametinib), as single agent, negatively impacts DC function, when combined with BRAF inhibitor (e.g. vemurafenib or dabrafenib), the functionality of DCs is restored, as well as their production of IL-12 [18,19].
One of the checkpoints on T cells is the membrane protein PD-1. Its ligand PD-L1 is expressed on activated effector T cells, on MDSCs and on cancer cells [2][3][4][5]. The complex PD-1-PD-L1 inhibits the function of effector T cells [1], but enhances the activation of Tregs [32] and thus promoting cancer.
The above interactions between cancer cells and the immune cells are summarized in Fig. 1. The mathematical model developed in the present paper is based on Fig. 1, and it includes BRAF/MEK and PD-1 inhibitors. Simulations of the model show that at low doses the two drugs are positively correlated, in the sense that the tumor volume decreases as each of the drugs is increased. However, at high doses the two drugs may become antagonistic, that is, an increase in dose of one of the drugs may actually result in an increase in the tumor volume.

Mathematical model
The mathematical model is based on the network shown in Fig. 1. The list of variables is given in Table 1. Since CCL2 and IL-6 are both produced by cancer cells and both recruit MDSCs and Tregs into tumor environment, we shall consider, for simplicity, only IL-6 in our model. We assume that the total density of cells within the tumor remains constant in space and time: We assume that the density of debris of dead cells from necrosis or apoptosis is constant. We also assume that the densities of immature dendritic cells and naive CD4 + and CD8 + T cells remain constant throughout the tumor tissue. Under the assumption (1), proliferation of cancer cells and immigration of immune cells into the tumor, give rise to internal pressure which results in cells movement. We assume that all the cells move with the same velocity, u; u depends on space and time and will be taken in unit of cm/day. We also assume that all the cells undergo dispersion (i.e., diffusion), and that all the cytokines and anti-tumor drugs are diffusing within the tumor.

Equation for DCs (D)
By necrotic cancer cells (N C ) we mean cancer cells undergoing the process of necrosis. Necrotic cancer cells release HMGB-1 (H) [33]. We model the dynamics of N C and H by the following equations: where λ N C C is the rate at which cancer cells become necrotic, d N is the rate at which necrotic cells turn into debris, and λ HN C is the rate at which necrotic cells produce HMGB-1. We note that although molecules like HMGB-1, or other proteins, may be affected by the velocity u, their diffusion coefficients are several order of magnitude larger than the diffusion coefficients of cells, hence their velocity term may be neglected. The degradation of HMGB-1 is fast (∼0.01/day) [34], and we assume that the process of necrosis is also fast. We may then approximate the two dynamical equations by the steady statenλ Dendritic cells are activated by HMGB-1 [35,36]. Hence, the activation rate of immature dendritic cells, with density D 0 , is proportional to D 0 Here, the Michaelis-Menten law is used to account for the limited rate of receptor recycling time which takes place in the process of DCs activation. Hence, the dynamics of DCs is given by where δ D is the diffusion coefficient and d D is the death rate of DCs.

Equation for CD4 + T cells (T 1 )
Naive CD4 + T cells differentiate into Th1 cells (T 1 ) under IL-12 (I 12 ) environment [20,21], while IL-10 and Tregs inhibit the differentiation of naive CD4 + T cells into T 1 cells [27,30]. The proliferation of activated T 1 cells is enhanced by IL-2. Both processes of activation and proliferation of T 1 are assumed to be inhibited by PD-1-PD-L1 (Q) by a factor 1 1+Q/K TQ . Hence T 1 satisfies the following equation: where T 10 is the density of the naive CD4 + T cells.

Equation for activated CD8 + T cells (T 8 )
Inactive CD8 + T cells are activated by IL-12 [20,21], and this process is resisted by IL-10 and Tregs [27,30]. IL-2 enhances the proliferation of activated CD8 + T cells. Similarly to the equation for T 1 , T 8 satisfies the following equation: where T 80 is the density of the inactive CD8 + T cells.

Equation for activated Tregs (T r )
Naive CD4 + T cells differentiate into Tregs (T r ) under activation by Fox3+ transcription factor. The complex PD-1-PD-L1 enhances the expression of PTEN, which results in upregulation of Fox3+, and hence in increased production of Tregs [32]. The production of T r is also enhanced by TGF-β (T β ) [30,31]. The activated Tregs are recruited into tumor by tumor-derived immunosuppressive cytokine IL-6 (and CCL2) [15,28,29]. Representing this chemoattraction by ∇ ·(χT r ∇I 6 ), we get the following equation for T r :

Equation for tumor cells (C)
Cancer cells are killed by T 1 and T 8 cells. We assume a logistic growth with carrying capacity (C M ) in order to account for competition for space among cancer cells. BRAF/MEK inhibitor (B), for example vemurafenib/dabrafenib, is used for treatment of advanced melanoma. Its mechanism of action involves selective inhibition of the mutated BRAF kinase that leads to reduced signaling through the aberrant RAS-RAF-MEK-ERK (MAPK) pathway. We assume that BRAF/MEK inhibitor suppresses the abnormal proliferation of tumor cells by a factor 1 1+B/K CB . Then, the equation for C takes the form: where η 1 and η 8 are the killing rates of cancer cells by T 1 and T 8 , and d C is the natural death rate of cancer cells.

Equation for IL-12 (I 12 )
The proinflammatory cytokine IL-12 is secreted by activated DCs [20,21], so that The maturation and activation of dendritic cells is interrupted by melanoma cells, which means that the production rate coefficient λ I 12 D is small. However, in BRAF mutant melanoma, BRAF inhibitor alone or in combination with MEK inhibitor, restores the compromised dendritic cells function, and in particular, the production of IL-12 by dendritic cells [18,19], and the corresponding equation for I 12 then takes the form:

Equation for IL-10 (I 10 )
IL-10 is produced by cancer cells and MDSCs [27]. Hence it satisfies the following equation:

Equation for PD-1 (P), PD-L1 (L) and PD-1-PD-L1 (Q)
PD-1 is expressed on the surface of activated CD4 + T cells, activated CD8 + T cells and Tregs. We assume that the number of PD-1 per cell is the same for T 1 and T 8 cells, but is smaller, by a factor ε T , for T r cells. If we denote by ρ P the ratio between the mass of one PD-1 protein to the mass of one T cell, then The coefficient ρ P is constant when no anti-PD-1 drug is administered. And in this case, to a change in T = T 1 + T 8 + ε T T r , given by ∂T ∂t , there corresponds a change of P, given by ρ P ∂T ∂t . For the same reason, ∇ ·(uP) = ρ P ∇ ·(uT) and ∇ 2 P = ρ P ∇ 2 T when no anti-PD-1 drug is injected. Hence, P satisfies the equation When anti-PD-1 drug (A) is applied, PD-1 is depleted (or blocked) by A. In this case, the ratio P T 1 +T 8 +ε T T r may change. In order to include in the model both cases of with and without anti-PD-1, we replace ρ P in the previous equation by where μ PA is the depletion rate of PD-1 by anti-PD-1. PD-L1 is expressed on the surface of activated CD4 + T cells, activated CD8 + T cells, MDSCs, and tumor cells. We assume that the number of PD-L1 per cell is the same for T 1 , T 8 and M cells, and denote the ratio between the mass of one PD-L1 protein to the mass of one cell by ρ L . Then where ε C depends on the specific type of tumor. PD-L1 from T cells or cancer cells combines with PD-1 on the plasma membrane of T cells, thus forming a complex PD-1-PD-L1 (Q) on the T cells [2,3]. Denoting the association and disassociation rates of Q by α PL and d Q , respectively, we can write The half-life of Q is less then 1 second (i.e. 1.16×10 −5 day) [43], so that d Q is very large. Hence we may approximate the dynamical equation for Q by the steady state equation,

Equation for anti-PD-1 (A)
We assume that anti-PD-1 is injected intradermally every three days for 30 days (as in mouse experiments [44]), providing a sourceÂ(t) of anti-PD-1: where γ A is the effective level of the drug; although the level of the drug varies between injections, for simplicity we take it to be constant. The drug A is depleted in the process of blocking PD-1. Hence,

Equation for BRAF/MEK inhibitor (B)
We assume that the BRAF/MEK inhibitor is injected intradermally every days for 30 days, providing a sourcê B(t) of BRAF/MKEi: Assuming that BRAF/MEKi is absorbed by C at a rate μ BC C B K B +B , we get the following equation for B:

Equation for cells velocity (u)
We assume that a part of the tumor consists of extracellular matrix, ECM (approximately, 0.4 g/cm 3 ), cancer cells (approximately, C = 0.4 g/cm 3 ) and MDSCs (approximately, M = 0.2 g/cm 3 ). We assume (in the section of parameter estimation) that the densities of the immune cells D, T 1 , T 8 and T r are approximately 4×10 −4 , 2×10 −3 , 1 × 10 −3 g/cm 3 and 5 × 10 −4 g/cm 3 , respectively, and, for consistency, take the constant in Eq. (1) to be 0.6039. We further assume that all cells have approximately the same volume and surface area, so that the diffusion coefficients of all the cells are the same. Adding Eqs. (2)-(7), we then get To simplify the computations, we assume that the tumor is spherical and denote its radius by r = R(t). We also assume that all the densities and concentrations are radially symmetric, that is, functions of (r, t), where 0 ≤ r ≤ R(t). In particular, u = u(r, t)e r , where e r is the unit radial vector.

Equation for free boundary (R)
We assume that the free boundary r = R(t) moves with the velocity of cells, so that

Boundary conditions
We assume that the naive CD4 + T cells and inactive CD8 + T cells that migrated from the lymph nodes into the tumor microenvironment have constant densitiesT 1 and T 8 at the tumor boundary, and that T 1 and T 8 are activated by IL-12 upon entering the tumor. We then have the following flux conditions at the tumor boundary: where σ T (I 12 ) = σ 0 We impose a no-flux boundary condition for all the remaining variables: No-flux for D, T r , M, C, I 12 , I 2 , T β , I 6 , I 10 , P, A, and B at r = R(t). (21) It is tacitly assumed here that the receptors PD-1 and ligands PD-L1 become active only after the T cells are already inside the tumor.

Initial conditions
Later on we shall compare the simulations of the model with experimental results for mice, for 60 days. Accordingly, we take initial values whereby the average density of cancer cells has not yet increased to its steady state. Then, by Eq. (1), the total density of the immune cells is initially above its steady state. We take (in unit of g/cm 3 ): Note that the initial conditions for the cells satisfy Eq. (1).
We assume that initially B = 0 and A = 0, and take the initial condition for I 12 , I 2 , T β , I 6 , I 10 and P to be close to their steady state values, which are computed in the section on parameter estimation. One choice of initial conditions is given as follows (in unit of g/cm 3 ): However, other choices of these initial conditions do not affect the simulations of the model after a few days.

Results and discussions
The simulations of the model were performed by Matlab based on the moving mesh method for solving partial differential equations with free boundary [45] (see the section on computational method). Figure 2 is a simulation of the model with no drugs (the control case) for the first 60 days. The average density or concentration of a species is computed as its total mass in the tumor divided by the tumor volume. The simulation shows consistency in the choice of the model parameters. Indeed, as can be quickly checked, the steady states of all the cytokines and cells are approximately equal to the half-saturation values that we assumed in estimating the parameters of the model. Furthermore, the volume of the tumor doubles approximately every 10 days, as was assumed in the choice of the parameter λ 0 (used in estimating some parameters of Eq. (7)). It is interesting to note that the initial increase in TGF-β more than compensates for the initial decrease in P and L, as evident by the initial increase in T r . This initial increase of T r results in initial decrease in the T 1 and T 8 cells. We also note that the initial increase in cancer cells results in an increase in the D cells. Figure 3 shows the growth of the tumor radius during 60 days when drug is administered. With no drugs, the radius increases from 0.01 cm to 0.037 cm. Treatment with BRAF/MEK inhibitor alone decreased the radius growth more than anti-PD-1 alone, and the combined therapy did better than anti-PD-1 alone. These results agree with mouse experiments reported in [44].
We next consider combination therapy for a range of values of BRAF/MEK inhibitor and anti-PD-1. We define the efficacy of a combination therapy, with (γ B , γ A ), by the formula: where the tumor radius R 60 = R 60 (γ B , γ A ) is computed at day 60; R 60 (0, 0) is the radius at day 60 in the control case (no drugs). The efficacy is a positive number, and its value lies between 0 and 1 (or between 0 and 100%). Figure 4 is the efficacy map of the combined therapy, with γ B in the range of 0 − 5 × 10 −9 g/cm 3 · day and γ A in the range of 0 − 1.4 × 10 −9 g/cm 3 · day. The color column shows the efficacy for any pair of (γ B , γ A ); the maximum efficacy is 0.97 (97%). As the number of cancer cells increases, the tumor radius increases. Hence, if T 1 and T 8 were monotone  Tables 2 and 3 Fig. 3 The growth of tumor radius R(t) during the administration of anti-PD-1 drug and BRAF/MEK inhibitors. Anti-PD-1 is administered at rate γ A = 0.3 × 10 −9 g/cm 3 · day and BRAF/MEK inhibitor is administered at rate γ B = 0.5 × 10 −9 g/cm 3 · day. All other parameter values are the same as in Tables 2 and 3 Fig . 4 Drug efficacy map. The color column shows the efficacy E(γ B , γ A ) when γ B varies between 0 − 5 × 10 −9 g/cm 3 · day and γ A varies between 0 − 1.4 × 10 −9 g/cm 3 · day. All other parameter values are the same as in Tables 2 and 3 increasing functions of γ A (or of γ B ), then we should see that R 60 (γ B , γ A ) is a decreasing function of γ A (or of γ B ), and E(γ B , γ A ) would then also be an increasing function of γ A (or of γ B ). But Fig. 4 shows that this is not generally the case; indeed there are small oscillations in "northeast" corner of the figure. This means that the functions T 1 and T 8 cannot be monotone increasing with respect to γ B for fixed γ A > 0.5 × 10 −9 g/cm 3 · day, and also cannot be monotone increasing in γ A for fixed γ B > 1.5 × 10 −9 g/cm 3 · day. Indeed, for example, Fig. 5a shows that the average densities of T 1 and T 8 are decreasing functions of γ B , for fixed γ A = 1.26 × 10 −9 g/cm 3 · day; however, for smaller values of γ A , T 1 and T 8 may become monotone increasing, as seen, for example, in Fig. 5b with γ A = 0.14×10 −9 g/cm 3 ·day. Similarly, Fig. 6a shows that, for fixed γ B = 3 × 10 −9 g/cm 3 · day, there is a γ A -interval where T 1 and T 8 are decreasing as γ A increases. The γ Ainterval where T 1 and T 8 are decreasing may shrink as we take a smaller fixed γ B , as seen, for example, in Fig. 6b with γ B = 0.1 × 10 −9 g/cm 3 · day.
A possible explanation for Fig. 5a is based on the antagonistic pathway shown in Fig. 7. When γ B increases, the population of cancer cells decreases, and then, by Eqs. (2)-(4) and (8), so does the signal to activate T cells by dendritic cells-derived IL-12 (since the number of activated dendritic cells decrease with decreased cancer cell density) and thus the densities of T 1 and T 8 decrease. As for Fig. 6a, when γ A begins to increase, T 1 and T 8 also begin to increase, which results in a decrease of cancer cells. Then, as explained in the case of Fig. 5a, this leads to a decrease in dendritic cells-derived IL-12 and, hence, the density of activated T 1 and T 8 cells will begin to decrease as γ A continues to increase for a while.
If we inject IL-12 directly into tumor (as an additional drug), the influence of dendritic cells-secreted IL-12 diminishes, and the antagonism between BRAF/MEKi and anti-PD-1 also diminishes and it disappears already at very small amount of injection, e.g., an injection of order of magnitude 10 −14 gcm 3 · day.

Sensitivity analysis
We performed sensitivity analysis, with respect to the rumor radius R at day 60 in the control case, with respect to some of the production parameters of the system (2)-(16), namely, λ DC , λ T 1 I 12 , λ T 8 I 12 , λ T r T β , λ T β C , λ I 6 C , λ I 10 C ,and the parameters K TQ , η 1 and η 8 which play important role in the dynamics of C. Following the method of [46], we performed Latin hypercube sampling and generated 1000 samples to calculate the partial rank correlation coefficients (PRCC) and the p-values with respect to the tumor radius at day 60. In sampling all the parameters, we took the range of each from 1/2 to twice its values in Tables 2 and 3. The results are shown in Fig. 8.
We see that the production/activation rates that promote effector T cells, namely, λ DC , λ T 1 I 12 and λ T 8 I 12 , are negatively correlated to the tumor radius, while the production/activation rates of the effector T cell-suppressors, such as λ T r T β , λ I 10 C , λ T β C and λ I 6 C , are positively correlated to the tumor radius. The killing rate of effector T cells, η 1 and η 8 are negatively correlated to the tumor radius, and the correlation with η 8 is higher than with η 1 .

Conclusion
BRAF mutation occurs in up to 66% of human malignant melanomas and for this reason BRAF has been one Fig. 5 Average densities of T 1 and T 8 . a Average densities of T 1 and T 8 decrease as γ B increases for fixed γ A = 1.26 × 10 −9 g/cm 3 · day; b. Average densities of T 1 and T 8 increase as γ B increases for fixed γ A = 0.14 × 10 −9 g/cm 3 · day. Here, γ B varies between 0 − 5 × 10 −9 g/cm 3 · day and all other parameter values are the same as in Tables 2 and 3 Fig. 6 Average densities of T 1 and T 8 . a There is a γ A -interval where average densities of T 1 and T 8 are decreasing as γ A increases for fixed γ B = 3 × 10 −9 g/cm 3 · day. b The γ A -interval where average T 1 and T 8 are decreasing may shrink as γ B is taken to be smaller, e.g. γ B = 0.1 × 10 −9 g/cm 3 · day. Here, γ A varies between n.4 × 10 −9 g/cm 3 · day and all other parameter values are the same as in Tables 2 and 3 of the primary targets in melanoma therapy. Treatment with BRAF inhibitors (such as vemurafenib or dabrafenib) encounters MAPK-driven resistance, but combining it with MEK inhibitor (e.g. trametinib) significantly reduces this resistance as well as toxicity. While the response to the combined BRAF/MEK inhibitor is significant, it is short lived. On the other hand, PD-1 antibody (nivolumab) has lower response rate but a far greater durability. It was therefore suggested that BRAF/MEK inhibitor should positively correlate with anti-PD-1.
In the present paper we developed a mathematical model to test this hypothesis, in silico, by computing the efficacy of the combined therapy. The model is represented by a system of partial differential equations within the tumor tissue. The model includes immune cells (Th1 and CD8 + T cells, Tregs, MDSCs and dendritic cells), cytokines (IL-12, IL-2, IL-6, IL-10 and TGF-β), and PD-1, PD-L1 and the complex PD-1-PD-L1. We simulated the model with combination of drugs, BRAF/MEK inhibitor at the 'level' γ B and PD-1 antibody at the 'level' γ A , and computed the tumor radius R 60 = R 60 (γ A , γ B ) at day 60, and the efficacy ; the efficacy is an expression that quantifies the reduction in tumor size compared to the control case (no drugs). Fig. 4 shows that for low levels of γ B and γ A , the two drugs are positively correlated, in the sense that tumor volume decreases as each of the drugs is increased. However, in the 'northeast' corner of Fig. 4 we see that for higher levels of γ B and γ A there are zones where the drugs are antagonistic in the sense that when γ B and γ A in these zones are increased, the efficacy actually decreases. The antagonism between the combined drugs can be explained by the pathway shown in Fig. 7. An increase in the number of effector T cells (Th1 and CD8 + ) results in decrease in cancer cells and necrotic cancer cells, hence in decreased signals to activate dendritic cells. This results in a decrease in IL-12 production by dendritic cells, and hence in a decrease in effector T cells.

The efficacy map in
The parameter λ I 12 B may be viewed as the immune system response to BRAF/MEK inhibitor. When this parameter is increased, the antagonism in the combined therapy is reduced, but it does not completely disappear (not shown here).
The mathematical model presented in this paper has several limitations: (i) In order to focus on the combined therapy of a BRAF/MEK inhibitor and an anti-PD-1 drug, we did   T 1 Density of CD4 + T cells from lymph node 4 × 10 −3 g/cm 3 Estimated T 8 Density of CD8 + T cells from lymph node 2 × 10 −3 g/cm 3 Estimated not include in the model the effect of angiogenesis, thus assuming that the tumor is avascular. We tacitly assumed that the effect of this omission is not significant in comparing the results of therapy to no therapy. (ii) We assumed that the densities of immature, or naive, immune cells remain constant throughout the progression of the cancer and that density of debris of dead cells is constant. (iii) We assumed that the process of necrosis is fast, and that the density of cancer cells undergoing necrosis is at steady state. (iv) In estimating parameters we made a steady state assumption in some of the differential equations.
(v) We did not make any direct connection between drugs administered to the patient, and their 'effective strengths' γ B and γ A , as they appear in the differential equations, since these data are not available.
A general study of synergistic and antagonistic networks in drug combinations appeared in [47]. Clinical records on combination therapy show that the number of drugs that are synergistic far exceeds the number of drugs that are antagonistic [48].
In our model, the combination (γ B , γ A ) is antagonistic when the drugs are administered in high doses, but not in low doses. For this reason it will be important to identify more carefully the zones of antagonism, by animal exper- Half-saturation of CD4 + T cells 2 × 10 −3 g/cm 3 Estimated

Half-saturation
In an expression of the form Y X K X +X where Y is activated by X, the half-saturation parameter K X is taken to be the approximate steady state concentration of species X.

Equation (2)
The number of DCs in various organs (heart, kidney, pancreas and liver) in mouse varies from 1.1 × 10 6 cells/cm 3 to 6.6 × 10 6 cells/cm 3 [60]. In the dermal tissue, the number of DCs is larger (600-1500 cells/mm 2 ) [61,62], but we do not specify where the melanoma cancer is located; it may be at its initial dermal tissue or in another organ where it metastasized.
Mature DCs are approximately 10 to 15 μm in diameter [63]. Accordingly, we estimate the steady state of DCs to be K D = 4 × 10 −4 g/cm 3 . We assume that there are always immature dendritic cells, some coming from the blood as tumor infiltrating dendritic cells (TID) [20,21,64]. We also assume that the density of immature DCs to be smaller than the density of active DCs, and take D 0 = 1 20 K D = 2 × 10 −5 g/cm 3 . From the steady statenof Eq. (2), we get λ DC = 2d D D/D 0 = 4/day, since d D = 0.1/day [65].
We take K C = 0.4 g/cm 3 .

Equation (3)
The number of lymphocytes is approximately twice the number of DCs [60]. T cells are approximately 14 to 20 μm in diameter. Assuming that the number of Th1 cells is 1/4 the number of lymphocytes, we estimate steady state density of Th1 cells to be K T 1 = 2 × 10 −3 g/cm 3 . We assume that the density of naive CD4 + T cells to be less than the density of Th1, and take T 10 = 1 5 K T = 4×10 −4 g/cm 3 . As in [65], we choose K TT r to be half-saturation of T r , that is, K TT r = 5 × 10 −4 g/cm 3 , and as in [66], we choose K TI 10 to be half-saturation of I 10 , namely, K TI 10 = 2 × 10 −7 g/cm 3 . We assume that in steady state, Q/K TQ = 2 (the value of K TQ is derived in the estimates of Eqs. (13)- (15)). From the steady state of Eq. (3), we get where λ T 1 I 2 = 0.25/day [65], d T 1 = 0.197/day [65], T 10 = 4 × 10 −4 g/cm 3 and T 1 = K T 1 = 2 × 10 −3 g/cm 3 . Hence λ T 1 I 12 = 18.64/day.

Equation (12)
The half-life of IL-10 ranges from 1.1 to 2.6 hours [78]; we take it to be 2 hours, that is, t 1/2 = 0.08 day, so that d I 10 = 8.32 day −1 . The concentration of serum IL-10 in melanoma is 8.75 × 10 −12 g/cm 3 [79]. We assume that the concentration of IL-10 in tissue is higher than in serum, and take I 10 = 8.75 × 10 −11 g/cm 3 . In melanoma, the tissue concentrations of IL-10 secreted by tumor cells and by macrophages are similar [80], and, accordingly, we assume that λ I 10 C C = λ I 10 M M in steady state. Hence, from the steady state of Eq. (12) we get, 2λ I 10 C C − d I 1 0 I 10 = 0, so that λ I 10 C = d I 10 I 10 /2C = 9.10 × 10 −10 /day, and λ I 10 M = λ I 10 C C/M = 1.82 × 10 −9 /day.

Equations (13)-(15)
In order to estimate the parameters K TQ (in Eqs. (3) and (4)) and K Q (in Eq. (5)), we need to determine the steady state concentrations of P and L in the control case (no drugs). To do that, we begin by estimating ρ P and ρ L .

Equations (16)-(17)
In mice experiments [44,86] different amounts of drugs were injected, and the amount of BRAF/MEK inhibitor was larger than the amount of anti-PD-1. It is difficult to compare the amounts injected into mice with the actual levels of the drugs which appear in Eqs. (16) and (17), since there is no information available on the PK/PD of the drugs. For the choice of γ A = 0.3 × 10 −9 g/cm 3 · day and γ B = 0.5 × 10 −9 g/cm 3 · day, we found that the simulations are in qualitative agreement with experiments reported in [44]. We shall accordingly take γ A in the range of n.4 × 10 −9 g/cm 3 · day and γ B in the range of 0 − 5 × 10 −9 g/cm 3 · day. By [87], the half-life of anti-PD-1 is 15 days, so that d A = ln2 15 = 0.046 day −1 . We assume that 10% of A is used in blocking PD-1, while the remaining 90% degrades naturally. Hence, μ PA PA/10% = d A A/90%, so that μ PA = d A 9P = 0.046 9 × (8.46 × 10 −10 ) = 6.04×10 6 cm 3 /g·day.

Computational method
We employ moving mesh method [45] to numerically solve the free boundary problem for the tumor proliferation model. To illustrate this method, we take Eq. (2) as example and rewrite it as the following form: where F represents the term in the right hand side of Eq. (2). Let r k i and D k i denote numerical approximations of i-th grid point and D(r k i , nτ ), respectively, where τ is the size of time-step. The discretization of Eq. (27) is derived by the fully implicit finite difference scheme: where D r = ,