 Research Article
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Combination therapy for melanoma with BRAF/MEK inhibitor and immune checkpoint inhibitor: a mathematical model
BMC Systems Biologyvolume 11, Article number: 70 (2017)
Abstract
Background
The Braf gene is mutated in up to 66% of human malignant melanomas, and its protein product, BRAF kinase, is a key part of RASRAFMEKERK (MAPK) pathway of cancer cell proliferation. BRAFtargeted 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 antiPD1, 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 PD1 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 PD1 inhibitor in melanoma therapy. The model includes dendritic and cancer cells, CD 4^{+} and CD 8^{+} T cells, MDSC cells, interleukins IL12, IL2, IL6, IL10 and TGF β, PD1 and PDL1, and the two drugs: BRAF/MEK inhibitor (with concentration γ _{ B }) and PD1 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
PD1 is an immunoinhibitory receptor predominantly expressed on activated T cells [1, 2]. Its ligand PDL1 is upregulated on the same activated T cells, and is also expressed by myeloidderived suppressor cells (MDSCs) [2–5] and in some human cancer cells, including melanoma, lung cancer, colon cancer, and leukemia [2, 3]. The complex PD1PDL1 is known to inhibit T cell function [1]. Immune checkpoints are regulatory pathways in the immune system that inhibit its active response against specific targets. In the case of cancer, the complex PD1PDL1 functions as an immune checkpoint for antitumor T cells. There has been much progress in recent years in developing checkpoint inhibitors, primarily antiPD1 and antiPDL1 [6].
The Braf gene is mutated in up to 66% of human malignant melanomas, and its protein product, BRAF kinase, is a key part of the RASRAFMEKERK (MAPK) pathway of cancer cell proliferation [7]. BRAFtargeted therapy induces significant response in the majority of patients but the response is short lived (about 6 months) [7–9]. Initial clinical trials indicate that concurrent inhibition of BRAF with MEK decreases MAPKdriven 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 [11–13]. It was therefore suggested that BRAF/MEKtargeted therapy may synergize with the PD1 pathway blockade to enhance antitumor immunity [4, 11, 14, 15]. MetaAnalysis of randomized clinical trials show that compared with other treatments of advanced BRAFmutated melanoma, combined BRAF/MEK and PD1 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 antiPD1 (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.
Melanomaderived factors alter the maturation and activation of tissueresident 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 IL12 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 IL12 [18, 19].
Dendritic cellderived IL12 activates effector T cells (Th1 and CD 8^{+} T cells) [20, 21]. Th1 produces IL2 which further promotes proliferation of effector T cells. CD 4^{+} T cells (Th1) can kill cancer cell directly, for example, through FAS or TRAILdependent pathway [22–25], while CD 8^{+} T cell is more effective in killing cancer cells [26]. Cancer cells suppress the functions of effector T cells by producing immunosuppressor cytokines TGF β, IL6, CCL2 and IL10 [27]. IL10 inhibits the activation of Th1 and CD 8^{+} T cells [27]. IL6 and CCL2 recruit MDSCs into tumor [19, 28, 29], and MDSCs produce TGF β and IL10. IL6 and CCL2 also recruit T regulatory T cells (Tregs) [15, 28, 29]. TGF β is produced not only by cancer cells and MDSCs, but also by Tregs [30], and Tregs become activated by TGF β [30, 31]. Tregs modulate Th1 and CD 8^{+} T cells [30], thus promoting tumor growth.
One of the checkpoints on T cells is the membrane protein PD1. Its ligand PDL1 is expressed on activated effector T cells, on MDSCs and on cancer cells [2–5]. The complex PD1PDL1 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 PD1 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.
Methods
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 IL6 are both produced by cancer cells and both recruit MDSCs and Tregs into tumor environment, we shall consider, for simplicity, only IL6 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 CD 4^{+} and CD 8^{+} 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 antitumor 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 HMGB1 (H) [33]. We model the dynamics of N _{ C } and H by the following equations:
where $\lambda _{N_{C}C}$ is the rate at which cancer cells become necrotic, d _{ N } is the rate at which necrotic cells turn into debris, and $\lambda _{HN_{C}}$ is the rate at which necrotic cells produce HMGB1. We note that although molecules like HMGB1, 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 HMGB1 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 staten$\lambda _{N_{C}C}Cd_{N_{C}}N_{C}=0$ and $\lambda _{HN_{C}}N_{C}d_{H}H=0$, so that H is proportional to C.
Dendritic cells are activated by HMGB1 [35, 36]. Hence, the activation rate of immature dendritic cells, with density D _{0}, is proportional to $D_{0}\frac {H}{K_{H}+H}$, or to $D_{0}\frac {C}{K_{C}+C}$, since H is proportional to C. Here, the MichaelisMenten 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 CD 4^{+} T cells (T _{1})
Naive CD 4^{+} T cells differentiate into Th1 cells (T _{1}) under IL12 (I _{12}) environment [20, 21], while IL10 and Tregs inhibit the differentiation of naive CD 4^{+} T cells into T _{1} cells [27, 30]. The proliferation of activated T _{1} cells is enhanced by IL2. Both processes of activation and proliferation of T _{1} are assumed to be inhibited by PD1PDL1 (Q) by a factor $\frac {1}{1+Q/K_{TQ}}$. Hence T _{1} satisfies the following equation:
where T _{10} is the density of the naive CD 4^{+} T cells.
Equation for activated CD 8^{+} T cells (T _{8})
Inactive CD 8^{+} T cells are activated by IL12 [20, 21], and this process is resisted by IL10 and Tregs [27, 30]. IL2 enhances the proliferation of activated CD 8^{+} T cells. Similarly to the equation for T _{1}, T _{8} satisfies the following equation:
where T _{80} is the density of the inactive CD 8^{+} T cells.
Equation for activated Tregs (T _{ r })
Naive CD 4^{+} T cells differentiate into Tregs (T _{ r }) under activation by Fox3+ transcription factor. The complex PD1PDL1 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 tumorderived immunosuppressive cytokine IL6 (and CCL2)[15, 28, 29]. Representing this chemoattraction by ∇·(χ T _{ r }∇I _{6}), we get the following equation for T _{ r }:
Equation for activated MDSCs (M)
Tumor recruits macrophages and “educates” them to become tumorassociatedmacrophages (TAMs), which behave like MDSCs [37, 38]. MDSCs are also chemotactically attracted to the tumor microenvironment by IL6 (and CCL2) [15, 19, 28, 29, 39]. As in [40], the Eq. of MDSCs is taken to be the following form:
where M _{0} is the source/influx of macrophages from the blood.
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 RASRAFMEKERK (MAPK) pathway. We assume that BRAF/MEK inhibitor suppresses the abnormal proliferation of tumor cells by a factor $\frac {1}{1+B/K_{CB}}$. 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 IL12 (I _{12})
The proinflammatory cytokine IL12 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 $\lambda _{I_{12}D}$ 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 IL12 by dendritic cells [18, 19], and the corresponding equation for I _{12} then takes the form:
Equation for IL2 (I _{2})
IL2 is produced by activated CD 4^{+} T cells (T _{1}) [21]. Hence,
Equation for TGF β (T _{ β })
TGF β is produced by tumor cells [27], MDSCs [31, 41, 42] and Tregs [30]:
Equation for IL6 (I _{6})
IL6 is produced by cancer cells [15, 19, 28], so that
Equation for IL10 (I _{10})
IL10 is produced by cancer cells and MDSCs [27]. Hence it satisfies the following equation:
Equation for PD1 (P), PDL1 (L) and PD1PDL1 (Q)
PD1 is expressed on the surface of activated CD 4^{+} T cells, activated CD 8^{+} T cells and Tregs. We assume that the number of PD1 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 PD1 protein to the mass of one T cell, then
The coefficient ρ _{ P } is constant when no antiPD1 drug is administered. And in this case, to a change in T=T _{1}+T _{8}+ε _{ T } T _{ r }, given by $\frac {\partial T}{\partial t}$, there corresponds a change of P, given by $\rho _{P}\frac {\partial T}{\partial t}$. For the same reason, ∇·(u P)=ρ _{ P }∇·(u T) and ∇^{2} P=ρ _{ P }∇^{2} T when no antiPD1 drug is injected. Hence, P satisfies the equation
Recalling Eqs. (3)(5) for T _{1},T _{8} and T _{ r }, we get
When antiPD1 drug (A) is applied, PD1 is depleted (or blocked) by A. In this case, the ratio $\frac {P}{T_{1}+T_{8}+\varepsilon _{T}T_{r}}$ may change. In order to include in the model both cases of with and without antiPD1, we replace ρ _{ P } in the previous equation by $\frac {P}{T_{1}+T_{8}+\varepsilon _{T}T_{r}}$. Hence,
where μ _{ PA } is the depletion rate of PD1 by antiPD1.
PDL1 is expressed on the surface of activated CD 4^{+} T cells, activated CD 8^{+} T cells, MDSCs, and tumor cells. We assume that the number of PDL1 per cell is the same for T _{1}, T _{8} and M cells, and denote the ratio between the mass of one PDL1 protein to the mass of one cell by ρ _{ L }. Then
where ε _{ C } depends on the specific type of tumor.
PDL1 from T cells or cancer cells combines with PD1 on the plasma membrane of T cells, thus forming a complex PD1PDL1 (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 halflife 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, α _{ PL } P L=d _{ Q } Q, or
where σ=α _{ PL }/d _{ Q }.
Equation for antiPD1 (A)
We assume that antiPD1 is injected intradermally every three days for 30 days (as in mouse experiments [44]), providing a source $\hat A(t)$ of antiPD1:
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 PD1. Hence,
Equation for BRAF/MEK inhibitor (B)
We assume that the BRAF/MEK inhibitor is injected intradermally every days for 30 days, providing a source $\hat B(t)$ of BRAF/MKEi:
Assuming that BRAF/MEKi is absorbed by C at a rate $\mu _{BC}C\frac {B}{K_{B}+B}$, 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 CD 4^{+} T cells and inactive CD 8^{+} T cells that migrated from the lymph nodes into the tumor microenvironment have constant densities $\hat T_{1}$ and $\hat T_{8}$ at the tumor boundary, and that T _{1} and T _{8} are activated by IL12 upon entering the tumor. We then have the following flux conditions at the tumor boundary:
where $\sigma _{T}(I_{12})=\sigma _{0} \frac {I_{12}}{I_{12}+K_{I_{12}}}$.
We impose a noflux boundary condition for all the remaining variables:
It is tacitly assumed here that the receptors PD1 and ligands PDL1 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 halfsaturation 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 antiPD1 alone, and the combined therapy did better than antiPD1 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 antiPD1. 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 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 _{ A }interval 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 cellsderived IL12 (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 cellsderived IL12 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 IL12 directly into tumor (as an additional drug), the influence of dendritic cellssecreted IL12 diminishes, and the antagonism between BRAF/MEKi and antiPD1 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 }, $\lambda _{T_{1}I_{12}}$, $\lambda _{T_{8}I_{12}}$, $\lambda _{T_{r}T_{\beta }}$, $\lambda _{T_{\beta } C}$, $\lambda _{I_{6}C}$, $\lambda _{I_{10}C}$,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 pvalues 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 }, $\lambda _{T_{1}I_{12}}$ and $\lambda _{T_{8}I_{12}}$, are negatively correlated to the tumor radius, while the production/activation rates of the effector T cellsuppressors, such as $\lambda _{T_{r}T_{\beta }}$, $\lambda _{I_{10}C}$, $\lambda _{T_{\beta } C}$ and $\lambda _{I_{6}C}$, 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 of the primary targets in melanoma therapy. Treatment with BRAF inhibitors (such as vemurafenib or dabrafenib) encounters MAPKdriven 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, PD1 antibody (nivolumab) has lower response rate but a far greater durability. It was therefore suggested that BRAF/MEK inhibitor should positively correlate with antiPD1.
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 CD 8^{+} T cells, Tregs, MDSCs and dendritic cells), cytokines (IL12, IL2, IL6, IL10 and TGF β), and PD1, PDL1 and the complex PD1PDL1. We simulated the model with combination of drugs, BRAF/MEK inhibitor at the ‘level’ γ _{ B } and PD1 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 $E(\gamma _{B},\gamma _{A})=\frac {R_{60}(0,0)R_{60}(\gamma _{B},\gamma _{A})}{R_{60}(0,0)}$; the efficacy is an expression that quantifies the reduction in tumor size compared to the control case (no drugs).
The efficacy map in 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 CD 8^{+}) results in decrease in cancer cells and necrotic cancer cells, hence in decreased signals to activate dendritic cells. This results in a decrease in IL12 production by dendritic cells, and hence in a decrease in effector T cells.
The parameter $\lambda _{I_{12}B}$ 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 antiPD1 drug, we did 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 experiments or by early clinical trials, in order to avoid those zones in more advanced clinical trials.
Appendix
Parameter estimation
Halfsaturation
In an expression of the form $Y\frac {X}{K_{X}+X}$ where Y is activated by X, the halfsaturation parameter K _{ X } is taken to be the approximate steady state concentration of species X.
Diffusion coefficients
By [49], we have the following relation for estimating the diffusion coefficients of a protein p:
where M _{ V } and δ _{ V } are respectively the molecular weight and diffusion coefficient of VEGF, M _{ p } is the molecular weight of p, and M _{ V }=24kDa [50] and δ _{ V }=8.64×10^{−2} cm^{2} day^{−1} [51]. Since, $M_{I_{2}}=17.6$kDa [52], $M_{I_{12}}=70$kDa [53], $M_{T_{\beta }}=25$kDa [54], $M_{I_{6}}=21$kDa [55, 56], $M_{I_{10}}=20.5$kDa [57], M _{ A }=32kDa [58] and M _{ B }=489.93Da [59], we get $\delta _{I_{2}}=9.58\times 10^{2}\ \text {cm}^{2}\:\text {day}^{1}$, $\delta _{I_{12}} =6.05\times 10^{2}\ \text {cm}^{2}\:\text {day}^{1}$, $\delta _{T_{\beta }}=8.52\times 10^{2}\ \text {cm}^{2}\:\text {day}^{1}$, $\delta _{I_{6}}=9.03\times 10^{2}\ \text {cm}^{2}\:\text {day}^{1}$, $\delta _{I_{10}}=9.11\times 10^{2}\ \text {cm}^{2}\:\text {day}^{1}$, δ _{ A }=7.85×10^{−2} cm^{2} day^{−1} and δ _{ B }=3.16×10^{−1} cm^{2} day^{−1}.
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 (6001500 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 $D_{0}=\frac {1}{20}K_{D}=2\times 10^{5}\ \mathrm {g}/\text {cm}^{3}$. 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 $K_{T_{1}}=2\times 10^{3}\ \mathrm {g}/\text {cm}^{3}$. We assume that the density of naive CD 4^{+} T cells to be less than the density of Th1, and take $T_{10}=\frac {1}{5}K_{T}=4\times 10^{4}\ \mathrm {g}/\text {cm}^{3}$. As in [65], we choose $K_{TT_{r}}$ to be halfsaturation of T _{ r }, that is, $K_{TT_{r}}=5\times 10^{4}\ \mathrm {g}/\text {cm}^{3}$, and as in [66], we choose $K_{TI_{10}}$ to be halfsaturation of I _{10}, namely, $K_{TI_{10}}=2\times 10^{7}\ \mathrm {g}/\text {cm}^{3}$. 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 $\lambda _{T_{1}I_{2}}=0.25$/day [65], $d_{T_{1}}=0.197$/day [65], T _{10}=4×10^{−4} g/cm^{3} and $T_{1}=K_{T_{1}}=2\times 10^{3}\ \mathrm {g}/\text {cm}^{3}$. Hence $\lambda _{T_{1}I_{12}}=18.64/\text {day}$.
Equation (4)
The CD4/CD8 ratio in the blood is 2:1. We assume a similar ratio in tissue, and take $T_{80}=\frac {1}{2}T_{10}=2\times 10^{4}\ \mathrm {g}/\text {cm}^{3}$. We also take steady state of T _{8} to be the half of steady state of T _{1}, i.e., $K_{T_{8}}=\frac {1}{2}K_{T_{1}}=1\times 10^{3}\ \mathrm {g}/\text {cm}^{3}$. From the steady state of Eq. (4), we have
where $\lambda _{T_{8}I_{2}}=0.25$/day [65], $d_{T_{8}}=0.18$/day [65], T _{80}=2×10^{−4} g/cm^{3}, $T_{8}=K_{T_{8}}=1\times 10^{3}\ \mathrm {g}/\text {cm}^{3}$. Hence $\lambda _{T_{8}I_{12}}=16.6/\text {day}$.
Equation (5)
We assume that TGF β activates Tregs more than PD1PDL1 does, and take $\lambda _{T_{r}T_{\beta }}=5\lambda _{T_{r}Q}$. From the steady state of Eq. (5), we get, $(\lambda _{T_{r}T_{\beta }}\cdot \frac {1}{2}+\lambda _{T_{r}Q}\cdot \frac {1}{2})T_{10}d_{T_{r}}T_{r}=0$, where T _{10}=1×10^{−3} g/cm^{3}, $T_{r}=K_{T_{r}}=5\times 10^{4}\ \mathrm {g}/\text {cm}^{3}$ [65], and $d_{T_{r}}=0.2$/day [65]. Hence $\lambda _{T_{r}Q}=0.083/\text {day}$ and $\lambda _{T_{r}T_{\beta }}=0.415$/day.
Equation (6)
The density of tumorassociated macrophages in melanoma can be up to 30% of the tumor tissue density [67]; we take MDSC density to be 20% of the tumor tissue density, so that M=0.2 g/cm^{3} in steady state. From the steady state of Eq. (6), we get, $\frac {1}{2}\lambda _{M}(M_{0}M)=d_{M}M$, where d _{ M }=0.015/day [40], λ _{ M }=20/19=1.05 [40], and M=K _{ M }=0.2 g/cm^{3}. Hence, M _{0}=0.21 g/cm^{3}.
Equation (7)
We take d _{ C }=0.17 day^{−1} and C _{ M }=0.8 g/cm^{3} [65]. In the control case (no antitumor drugs), the tumor grows according to
Mouse experiments show that tumor volume doubles within 5 15 days [44, 68–70]. Assuming a linear growth
during the volume doubling time in the control case, we conclude from Eq. (23) that
where $\lambda _{0}\in \left (\frac {\text {ln}2}{15},\frac {\text {ln}2}{5}\right)$. We assume that without immune responses and BRAF/MEK inhibitor,
so that
We further assume that with immune response and BRAF/MEK inhibitor, the density of cancer cell still grows,
so that
We take λ _{0}=0.069/day, and assume that in steady state, C is approximately 0.4 g/cm^{3}, so that from Eq. (25) we get $\frac {1}{2}\lambda _{C}d_{C}=2\lambda _{0}$, or λ _{ C }=2(2λ _{0}+d _{ C })=0.616/day. By comparing Eq. (24) to Eq. (25), we see that η _{1} T _{1}+η _{8} T _{8}=λ _{0}. Noting that T _{8} cells kill cancer cells more effectively than T _{1} cells, we take η _{8}=4η _{1}, so that (with $T_{1}=K_{T_{1}}=2\times 10^{3}\ \mathrm {g}/\text {cm}^{3}$ and $T_{8}=K_{T_{8}}=1\times 10^{3}\ \mathrm {g}/\text {cm}^{3}$) $\eta _{1}=\frac {\lambda _{0}}{T_{1}+4T_{8}}=11.5\ \text {cm}^{3}/\mathrm {g}\cdot \text {day}$ and η _{8}=46 cm^{3}/g·day. From Eq. (26), we have $\frac {1}{2}\lambda _{C}\cdot \frac {1}{1+B/K_{CB}}(\eta _{1}T_{1}+\eta _{8}T_{8})d_{C}=\frac {1}{5}\lambda _{0}$. Since λ _{ C }=2(2λ _{0}+d _{ C }) and η _{1} T _{1}+η _{8} T _{8}=λ _{0}, we get $(2\lambda _{0} +d_{C})\cdot \frac {1}{1+B/K_{CB}}\lambda _{0}d_{C}=\frac {1}{5}\lambda _{0}$, so that (with B=K _{ B }=6.69×10^{−10} g/cm^{3}) $K_{CB}=B\frac {5d_{C}+6\lambda _{0} }{4\lambda _{0}}=3.06\times 10^{9}\ \mathrm {g}/\text {cm}^{3}$.
Equation (8)
The serum level of IL12 in melanoma patients varies from 7.5×10^{−11}−9.6×10^{−11} g/cm^{3} [71, 72]. We assume that the IL12 level in tissue is higher, and take $I_{12}=K_{I_{12}}=8\times 10^{10}\ \mathrm {g}/\text {cm}^{3}$. In the control case (no drugs), from the steady state of Eq. (8), we get $\lambda _{I_{12}D}Dd_{I_{12}}I_{12}=0$, where $d_{I_{12}}=1.38$/day [65] and D=K _{ D }=4×10^{−4} g/cm^{3}. Hence, $\lambda _{I_{12}D}=2.76\times 10^{6}$/day. In the simulations we take $\lambda _{I_{12}B}=1$, but simulations do not change qualitatively with smaller or larger values of $\lambda _{I_{12}B}$.
Equation (9)
From the steady state of Eq. (9), we get $\lambda _{I_{2}T_{1}}T_{1}d_{I_{2}}I_{2}=0$, where $d_{I_{2}}=2.376$/day [65] and $I_{2}=K_{I_{2}}=2.37\times 10^{11}\ \mathrm {g}/\text {cm}^{3}$ [65], and $T_{1}=K_{T_{1}}=2\times 10^{3}\ \mathrm {g}/\text {cm}^{3}$. Hence, $\lambda _{I_{2}T_{1}}=2.82\times 10^{8}$/day.
Equation (10)
The halflife of TGF β is about 2 min [73], that is, t _{1/2}=0.0014 day, so that $d_{T_{\beta }}=\text {ln}2/t_{1/2}=499.07\ \text {day}^{1}$. The concentration of serum TGF β in melanoma is 2.68×10^{−14} g/cm^{3} [74]. We assume that the concentration of TGF β in tissue is higher than in serum, and take T _{ β }=2.68×10^{−13} g/cm^{3}. By [75], $\lambda _{T_{\beta } T_{r}}=5.57\times 10^{9}$/day. According to [27, 42], melanoma cells secrete more TGF β than MDSC, and we assume that $\lambda _{T_{\beta } C}C=2\lambda _{T_{\beta } M}$M. Hence, from the steady state of Eq. (10) we have, $\lambda _{T_{\beta } C}C+\lambda _{T_{\beta } M}M+\lambda _{T_{\beta } T_{r}}T_{r}=d_{T_{\beta }}T_{\beta }$, or $3\lambda _{T_{\beta } M}M+\lambda _{T_{\beta } T_{r}}T_{r}=d_{T_{\beta }}T_{\beta }$. Thus $\lambda _{T_{\beta } M}=(d_{T_{\beta }}T_{\beta }\lambda _{T_{\beta } T_{r}}T_{r})/ (3M)=2.18\times 10^{10}$/day, and $\lambda _{T_{\beta } C}=2\lambda _{T_{\beta } M}M/C= 2.18\times 10^{10}$/day.
Equation (11)
The halflife of IL6 is less than 6 hours [76], and we take it to be 4 hours, that is, t _{1/2}=0.17 day, so that $d_{I_{6}}=\text {ln}2/t_{1/2}=4.16\ \text {day}^{1}$. The concentration of serum IL6 in melanoma is 3.4×10^{−12} g/cm^{3} [77]. We assume that the concentration of IL6 in tissue is higher than in serum, and take I _{6}=3.4×10^{−11} g/cm^{3}. From the steady state of Eq. (11), we get $\lambda _{I_{6}C}=d_{I_{6}}I_{6}/C=3.54\times 10^{10}$/day.
Equation (12)
The halflife of IL10 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 $d_{I_{10}}=8.32\ \text {day}^{1}$. The concentration of serum IL10 in melanoma is 8.75×10^{−12} g/cm^{3} [79]. We assume that the concentration of IL10 in tissue is higher than in serum, and take I _{10}=8.75×10^{−11} g/cm^{3}. In melanoma, the tissue concentrations of IL10 secreted by tumor cells and by macrophages are similar [80], and, accordingly, we assume that $\lambda _{I_{10}C}C=\lambda _{I_{10}M}M$ in steady state. Hence, from the steady state of Eq. (12) we get, $2\lambda _{I_{10}C}Cd_{I_{1}0}I_{10}=0$, so that $\lambda _{I_{10}C}=d_{I_{10}}I_{10}/2C=9.10\times 10^{10}$/day, and $\lambda _{I_{10}M}=\lambda _{I_{10}C}C/M=1.82\times 10^{9}$/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 }.
By [81], the mass of one PD1 is m _{ P }=8.3×10^{−8} pg= 8.3×10^{−20} g, and by [1] the mass of one PDL1 is m _{ L }=5.8×10^{−8} pg= 5.8×10^{−20} g. We assume that the mass of one T cell is m _{ T }=10^{−9} g. By [82], there are 3000 PD1 proteins and 9000 PDL1 proteins on one T cell (T _{1} or T _{8}). Since ρ _{ P } T is the density of PD1 (without antiPD1 drug), we get $\rho _{P}=3000\times \frac {m_{P}}{m_{T}}=\frac {3000\times (8.3\times 10^{20})}{10^{9}}=2.49\times 10^{7}$, and $\rho _{L}=9000\times \frac {m_{L}}{m_{T}}=\frac {9000\times (5.8\times 10^{20})}{10^{9}}=5.22\times 10^{7}$.
In order to estimate steady state concentration of P, we assume that the average densities of T _{1}, T _{8} and T _{ r } are approximately 2×10^{−3}, 1×10^{−3} and 5×10^{−4} g/ cm^{3}, respectively. PD1 is expressed by Tregs at higher or lower level than in T _{1} and T _{8} cells depending on the type of the cancer [83]; we assume that ε _{ T }=0.8. Hence, in steady state,
The parameter ε _{ C } in Eq. (14) depends on the type of cancer. We take ε _{ C }=0.01 [84]. MDSCs express PDL1 at lower level than tumor cells [85], and accordingly, we assume that $\varepsilon _{M}M=\frac {1}{4}\varepsilon _{C}C$, so that ε _{ M }=ε _{ C } C/4M=ε _{ C }/2=0.005. Then, by Eq. (14), we get
In steady state with $P=\bar P$, $L=\bar L$ and $Q=\bar Q$, we have, by Eq. (15), $\bar Q=\sigma \bar P\bar L$. We take $K_{TQ}=\frac {1}{2}\bar Q=\frac {1}{2}\sigma \bar P\bar L$. Hence, $Q/K_{TQ}=PL/(\frac {1}{2}\bar P\bar L)$ and
where $K^{\prime }_{TQ}:=\frac {1}{2}\bar P\bar L=\frac {1}{2}\times (8.46\times 10^{10})\times (4.176\times 10^{9})=1.77\times 10^{18}\ \mathrm {g}^{2}/\text {cm}^{6}$. Similarly, $K_{Q}=\bar Q=\sigma \bar P\bar L$, so that in Eq. (5),
where $K^{\prime }_{Q}:=\bar P\bar L=3.54\times 10^{18}\ \mathrm {g}^{2}/\text {cm}^{6}$.
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 antiPD1. 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 halflife of antiPD1 is 15 days, so that $d_{A}=\frac {\text {ln} 2}{15}=0.046\ \text {day}^{1}$. We assume that 10% of A is used in blocking PD1, while the remaining 90% degrades naturally. Hence, μ _{ PA } P A/10%=d _{ A } A/90%, so that
The halflife of BRAF inhibitor (dabrafenib) is 8 hours [88], and the halflife of MEK inhibitor (trametinib) is 33 h [89]. In the combination of BRAF/MEKi, the proportion of MEKi is smaller than BRAFi [44], and accordingly we take the halflife of BRAF/MEKi to be 10 h, so that $d_{B}=\frac {\text {ln} 2}{10/24}=1.66\ \text {day}^{1}$. We assume that 10% of B is absorbed by cancer cells, while the remaining 90% degrades naturally, so that $\mu _{BC}C\frac {B}{K_{B}+B}/10\%=d_{B}B/90\%$. From Eq. (17), we get B≥γ _{ B }/d _{ B }, and we assume that
where d _{ B }=1.66/day. We take γ _{ B } to be order of magnitude 10^{−9} g/cm^{3}·day in the simulations. Hence, B=K _{ B }=6.69×10^{−10} g/cm^{3} in steady state, and μ _{ BC }=2d _{ B } B/9C=6.17×10^{−10}/day.
Eqs. (20): We assume that $\hat T_{1}$ is larger than $K_{T_{1}}$ and take $\hat T_{1}=4\times 10^{3}\ \mathrm {g}/\text {cm}^{3}$. Similarly, we also assume that $\hat T_{8}$ is larger than $K_{T_{8}}$ and take $\hat T_{8}=2\times 10^{3}\ \mathrm {g}/\text {cm}^{3}$.
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 $r_{i}^{k}$ and $D_{i}^{k}$ denote numerical approximations of ith grid point and $D(r_{i}^{k},n\tau)$, respectively, where τ is the size of timestep. The discretization of Eq. (27) is derived by the fully implicit finite difference scheme:
where $D_{r}=\frac {h_{1}^{2}D_{i+1}^{k+1}h_{1}^{2}D_{i1}^{k+1}(h_{1}^{2}h_{1}^{2})D_{i}^{k+1}}{h_{1}(h_{1}^{2}h_{1}h_{1})}$, $D_{rr}=2\frac {h_{1}D_{i+1}^{k+1}h_{1}D_{i1}^{k+1}+(h_{1}h_{1})D_{i}^{k+1}}{h_{1}(h_{1}h_{1}h_{1}^{2})}$,$u_{r}=\frac {h_{1}^{2}u_{i+1}^{k+1}h_{1}^{2}u_{i1}^{k+1}(h_{1}^{2}h_{1}^{2})u_{i}^{k+1}}{h_{1}(h_{1}^{2}h_{1}h_{1})}$, $h_{1}=r_{i1}^{k+1}r_{i}^{k+1}$ and $h_{1}=r_{i+1}^{k+1}r_{i}^{k+1}$. The mesh moves by $r_{i}^{k+1}=r_{i}^{k}+u_{i}^{k+1}\tau $, where $u_{i}^{k+1}$ is solved by the velocity equation.
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Acknowledgements
This work is supported by the Mathematical Biosciences Institute and the National Science Foundation (Grant DMS 0931642), and by the Renmin University of China and the National Natural Science Foundation of China (Grant No. 11501568), and the International Postdoctoral Exchange Fellowship Program 2016 by the Office of China Postdoctoral Council.
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The dataset supporting the conclusions of this article is included within the article.
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XL and AF developed and simulated the model, and wrote the final manuscript. Both authors read and approved the final manuscript.
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Correspondence to Xiulan Lai.
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Keywords
 Melanoma
 Mathematical modeling
 BRAF/MEK inhibitor
 PD1 inhibitor
 Combination therapy