Positive feedback loops drive immune cell polarization toward a pro-tumor phenotype that accentuates immunosuppression and tumor angiogenesis. This phenotypic switch leads to the escape of cancer cells from immune destruction. These positive feedback loops are generated by cytokines such as TGF-β, Interleukin-10 and Interleukin-4, which are responsible for the polarization of monocytes and M1 macrophages into pro-tumor M2 macrophages, and the polarization of naive helper T cells intopro-tumor Th2 cells. In this article, we present a deterministic ordinary differential equation (ODE) model that includes key cellular interactions and cytokine signaling pathways that lead to immune cell polarization in the tumor microenvironment. The model was used to simulate various cancer treatments in silico. We identified combination therapies that consist of M1 macrophages or Th1 helper cells, coupled with an anti-angiogenic treatment, that are robust with respect to immune response strength, initial tumor size and treatment resistance. We also identified IL-4 and IL-10 as the targets that should be neutralized in order to make these combination treatments robust with respect to immune cell polarization. The model simulations confirmed a hypothesis based on published experimental evidence that a polarization into the M1 and Th1 phenotypes to increase the M1-to-M2 and Th1-to-Th2 ratios plays a significant role in treatment success. Our results highlight the importance of immune cell reprogramming as a viable strategy to eradicate a highly vascularized tumor when the strength of the immune response is characteristically weak and cell polarization to the pro-tumor phenotype has occurred.
The differentiation and polarization of certain immune cells into pro-inflammatory and anti-inflammatory cells confer the immune system with versatility to exhibit a strong, but controlled, response against invading pathogens and foreign antigens. This is made possible by the initial generation of a strong inflammatory response that is subsequently regulated and attenuated by an anti-inflammatory response once the invading agents have been destroyed. A phenotypic switch by immune cells during an infection and after its resolution makes it possible for the immune system to regulate its own activity and return to a state of homeostasis [
When healthy cells mutate and become cancerous, immune cells such as natural killer cells and cytotoxic T lymphocytes (CTL) try to eliminate these anomalous cells. They do so by infiltrating the tumor site and releasing proteins that destroy the cancer cell membrane (Perforin), and release enzymes that lead to cancer cell apoptosis (Granzyme B) [
Experimental work has shown that it is possible to reprogram M2 macrophages to develop an M1 phenotype by increasing the environmental concentration of anti-tumor cytokines. Monocytes and naïve helper T cells develop an M1 and Th1 phenotype, respectively, if the concentration of anti-tumor cytokinesINF-γ and TNF-α is high. Given this phenotypic plasticity of immune cells, it is no surprise that cancer cells have evolved ways to hijack the mechanisms of cell polarization for their own advantage. Cancer cells can escape immune destruction by releasing pro-tumor cytokines, such as TGF-β, that decrease the M1-to-M2 macrophage ratio and the Th1-to-Th2 helper cell ratio. What makes this hijacking relevant, in the context of cancer treatments, is that macrophages and helper T cells are found in the tumor site in relatively high concentrations [
A tumor is a complex dynamical system, and its survival depends on a diverse set of signaling networks characterized by cytokine-driven positive feedback loops that can reinforce the anti-tumor phenotype or the pro-tumor phenotype of tumor-infiltrating immune cells. For example, M1 macrophages secrete IL-12 which leads to the differentiation of immature helper T cells into Th1 cells. Th1 cells secrete IFN-ϒ which reinforces the M1 macrophage phenotype. This positive feedback loop perpetuates the M1 and Th1 anti-tumor polarization of these cells, which can lead to tumor destruction. On the other hand, M2 macrophages secrete IL-4 and IL-6 [
More complex immune cell interactions exist. M2 macrophages and Th2 cells secrete TGF-β which converts naïve helper T cells into pro-tumor regulatory T cells (Tregs) [
This article is organized as follows. In Section 2 we describe the mathematical model that was formulated to investigate potential ways to break up pro-tumor feedback loops that lead to treatment failure. We also used the model to assess the relative effectiveness of various combination treatments. In Section 3 we describe the predictions of the model, including the combination treatments that were found to be robust with respect to the level of immune response strength, the tumor size at the start of treatment, treatment resistance, and immune cell polarization. In Section 4 we discuss the implications of the model predictions and comment on ways that the model can be improved. We conclude by highlighting the importance of increasing the M1-to-M2 ratio and the Th1-to-Th2
ratio to boost the anti-tumor immune response, in conjunction with a reduction of TGF-β-driven angiogenesis. This approach is predicted to lead to treatment synergy and robustness, and to the effective disruption of the pro-tumor cytokine-driven signaling networks that lead to tumor survival.
In [
The ODE equations were coded in Scilab (http://www.scilab.org/) and were solved by using a built-in 4th-order explicit Runge-Kutta method with fixed step size. This numerical method has a fast rate of convergence of O(h4) and guarantees a stable computation time. Many of the model parameter values were obtained from published literature pertaining to experimental or mathematical modeling work involving various murine and human cancers, as well as infection models that describe a pathogen-immune system interaction that is similar to a tumor-immune system interaction.
It is known that the phenotypes of macrophages vary widely and, thus, they cannot be classified as being purely anti-tumor or purely pro-tumor [
The Th1 and Th2 phenotypes of helper T cells tend to be terminally-differentiated states [
The equations of the model are presented in Appendix A. The ODEs that describe the population dynamics of treatment-sensitive and treatment-resistant cancer cells include terms that represent cancer cell destruction by NK cells, CTL, M1 macrophages and Th1 helper T cells. Chemotherapy kills cancer cells and immune system cells. We did not incorporate an anti-tumor humoral response. In the cytokine equations, we included terms that represent antagonistic interactions between certain cytokines. For example, an increase in the concentration of IL-4 and IL-10 decreases the production rate of IL-12 by M1 macrophages and by Th1 helper cells. Similarly, an increase in the concentration of IL-12 decreases the production rate of IL-4 by M2 macrophages and by Th2 helper cells. Antagonistic effects between IFN-α and TNF-α, and between IFN-α and IFN-γ were also included. Supplementary
To investigate the joint effect of immunosuppression, tumor angiogenesis and immune cell polarization on treatment success, we considered three different scenarios at the start of treatment. These scenarios are listed below in order from most favorable to least favorable:
1) Low immunosuppression, low angiogenesis and no pro-tumor polarization.
2) High immunosuppression, high angiogenesis and no pro-tumor polarization.
3) High immunosuppression, high angiogenesis and high pro-tumor polarization.
To simulate scenario 1, the initial conditions were chosen by assuming a low concentration of Tregs, TGF-β and activated endothelial cells, and by assuming that no M2 macrophages and no Th2 cells are initially present. To simulate scenario 2, the initial conditions were chosen by assuming a high concentration of Tregs, TGF-β and activated endothelial cells, but no M2 macrophages and no Th2 cells are initially present. To simulate scenario 3, the initial conditions were chosen to reflect a high concentration of Tregs, TGF-β, activated endothelial cells, M2 macrophages and Th2 cells. Supplementary
Supplementary
As was done in [
To investigate the effect of cytokine concentration on immune cell polarization and tumor growth, we used a partial differential equation (PDE) model of a granuloma that develops in response to a lung infection [
Supplementary
The parameters d, l and s together determine the level of strength of a patient’s immune response D by cytotoxic T lymphocytes (CTL), as defined by Equation (28). The 3 levels of immune response strength that we considered were the same as those defined in [
1) Strong Immune Response: ( d = 2.1 , l = 1.1 , s = 5 × 10 − 3 )
a) Best case: Low immunosuppression, low angiogenesis and no polarization.
b) High immunosuppression, high angiogenesis and no polarization.
c) High immunosuppression, high angiogenesis and high pro-tumor polarization.
2) Moderate Immune Response: ( d = 1.6 , l = 1.4 , s = 8 × 10 − 3 )
a) Low immunosuppression, low angiogenesis and no polarization.
b) High immunosuppression, high angiogenesis and no polarization.
c) High immunosuppression, high angiogenesis and high pro-tumor polarization.
3) Weak Immune Response: ( d = 1.3 , l = 2 , s = 4 × 10 − 2 )
a) Low immunosuppression, low angiogenesis and no polarization.
b) High immunosuppression, high angiogenesis and no polarization.
c) Worst case: High immunosuppression, high angiogenesis and high pro-tumor polarization.
To ensure the safety in the clinic of all the simulated treatments, we followed the safety criteria described in [
We first considered the case of initially low immunosuppression, low angiogenesis and no initial pro-tumor cell polarization (refer to Supplementary
Under the initial conditions described above, the model predicts that a5-cycle Sunitinib + NK cell treatment given at their regular rates will fail to reduce the size of the tumor despite the significant reduction of the Treg population caused by the Sunitinib injections. This treatment fails due to the high concentration of immunosuppressive TGF-β at the start of the treatment and that remains high throughout the treatment period. Additionally, the model predicts that a Fresolimumab monotherapy consisting of 20 injections at the regular rate will also fail to eliminate the tumor. Similarly, 20 injections of M1 macrophages administered concurrently with 20 injections of Th1 cells at their regular rates is not sufficient to reduce the size of the tumor. An important result of this particular simulation is that injecting M1 macrophages and Th1 helper cells concurrently at their regular rates may not be sufficient to significantly alter the M1-to-M2 ratio and the Th1-to-Th2 ratio, if immunosuppression and angiogenesis are initially high. Other measures must be taken to modify these ratios, such as increasing the treatment dosage or disrupting additional signaling pathways. For example, one M1macrophage injection + one Fresolimumab (anti-TGF-β) injection administered concurrently at their regular rates eliminated the tumor in approximately 90 days. This success was due in part to the strong CTL response that the immune system is capable of exhibiting in this scenario, and the complementary
nature at work of the two treatment modalities: a boosting of the CTL response coupled with a reduction of tumor angiogenesis and immunosuppression exerted by TGF-β.
A more practical combination treatment that eliminates the tumor consists of giving 9 M1 macrophage injections + 5 Fresolimumab injections both administered at 2 times their regular rate. The need for an increased dosage was due to the high pro-tumor immune cell polarization that existed at the start of treatment. The model predicts that a monotherapy consisting of17 M1 macrophage injections administered at three times the regular rate will also eliminate the tumor. These predictions highlight the importance of reducing tumor angiogenesis and immunosuppression as a treatment strategy. These results also suggest that a lack of reduction of tumor angiogenesis can be compensated by significantly increasing the M1-to-M2 macrophage polarization ratio by increasing the infusion rates of M1 macrophages into the tumor site. We surmise that a high M1-to-M2 polarization ratio leads to a rate of cancer cell destruction by immune cells that is greater than the rate of cancer cell replication, even when this replication rate is enhanced by angiogenesis.
In the no treatment case with low immunosuppression, low angiogenesis and no initial immune cell polarization favorable to the tumor, the immune system initially decreases the size of the tumor. However, due to its moderate response, the immune system is unable to eliminate all the cancer cells and the tumor grows again to its maximum carrying capacity. With treatment, there were multiple treatment combinations that eliminated the tumor. A moderate immune response, without an initial pro-tumor immune cell polarization, made it possible for a single injection of Irinotecan or a single injection of Panitumumab administered at their regular rate to eliminate the tumor, including the Irinotecan-resistant and Panitumumab-resistant cells. This result is shown in
tumor by giving a single M1 macrophage or Th1 cell injection at a lower infusion rate (1 × 107 cells L−1 day−1) than the regular rate (5.627 × 108 cells L−1 day−1). However, a monotherapy consisting of 5 Sunitinib cycles administered at the regular rate had no effect on tumor growth. A reasonable explanation for this outcome is the fact that in this scenario, the Treg population was already low at the start of treatment and, hence, the Treg-targeted Sunitinib injections served no useful purpose.
In the case of high immunosuppression, high angiogenesis and no initial pro-tumor immune cell polarization, there were several combinations treatments that led to a complete response, as can be seen in
・ 6 M1 macrophage injections + 3 Fresolimumab injections given at their regular rates
・ 8 Th1 cell injections + 4 Fresolimumab injections given at their regular rates
・ 1 cycle of Sunitinib administered at its regular ratein combination with 3 injections of Fresolimumab given at 1.28 times its regular rate
・ 9 Fresolimumab injections administered at 1.45 times the regular rate
Compared to Fresolimumab monotherapy, the combination treatments eliminated the tumor in a significantly shorter amount of time.
In the case of low immunosuppression, low angiogenesis and no initial pro-tumor cell polarization (see
In the case of high immunosuppression, high angiogenesis and no initial pro-tumor immune cell polarization, there were several combinations treatments that led to tumor elimination. The most notable case, shown in
The M1 macrophage + Fresolimumab combination treatment was one of several treatments that were identified as being robust with respect to the level of immune response strength by CTL, with respect to the initial tumor size and with respect to resistance to Irinotecan and Panitumumab. These robust treatments, which are administered at their regular rates, are listed in
In the worst-case scenario of high immunosuppression, high angiogenesis and high pro-tumor cell polarization with a weak immune response, there were no treatment combinations that could be administered at feasible rates that led to tumor elimination. Therefore, none of the treatments listed in
The lack of treatment robustness with respect to cell polarization, when assuming a weak immune response, motivated the authors to investigate the effect of directly disrupting the cytokine-driven feedback loops that are responsible for treatment failure. In general, when human cancers are diagnosed, they are well established and have already developed strong immunosuppressive mechanisms [
The gene knockout simulation showed that the tumor is eliminated when 4 M1 macrophage injections are administered together with 2Fresolimumab injections at their regular injection rates (see
or a Th1 cell + Fresolimumab treatment. This multi-pronged strategy confers these combination treatments with a robustness with respect to anti-tumor cell polarization, making them the most effective protocols that were identified.
We proceeded to investigate the extent to which the predicted treatment outcomes depend on the values of the model parameters. To that end, we conducted a comprehensive sensitivity analysis of the weak immune response case with
Treatment Combinations | Treatment Description | Time to Elimination |
---|---|---|
M1 macrophages + Fresolimumab | 8 weekly injections of M1 macrophages and 4 biweekly injections of F. | 102 days |
Th1 helper cells + Fresolimumab | 17 weekly injections of Th1 helper cells and 9 biweekly injections of F. | 160 days |
M1 macrophages + Fresolimumab + Sunitinib | 1 cycle of S concurrent with 3 biweekly injections of M1 and 3 biweekly injections of F. | 56 days |
Th1 helper cells + Fresolimumab + Sunitinib | 2 cycles of S concurrent with 6 biweekly injections of Th1 and 6 biweekly injections of F (Th1 at a reduced dose of v T 1 = 5 × 10 6 cells / L ⋅ day ). | 108 days |
high immunosuppression, high angiogenesis and no cell polarization. The analysis consisted of decreasing (and increasing) each parameter value by 5% and keeping track of the percent change in the predicted tumor size at steady state.
Of note is the fact that our model is significantly less sensitive to the production rate of IL-4 and IL-10 by M2 macrophages and by Th2 cells. This means that the predicted treatment outcomes are not affected by slight changes to the rate of production and decay of these cytokines.
In this work, we focused on increasing the M1-to-M2 macrophage and the Th1-to-Th2 helper cell ratios by injecting anti-tumor polarized cells (M1 macrophages and Th1 helper cells) and by disrupting pro-tumor positive feedback loops driven by TGF-β, IL-4 and IL-10. An alternative approach to increase the M1-to-M2 macrophage and the Th1-to-Th2 helper cell ratios is to inject anti-tumor cytokines, such as IL-12 and IFN-γ, to reinforce the anti-tumor positive feedback loops that polarize macrophages and naïve helper T cells into the M1 and Th1 phenotypes, respectively. We did not include in our model an immunosuppressive positive feedback loop that involves the humoral immune system.
Parameter | Description | % Change in Tumor Size | |
---|---|---|---|
Parameter decreased by 5% | Parameter increased by 5% | ||
a w | Growth rate of wild-type tumor cells. | −0.0001000% | 0.0000905% |
T K | Carrying capacity of wild-type and mutant tumor cells combined in the absence of tumor angiogenesis. | −0.0022626% | 0.0022626% |
α | Rate of circulating lymphocyte production. | 0.0000786% | −0.0000782% |
p 1 | Maximum rate of production of TGF-β by hypoxic tumor cells. | −1.8985642% | 1.7851783% |
b 1 | Critical tumor size at which the angiogenic switch occurs. | 0.0000014% | −0.0000014% |
S 1 | Concentration of TGF-β necessary to reduce the CD8+ T cell killing rate of tumor cells by half. | 0.0000000% | −0.0000000% |
w | Rate of Treg cell production. | −0.0000055% | 0.0000054% |
b K | Proliferation rate of angiogenic endothelial cells. | −2.0499774% | 1.9076866% |
K m a x | Rate at which TGF-β stimulates tumor growth. | −3.7233483% | 3.6382439% |
e | Rate of NK synthesis. | 0.0000937% | −0.0000938% |
r 1 | Rate of activation of CD8+ T cells due to NK cell-lysed tumor cell debris. | 0.0000003% | −0.0000003% |
p R | Rate of IL-2-induced Treg cell proliferation. | −0.0000097% | 0.0000097% |
d | Immune strength coefficient. | 0.0000004% | −0.0000004% |
l | Immune system strength scaling coefficient. | −0.0000097% | 0.0000042% |
s | Value describing how quickly CD8+ T cells respond to the presence of a tumor. | −0.0000004% | 0.0000004% |
λ M o | Differentiation rate of monocytes to M1 macrophages and to M2 macrophages. | −0.0000004% | 0.0000004% |
λ M 2 | Maximal rate at which M1 macrophages are activated to become M2 macrophages. | −0.0000000% | 0.0000000% |
M 0 | Source term of monocytes. | −0.0000004% | 0.0000004% |
T 0 | Source term of naïve helper T cells. | −0.0000721% | 0.0000694% |
λ T 2 | Production rate of Th2 cells. | −0.0000001% | 0.0000001% |
λ I 4 M 2 | Production rate of IL-4 by M2 macrophages. | −0.0000000% | 0.0000000% |
λ I 10 M 2 | Production rate of IL-10 by M2 macrophages. | −0.0000003% | 0.0000002% |
λ I 10 T 2 | Production rate of IL-10 by Th2 cells. | −0.0000001% | 0.0000001% |
It is known that Tregs and Bregs can reinforce each other’s pro-tumor phenotype through secretion of TGF-β and IL-10. We plan to include additional feedback loops into the model that involve the humoral immune response to assess their effect on treatment robustness. We will also consider additional sources of IL-10, such as Tregs and Bregs [
The cell-cell, cell-cytokine and cytokine-cytokine interactions were modeled according to previously published models by using first-order kinetics and by using Hill functions to account for rate saturation. Although there is not a prescribed way to simulate a given interaction, it is important to simulate an interaction in the most biologically-realistic manner. Therefore, it is worth checking whether the model predictions are sensitive to the approach taken when modeling certain interactions. In the future, we plan to undertake such an analysis by introducing Hill Functions of different orders to quantify their effect on treatment success.
In our model, we chose to make the maximum tumor size and tumor angiogenesis depend on TGF-β due to its multiple pro-tumor functions [
In the weak immune response case with high immunosuppression, high angiogenesis, and high pro-tumor cell polarization, it is possible to eliminate the tumor by administering a smaller number of Th1 helper cell injections. However, the drawback is that it would take longer to eliminate the tumor (over 300 days). This means that we would need to keep IL-4 and IL-10 at minimal concentration during this entire treatment period. This may not be advisable, especially if there are serious safety concerns of a prolonged reduced concentration of IL-4 and IL-10. For example, it has been shown that IL-10-deficient mice develop lethal uncontrolled inflammation of the intestine [
Treatment effectiveness does not necessarily transfer from one cancer type to another. For example, it is quite possible that a treatment that can eliminate a slow-growing tumor will fail to stop the growth of a more aggressive tumor, such as glioblastoma. This aspect will be considered when we parametrize our model to identify the treatments that are most likely to eradicate a specific type of cancer. The purpose of our modeling project was primarily to assess the extent to which cytokine-driven feedback loops that perpetuate macrophage and helper T cell polarization into a pro-tumor phenotype determine treatment outcome. Second, we wanted to quantify the extent to which disrupting such feedback loops increases the likelihood of treatment success. We conducted a relative comparison between treatments of the time required for tumor elimination, the number of required injections and the required dose. The model results predicted that injecting M1 macrophages or Th1 cells, administering anti-TGF-β to reduce angiogenesis and simultaneously reducing the production of IL-4 and IL-10 is a promising strategy to eliminate a tumor.
Cytokine-driven feedback loops play an essential role in determining the steady-state dynamics of tumor-immune cell interactions, and the likelihood of treatment success. M1 macrophage + Fresolimumab and Th1 cell + Fresolimumab combination treatments were found to be robust with respect to the level of immune response strength, the initial tumor size and treatment resistance. The model predicts that treatments that simultaneously decrease tumor angiogenesis and boost the concentration of M1 macrophages or Th1 cells will be most effective, since they boost the M1-to-M2 and Th1-to-Th2 ratios. These treatments can be made robust with respect to pro-tumor immune cell polarization if coupled with antibodies that neutralize IL-4 and IL-10. Novel cancer treatments based on IL-10 and IL-4 antibodies could pave the way for tumor elimination despite a worst-case scenario at the start of treatment consisting of a highly immunosuppressive, angiogenic and polarized tumor microenvironment.
The authors would like to thank the BUILD-PODER program at California State University-Northridge for providing the computational equipment and financial support that made this work possible.
LS conceived and managed the project. LS designed the computational experiments. LS and VM conducted the simulations. LS and VM analyzed the results. LS and VM contributed to manuscript preparation.
The authors declare that there are no conflicts of interest associated with the publication of this article.
Morales, V. and Soto-Ortiz, L. (2018) Modeling Macrophage Polarization and Its Effect on Cancer Treatment Success. Open Journal of Immunology, 8, 36-80. https://doi.org/10.4236/oji.2018.82004
Equations (1)-(27) represent the ODEs of the mathematical model that we used to investigate the role of cytokine-driven polarization of immune cells on treatment success. Equation (28) was defined in [
Wild-type Cancer Cells (sensitive to all the treatments):
d T w d t = a w T w ( 1 − T w + T c p + T i T K + g 2 E ) − μ C 1 K M + C 1 T w − ( c + ξ A A + h 1 ) ( e − λ T R ) N T w − D T w − ( K T + K A T A ) ( T w T w + α 1 T c p ) ( 1 − e − δ T ( C 1 + C 2 ) ) T w − ψ A T w − δ M 1 M 1 T w − δ T 1 T 1 T w (1)
KRAS-Mutant Cancer Cells (resistant to Panitumumab and Cetuximab):
d T c p d t = a m T c p ( 1 − T w + T c p + T i T K + g 2 E ) − ( c + ξ A A + h 1 ) ( e − λ T R ) N T c p − D T c p − K T ( T w T w + α 1 T c p ) ( 1 − e − δ T R ( C 1 + C 2 ) ) T c p − δ M 1 M 1 T c p − δ T 1 T 1 T c p (2)
Chemotherapy-Resistant Cancer Cells (resistant to Irinotecan):
d T i d t = a m T i ( 1 − T w + T c p + T i T K + g 2 E ) + μ C 1 K M + C 1 T w − ( c + ξ A A + h 1 ) ( e − λ T R ) N T i − D T i − ( K T + K A T A ) ( T w T w + α 1 T c p ) ( 1 − e − δ T R C 2 ) T i − ψ A T i − δ M 1 M 1 T i − δ T 1 T 1 T i (3)
Natural Killer Cells:
d N d t = e C − f N − ( p + p A A A + h 1 ) N ( T w + T c p + T i ) + p N N I 2 g N + I 2 − K N ( 1 − e − δ N ( C 1 + C 2 ) ) N + v N K (4)
Cytotoxic T Lymphocytes:
d L d t = − θ m L θ + I 2 + j T w + T c p + T i k + T w + T c p + T i L − q L ( T w + T c p + T i ) + ( r 1 N + r 2 C ) ( T w + T c p + T i ) − u L 2 R I 2 κ + I 2 − K L ( 1 − e − δ L ( C 1 + C 2 ) ) L + p I 2 L I 2 g I 2 + I 2 + v C T L (5)
Circulating T Lymphocytes in the Blood:
d C d t = α − β C − K C ( 1 − e − δ C ( C 1 + C 2 ) ) C (6)
Activated Endothelial Cells:
d E d t = b K E ( 1 − E K m a x B g 1 + B ) − d K E ( T w + T c p + T i ) 2 3 (7)
Regulatory T Cells
d R d t = w C − u R R + ( p R R I 2 g R + I 2 ) ( B B + g T G F β ) + λ T r e g T 0 ( B B + K T G F β ) − h R ( 1 − e − λ R S ) R − K R ( 1 − e − δ R ( C 1 + C 2 ) ) R (8)
Chemotherapy Drug #1 (Irinotecan―Dose-dependent resistance emerges)
d C 1 d t = − γ 1 C 1 + v C 1 (9)
Chemotherapy Drug #2 (Hypothetical―No resistance emerges)
d C 2 d t = − γ 2 C 2 + v C 2 (10)
Interleukin-2:
d I 2 d t = − μ I 2 I 2 + ϕ C + ω L I 2 ς + I 2 + λ I 2 T 1 T 1 + v I 2 (11)
Panitumumab and Cetuximab (monoclonal antibodies):
d A d t = − η A − λ ( T w + T c p + T i ) ( A A + h 2 ) + v A (12)
TGF-β:
d B d t = p 1 ( T w + T c p + T i ) 2 b 1 2 + ( T w + T c p + T i ) 2 − u 1 B − b 2 F B (13)
Sunitinib (receptor tyrosine kinase inhibitor):
d S d t = − η S S + v S (14)
Fresolimumab (anti-TGF-β):
d F d t = − η F F − b 3 B F + v F (15)
M1 Macrophages (anti-tumor):
d M 1 d t = λ M 0 ε 1 ε 1 + ε 2 M 0 + λ M 1 ε 1 ε 1 + ε 2 M 2 − λ M 2 ε 2 ε 2 + ε 1 M 1 − K C ( 1 − e − δ C ( C 1 + C 2 ) ) M 1 − d M 1 M 1 + v M 1 (16)
M2 Macrophages (pro-tumor):
d M 2 d t = λ M 0 ε 2 ε 2 + ε 1 M 0 − λ M 1 ε 1 ε 1 + ε 2 M 2 + λ M 2 ε 2 ε 2 + ε 1 M 1 − K C ( 1 − e − δ C ( C 1 + C 2 ) ) M 2 − d M 2 M 2 (17)
Th1 Helper T Cells (anti-tumor):
d T 1 d t = λ T 1 M 1 T 0 M 1 M 1 + K M 1 I 12 I 12 + K I 12 K I 10 K I 10 + I 10 + λ T I 2 I 2 I 2 + K I 2 T 1 − K C ( 1 − e − δ C ( C 1 + C 2 ) ) T 1 − d T 1 T 1 + v T 1 (18)
Th2 Helper T Cells (pro-tumor):
d T 2 d t = λ T 2 T 0 M 2 M 2 + K M 2 I 4 I 4 + K I 4 K T 1 K T 1 + T 1 − K C ( 1 − e − δ C ( C 1 + C 2 ) ) T 2 − d T 2 T 2 (19)
Interleukin-1β (anti-tumor):
d I 1 β d t = ( λ 1 β M 1 M 1 + λ I 1 β T 1 T 1 ) ( K I α K I α + I α ) − d I 1 β I 1 β (20)
Interleukin-4 (pro-tumor):
d I 4 d t = ( λ I 4 M 2 M 2 + λ I 4 T 2 T 2 ) ( K I 12 K I 12 + I 12 ) − d I 4 I 4 (21)
Interleukin-10 (pro-tumor):
d I 10 d t = λ I 10 M 2 M 2 + λ I 10 T 2 T 2 − b 4 A I 10 I 10 − d I 10 I 10 (22)
Interleukin-12 (anti-tumor):
d I 12 d t = λ I 12 M 1 M 1 + λ I 12 T 1 T 1 ( 1 + I 4 K I 4 ) ( 1 + I 10 K I 10 ) − d I 12 I 12 + v I 12 (23)
Tumor Necrotic Factor-α (anti-tumor):
d T α d t = λ T α M 1 M 1 + λ T α T 1 T 1 ( 1 + I 10 K I 10 ) ( 1 + I α K I α ) − d T α T α (24)
Interferon-α (pro-tumor):
d I α d t = λ I α M 2 M 2 ( K I 1 β K I 1 β + I 1 β ) − d I α I α (25)
Interferon-γ (anti-tumor):
d I γ d t = λ I γ T 1 T 1 K I α K I α + I α − d I γ I γ + v I γ (26)
Anti-Interleukin-10 (anti-tumor):
d A I 10 d t = − η A I 10 A I 10 − b 5 I 10 A I 10 + v A I 10 (27)
Immune Response Strength by CTL:
D = d ( 1 + B S 1 ) ( 1 + I 10 T I 10 ) ⋅ ( L T w + T c p + T i ) l s + ( L T w + T c p + T i ) l (28)
Polarization into Anti-Tumor Cells Driven by IFN-γ and TNF-α:
ε 1 = ( λ M I γ I γ I γ + K I γ + λ M T α T α T α + K T α ) ( K I 10 K I 10 + I 10 ) ( K T G F β K T G F β + B ) (29)
Polarization into Pro-Tumor Cells Driven by IL-4 and IL-10:
ε 2 = ( λ M I 4 I 4 I 4 + K I 4 + λ M I 10 I 10 I 10 + K I 10 ) ( B B + K T G F β ) (30)
Variable | Definition | Units |
---|---|---|
Tw | Number of wild-type cancer cells that are sensitive to all treatments | cells |
Tcp | Number of mutant cancer cells resistant to cetuximab and panitumumaband that are sensitive to irinotecan and the hypothetical chemotherapy drug C2 | cells |
Ti | Number of mutant cancer cells resistant to irinotecan and that are sensitive to cetuximab, panitumumab and the hypothetical chemotherapy drug C2 | cells |
N | Concentration of NK cells per liter of blood | cells/L |
L | Concentration of CD8+ T cells per liter of blood | cells/L |
C | Concentration per liter of blood of other circulating lymphocytes not including NK cells, CD8+ T cells or regulatory T cells | cells/L |
E | Number of activated endothelial cells | cells |
R | Concentration of regulatory T cells per liter of blood | cells/L |
C1 | Concentration of the chemotherapy agent Irinotecan per liter of blood to which cancer cells become resistant | mg/L |
C2 | Concentration of a hypothetical chemotherapy agent per liter of blood to which cancer cells do not develop resistance | mg/L |
I2 | Concentration of IL-2 per liter of blood | IU/L |
A | Concentration of the monoclonal antibodies Cetuximaband Panitumumab per liter of blood | mg/L |
B | Concentration of TGF-βper liter of blood | IU/L |
S | Concentration of the tyrosine kinase inhibitor Sunitinibper liter of blood | mg/L |
F | Concentration of the monoclonal antibody Fresolimumabper liter of blood | mg/L |
M1 | Concentration of M1 Macrophages per liter of blood | cells/L |
M2 | Concentration of M2 Macrophages per liter of blood | cells/L |
T1 | Concentration of Th1 cells per liter of blood | cells/L |
T2 | Concentration of Th2 cells per liter of blood | cells/L |
I1β | Concentration of IL-1β per liter of blood | IU/L |
I4 | Concentration of IL-4 per liter of blood | IU/L |
I10 | Concentration of IL-10 per liter of blood | IU/L |
I12 | Concentration of IL-12 per liter of blood | IU/L |
Tα | Concentration of TNF-α per liter of blood | IU/L |
Iα | Concentration of IFN-α per liter of blood | IU/L |
Iγ | Concentration of IFN-γ per liter of blood | IU/L |
AI10 | Concentration of anti-IL-10 per liter of blood | mg/L |
Variable | Units | Scenario 1 LI, LA, NP | Scenario 2 HI, HA, NP | Scenario 3 HI, HA, HP |
---|---|---|---|---|
Tw | cells | 1 × 109 | 1 × 109 | 1 × 109 |
Tcp | cells | 35 | 35 | 35 |
Ti | cells | 0 | 0 | 0 |
N | cells/L | 9 × 107 | 9 × 107 | 9 × 107 |
L | cells/L | 1.8 × 105 | 1.8 × 105 | 1.8 × 105 |
C | cells/L | 9 × 108 | 9 × 108 | 9 × 108 |
E | cells | 1 | 2 × 109 | 2 × 109 |
R | cells/L | 1 | 4 × 108 | 4 × 108 |
C1 | mg/L | 0 | 0 | 0 |
C2 | mg/L | 0 | 0 | 0 |
I2 | IU/L | 1173 | 1173 | 1173 |
A | mg/L | 0 | 0 | 0 |
B | IU/L | 1 | 1 × 104 | 1 × 104 |
S | mg/L | 0 | 0 | 0 |
F | mg/L | 0 | 0 | 0 |
M1 | cells/L | 0 | 0 | 0 |
M2 | cells/L | 0 | 0 | 1 × 105 |
T1 | cells/L | 0 | 0 | 0 |
T2 | cells/L | 0 | 0 | 9 × 107 |
I1β | IU/L | 1 | 1 | 1 |
I4 | IU/L | 1 | 1 | 2.21 × 105 |
I10 | IU/L | 1 | 1 | 2.235 × 103 |
I12 | IU/L | 1 | 1 | 1 |
Tα | IU/L | 1 | 1 | 1 |
Iα | IU/L | 1 | 1 | 1 |
Iγ | IU/L | 1 | 1 | 1 |
AI10 | mg/L | 0 | 0 | 0 |
Treatment | Agent Type | Dose and Frequency | Infusion Time | Infusion Rate |
---|---|---|---|---|
Chemotherapy | Irinotecan [ | 125 mg/m2 given weekly. This cycle may be repeated. | 1.5 hr | v C 1 = 57.947 mg / L ⋅ day |
Chemotherapy | (Hypothetical) | 125 mg/m2 given weekly. This chemotherapy agent was assumed to have the same cytotoxic properties on the tumor and immune cells as Irinotecan. It was assumed that tumor cells never develop resistance to this drug. This cycle may be repeated. | 1.5 hr | v C 2 = 57.947 mg / L ⋅ day |
Monoclonal antibody | Cetuximab [ | Loading dose (LD): 400 mg/m2 (a one-time injection before giving the maintenance doses) Maintenance dose (MD): 250 mg/m2 (given weekly ? start one week after giving LD and rest 2 weeks every 4 weeks). This cycle may be repeated. | LD: 2 hr MD: 1 hr | LD : v A = 139.072 mg / L ⋅ day MD : v A = 173.840 mg / L ⋅ day Note: the same infusion term vA is used for cetuximab and for panitumumab, but their vA values are different. |
Monoclonal antibody | Panitumumab [ | 6 mg/kg every two weeks. No loading dose is required. This cycle may be repeated. | 1 hr | v A = 168.816 mg / L ⋅ day |
---|---|---|---|---|
Monoclonal antibody | Fresolimumab (anti-TGF-β) [ | 3 mg/kg every two weeks. This cycle may be repeated. | 1.5 hr | v F = 56.272 mg / L ⋅ day |
Monoclonal antibody | Anti-IL-10 Est. from [ | 3 mg/kg every two weeks. This cycle may be repeated. | 1.5 hr | v A I 10 = 21.0 mg / L ⋅ day |
RTK inhibitor | Sunitinib (anti-Treg) [ | A Sunitinib capsule is given daily for 28 straight days followed by two weeks of rest. In total, 23.447 mg of sunitinib are administered per each 6-week cycle. This cycle may be repeated. | None | v S = 0.8374 mg / L ⋅ day |
Adoptive cell transfer | NK cells [ | An intravenous injection of 2 × 107 NK cells per kg is given weekly. This cycle may be repeated. | 1 hr | v N K = 5.627 × 10 8 cells / L ⋅ day |
Adoptive cell transfer | CTL [ | Five intravenous injections of 1 × 1010 CTL per m2 are given every 5 days and they are followed by 45 days of rest. This 65-day cycle may be repeated. | 1 hr | v C T L = 6.9536 × 10 9 cells / L ⋅ day |
Cytokine Treatment | IL-2 [ | 180,000 IU/kg injected over 15 minutes every 8 hours. | 0.25 hr | v I L 2 = 2.0258 × 10 7 IU / L ⋅ day |
---|---|---|---|---|
Adoptive cell transfer | M1 macrophages | An intravenous injection of 2 × 107 M1 macrophages per kg is given weekly. This cycle may be repeated. | 1 hr | v M 1 = 5.627 × 10 8 cells / L ⋅ day |
Adoptive cell transfer | Th1 helper cells | An intravenous injection of 2 × 107 Th1 cells per kg is given weekly. This cycle may be repeated. | 1 hr | v T 1 = 5.627 × 10 8 cells / L ⋅ day |
Parameter | Description | Value | Units | Ref. |
---|---|---|---|---|
a w | Tumor growth rate (colorectal cancer) | 2.31 × 10 − 1 | day−1 | [ |
c | Rate of NK cell-induced tumor death | 5.156 × 10 − 14 | L cell−1 day−1 | [ |
d | Immune strength coefficient | {1.3, 1.6, 2.1} | day−1 | [ |
l | Immune system strength scaling coefficient | {1.1, 1.4, 2} | unitless | [ |
s | Value describing how quickly CD8+ T cells respond to the presence of a tumor | { 5 × 10 − 3 , 8 × 10 − 3 , 4 × 10 − 2 } | L | [ |
ξ | Rate of NK cell-induced tumor death through antibody-dependent cellular cytotoxicity (ADCC) | 6.5 × 10 − 10 (0 for panitumumab) | L cell−1 day−1 | [ |
h 1 | Concentration of mAbs necessary for half-maximal increase in ADCC activity | 1.25 × 10 − 6 (0 for panitumumab) | mg L−1 | [ |
X | Determines the percent level of the maximum rate of chemotherapy-induced tumor death of wild-type and mutant cells | 0.75 | unitless | [ |
K T | Death rate of wild-type and mutant tumor cells due to chemotherapy | ( 8.1 × 10 − 1 ) ⋅ X | day−1 | [ |
---|---|---|---|---|
K A T | Chemotherapy-induced death rate of wild-type and mutant tumors due to mAbs cetuximab and panitumumab | 4 × 10 − 4 | L mg−1 day−1 | [ |
δ T | Chemotherapy efficacy coefficient on wild-type and mutant cancer cells | 2 × 10 − 1 | L mg−1 | [ |
Y | Determines the percent level of the maximum rate of mAb-induced tumor death of wild-type and mutant cells | 0.75 | unitless | [ |
ψ | Rate of mAb-induced wild-type and mutant tumor cell death | ( 2.28 × 10 − 2 ) ⋅ Y for cetuximab ( 3.125 × 10 − 2 ) ⋅ Y for panitumumab | L mg−1 day−1 | [ |
f | Rate of NK cell turnover | 1 × 10 − 2 | day−1 | [ |
e | Rate of NK synthesis from circulating lymphocytes | 1 9 ⋅ f | day−1 | [ |
g N | IL-2 concentration needed for half-maximal NK cell proliferation | 2.5036 × 10 5 | IU L−1 | [ |
p N | Rate of IL-2-induced NK cell proliferation | 5.13 × 10 − 2 | day−1 | [ |
p | Rate of NK cell death due to interaction with the tumor | 5.156 × 10 − 14 | cell−1 day−1 | [ |
p A | Rate of NK cell death due to interaction with mAbs complexes | 6.5 × 10 − 10 (0 for panitumumab) | cell−1 day−1 | [ |
K N | Rate of NK cell depletion due to chemotherapy toxicity | 9.048 × 10 − 1 | day−1 | [ |
δ N | Coefficient of chemotherapy toxicity on NK cells | 2 × 10 − 1 | L mg−1 | [ |
m | Rate of turnover of activated CD8+ T cells | 5 × 10 − 3 | day−1 | [ |
θ | IL-2 concentration required to halve the CD8+ T cell turnover rate | 2.5036 × 10 − 3 | IU L−1 | [ |
q | Rate of CD8+ T cell death due to interaction with the tumor | 5.156 × 10 − 17 | cell−1 day−1 | [ |
r 1 | Rate of activation of CD8+ T cell due to NK cell-lysed tumor cell debris | 5.156 × 10 − 12 | cell−1 day−1 | [ |
r 2 | Rate of CD8+ T cell production from circulating lymphocytes | 1 × 10 − 15 | cell−1 day−1 | [ |
p I 2 | Rate of CD8+ T cell activation induced by IL-2 | 2.4036 | day−1 | [ |
---|---|---|---|---|
g I 2 | Concentration of IL-2 necessary for half-maximal CD8+ T cell activation | 2.5036 × 10 3 | IU L−1 | [ |
u | Rate of inhibition of surplus CD8+ T cells induced by Treg cells in the presence of IL-2 | 2.3085 × 10 − 13 | L2 cell−2 day−1 | [ |
κ | Concentration of IL-2 required to halve the immunosuppressive effect of Treg cells on CD8+ T cells | 2.5036 × 10 3 | IU L−1 | [ |
j | Rate of activation of CD8+ T cells due to CD8+ T cell-lysed tumor cell debris | 1.245 × 10 − 4 | day−1 | [ |
k | Tumor size required for half-maximal CD8+ T cell activation by CD8+ T cell-lysed tumor cell debris | 2.019 × 10 7 | cells | [ |
K L | Rate of CD8+ T cell depletion from chemotherapy toxicity | 4.524 × 10 − 1 | day−1 | [ |
δ L | Coefficient of chemotherapy toxicity on CD8+ T cells | 2 × 10 − 1 | L mg−1 | [ |
β | Rate of circulating lymphocyte turnover | 6.3 × 10 − 3 | day−1 | [ |
α | Rate of circulating lymphocyte production | ( 3 × 10 9 ) ⋅ β | cells L−1 day−1 | [ |
K C | Rate of lymphocyte depletion from chemotherapy toxicity | 5.7 × 10 − 1 | day−1 | [ |
δ C | Coefficient of chemotherapy toxicity on circulating lymphocytes | 2 × 10 − 1 | L mg−1 | [ |
μ I 2 | Rate of excretion and elimination of IL-2 | 11.7427 | day−1 | [ |
ω | Rate of IL-2 production from CD8+ T cells | 7.88 × 10 − 2 | IU cell−1 day−1 | [ |
ϕ | Rate of IL-2 production from circulating CD4+ and naive CD8+ T cells | 1.788 × 10 − 7 | IU cell−1 day−1 | [ |
ζ | Concentration of IL-2 for half-maximal CD8+ T cell IL-2 production | 2.5036 × 10 3 | IU L−1 | [ |
γ 1 | The rate of excretion and elimination of irinotecan | 4.077 × 10 − 1 | day−1 | [ |
η | Rate of cetuximab and panitumumab turnover and excretion | 1.386 × 10 − 1 for cetuximab 9.242 × 10 − 2 for panitumumab | day−1 | [ |
λ | Rate of mAb-tumor cell complex formation | 8.9 × 10 − 14 for cetuximab 8.6 × 10 − 14 for panitumumab | mg cell−1 L−1 day−1 | [ |
---|---|---|---|---|
h 2 | Concentration of cetuximab or panitumumab for half-maximal EGFR binding | 4.45 × 10 − 5 for cetuximab 4.3 × 10 − 5 for panitumumab | mg L−1 | [ |
T K | Carrying capacity of wild-type and mutant tumor cells combined in the absence of tumor angiogenesis | 1 × 10 6 | cells | [ |
g 2 | Conversion factor from number of activated endothelial cells to the increase in tumor carrying capacity | 1 | tumorcells endothelialcell | Est. |
p 1 | Maximum rate of production of TGF-β by hypoxic tumor cells | 1 × 10 5 | IU L−1 day−1 | Est. from [ |
b 1 | Critical tumor size at which the angiogenic switch occurs | 1 × 10 6 | cells | [ |
S 1 | Concentration of TGF-β necessary to reduce the CD8+ T cell killing rate of tumor cells by half | 7 × 10 4 | IU L−1 | [ |
u 1 | Decay rate of TGF-β | 10 | day−1 | [ |
b K | Proliferation rate of angiogenic endothelial cells | 0.198 | day−1 | [ |
K m a x | Maximum carrying capacity for blood vessel growth stimulated by TGF-β secreted by tumor cells | 5 × 10 9 | cells | Est. |
g 1 | TGF-β concentration that gives a half-maximal proliferation rate of endothelial cells | 1 × 10 4 | IU L−1 | Est. |
d K | Growth inhibition coefficient of endothelial cells by tumor cells | 5 × 10 − 8 | cell−2/3 day−1 | Est. from [ |
λ T | Suppressive effect of Treg cells on wild-type and mutant tumor cell kill rate by NK cells | 1.59 × 10 − 9 | L cell−1 | [ |
w | Rate of Treg cell production from circulating lymphocytes | 4.698 × 10 − 4 | day−1 | [ |
u R | Rate of Treg cell turnover | 3.851 × 10 − 2 | day−1 | [ |
p R | Rate of IL-2-induced Treg cell proliferation | 3.598 × 10 − 2 | day−1 | [ |
g R | Concentration of IL-2 necessary for half-maximal activation of Treg cells | 11.027 | IU L−1 | [ |
h R | Rate of Treg cell inhibition by Sunitinib | 0.227 | day−1 | [ |
---|---|---|---|---|
K R | Rate of Treg cell depletion from chemotherapy toxicity | 5.7 × 10 − 1 | day−1 | [ |
α 1 | Determines the scope of influence of KRAS-mutant tumor cells Tcp in making tumor cells resistant to chemotherapy and in reducing the sensitizing role of Cetuximab and Panitumumab to chemotherapy | 1 × 10 7 | unitless | [ |
λ R | Efficacy of Sunitinib in inhibiting the immunosuppressive activity of Tregs | 50.02 | L mg−1 | [ |
δ R | Chemotherapy toxicity on Tregs | 2 × 10 − 1 | L mg−1 | [ |
η S | Rate of excretion and elimination of Sunitinib | 0.277 | day−1 | [ |
a m | Growth rate of mutant tumor cells (colorectal cancer) | 2.31 × 10 − 1 | day−1 | [ |
μ | Maximum mutation rate of wild-type tumor cells | 4 × 10 − 5 | day−1 | [ |
K M | Concentration of Irinotecan chemotherapy that leads to a half-maximal rate of mutation of wild-type tumor cells Tw into irinotecan-resistant tumor cells Ti | 1 × 10 3 | mg L−1 | Est. |
δ T R | Efficacy of the second type of chemotherapy drug of killing irinotecan-resistant tumor cells | 2 × 10 − 1 | L mg−1 | [ |
γ 2 | Rate of excretion and elimination of the hypothetical chemotherapy drug | 4.077 × 10 − 1 | day−1 | [ |
η F | Degradation rate of anti-TGF-beta (Fresolimumab) | 0.033 | day−1 | Est. from [ |
b 2 | Rate of loss of free TGF-β due to binding with anti-TGF-β | 100 | L mg−1 day−1 | Est. from [ |
b 3 | Rate of loss of free anti-TGF-β due to binding with TGF-β | 2.5 × 10 − 13 | L IU−1 day−1 | Est. from [ |
λ M 0 | Differentiation rate of M0 macrophages into an M1 or M2 macrophage | 1 × 10 − 4 | day−1 | Est. from [ |
λ M 1 | Maximal rate at which M2 macrophages switch phenotype to become M1 macrophages | 6 × 10 − 3 | day−1 | [ |
---|---|---|---|---|
λ M 2 | Maximal rate at which M1 macrophages switch phenotype to become M2 macrophages | 6 × 10 − 3 | day−1 | Est. from [ |
λ M I 4 | Production rate of M2 macrophages due to IL-4 | 1 × 10 − 3 | day−1 | [ |
λ M I γ | Production rate of M1 macrophages due to IFN-γ | 1 × 10 − 3 | day−1 | [ |
λ M T α | Production rate of M1 macrophages due to TNF-α | 1 × 10 − 3 | day−1 | [ |
λ T 1 M 1 | Production rate of Th1 cells by M1 macrophages and IL-12 | 0.23 | day−1 | [ |
λ T I 2 | Production rate of Th1 cells by IL-2 | 1 | day−1 | [ |
λ T 2 | Production rate of Th2 cells | 0.8 | day−1 | [ |
λ I γ T 1 | Production rate of IFN-γ by Th1 cells | 3.731 × 10 − 3 | IU cell−1 day−1 | [ |
λ I 12 M 1 | Production rate of IL-12 by M1 macrophages | 0.13 | IU cell−1 day−1 | Est. from [ |
λ I 2 T 1 | Production rate of IL-2 by Th1 cells | 0.0672 | IU cell−1 day−1 | [ |
λ T α M 1 | Production rate of TNF-α by M1 macrophages | 13.91 | IU cell−1 day−1 | [ |
λ I 1 β M 1 | Production rate of IL-1β by M1 macrophages | 0.1022 | IU cell−1 day−1 | Est. from [ |
λ I 10 M 2 | Production rate of IL-10 by M2 macrophages | 0.02 | IU cell−1 day−1 | [ |
λ I 4 T 2 | Production rate of IL-4 by Th2 cells | 0.0775 | IU cell−1 day−1 | [ |
λ I 4 M 2 | Production rate of IL-4 by M2 macrophages | 0.3094 | IU cell−1 day−1 | [ |
λ I α M 2 | Production rate of IFN-alpha by M2 macrophages | 1 × 10 − 5 | IU cell−1 day−1 | [ |
λ T r e g | Production rate of Tregs from Th0 cells due to TGF-β | 0.001 | day−1 | Est. |
λ M I 10 | Production rate of M2 macrophages due to IL-10 | 1 × 10 − 3 | day−1 | Est. from [ |
λ I 1 β T 1 | Production rate of IL-1beta by Th1 cells | 0.1022 | IU cell−1 day−1 | Est. from [ |
λ T α T 1 | Production rate of TNF-alpha by Th1 cells | 13.91 | IU cell−1 day−1 | Est. from [ |
---|---|---|---|---|
λ I 10 T 2 | Production rate of IL-10 by Th2 cells | 0.02 | IU cell−1 day−1 | Est. from [ |
λ I 12 T 1 | Production rate of IL-12 by Th1 cells | 0.13 | IU cell−1 day−1 | Est. from [ |
d M 1 | Death rate of M1 macrophages | 0.02 | day−1 | [ |
d M 2 | Death rate of M2 macrophages | 0.008 | day−1 | [ |
d T 1 | Death rate of Th1 cells | 1.97 × 10 − 1 | day−1 | [ |
d T 2 | Death rate of Th2 cells | 1.97 × 10 − 1 | day−1 | [ |
d I γ | Degradation rate of IFN-γ | 2.16 | day−1 | [ |
d T α | Degradation rate of TNF-α | 55.45 | day−1 | [ |
d I 1 β | Degradation rate of IL-1β | 6.65 | day−1 | [ |
d I 4 | Degradation rate of IL-4 | 50 | day−1 | [ |
d I 10 | Degradation rate of Il-10 | 8.32 | day−1 | [ |
d I 12 | Degradation rate of IL-12 | 1.38 | day−1 | [ |
d I α | Degradation rate of IFN-α | 2.16 | day−1 | Est. from [ |
M 0 | Constant source of monocytes | 5 × 10 10 | cells L−1 | [ |
T 0 | Constant source of naïve T cells | 2 × 10 10 | cells L−1 | [ |
K T 1 | Th1 cell saturation | 1 × 10 10 | cells L−1 | [ |
K I γ | IFN-γ saturation | 2.6 × 10 6 | IU L−1 | [ |
K I 2 | IL-2 saturation | 8 × 10 6 | IU L−1 | [ |
K I 4 | IL-4 saturation | 2.6 × 10 6 | IU L−1 | [ |
K I 10 | IL-10 saturation | 3 × 10 4 | IU L−1 | Est. from [ |
K I 12 | IL-12 saturation | 1.95 × 10 8 | IU L−1 | [ |
K M 1 | M1 saturation | 5 × 10 10 | cells L−1 | [ |
K M 2 | M2 saturation | 1 × 10 11 | cells L−1 | [ |
K I 1 β | IL-1β saturation | 1.3 × 10 5 | IU L−1 | [ |
K I α | IFN-α saturation | 2.6 × 10 6 | IU L−1 | Est. from [ |
K T α | TNF-α saturation | 1.3 × 10 7 | IU L−1 | [ |
K T G F β | TGF-beta saturation | 2600 | IU L−1 | Est. from [ |
g T G F β | Concentration of TGF-β for half-maximal activation of Tregs | 7 × 10 4 | IU L−1 | Est. |
δ M 1 | Death rate of cancer cells by M1 macrophages | 1 × 10 − 9 | L cell−1 day−1 | Est. from [ |
---|---|---|---|---|
δ T 1 | Death rate of cancer cells by Th1 cells. | 1 × 10 − 9 | L cell−1 day−1 | Est. from [ |
T I 10 | IL-10 concentration that reduces CTL anti-tumor response by half | 3 × 10 4 | IU L−1 | Est. |
η A I 10 | Degradation rate of anti-IL-10 | 3.3 × 10 − 2 | day−1 | Est. from [ |
b 4 | Rate of loss of free IL-10 due to binding with anti-IL-10 | 15 | L mg−1 day−1 | Est. from [ |
b 5 | Rate of loss of free anti-IL-10 due to binding with IL-10 | 1.665 × 10 − 12 | L IU−1 day−1 | Est. from [ |