Mathematical modeling plays a pivotal role in advancing our understanding of tumors and their treatment. By leveraging mathematical principles and computational tools, these models uncover insights into tumor dynamics, growth patterns, and responses to treatment and also enable the optimization of personalized treatment strategies.
Mathematics can be used to guide laboratory research by anticipating the properties and behavior of components and systems. So, in order to forecast how certain components of our design would behave, we utilized a tumor growth mathematical framework. Our model takes inspiration from an in vivo experiment
[4] that includes a mixture of target antigen-positive and target antigen-negative tumor cells.
Our model involves the progression of four separate cellular groups throughout the treatment period. In particular, we analyze the entire tumor population as comprising two subgroups labeled as T
s and T
r, representing the antigen-positive and antigen-negative tumor cells present in the solid tumor, respectively. The other two cell populations include the CAR-T cells (C) and the Enhanced Immune Response cells (E).
The term enhanced immune response refers to the phenomenon where CAR-T cells trigger CD8 T cells, innate immune cells (such as macrophages, neutrophils, or natural killer (NK) cells), or even antigen-presenting cells (APCs) involved in antigen cross-presentation, allowing CAR T-cells to exert their therapeutic impact beyond the targeted antigen-expressing cells. This enhanced immune response enables CAR-T cells not only target cancer cells expressing a specific antigen but also stimulate cytotoxic activity against nearby cancer cells that lack the targeted antigen expression.
The equations:
-
\[ \frac{\mathrm{dT_{s}} }{\mathrm{d} t}=r_{1}T_{s}(1-\frac{T_{s}+T_{r}}{K_{1}})- D_{E}T_{s}-D_{C}T_{s}\]
-
\[ \frac{\mathrm{dT_{r}} }{\mathrm{d} t}=r_{2}T_{r}(1-\frac{T_{s}+T_{r}}{K_{1}})- D_{R}T_{r}\]
-
\[ \frac{\mathrm{dC} }{\mathrm{d} t}=I_{C}-\gamma_{C}C-\mu _{C}log(\frac{E+C}{K_{2}})\frac{D_{C}^{2}}{k+D_{C}^{2}}C - \omega_{C}CT{s}\]
-
\[ \frac{\mathrm{dE} }{\mathrm{d} t}=e-\gamma_{E}E-\mu _{E}log(\frac{E+C}{K_{2}})\frac{D_{E}^{2}}{k+D_{E}^{2}}E - \omega_{E}E(T{s}+T_{r})\]
where
\[D_{E}=d_{E}\frac{(E/T_{s})^l}{s+(E/T_{s})^l}\]
\[D_{C}=d_{C}\frac{(C/T_{s})^l}{s+(C/T_{s})^l}\]
\[D_{R}=d_{E}\frac{(E/T_{r})^l}{s+(E/T_{r})^l}\]
The first two equations (1) and (2) exhibit the logistic growth of tumor populations T
s and T
r, each with its own growth rate. The killing of the antigen-positive tumor population is modelled through two ratio-dependent cell-kill terms denoted by D
E and D
E. Those terms were originally proposed to capture the interaction between tumor cells and immunological components, including CD8+ T cells and natural killer cells. The effect on antigen-negative tumour population, T
r follows the same ratio-dependent cell-kill term denoted by D
R [5].
In the equations (3) and (4), the parameters γ
Cand γ
E represent the natural death rates of CAR T-cells and enhanced response cells, respectively. The parameter ω
C represents the rate of exhaustion of CAR T-cells resulting from exposure to the antigen-positive tumor population, while ω
E represents the rate of exhaustion of enhanced response cells due to their interaction with both tumor cell populations. We assume that the recruitment of CAR T-cells and enhanced response cells follows the Michaelis-Menten dynamics
[2]. The recruitment is represented by the terms D
C2/(k + D
C2) and D
E2/(k + D
E2), respectively. The recruitment of CAR-T cells and enhanced response cells are scaled by the immune cell competition term log((E+C)/K
2), which limits the proliferation of immune cell components beyond the carrying capacity K
2, with their maximum recruitment being denoted by μ
C and μ
E, respectively
[3].
CAR T-cell injection is modelled as a one-time increase in the amount of CAR T-cells on the days of injection. In equation (4), I
c represents the amount of CAR T-cell reaching the tumor site.
In our model, we assume that 10
6 cells can be approximated by 1mm
3 of cells. Initial conditions are set as T
s(0) = 50p, T
r(0) = 50(1−p), C(0) = 0 and E(0) = 0.05 where p is the percentage of the antigen-positive tumor cell population. All units are given in mm
3.
Parameters
The parameters used in the above set of equations along with their description and numerical values are tabulated below. In the absence of literature on the parameters, we have made educated guesses about some parameter values.
Paramter |
Description |
Value |
Units |
r1 |
Antigen postive cells proliferation rate |
0.151 |
day-1 |
r2 |
Antigen negative cells proliferation rate |
0.180 |
day-1 |
K1 |
Carrying capacities of tumor |
5.9 x 103 |
mm3 |
l |
exponent of tumor lysis by either C or E
|
1.291 |
unit-less |
μC |
max recruitment rate of CARTs by antigen-positive tumor lysis |
0.6 |
day-1 |
dC |
maximum killing rate of antigenpositive cells via CAR-T cells |
0.27 |
day-1 |
s |
steepness of fractional antigen-negative tumor kill by E cells |
3.05 x 10-1 |
unit-less |
γC |
CAR-T's death rate |
2.93 x 10-2 |
day-1 |
k |
steepness of CAR-T/Enhanced cell recruitment |
2.019 x 10-7 |
day-2 |
ωC |
CAR-T exhaustion due to antigen-positive cells |
3 x 10-5 | mm-3 day-1 |
K2 |
immune cell carrying capacity |
1.65 x 103 |
mm3 |
dE |
maximum killing rate of antigen-positive/antigen-negative cells by enhanced response cells
|
0.27 |
day-1 |
μE |
max recruitment rate of E by antigen-positive tumor lysis |
0.82 |
day-1 |
e |
base recruitment rate of E cells |
5 x 10-2 |
mm3 day-1 |
γE |
E cell death rate |
2 x 10-2 |
day-1 |
ωE |
E cell exhaustion due to antigen-positive/antigen-negative
cells |
3.42 x 10-6 |
mm3 day-1 |
u |
Concentrion of drug |
25 |
μM |
a1 |
Fraction of antigen postive cells killed by drug |
0.3 |
unit-less |
a2 |
Fraction of antigen negative cells killed by drug |
0.3 |
unit-less |
a3 |
Fraction of C cells killed by drug |
<a1,2 |
unit-less |
a4 |
Fraction of E cells killed by drug |
<a1,2 |
unit-less |
Effect Of Drug:
We further improve our model by including the drug Honokiol and obtain its effect on the progression of the tumor populations of T
sand Tr. The amount of drug present at the tumour location at time t is designated as u(t). With the response curve in all cases being represented by an exponential F(u)= a(1- e
-ku)
[5] and F(u) being the fraction of cells killed for a given dose of drug, u, at the tumour site. Since the details of the pharmacokinetics are unknown, we let k = 1 in these preliminary studies. We assume that the drug kills all types of cells but the kill rate varies for each type of cell.
The improved Equations are:
-
\[\frac{\mathrm{dT_{s}} }{\mathrm{d} t}=r_{1}T_{s}(1-\frac{T_{s}+T_{r}}{K_{1}})- D_{E}T_{s}-D_{C}T_{s} - a_{1}(1-e^{-u})T_{s}\]
-
\[\frac{\mathrm{dT_{r}} }{\mathrm{d} t}=r_{2}T_{r}(1-\frac{T_{s}+T_{r}}{K_{1}})- D_{R}T_{r}-a_{2}(1-e^{-u})T_{r}\]
-
\[\frac{\mathrm{dC} }{\mathrm{d} t}=I_{C}-\gamma_{C}C-\mu _{C}log(\frac{E+C}{K_{2}})\frac{D_{C}^{2}}{k+D_{C}^{2}}C - \omega_{C}CT{s}-a_{3}(1-e^{-u})C\]
-
\[\frac{\mathrm{dE} }{\mathrm{d} t}=e-\gamma_{E}E-\mu _{E}log(\frac{E+C}{K_{2}})\frac{D_{E}^{2}}{k+D_{E}^{2}}E - \omega_{E}E(T{s}+T_{r})-a_{4}(1-e^{-u})E\]
Our model centers on the tumor site subsequent to the allosteric displacement of Honokiol by AM-6538 from the 'seeker' T-cell, which occurs once these seeker T-cells have successfully reached the tumor site. CAR T cells can then be administered intravenously.
Analysis:
- Tumor growth Model:
We modeled the progression of tumor growth in a scenario where no treatment was applied. In this context, without the presence of CAR-T cells and the medication, the tumor mass exhibited continuous growth.
Figure 5. Tumor progression without treatment.
-
CAR-T cell administration:
This plot illustrates the impact of CAR-T cells and the resulting enhanced immune response on the tumor population. CAR-T cells demonstrate efficacy against antigen-positive tumor cells, but the antigen-negative cell population persists for a longer duration.
Figure 6: Tumor progression with CAR-T cell administration.
-
Honokiol administration:
Honokiol, when used in isolation, does not exhibit any significant positive impact on the tumor population. While a higher concentration of honokiol might lead to a reduction in tumor mass, it's important to note that the cytotoxic effects of honokiol impose an upper limit on the concentration that can be safely administered.
Figure 7: Tumor progression with Honokiol administration.
-
CAR-T cells along with honokiol administration:
The synergistic effect of CAR-T cells and honokiol results in a significant reduction in tumor mass within a shorter timeframe.
Figure 8: Tumor progression with CAR-T cells along with honokiol.
Insights:
From our model, we derived the following insights:
Independent CAR-T cells and honokiol treatment is not enough to control the tumor volume. The combined effect of CART and honokiol, however, leads to limiting the tumor mass and also significantly decreasing tumor volume in lesser time, showing more effectiveness.
These findings can be analyzed to potentially reduce the required CAR-T cell concentration by regulating the honokiol concentration, thereby enhancing the cost-effectiveness of the treatment.
References:
-
Kara, E., Jackson, T.L., Jones, C., McGee II, R.L. and Sison, R., 2023. Mathematical Modeling Insights into Improving CAR T cell Therapy for Solid Tumors: Antigen Heterogeneity and Bystander Effects. arXiv preprint arXiv:2307.05606. Link
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Kuznetsov, V.A., Makalkin, I.A., Taylor, M.A. and Perelson, A.S., 1994. Nonlinear dynamics of immunogenic tumors: parameter estimation and global bifurcation analysis. Bulletin of mathematical biology, 56(2), pp.295-321.Link
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Kimmel, G.J., Locke, F.L. and Altrock, P.M., 2019. Response to CAR T cell therapy can be explained by ecological cell dynamics and stochastic extinction events. bioRxiv, p.717074. Link
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Klampatsa, A., Leibowitz, M.S., Sun, J., Liousia, M., Arguiri, E. and Albelda, S.M., 2020. Analysis and augmentation of the immunologic bystander effects of CAR T cell therapy in a syngeneic mouse cancer model. Molecular Therapy-Oncolytics, 18, pp.360-371. Link
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De Pillis, L.G. and Radunskaya, A., 2001. A mathematical tumor model with immune resistance and drug therapy: an optimal control approach. Computational and Mathematical Methods in Medicine, 3(2), pp.79-100. Link
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Qu, J., Mei, Q., Chen, L. and Zhou, J., 2021. Chimeric antigen receptor (CAR)-T-cell therapy in non-small-cell lung cancer (NSCLC): current status and future perspectives. Cancer immunology, immunotherapy, 70, pp.619-631. Link
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Cheng, X., Wang, F., Qiao, Y., Chen, T., Fan, L., Shen, X., Yu, D., Huang, Y. and Wei, M., 2023. Honokiol inhibits interleukin-induced angiogenesis in the NSCLC microenvironment through the NF-κB signaling pathway. Chemico-Biological Interactions, 370, p.110295. Link