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 antigenpositive and target antigennegative 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 antigenpositive and antigennegative tumor cells present in the solid tumor, respectively. The other two cell populations include the CART cells (C) and the Enhanced Immune Response cells (E).
The term enhanced immune response refers to the phenomenon where CART cells trigger CD8 T cells, innate immune cells (such as macrophages, neutrophils, or natural killer (NK) cells), or even antigenpresenting cells (APCs) involved in antigen crosspresentation, allowing CAR Tcells to exert their therapeutic impact beyond the targeted antigenexpressing cells. This enhanced immune response enables CART 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 antigenpositive tumor population is modelled through two ratiodependent cellkill 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 antigennegative tumour population, T
_{r} follows the same ratiodependent cellkill term denoted by D
_{R} ^{[5]}.
In the equations (3) and (4), the parameters γ
_{C}and γ
_{E} represent the natural death rates of CAR Tcells and enhanced response cells, respectively. The parameter ω
_{C} represents the rate of exhaustion of CAR Tcells resulting from exposure to the antigenpositive 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 Tcells and enhanced response cells follows the MichaelisMenten dynamics
^{[2]}. The recruitment is represented by the terms D
_{C}^{2}/(k + D
_{C}^{2}) and D
_{E}^{2}/(k + D
_{E}^{2}), respectively. The recruitment of CART 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 Tcell injection is modelled as a onetime increase in the amount of CAR Tcells on the days of injection. In equation (4), I
_{c} represents the amount of CAR Tcell 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 antigenpositive 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 
r_{1} 
Antigen postive cells proliferation rate 
0.151 
day^{1} 
r_{2} 
Antigen negative cells proliferation rate 
0.180 
day^{1} 
K_{1} 
Carrying capacities of tumor 
5.9 x 10^{3} 
mm^{3} 
l 
exponent of tumor lysis by either C or E

1.291 
unitless 
μ_{C} 
max recruitment rate of CARTs by antigenpositive tumor lysis 
0.6 
day^{1} 
d_{C} 
maximum killing rate of antigenpositive cells via CART cells 
0.27 
day^{1} 
s 
steepness of fractional antigennegative tumor kill by E cells 
3.05 x 10^{1} 
unitless 
γ_{C} 
CART's death rate 
2.93 x 10^{2} 
day^{1} 
k 
steepness of CART/Enhanced cell recruitment 
2.019 x 10^{7} 
day^{2} 
ω_{C} 
CART exhaustion due to antigenpositive cells 
3 x 10^{5}  mm^{3} day^{1} 
K_{2} 
immune cell carrying capacity 
1.65 x 10^{3} 
mm^{3} 
d_{E} 
maximum killing rate of antigenpositive/antigennegative cells by enhanced response cells

0.27 
day^{1} 
μ_{E} 
max recruitment rate of E by antigenpositive tumor lysis 
0.82 
day^{1} 
e 
base recruitment rate of E cells 
5 x 10^{2} 
mm^{3} day^{1} 
γ_{E} 
E cell death rate 
2 x 10^{2} 
day^{1} 
ω_{E} 
E cell exhaustion due to antigenpositive/antigennegative
cells 
3.42 x 10^{6} 
mm^{3} day^{1} 
u 
Concentrion of drug 
25 
μM 
a_{1} 
Fraction of antigen postive cells killed by drug 
0.3 
unitless 
a_{2} 
Fraction of antigen negative cells killed by drug 
0.3 
unitless 
a_{3} 
Fraction of C cells killed by drug 
<a_{1,2} 
unitless 
a_{4} 
Fraction of E cells killed by drug 
<a_{1,2} 
unitless 
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
_{s}and 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:

\[\frac{\mathrm{dT_{s}} }{\mathrm{d} t}=r_{1}T_{s}(1\frac{T_{s}+T_{r}}{K_{1}}) D_{E}T_{s}D_{C}T_{s}  a_{1}(1e^{u})T_{s}\]

\[\frac{\mathrm{dT_{r}} }{\mathrm{d} t}=r_{2}T_{r}(1\frac{T_{s}+T_{r}}{K_{1}}) D_{R}T_{r}a_{2}(1e^{u})T_{r}\]

\[\frac{\mathrm{dC} }{\mathrm{d} t}=I_{C}\gamma_{C}C\mu _{C}log(\frac{E+C}{K_{2}})\frac{D_{C}^{2}}{k+D_{C}^{2}}C  \omega_{C}CT{s}a_{3}(1e^{u})C\]

\[\frac{\mathrm{dE} }{\mathrm{d} t}=e\gamma_{E}E\mu _{E}log(\frac{E+C}{K_{2}})\frac{D_{E}^{2}}{k+D_{E}^{2}}E  \omega_{E}E(T{s}+T_{r})a_{4}(1e^{u})E\]
Our model centers on the tumor site subsequent to the allosteric displacement of Honokiol by AM6538 from the 'seeker' Tcell, which occurs once these seeker Tcells 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 CART cells and the medication, the tumor mass exhibited continuous growth.
Figure 5. Tumor progression without treatment.

CART cell administration:
This plot illustrates the impact of CART cells and the resulting enhanced immune response on the tumor population. CART cells demonstrate efficacy against antigenpositive tumor cells, but the antigennegative cell population persists for a longer duration.
Figure 6: Tumor progression with CART 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.

CART cells along with honokiol administration:
The synergistic effect of CART cells and honokiol results in a significant reduction in tumor mass within a shorter timeframe.
Figure 8: Tumor progression with CART cells along with honokiol.
Insights:
From our model, we derived the following insights:
Independent CART 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 CART cell concentration by regulating the honokiol concentration, thereby enhancing the costeffectiveness 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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Cheng, X., Wang, F., Qiao, Y., Chen, T., Fan, L., Shen, X., Yu, D., Huang, Y. and Wei, M., 2023. Honokiol inhibits interleukininduced angiogenesis in the NSCLC microenvironment through the NFκB signaling pathway. ChemicoBiological Interactions, 370, p.110295. Link