Modeling tumour heterogeneity of PD-L1 expression in tumour progression and adaptive therapy

Abstract

Although PD-1/PD-L1 inhibitors show potent and durable anti-tumour effects in some refractory tumours, the response rate in overall patients is unsatisfactory, which in part due to the inherent heterogeneity of PD-L1. In order to establish an approach for predicting and estimating the dynamic alternation of PD-L1 heterogeneity during cancer progression and treatment, this study establishes a comprehensive modelling and computational framework based on a mathematical model of cancer cell evolution in the tumour-immune microenvironment, and in combination with epigenetic data and overall survival data of clinical patients from The Cancer Genome Atlas. Through PD-L1 heterogeneous virtual patients obtained by the computational framework, we explore the adaptive therapy of administering anti-PD-L1 according to the dynamic of PD-L1 state among cancer cells. Our results show that in contrast to the continuous maximum tolerated dose treatment, adaptive therapy is more effective for PD-L1 positive patients, in that it prolongs the survival of patients by administration of drugs at lower dosage.

Appendices

Appendix A: Parameter estimation

Parameters in the equation of C

In modelling tumour-immune surveillance (Mahasa et al. 2016), the per capita growth rate of tumour cells was estimated to be $0.5822\mathrm{day}^{-1}$. We accordingly take the cancer cell basic production rate to be $\bar{\beta }_{C}=0.5822/24\mathrm{h}^{-1}=0.0243\mathrm{h}^{-1}$. In modelling anti-tumour T cells response (Liao et al. 2014), the death rate of tumour cells was estimated to be $0.173\mathrm{day}^{-1}$. We accordingly take the apoptosis rate of proliferating cancer cells to be $\mu _{C}=0.173/24\mathrm{h}^{-1}=0.0072\mathrm{h}^{-1}$. We further assume that the apoptosis rate of non-proliferating cancer cells is much lower than that of the proliferating ones, and take $\bar{\kappa }_{C}=\mu _{C}/10 = 7.2\times 10^{-4}\mathrm{h}^{-1}$.

In studying the impact of intra-tumour heterogeneity on anti-tumour CD$8^+$ T cell immune response (Leschiera et al. 2022), the mean cell cycle time of tumour cells was estimated to be $24\mathrm{h}$, where the duration interval was $17-48\mathrm{h}$ (Tubiana 1989; Gordon and Lane 1980). Hence we take $\tau _{C}=24\mathrm{h}$. In modelling tumour-immune surveillance (Mahasa et al. 2016), the reciprocal carrying capacity of the tumour cells was estimated to be $2.33\times 10^{-8}{\mathrm{cell}}^{-1}$. We take tumor carrying capacity as ${\hat{C}}_{*}=1/(2.33\times 10^{-8}\mathrm{cell}^{-1})=4.3\times 10^7{\mathrm{cell}}$.

Parameters in the equations of $T_0$ and T

In metastatic melanoma microenvironment (Tsur et al. 2019), activation rate of naive antigen-specific CD$8^+$ T cells was estimated to be $0.8318\mathrm{day}^{-1}$. In the tumour microenvironment (Dritschel et al. 2018), the death rate of helper T cells was estimated to be $0.1\mathrm{day}^{-1}$. Hence we take the resting T cell basic proliferation rate as $\beta _{T}=0.8318/24\mathrm{h}^{-1}=0.0347\mathrm{h}^{-1}$, and the apoptosis rate of proliferating T cell as $\mu _{0}=0.1/24\mathrm{h}^{-1}=0.0042\mathrm{h}^{-1}$. By Kinjyo et al. (2015), the T cell cycle time is $14.3\pm 4.4\mathrm{h}$. We accordingly take $\tau _{0}=14.3\mathrm{h}$.

In the estimation of T cell kinetics in humans (Macallan et al. 2019), the proliferation rate of memory T cell ranges $0.006-0.16{\mathrm{day}}^{-1}$. Here we take the coefficient of resting T cell differentiation rate as $\bar{\kappa }_T = 0.104/24\mathrm{h}^{-1}=0.0043 \mathrm{h}^{-1}$. In modelling tumour-immune surveillance (Mahasa et al. 2016), the per capita death rate of CTLs was estimated to be $0.02\mathrm{day}^{-1}$; the binding rate of CTLs to tumour cells was $1.3\times 10^{-7}{\mathrm{day}}^{-1}$. We accordingly take the apoptosis rate of effector T cell as $\mu _{T}=0.02/24{\mathrm{h}}^{-1}=8.3\times 10^{-4}{\mathrm{h}}^{-1}$; the killing rate of effector T cells as $\eta _{0}=1.3\times 10^{-7}/24\mathrm{h}^{-1}$ $=5.4\times 10^{-9}$ ${\mathrm{h}}^{-1}$.

Appendix B: Cell-based stochastic simulation

We have three epigenetic states, $x_0$, $x_1$ and $x_2$, in the differential-integral equations model (2). It is very expensive to solve the system numerically, such as using the Euler method, due to the high dimensional integration. Therefore, we apply the method of cell-based stochastic simulation proposed in Lei (2020a). By this approach, we model the growth process of a multiple-cell system with a collection of epigenetic states. The cell-based stochastic simulation tracks the behaviours of each cell according to its own epigenetic states. The sketch of the numerical scheme is given as follows.

$\mathbf {Initialize}$ the time $t = 0$, the cancer cell number $Q_C$ (cancer cells pool: $\Sigma _C=\left\{ C_i(\mathbf {x}_i,A_i) \right\} _{i=1}^{Q_C}$), the $T_0$ cell number $Q_{T_0}$ (resting T cells pool: $\Sigma _{T_0}=\left\{ T_{0i}(\mathbf {x}_i,A_i) \right\} _{i=1}^{Q_{T_0}})$, the T cell number $Q_T$ (T cell pool: $\Sigma _T=\left\{ T_i(\mathbf {x}_i,A_i) \right\} _{i=1}^{Q_{T}}$). At the initial state, all cells are at the resting phase, and the corresponding age at the proliferating phase is $A_i=0$.

$\mathbf {for}$ t from 0 to T with step $\bigtriangleup t$ $\mathbf {do}$

$\mathbf {for}$ cancer cells in $\Sigma _C$ $\mathbf {do}$

• Calculate the proliferation rate $\beta _C$, the apoptosis rate of proliferating cells $\mu _C$, and the death rate $\kappa _C$, the killing rate of cancer cells by effector T cells $\eta$.

• Determine the cell fate during the time interval $(t,t+\Delta t)$:

• When the cell is at the resting phase, undergo death with a probability $\kappa _C\Delta t$, be killed by effector T cell with a probability $\eta \Delta t$ or enter the proliferation phase with a probability $\beta _C\Delta t$. If the cell enters the proliferation phase, set the age $A_i=0$.

• When the cell is at the proliferating phase, if the age $A_i<\tau$, the cell is either removed (through apoptosis) with a probability $\mu _C\Delta t$ or remains unchanged and $A_i=A_i+\Delta t$; if the age $A_i \ge \tau$, the cell undergoes mitosis and divides into two cells. When mitosis occurs, the epigenetic state of each daughter cell is determined according to the inheritance probability functions $p_0\left( {x_{0}}, {y}\right)$ and $p_1\left( {x_{1}}, {y}\right)$.

$\mathbf {end ~ for }$

$\mathbf {for}$ resting T cells in $\Sigma _{T_0}$ $\mathbf {do}$

• Calculate the proliferation rate $\beta _{T}$, the apoptosis rate $\mu _{0}$, and the differentiation rate $\kappa _{T}$.

• Determine the cell fate during the time interval $(t, t+\Delta t)$:

• When the cell is at the resting phase, undergo differentiation with a probability $\kappa _T\Delta t$ or enter the proliferation phase with a probability $\beta _T\Delta t$. If the cell enters the proliferation phase, set the age $A_i=0$.

• When the cell is at the proliferating phase, if the age $A_i<\tau$, the cell is either removed (through apoptosis) with a probability $\mu _0\Delta t$ or remains unchanged and $A_i=A_i+\Delta t$; if the age $A_i \ge \tau$, the cell undergoes mitosis and divides into two cells. When mitosis occurs, the epigenetic state of each daughter cell is determined according to the inheritance probability function $p_2\left( \mathbf {x}, \mathbf {y}\right)$.

$\mathbf {end ~ for }$

$\mathbf {for}$ effector T cells in $\Sigma _{T}$ $\mathbf {do}$

• Calculate the apoptosis rate $\mu _T$.

• Determine the cell fate during the time interval $(t, t+\Delta t)$:

• The cell is removed (through apoptosis) with a probability $\mu _T\Delta t$.

$\mathbf {end ~ for }$

$\mathbf {Update}$ the system with the caner cell number, the resting T cell number, the effector T cell number, the epigenetic states of all surviving cells, and the ages of the proliferating phase cells, and set $t=t+\Delta t$.

$\mathbf {end ~ for }$