Role of senescent tumor cells in building a cytokine shield in the tumor microenvironment: mathematical modeling

Abstract

Cellular senescence can induce dual effects (promotion or inhibition) on cancer progression. While immune cells naturally respond and migrate toward various chemotactic sources from the tumor mass, various factors including senescent tumor cells (STCs) in the tumor microenvironment may affect this chemotactic movement. In this work, we investigate the mutual interactions between the tumor cells and the immune cells that either inhibit or facilitate tumor growth by developing a mathematical model that consists of taxis–reaction–diffusion equations and receptor kinetics for the key players in the interaction network. We apply a mathematical model to a transwell Boyden chamber invasion assay used in the experiments to illustrate that STCs can play a pivotal role in negating immune attack through tight regulation of intra- and extra-cellular signaling molecules. In particular, we show that senescent tumor cells in cell cycle arrest can block intratumoral infiltration of CD8+ T cells by secreting a high level of CXCL12, which leads to significant reduction its receptors, CXCR4, on T cells, and thus impaired chemotaxis. The predictions of nonlinear responses to CXCL12 were in good agreement with experimental data. We tested several hypotheses on immune-tumor interactions under various biochemical conditions in the tumor microenvironment and developed new concepts for anti-tumor strategies targeting senescence induced immune impairment.

Appendices

Appendix

Parameter estimation

1.1 Parameter values

$D_T$ (Diffusion coefficient of T cell): From the measured value of the random motility of T cells in a nest of the tumor in Salmon et al. (2012), Li et al. calculated the motility coefficient of 5 $\upmu$m$^2/$min (Li et al. 2019). We take $D_T = 5\,\upmu$m$^2/$min $= 8.33\times 10^{-10}$ cm$^2/$s.

$D_n, D_{ns}$ (Diffusion coefficient of tumor cells and STCs): Measured random motility of typical tumor cells was used in various experimental and theoretical studies including (Kim et al. 2018b, 2019b). We take this value and assume that both tumor cells and STCs have the same diffusion coefficient, leading to $D_n=D_{ns} = 1.0\times 10^{-11}$ cm$^2/$s.

$D_L$ (Diffusion coefficient of CXCL12): Veldkamp et al. (2009) obtained the measurement of the self-diffusion coefficient of CXCL12 in an experiment involving a water-suppressed longitudinal coding-decoding (water-sLED) system. The diffusion constant of CXCL12 in water was measured to be $1.74\times 10^{-6}$ cm$^2/$s in Szatmary et al. (2014) and Veldkamp et al. (2009). We take $D_L = 1.74\times 10^{-6}$ cm$^2/$s.

$D_A$ (Diffusion coefficient of CXCL12 antibody): In an experimental study of the inhibitory effect of tannic acid on CXCL12/CXCR4 axis (Chen et al. 2003), the diffusion coefficient of tannin, the CXCL12 antibody, has been reported to be in a range of (3.8–18.4) $\times 10^{-5}$ cm$^2/$s (Mhessn et al. 2011). On the other hand, the diffusion coefficient of tannin was measured to be [6.89 $\times 10^{-11}$–1.66 $\times 10^{-4}$) cm$^2/$s in various studies Tasheva et al. (2019)] [also see refs in Tasheva et al. (2019)]. We take $D_A = 3.8\times 10^{-7}$ cm$^2/$s.

$\mu _{L}$ (Decay rate of CXCL12): The half-life of CXCL12 was reported to be around 26.5 ± 8.7 min (Takekoshi et al. 2012; Ziarek et al. 2013). In addition, the half-life of various types of CXCL were reported to be 2–15.5 h (Orlikowsky et al. 2004; Redl et al. 1991b, a). By taking 26.5 min for the half-life, we get $\mu _L = ln(2)/(26.5/60)$ h$^{-1} = 1.5694\times 10^{-1}$ h$^{-1} = 4.36 \times 10^{-4}$ s$^{-1}$.

$\mu _{D}$ (Decay rate of binding inhibitors): The half-life of AMD3100, inhibitor of CXCL12-CXCR4 binding, was reported to be 3.6$\;h$ (Hendrix et al. 2000). On the other hand, the half-life was measured to be 1$\;h$ in mice after injecting AMD3100 (6 mg/kg, SC). By taking the half-life of 3.6 h, we get $\mu _{D} = ln(2)/(3.6)$ h$^{-1} = 5.35 \times 10^{-5}$ s$^{-1}$.

$\mu _{A}$ (Decay rate of CXCL12 inhibitors): The half-life of CXCL12 inhibitors such as tannic acid was estimated to be 0.96 h (Chen et al. 2003), leading to the decay rate $\mu _{A} = ln(2)/(0.96)$ h$^{-1} = 2.01\times 10^{-4}$ s$^{-1}$.

$k_{-1}$ (Dissociation rate): Larson et al. (2015) calculated the dissociation rate of CXCL12 to be about (0.031–0.045) s$^{-1}$ by adding more than 100 times AMD3100 to the previous mixture. We take $k_{-1} = 0.031$ s$^{-1}$.

$k_{1}$ (Association rate): Larson et al. (2015) calculated the $K_D$ value of the labeled ligand from the HTRF ratio generated after reaching binding equilibrium in response to different concentrations of ligands. In the saturated binding empirical system, the binding activities increase as the concentration of labeled CXCL12 increases. The level of ligands binding to half of the receptor site at equilibrium or $K_D$ was 29.3 nM. We take $K_D = 29.3$ nM, leading to the association rate $k_1 = \frac{k_{-1}}{K_D} = 0.011$ nM$^{-1}$s$^{-1}$.

$r, r_s$ (Proliferation rates of tumor cells): We take the proliferation rate of non-senescent tumor cells, $r = 3.33\times 10^{-5}$ s$^{-1}$, based on experimental observations in (Choi et al. 2021). We assume that senescent tumor cells have a lower growth rate than non-senescent cells. Thus, we take $r_s = 2.22\times 10^{-5}$ s$^{-1}$.

$\lambda _L,\lambda _s$ (Production rate of CXCL12 from two types of tumor cells): Choi et al. (2021) observed that the cytokine shield was formed when the value of CXCL12 was high (> 20 times), and STCs secrete more CXCL12 than non-senescent tumor cells ($\lambda _L < \lambda _s$). Thus, we set the CXCL12 production rates, $\lambda _L=2.78 \times 10^{-1}$ nM cm$^3$ g$^{-1}$ s$^{-1}$ for non-senescent tumor cells and $\lambda _s = 30 \lambda _L = 8.33$ nM cm$^3$ g$^{-1}$ s$^{-1}$ for STCs.

$K_\delta , K_2$ (Hill-type coefficients in CXCL12-induced switching): Choi et al. (2021) showed that the overexpressed CXCL12 (> 20-fold) can downregulate the CXCR4 activities and inhibit the chemotaxis of T cells, forming the cytokine shield. In this work, this suppression is controlled by the inhibition term $\delta _L(L) = \bigg ( \delta _{0} + \delta _{1} \frac{L^m}{K_\delta ^m+ L^m} \bigg )$ in the chemotaxis term of the form $\frac{\partial T}{\partial t} = -\nabla \cdot \bigg ( \chi _L([\overline{L \cdot R}]) T \frac{\nabla L}{\delta _L+ \sigma _L |\nabla L| } \bigg )$ in Eq. (6) and the degradation of CXCR4 in response to the high CXCL12 level is regulated in the L-dependent degradation term $k_{2}(L) = k_{20} \frac{L^{n_2}}{K_2^{n_2}+ L^{n_2}}$ in Eq. (11). Based on these experimental results, we assume that the suppression of the chemotactic movement and degradation of CXCR4 of T cells are activated when the CXCL12 level exceeds 7 times the reference value. Thus, we take $K_\delta = K_2 = 700$ nM.

1.2 Reference concentrations

T cells ($T^*$): Wong et al. reported the 95$\%$ range for the absolute counts of CD8+ T cells in several regional groups: (224–1014) cells/$\upmu$l (Hong Kong), (201–931) cells/$\upmu$l (Beijing), (243–1206) cells/$\upmu$l (Asian), (170–880) cells/$\upmu$l (European), and (145–884) cells/$\upmu$l (African) (Wong et al. 2013). We take $T^* = 1000$ cells/$\upmu$l $= 10^6$ cells/cm$^3$.

Tumor cells ($n^*, n_s^*$): We assume that tumor cells are uniformly distributed in the tumor. We also take same reference values for two types of tumor cells (tumor cells and STCs), leading to $n^* = n_s^* = 10^9~$cells/cm$^3$ as in ODonoghue et al. (1995), Friedman et al. (2006) and Kim et al. (2018b, 2019b).

CXCL12 ($L^*$) and CXCL12 antibody ($A^*$): Larson et al. (2015) performed an associative binding experiment using a range of the CXCL12 concentration, 0–100 nM. The steady state of the complex variables in the associated experiment was positively correlated with the CXCL12 input. Chen et al. (2003) reported that the inhibitory effect of CXCL12 antibody (tannic acid) on CXCL12-induced cell migration depends on dose, with an $IC_{50} = 7.5~\upmu$g/ml. In this experiment of the CXCL12/CXCR4 axis, tannic acid (10 $\upmu$g/ml) was shown to block cell migration in the presence of CXCL12 (100 ng/ml) in a chemotaxis chamber assay. Aqueous and organic extracts of Lianquiao were able to inhibit the CXCL12-induced cell migration (Chen et al. 2003). For example, organic extracts of Lianquiao can fully inhibit the chemotactic movement of CXCL12-induced infiltration at a concentration 200 $\upmu$g/ml. In a study of CXCL12-induced T cell infiltration (Choi et al. 2021), CXCL12 neutralizing antibody of the concentration of 5 $\upmu$g/ml was applied to the medium in the lower chamber of boyden chamber system for 24 h. Median serum levels of CXCL12 in control and various cancer patients were 0.865 ng/ml (control), 1.277 ng/ml (esophageal cancer), 1.118 ng/ml (adenocarcinoma of esophagus), and 1.501 ng/ml (squamous cell cancer of esophagus) (Lukaszewicz-Zajac and Mroczko 2016). In a study of CXCL12-CXCR4 axis, Song et al. (2021) reported that CXCL12 of concentration 10 nM can enhance the invasiveness of HCA-1 cells (> fivefold) relative control (CXCL12-) but its inhibitor BPRCX807 can negate this aggressive migration by 25% and 80% with inhibitor concentrations of 100, 1000 nM, respectively. We take $A^* = 7.5 \times 10^{-6}$ g/cm$^3$ and $L^* = 100$ nM.

CXCR4 ($[R]^*$), CXCL12-CXCR4 complex ($[\overline{L \cdot R}]^*$): Median serum levels of CXCR4 in control and various cancer patients were 0.932 ng/ml (control), 0.443 ng/ml (esophageal cancer), 0.403 ng/ml (adenocarcinoma of esophagus), and 0.534 ng/ml (squamous cell cancer of esophagus) (Lukaszewicz-Zajac and Mroczko 2016), while the $IC_{50}$ of CXCR4 antagonist has been reported to be in a range of (2.1–23.8) nM (Hanes et al. 2015; Hoffmann et al. 2012; Szpakowska et al. 2018). Considering the average values of CXCL12 concentration relative to the median CXCR4 concentration in the control group (Lukaszewicz-Zajac and Mroczko 2016), we take $[R]^* = 100$ nM. We take the same reference value for the CXCL12-CXCR4 complex, $[\overline{L \cdot R}]^* = 100$ nM.

The computational algorithm in the multi-scale model

• Step 0. Initialization.

  • Step 0.1. Fix a uniform grid for $\varOmega$ and initialize concentrations of CXCL12 (L) and densities of non-senescent tumor cells (n) and STCs ($n_s$) at each lattice point.

  • Step 0.2. Set the locations of the blood vessels outside the tumor based on the spatial distribution of tumor cell density ($n,n_s$). Initialize cell-based component by randomly placing T cells outside the tumor.

  • Step 0.3. Set the initial condition of the receptor ($[R]_i(t)$) and receptor complex $[\overline{L \cdot R}]_i(t)$ at each T cell i.

• Step 1. Generate discrete T cells at the blood vessels in a random time and random location of blood vessels and set initial conditions for the receptor and receptor complex at the T cell i.

• Step 2. Update the CXCL12 concentration at each T cell by interpolation from grid point to the cell site (see Dallon and Othmer 1997; Kim et al. 2011, 2018a; Kim and Othmer 2015). Solve the intracellular receptor binding equations (13)–(14) to update the receptor complex value at each T cell i.

• Step 3. Use the level of the receptor complex ($[\overline{L \cdot R}]_i(t)$) in order to determine movement of T cell i based on the switching behavior $\chi _0 \bigg ( \frac{[\overline{L \cdot R}]^{n_1}}{K_\chi ^{n_1}+ [\overline{L \cdot R}]^{n_1}} \bigg )$ in Eq. (4) and gradient of CXCL12, $\bigg ( \dfrac{ \nabla L}{\delta _L(L)+\sigma _L |\nabla L| } \bigg )$.

• Step 4. Determine the direction of motion for a T cell when its invasion switch is activated. The migration direction is based on up-gradient of CXCL12 ($\nabla L$) in the chemotaxis mode and randomly determined with a random number in the random walk mode.

• Step 5. Translation of T cells: Update the location of T cells based on mobility switch and movement direction in Steps 3–4.

• Step 6. Solve the reaction–diffusion equations (7), (8) and (15) of densities of non-senescent tumor cells and STCs, and CXCL concentration on a regular grid, using the ADI method, lagging the consumption term. Interpolation of receptor and receptor complex from the cell site to grid point is done as in Dallon and Othmer (1997), Kim et al. (2011, 2018a) and Kim and Othmer (2015). Replace Eq. (15) with Eqs. (23)–(24) in the presence of CXCL12 antibody for therapeutic approaches.

• Step 7. Go to Step 1.