The role of the microenvironment in regulation of CSPG-driven invasive and non-invasive tumor growth in glioblastoma

Abstract

Glioblastoma (GBM) is one of the most lethal type of brain cancer with poor survival time. GBM is characterized by infiltration of the cancer cells through the brain tissue while lower grade gliomas and other non-neural metastatic types form self-contained non-invasive lesions. Glycosylated chondroitin sulfate proteoglycans (CSPGs), acting as critical regulators of the tumor microenvironment, dramatically govern the spatiotemporal status of resident reactive astrocytes and activation of tumor-associated microglia. In this paper we develop a mathematical model to investigate the effect of the CSPG distribution on regulation of a fundamental switch between two distinct patterns: invasive and non-invasive tumors. We show that the model’s predictions agree with experimental results for a spherical glioma. The model specifically predicts that non-invasive tumor lesions are highly associated with a thick extracellular matrix (ECM) containing rich CSPGs, while the absence of glycosylated CSPGs results in diffusely infiltrative tumors. It is also shown that heavy CSPGs can drive the exodus of resident reactive astrocytes from the main tumor mass, and direct inhibition of tumor invasion by the astrogliotic capsule, leading to encapsulation of non-invasive lesions. However, stable residence of reactive astrocytes from GBM in the absence or low level of CSPGs presents a microenvironment favorable to diffuse infiltration due to loss of the primary (CSPG-induced cell-ECM bonding) and secondary (astrogliotic capsule) inhibitors. The mathematical model presents the clear role of the key tumor microenvironment in brain tumor invasion.

Appendices

Appendix 1: Parameter estimation

Random motility of invasive glioma cells ($D_i$) The experimental results in Hegedus et al. [52] and the experimental data of Demuth et al. [29] suggest a $D_i$ between $1.0 \times 10^{-5}$ and $2.0 \times 10^{-4}\,\mathrm{cm}^2/\mathrm{day}$, which was also validated in the experimental study of motility of glioma cell lines U87$\Delta$EGFR and U87WT [112]. We take $D_i = 1.1 \times 10^{-5}\,\mathrm{cm}^2/\mathrm{day}$ (=4.6$\times 10^{-5}\,\mathrm{mm}^2/\mathrm{h}$) within this experimental range for our work.

Diffusion coefficient of nutrient (glucose) ($D_G$) While the diffusion coefficient of G in the brain is estimated to be $6.7 \times 10^{-7}\,\mathrm{cm}^2/\mathrm{s}$ [59], the experiments in collagen gel suggest the diffusion coefficient of G, $1.3 \times 10^{-6}\,\mathrm{cm}^2/\mathrm{s}$ [103]. Effective diffusion coefficients in artificial biofilms were measured to be $6.46 \times 10^{-6}\,\mathrm{cm}^2/\mathrm{s}$ [130]. We take $D_G = 9.16 \times 10^{-7}\,\mathrm{cm}^2/\mathrm{s}$ (=3.3 $\times 10^{-1}\,\mathrm{mm}^2/\mathrm{h}$).

Diffusion coefficient of Chase-ABC ($D_C$) The diffusion coefficient of typical ECM proteinases such as MMPs is known to be small [71]. For example, the diffusion coefficient of MMP-1 was estimated to be $(8\pm 1.5)\times 10^{-9}\,\mathrm{cm}^2/\mathrm{s}$ for wide-type (activated) MMP-1 and $(6.7\pm 1.5)\times 10^{-9}\,\mathrm{cm}^2/\mathrm{s}$ for inactive MMP-1 [105]. From these experimental measurement, we take $D_C = 9.17\times 10^{-10} ~cm^2/s$ (=3.3 $\times 10^{-4}\,\mathrm{mm}^2/\mathrm{h}$).

Proliferation rate of tumor cells ($\lambda$) and Removal rate of necrotic cells ($\mu$) A large range of doubling times of glioma cells were reported. For example, the volume doubling time (Vd) of the ethyl-nitrosourea-induced rat glioma by MBI imaging ranged from 3.3 to 29.2 days ($11.3\pm 7.74$) and potential doubling time (Tp) determined immunohistochemically ranged from 2.3 to 13.3 days ($6.81\pm 3.33$) [93]. In a secreted Frizzled-related proteins study on the motility and growth of human malignant glioma cells, the doubling time of glioma cell lines (U87MG, LN-229, and U373MG) was measured to be in the range of 29–49 h among Neo, sFRP-1, sFRP-2 cases [104]. However, the percentage of dead cells (Neo case) were 22 and 70 % at 96 and 144 h, respectively, in this study [104], indicating the smaller doubling time (larger growth rate). The doubling times of the glioma cell lines were measured to be in the range of 22–39 h (39 h for U87MG, 24 h for LN-229, 22 h for T98G, and 33 h for LN-18 cell lines) [99, 128]. In a morphological characterization study, the doubling time of NG97 glioma cell line was 24 h [85]. In an in vitro experimental study [112] of growing/invasive behaviors of glioma cell lines, U87$\Delta$EGFR and U87WT, the doubling time of growing cells in the tumor core was observed to be 20 h, leading to the proliferation rate of $0.83/\mathrm{day}$ (=3.46 $\times 10^{-2}/\mathrm{h}$), but the proliferation rate of the invasive glioma cell lines were estimated to be in the range of 0–0.3/day (=0–1.25 $\times 10^{-2}/\mathrm{h}$) for mutant (U87$\Delta$EGFR) and 0.04–0.3/day (=($1.7\times 10^{-3}-1.25 \times 10^{-2})/\mathrm{h}$) for the wild type (U87WT). On the other hand, the removal rate of necrotic cells was estimated to be $(1.2 - 2.08) \times 10^{-2}\,\mathrm{h}^{-1}$ in studies of OV therapy in glioblastoma [37, 72]. We take the removal of necrotic cells, $\mu = 5.7\times 10^{-3}\,\mathrm{h}^{-1}$. The overall growth rate of tumor cells ($=x+n+x_i$) can be approximated from the sum of the equations of growing, dead, invasive glioma cells (Eqs. (8), (9), and (23)): $d(x+n+x_i)/dt = \lambda x(1-x/x_0) - \mu n$. Based these experimental data of doubling time and taking into account the death of cells in the experiments, we take the proliferation rate of growing tumor cells, $\lambda = 7.33 \times 10^{-2}\,\mathrm{h}^{-1}$.

Sensing radius of microglia for CSPGs ($r_s$) A cell can sense its microenvironmental conditions within a neighboring region, which was estimated to be a ball with a sending radius ($r_s$) in our work. Kim et al. [71] took the sensing radius of 25 $\upmu \mathrm{m}$ based on experimental observation of 9 L gliosarcoma cells to C6 glioma cells [131]. In a study of accelerated microglial pathology in Alzheimer’s disease [7], the tree area of was estimated to be 1800 $\upmu \mathrm{m}^2$, leading to the average radius of 24 $\upmu \mathrm{m}$ cell trees. In general, microglia were shown to be exceptional sensors of their microenvironment, responding to even subtle changes in their milieu on a short time scale of minutes, by undergoing coordinated changes in gene expression and morphology [1, 64, 108]. By taking into these observations, we take 60 $\upmu \mathrm{m}$ (=6.0 $\times 10^{-2}\,\mathrm{mm}$).

Table 1 Model parameters

ECM production rate from tumor cells ($\lambda _{42}$) The remodeling/production rate of ECM was estimated to be $5.0\times 10^{-6}\,\mathrm{s}^{-1}=0.018\,\mathrm{h}^{-1}$ in a study of the pattern formation of glioma cells [71]. In a study of oncolytic viral therapy on glioma cells [72], the rate was taken to be $6.0 \times 10^{-3}\,\mathrm{h}^{-1}$. In this work, we take a slightly smaller rate, $\lambda _{42} = 2.9\times 10^{-3}\,\mathrm{h}^{-1}$, leading to the effective production rate by tumor cells $\lambda _{42}\frac{x^{m_2}}{K_x^{m_2}+x^{m_2}}=(2.87\times 10^{-5} - 1.4\times 10^{-3})\,\mathrm{h}^{-1}$ by assuming cell density $x=(10^4-10^6)\,\mathrm{cells}/\mathrm{mm}^3$ [37, 72, 95] and $K_x=x^*, m_2=1$ in Table 1, and $\lambda _{42}$ above.

CSPG degradation rate ($\lambda _{41}$) Kim et al. [71] used $3.0\times 10^8\,\mathrm{cm}^3/(\mathrm{g}\cdot \mathrm{s})$ for the MMP-mediated degradation rate of the ECM (mainly collagen) by tumor cells. The degradation rate of ECM has a wild range. For instance, there were only 36.6, 24.5, 38.9 % reduction of ECM from degradation by trypsin, elastase, collagenase [107] over the long period of time, 5 days, leading to very small values of the degradation rate. In a study of the effect of Chase-ABC on CSPG content in the injured rate brain [83], Lin et al. found that there was an approximately 50 % decrease in GAG content (main constituent of CSPGs (E)) compared to normal tissue after 1 h of Chase-ABC treatment, indicating the approximated decay rate of 0.6931 $\mathrm{h}^{-1}$. In a study of components of perineuronal net ECM in rat brain [26], Deepa et al. found that various CSPGs act differently with Chase-ABC treatment for 1 hour: neurocan and versican completely removed (large degradation rate $>$ 10 h$^{-1}$) but aggrecan and phosphacan were not removed (very small degradation rate) while some others are partly removed. On the other hand, the rapid digestion of CSPGs followed by OV spread was observed in the experimental study of Chase-ABC in OV treatment of gliomas [30]. Based on the experimental data [30], the Chase-ABC-mediated degradation rate of CSPGs was estimated to be $\lambda _{41} = 9.0\times 10^{1}\,\mathrm{h}^{-1}$ in a Chase-ABC-assisted OV spread study [72], which generated good agreement between the in vitro experiment and the simulation results. In order to take into account the CSPG-dominant compact microenvironment in the in vivo system in our model, we take a slightly smaller value $\lambda _{41} =4.3 \times 10^{1}\,\mathrm{h}^{-1}$ in this work. By assuming $C=(0.5-250)\,\mathrm{mU}/\mathrm{ml}$ [12, 49, 72, 83], $K_C = C^*$, and $m_1=1$ in Table 1, our choice of $\lambda _{41}$ above leads to the effective CSPG degradation rate $\lambda _{41}\frac{C^{m_1}}{K_C^{m_1}+C^{m_1}} =(0.42 - 35.8)\,\mathrm{h}^{-1}$.

Natural decay rate of Chase-ABC ($\lambda _{62}$) The natural decay rate of MMPs was estimated to be $5.0\times 10^{-6}~s^{-1}=1.8 \times 10^{-2}\,\mathrm{h}^{-1}$ [71]. The half-life of Chase-ABC (at 37 $^\circ$C) varies significantly [132]; The relatively short half-life (6 days) was reported in the brain of the injected rat [83] but it may be 2–3 weeks with albumin or trehalose (stabilizers) [16, 82]. In the simulation, assuming the half-life of 230 h, we take $\lambda _{62} = ln(2)/(230\,\mathrm{h}) = 6.0\times 10^{-3}\,\mathrm{h}^{-1}$.

Threshold values ($G_{0}$) In vitro experiments suggest that tumor cells enter quiescent/necrotic state in response to nutrient withdrawal [lower than 40 % of normal glucose levels (25  mM) [14, 54] and 25 % of normal oxygen levels (0.2 mM) [14, 36]] due to growing mass of the tumor [76]. For the transition of tumor cells to hypoxic/necrotic cells, we take $G_{0} = 1.35 \times 10^{-3}\,\mathrm{g}/\mathrm{cm}^3$.

Chemotactic sensitivity of tumor cells ($\chi _i$) Glioma cells may travel a distance 0.4–0.5 cm in 150 h with EGF treatment [18]; the cells was shown to travel faster, up to a distance of 1.25 cm (0.75  cm) in 150 h in agar containing EGF (plain agar). The cell velocity was in the range of of 50–110 $\upmu \mathrm{m}/24\,\mathrm{h}$ in U87MGmEGFR spheroids experiments [27]. In a study of miR-451-mediated cell migration and proliferation in response to high and low glucose levels in a glioma, Kim et al. [75] took the chemotactic sensitivity, $1.86\times 10^{-7}\,\mathrm{cm}^5/(\mathrm{g}\cdot \mathrm{s})$, to glucose gradients in a slightly modified form with a upper bound. Assuming that gradient of the glucose concentration was in the range of $(10^{-4} - 3\times 10^{-2})\,\mathrm{g}/\mathrm{cm}^4$ and using drift velocity 25–110 $\upmu \mathrm{m}/24\,\mathrm{h}$ of migratory glioma cells, we compute $\chi _i=\frac{velocity}{gradient}$ to be $\chi _n$= ($1.9\times 10^{-6} - 1.3\times 10^{-3} )\,\mathrm{cm}^5/(\mathrm{g}\cdot \mathrm{s})$. In this work, we take $\chi _i= 2.9\times 10^{-6}\,\mathrm{cm}^5/(\mathrm{g}\cdot \mathrm{s})$ (=$1.04\times 10^{3}$ $\mathrm{mm}^5 \mathrm{g}^{-1}\mathrm{h}^{-1}$).

Appendix 2: Nondimensionalization

Table 2 lists the reference values in the model. We define $L=4\,\mathrm{mm}$ following the experimental setup in [109] and take the characteristic diffusion coefficient $D=1.27\times 10^{-5}\,\mathrm{cm}^2/\mathrm{s}$ so that $T = 3.5\,\mathrm{h}$. We determine the reference values for $x,n,A,M_r, M_a, E, \rho , C$ as follows:

Cell density ($x^*, n^*, A^*,M_r^* = M_a^*$): various cell densities can be used in the experiments and modeling: $5\times 10^5\,\mathrm{cells}/\mathrm{cm}^3$ [18], $7.23 \times 10^8\,\mathrm{cells}/\mathrm{ml}$ [89, 118], $10^9\,\mathrm{cells}/\mathrm{cm}^3$ [37, 72, 95]. In this study, we take $x^* = n^* = 10^6\,\mathrm{cells}/\mathrm{mm}^3$ from [37, 72, 95]. Density of fibrous astrocytes was estimated to be in the range of $2.0\times 10^5\,\mathrm{cells}/\mathrm{mm}^3$ in the brain [65]. We take $A^* = 2.0\times 10^5\,\mathrm{cells}/\mathrm{mm}^3$. Density of microglia was measured to be in the range of (0.5–2.5) $\times 10^{-5}\,\mathrm{cells}/(\upmu \mathrm{m}^3) =$ 0.5–2.5 $\times 10^4\,\mathrm{cells}/(\mathrm{mm}^3)$ in response to Cx3cr1GFP/Cx3cr1KO reporter knockout in a study of synaptic pruning by microglia in normal brain development [96, 120]. We take $M_r^* = M_a^* = 1.0 \times 10^4\,\mathrm{cells}/(\mathrm{mm}^3)$.

Table 2 Reference value

ECM density [CSPG ($E^*$), Tumor ECM ($\rho ^*$)] In a study of cell infiltration in glioma, Silver et al. [109] investigated the role of various CSPG concentrations (0–500 $\upmu \mathrm{g}/\mathrm{ml}$) and laminin concentration (5 $\upmu \mathrm{g}/\mathrm{ml}$) in regulation of glioma infiltration using both invasive and non-invasive co-culture assays. They found that while high concentrations of CSPGs (250, 500 $\upmu \mathrm{g}/\mathrm{ml}$) induce non-invasive glioma with astrocytes on the periphery of the well-defined tumor core, low levels of CSPG (0–10 $\upmu \mathrm{g}/\mathrm{ml}$) can allow glioma cell infiltration in the presence of astrocytes in the tumor aggregates [109]. It was shown that Cat-301 CNS CSPG with high molecular weight from brain in the density of 1.4 g/ml [38] can have similar biological properties as aggrecan, CSPG with the high molecular weight from cartilage in the density of 1.35–1.4 g/ml [53]. Dutt et al. [31, 32] investigated effects of versican, one of major CSPG components isolated from human glioma cell line (U251MG) and others (0–100 $\upmu \mathrm{g}/\mathrm{ml}$), on neural crest cell migration. They found that low concentrations of version V0/V1 ($\sim$25 $\upmu \mathrm{g}/\mathrm{ml}$) can block active migration of neural crest stem cells [31, 32]. The expression level of noncleaved isoform of Brevican/BEHAB was $>$4-fold higher in human gliomas compared to normal brain tissue [127]. Isolated versican levels in brain tissue were estimated to be 3 mg/100 g wet tissue components [9]. Glial HA-binding protein (GHAP) usually localized in white matter is present in the concentration of 8.2 mg/100 g in human white matter [9]. We take $E^*=250\,\upmu \mathrm{g}/\mathrm{cm}^3$ from [109]. In a OV spread study [72], the tumor ECM concentration was estimated to be $1.0\,\mathrm{mg}/\mathrm{cm}^3$ based on the experimental observation in the brain [63, 71, 112]. We take $\rho ^*=1.0\,\mathrm{mg}/\mathrm{cm}^3$ from [72].

Concentration of nutrients (glucose) ($G^*$): Sander and Deisboeck [106] used the characteristic glucose concentration $2\times 10^{-4}\,\mathrm{g}/\mathrm{cm}^3$, and the boundary condition $6\times 10^{-4}\,\mathrm{g}/\mathrm{cm}^3$ (cf. [27]). Kim et al. [71] took $G^* = 6\times 10^{-4}\,\mathrm{g}/\mathrm{cm}^3$ as a reference value in a study of pattern formation of glioma cells in response to glucose withdrawal in vitro experiments. The miR-451 expression was up-regulated in response to a high glucose concentration ($4.5\times 10^{-3}\,\mathrm{g}/\mathrm{cm}^3$) while the expression level of its counterpart, AMPK complex, was up-regulated when the tumor was exposed to glucose withdrawal condition ($1.0\times 10^{-4}\,\mathrm{g}/\mathrm{cm}^3$) in experimental studies of miR-451 regulation in glioblastoma [43, 44]. We take $G^* = 4.5\times 10^{-3}\,\mathrm{g}/\mathrm{cm}^3$ from a study of the miR-451-AMPK core control in regulation of cell migration and proliferation in GBM [75].

Concentration of Chase-ABC ($C^*$): High-dose infusions of Chase ($\sim$1000 U/ml) is necessary for delivery of Chase-ABC into deep regions of the spinal cord due to overflow and dilution beyond the intrathecal space [82]. A low (2 U/0.5 ml) dose of trehalose-assisted Chase was enough to digest CSPG decorin for functional recovery after injury [82]. A wide range of Chase ABC levels (0.5–50 mU/ml) were used in addition to 20 ng/ml EGF and 20 ng/ml bFGF in a study of biochemical effect of Chase-ABCs on the morphological changes in neural precursor cells [49]. Bruckner et al. found that a low dose of $0.25\,\mathrm{U}/\upmu \mathrm{l}$ of Chase-ABCs was enough for significant breakdown of CSPGs in the rat brain, leading to long-lasting and acute changes in CSPGs [12]. The protease-free Chase-ABC treatment (50 U/ml) led to digestion of CSPGs around the injection regions (resulting in low GAG content (1200–2000 $\upmu \mathrm{g}/\mathrm{mg}$) and high content ($4000\,\upmu \mathrm{g}/\mathrm{mg}$) with penicillinase-treatment), promoting axon regeneration [83]. In a OV spread study in glioma [72], Kim et al. took 50 mU/ml for the reference value of Chase-ABC. We take $C^* = 50\,\mathrm{mU}/\mathrm{ml}$.

We nondimensionalize the variables and parameters in the partial differential Eqs. (8)–(38) as follows:

$$\begin{aligned}&\bar{t} = \frac{t}{T}, ~\bar{{\mathbf x}} = \frac{{\mathbf x}}{L}, ~\bar{x} = \frac{x}{x^*}, ~\bar{n} = \frac{n}{n^*}, ~\bar{x}_i = \frac{x_i}{x_i^*}, ~\bar{E}=\frac{E}{E^*}, ~\bar{\rho }=\frac{\rho }{\rho ^*}, ~\bar{M}_r=\frac{M_r}{M_r^*}, \nonumber \\&~\bar{M}_a=\frac{M_a}{M_a^*}, ~\bar{A} = \frac{A}{A^*}, ~\bar{G}=\frac{C}{G^*}, ~\bar{C}=\frac{C}{C^*}, ~\bar{D}_{i} = \frac{D_{i}}{D}, ~\bar{D}_G = \frac{D_G}{D}, ~\bar{D}_C = \frac{D_C}{D}, \nonumber \\&~\bar{\lambda } = T \lambda , ~\bar{x}_0 = \frac{\bar{x}_0}{x^*}, ~\bar{\lambda }_{11}= T \lambda _{11}, ~\bar{\lambda }_{11}^\dagger = T \lambda _{11}\frac{x^*}{n^*}, ~\bar{\mu } = T \mu , ~\bar{\lambda }_{41} = T \lambda _{41}, \\&~\bar{K}_C = \frac{K_C}{C^*}, ~\bar{\lambda }_{42} = T \lambda _{42}, ~\bar{K}_x = \frac{K_x}{x^*}, \bar{\rho }_0 = \frac{\bar{\rho }_0}{\rho ^*}, \bar{k}_{1b} = k_{1b}\frac{E^*}{M_r^*}, \bar{\mu }_G = T \mu _G x^*, \nonumber \\&~\bar{\lambda }_{63} = T E^* \lambda _{63}, ~\bar{\lambda }_{62} = T \lambda _{62}, ~\bar{\chi }_i = \frac{T \chi _i G^*}{L^2}. \nonumber \end{aligned}$$

(39)