Why one-size-fits-all vaso-modulatory interventions fail to control glioma invasion: in silico insights

Abstract

Gliomas are highly invasive brain tumours characterised by poor prognosis and limited response to therapy. There is an ongoing debate on the therapeutic potential of vaso-modulatory interventions against glioma invasion. Prominent vasculature-targeting therapies involve tumour blood vessel deterioration and normalisation. The former aims at tumour infarction and nutrient deprivation induced by blood vessel occlusion/collapse. In contrast, the therapeutic intention of normalising the abnormal tumour vasculature is to improve the efficacy of conventional treatment modalities. Although these strategies have shown therapeutic potential, it remains unclear why they both often fail to control glioma growth. To shed some light on this issue, we propose a mathematical model based on the migration/proliferation dichotomy of glioma cells in order to investigate why vaso-modulatory interventions have shown limited success in terms of tumour clearance. We found the existence of a critical cell proliferation/diffusion ratio that separates glioma responses to vaso-modulatory interventions into two distinct regimes. While for tumours, belonging to one regime, vascular modulations reduce the front speed and increase the infiltration width, for those in the other regime, the invasion speed increases and infiltration width decreases. We discuss how these in silico findings can be used to guide individualised vaso-modulatory approaches to improve treatment success rates.

Figure 1. Histological images of functional and occluded blood vessels in gliomas.

(A) From right to left brain tissue infiltrated by glioma cells with meningeal blood vessels of normal size and anatomy. (B) Atypical but not occluded intratumoural blood vessels with activated endothelium and thicker/plumper muscular layers than the normal brain blood vessels. (C) A longitudinal section of a large intratumoural blood vessel with a no-obliterated part filled with blood (left) and an occluded thrombotic part (right). (D) Thrombotic occlusion in small intratumoural blood vessels. The arrowheads point to blood vessels which are magnified in the corresponding subfigures.

Figure 2. Modelling logic and hierarchy.

(A) Diagram of the interactions between glioma cells, oxygen availability, functional tumour vasculature and pro-angiogenic factors. (B) From left to right model complexity increases with respect to the interactions between system variables: density of glioma cells ρ(x, t), density of functional tumour vasculature v(x, t) and oxygen concentration σ(x, t). The parameters σ₀ and v₀ represent constant oxygen concentration and functional tumour vascularisation, respectively. The parameters h₂ and g₂ are the glioma cell oxygen consumption and vaso-occlusion rates, respectively (see equations (12)–(13).

Figure 3. Glioma cell oxygen consumption effects on tumour invasion for constant functional vascularisation (Model II).

Simulation maps with respect to the intrinsic proliferation b ∈ [2.73 × 10⁻⁴, 2.73 × 10⁻²] days⁻¹ and diffusion D ∈ [2.73 × 10⁻³, 2.73 × 10⁻¹] mm² days⁻¹ rates of glioma cells. (A) Effective diffusion, (B) effective proliferation, (C) tumour front speed and (D) infiltration width for different glioma cell oxygen consumption rates h₂ = {5.73 × 10⁻⁴, 5.73 × 10⁻³, 5.73 × 10⁻²} mm cell⁻¹ day⁻¹ in simulation maps I-III, respectively. (A–D) Differences between the simulation maps I-III. The other parameters are as in Table 1.

Figure 4. Vaso-occlusion effects on glioma invasion (Model III).

Simulation maps with respect to the intrinsic proliferation b ∈ [2.73 × 10⁻⁴, 2.73 × 10⁻²] days⁻¹ and diffusion D ∈ [2.73 × 10⁻³, 2.73 × 10⁻¹] mm² days⁻¹ rates of glioma cells. (A) Effective diffusion, (B) effective proliferation, (C) tumour front speed and (D) infiltration width for a constant glioma cell oxygen consumption rate h₂ = 5.73 × 10⁻³ mm cell⁻¹ day⁻¹ and different vaso-occlusion rates g₂ = {5.0 × 10⁻¹³, 5.0 × 10⁻¹², 1.5 × 10⁻¹¹} cells⁻ⁿ mmⁿ day⁻¹ in simulation maps I-III, respectively. (A–D) Differences between the simulation maps I-III. The other parameters are as in Table 1.

Figure 5. An overview of model simulation results.

(A) Variations in the glioma cell oxygen consumption rate, under the assumption of constant functional vascularisation, reveal a critical proliferation rate b^* that separates tumour invasive behaviors in different regimes (Model II). (B) Variations in the vaso-occlusion rate reveal a critical proliferation/diffusion ratio Λ⁺ = b/D for b > b⁺ that separates tumour invasive behaviors in different regimes (Model III). Colour gradients from low to high represent the increase of glioma cell oxygen consumption and vaso-occlusion. The purple and black wedges/bars represent the corresponding effects on the tumour front speed and infiltration width for increasing/decreasing glioma cell oxygen consumption and vaso-occlusion rates.