Phase-field model of vascular tumor growth: Three-dimensional geometry of the vascular network and integration with imaging data

Biologically-Based Mathematical Modeling of Tumor Vasculature and Angiogenesis via Time-Resolved Imaging Data

“…Xu et al [ 81 , 82 ] developed a 3D hybrid model of tumor angiogenesis coupled to TAF, interstitial flow, and blood flow, which was initialized with photoacoustic imaging data. TAF dynamics were modeled with a reaction-diffusion PDE describing the secretion of TAF by hypoxic tumor cells, diffusion through the extracellular space, uptake by endothelial cells, and decay of TAF over time.…”

“…Photoacoustic images were scaled between −1 and 1 to segment extravascular space and capillaries, respectively. The map of capillaries and extravascular space was then used as the initial vasculature network in the model by Xu et al [ 81 ]. While the model was not calibrated by time-resolved data, their image processing and modeling framework demonstrated an approach to readily utilize photoacoustic imaging data directly in the model without extensive image processing or manual adjustments.…”

“…In this section, we identify four major areas of research at the tissue scale (shown in Figure 4 ) and discuss the modeling strategy or strategies applied to these areas. Broadly, these areas include: (1) representing the evolving geometry of the tumor’s vascular network (panel a in Figure 4 ) [ 29 , 82 , 87 , 88 , 119 , 120 , 121 , 122 , 123 ], (2) estimating the associated blood flow and vascular transport of substances (panel b in Figure 3 ) [ 81 , 88 , 122 , 123 , 124 , 125 , 126 , 127 ], (3) describing the mechanisms underlying the complex interplay between tumor growth and vasculature dynamics (panel c in Figure 4 ) [ 32 , 81 , 87 , 88 , 121 , 123 , 128 , 129 ], and (4) determining the effect of cytotoxic, targeted, and anti-angiogenic therapies on the tumor-associated vascular network as well as the tumor itself (panel d in Figure 4 ) [ 28 , 85 , 86 , 124 ].…”

“…The third and fourth areas focus on describing the mechanisms underlying the complex interplay between tumor growth and vasculature in the absence of treatment (panel c in Figure 4 ) [ 32 , 81 , 87 , 88 , 121 , 123 , 128 , 129 ] and during treatment (panel d in Figure 4 ). Many of the same discrete, continuum, and hybrid models of angiogenesis and vasculature network mentioned in the previous two areas are also applied to study the interplay between the tumor and vasculature with an increased emphasis on modeling the tumors themselves, as they trigger the angiogenic cascade, influence the development of the neovasculature, and are the ultimate beneficiaries of the angiogenic blood supply.…”

“…Alternatively, phase field models may be used to describe the coexistence of a number of phases representing different tissue types (e.g., tumor and normal tissue) [ 140 ] and their interactions or transitions between each other. The spatial and temporal evolution is dictated by a free energy potential, which restricts mixing and penalizes spatial variation in individual phases, and several source terms describing the growth, death, response to treatment, and transition from one species to another (e.g., a proliferative tumor cell may transition to hypoxic tumor cell in response to scarce nutrients) [ 81 , 87 , 121 , 137 ]. The PDE governing the dynamics of each cell species is obtained by combining the mass flux, which is defined in terms of the gradient of the free energy potential [ 140 ], and the aforementioned source terms [ 81 , 87 , 121 , 137 ].…”