Vascular Neoplasms

We describe the haemodynamics in the individual capillaries using Poiseuille’s equation [29 (link)], where the volumetric flow rate in a vessel is related to the pressure drop across it via
${\dot{Q}}_{\text{vsc}}=-\frac{\pi R^4\left(t\right)}{8{\mu }_{\text{B}}{L}_{\text{vsc}}\left(t\right)}\Delta p_{\text{vsc}},$
where μ_B the blood viscosity, R is the capillary lumen radius with cross-sectional area πR², and L_vsc is the length of a capillary segment. Note that R and L_vsc change in time since vessels are deformed under solid stresses. The blood viscosity is assumed homogeneous and constant in time; this is in order to remove an additional model parameter in lieu of suitable data with which to inform it. Interstitial fluid flow is described using Darcy’s law [33 (link), 34 (link)] so that the volumetric fluid flow rate in the extracellular space is given by
${\dot{Q}}_{\text{int}}=-\frac{K_{\text{int}}{A}_{\text{int}}}{L_{\text{int}}\left(t\right)}\Delta p_{\text{int}},$
where K_int is the hydraulic conductivity of the interstitium, A_int is the interstitium cross-sectional area, and L_int is the length of a tissue segment whose interstitial fluid pressure difference is denoted by Δp_int. The cross-sectional area can be expressed with respect to the mean capillary radius and the vascular density, S_vsc, as $A_{\text{int}}=2\pi \overline{R}/S_{\text{vsc}}$ [35 (link)]. Here $\overline{R}$ is the average capillary radius in the local neighbourhood of the connective tissues under consideration.
To model the fluid movement across the capillary barrier that occurs as a result of filtration, we use Starling’s equation. Similarly to Baish at al. [35 (link)], the volumetric transvascular flow rate across the permeable endothelium is expressed through Starling’s law
${\dot{Q}}_{\text{trv}}=K_{\text{vsc}}\left(t\right)A_{\text{vsc}}\left(t\right)\left(p_{\text{eff}}-p_{\text{int}}\right).$
Here K_vsc is the hydraulic conductivity of the endothelial barrier, which can be expressed as a function of the size of the fenestrations on the vessel (pores’ average radius), r_p, the fraction of vessel-wall surface occupied by pores, γ_p, and the blood viscosity, μ_B, via [36 (link)]:
$K_{\text{vsc}}=\frac{{\gamma }_{\text{p}}{r}_{\text{p}}^2\left(t\right)}{8{\mu }_{\text{B}}h\left(t\right)}.$
Finally, A_vsc is the surface area of the blood vessel wall and the “effective” pressure is given by
$p_{\text{eff}}=p_{\text{vsc}}-\left({\pi }_{\text{vsc}}-{\pi }_{\text{int}}\right){\sigma }_{\text{o}},$
where σ_o is the average osmotic reflection coefficient of the plasma proteins, π_vsc is the osmotic pressure of the plasma at the permeable vascular wall, while π_int is the corresponding osmotic pressure of the interstitial fluid. This modelling approach accounts for the contribution of the colloid osmotic pressure of plasma and interstitial fluid. Including those features are important for a complete modelling description of the micro-circulation system. Nonetheless, numerical experiments have revealed that the omission of the rightmost term in Eq (8) in the vascular–interstitium interaction model has only marginally affected the qualitative predictions of the proposed tumour-growth angiogenesis model. This is also supported by the experimental findings of Tong and colleagues [37 (link)], who showed that the osmotic pressure difference across the wall of tumour vessels is negligible.
It is important to note here that we do not include the lymphatic system in the current model, given that it is generally assumed to be compromised in tumour tissues. However, this would be a straightforward extension to the current framework.
The flow rate Eqs (4)–(6) are coupled to a model for the vascular and interstitial pressures. In the vascular network, conservation of fluid flux at vessel junctions provides a linear system of equations to solve for the nodal pressures, subject to pressure/flow boundary conditions on the terminal nodal points of the network. The interstitial pressure, p_int, satisfies the Poisson equation, where the source term captures both vascular and osmotic contributions, following the approach of Stylianopoulos and Jain [26 (link)]. Quasi-steady state fluid flow is solved numerically for the vascular and interstitial pressures (defined on nodal points in the vascular network, and extravascular-space points, respectively). An interconnected grid of tissue and vascular nodes is considered. Tissue nodes are connected to each other via the 3D FE mesh, where each tissue element corresponds to a two node edge element of the FE grid. Vascular nodes are connected to each other according to the network structure generated by the vascular network module, defined in the Angiogenesis model subsection. Also, as shown in the 2D illustration of Fig 1C, each vascular node is contained in a FE of the discrete three-dimensional domain of analysis. Thus, in order to describe transvascular flow through Eq (6), each vascular node is associated with the corresponding vertices (i.e. tissue nodes) of the FE.
After every simulation of the flow model, the magnitude of the average wall shear stress (WSS) distribution, τ_f, the axial blood-flow mean velocity in a vascular segment, v_vsc, and the fluid velocity at the interstitium, v_int, can be evaluated using the following relationships:
${\tau }_{\text{f}}=R|\Delta p_{\text{vsc}}|/L_{\text{vsc}},$
$v_{\text{vsc}}={\dot{Q}}_{\text{vsc}}/\pi R^2,$
$v_{\text{int}}=-K_{\text{int}}\Delta p_{\text{int}}/L_{\text{int}},$
where Δp_vsc and Δp_int is the pressure difference between two vascular and two interstitial nodes respectively of the corresponding discretised domains of analysis (i.e. the vascular network and the ECM) respectively. The value set for the material parameters of the above equations are provided separately in S2 Table.

All patients included were treated with TACE, including conventional TACE (cTACE) and drug-eluting bead TACE (DEB-TACE). Interventional physicians choose cTACE or DEB-TACE based on tumor burden and patient characteristics. The basic treatment process of DEB-TACE resembles that of cTACE except for the embolic agents. cTACE uses lipiodol (Guerbet), gelatin sponge particles, and polyvinyl alcohol as embolic agents. Selective or super-selective embolization of tumor-supplying vessels is performed whenever technically justified [23 (link)]. For DEB-TACE, 100–300 μm diameter CalliSpheres® Beads (CB; Jiangsu Hengrui Pharmaceutical Co., Ltd.) were used as carriers, loaded with 60–80 mg epirubicin, pirarubicin, or doxorubicin. All procedures were administered by interventional physicians with at least 10 years of experience. All patients were admitted for postoperative supportive care after TACE procedure and were managed routinely.
Study cohort judgment of TACE response was performed according to the modified Response Evaluation Criteria in Solid Tumors (mRECIST) [24 (link)] criterion. In brief, the therapeutic response of TACE was stratified into four grades: (a) complete response (CR): complete disappearance of the lesion; (b) partial response (PR): a minimum 30% reduction in the sum of diameters of viable target lesions (enhancement in the arterial phase); (c) progressive disease (PD): at least 20% extension in the sum of the diameters of viable (enhancing) target lesions; and (d) stable disease (SD): neither PR nor PD. Based on mRECIST, CR and PR patients were categorized as objective response (OR) cohort, and PD and SD patients as non-objective response (NOR) group. This assessment was determined by two professional abdominal radiologists based upon the follow-up MR images. Among the 144 patients enrolled, 75 were assigned to the NOR group and 69 to the OR group. In the independent external validation set, 14 patients were in the NOR group and 14 in the OR group.

Endovascular treatment was performed under general anaesthesia on a total of 16 patients, using Allura Xper FD 20 angiographic system (Philips Medical System, Eindhoven, the Netherlands). In all cases, a catecolamine secretion test and preoperative angiographic study were performed to exclude major contraindications and precisely assess tumour vascularization characteristic (both feeding arteiries and draining veins). Eight patients were treated, between 2010 and 2016 with Polivinil Alcohol (PVA) Contour micro particles (Boston Scientific, Boston MA) while 8 patients, treated between 2016 and 2020, PPS tumors were embolized through SQUID12 direct injection. Puncture procedure was done under ultrasound and fluoroscopic guide, inserting a 19–22 gauge, 10 mm needle directly in the specific tumor target vessels. Needle hub was then connected to DMSO-compatible tube, to allow the injection of embolic liquid agent. A specific balloon micro catheter (SCEPTER 4 × 11 mm XC, Microvention CA, USA) was positioned, to prevent external carotid artery reflux during SQUID12 injection. A post procedure angiography was performed to assess the extent of tumor devascularization, graded as total (100%), near total (95–99%), sub-total (70–95%), moderate (30–70%), low (up to 30%).