Effect of wall compliance and permeability on blood-flow rate in counter-current microvessels formed from anastomosis during tumor-induced angiogenesis.

Tumor blood flow controls tumor growth by delivering nutrients and oxygen to tumors and carrying metabolic waste away from tumors. Insufficient delivery of oxygen leads to hypoxic and acidic regions in solid tumors [1]. It also affects the efficacy of treatment by delivering blood-borne therapeutic agents to tumors. Inability of delivering a sufficient amount to all tumor cells results in residual tumor cells leading to tumor regrowth and development of resistant cells [2–5]. Hence, investigation of the blood flow in tumor vasculature may lead to a better understanding of tumor pathology and development of new treatment methods. Compared to the blood flow in normal tissues, tumor blood flow is inhomogeneous because of heterogeneity in tumor vasculature, vessel-wall leakiness, and compliance. In general, it is reduced and thus hinders delivery of nutrients, oxygen, as well as therapeutic drugs [5–12]. Blood flow in a microvessel, if approximated by the Hagen-Poiseuille’s law, is determined by the driving force, the difference between arterial and venous pressures, and by the resistance controlled by the blood viscosity, blood vessel size, and geometry. In the central region of a tumor, the arteriole–venule pressure difference is reduced compared to the surrounding normal tissues and the tumor vasculature is also more tortuous [5,13,14]. These factors can partially explain the reduced tumor blood flow compared to that in the surrounding normal tissue [13,14]. In addition, other well-documented tumor characteristics, such as tumor vessel leakiness, elevated interstitial fluid pressure (IFP), the coupling between intravascular, and interstitial fluid exchange, might also contribute to the reduced tumor blood flow [8,15–17]. Evidence for tumor vessel leakiness or increased vessel permeability has come from the observation of extravasation of soluble tracers and blood-borne particles by scanning electron microscopy and magnetic resonance imaging [18]. Highly active but less regulated angiogenesis and microvascular remodeling are believed to attribute to tumor vessel leakiness and increased permeability. Elevated IFP is derived from the accumulated water caused by tumor vessel leakiness and the lack of functional lymphatic vessels [5,10]. The IFP can affect the vessel radius if the vessel is compliant, and, as a consequence, affect the tumor blood flow. Much effort has been made to elucidate the relationship among tumor vessel leakiness, the IFP, the interstitial fluid velocity (IFV), and the drug delivery to tumor using both experimental and theoretical approaches [16,17,19–26]. Jain and his co-workers developed a series of mathematical models to predict the elevated IFP and study the effect of tumor vessel leakiness and compliance on the pressure–flow relation and pressure profiles in solid tumors or along a vessel. Pertinent to this study, Netti et al. [16] and Baish et al. [17] modeled the tumor vasculature as one or a pair of equivalent permeable and compliant vessels embedded in an isotropic porous medium, Other researchers primarily focused on the interactions between intravascular, transmural, and interstitial fluid flow in a complex tumor vessel network formed by tumor-induced angiogenesis [22,24,25]. However, few investigated the combined effects of vessel-wall compliance and permeability on the blood-flow rate in the tumor vessels, and the normalized tumor vessels after anti-angiogenic therapy [27,28]. The normalized vasculature is characterized by the reduced vessel wall permeability, normalized vessel diameter, and normalized vessel density and organization [29–31]. Therefore, to elucidate the quantitative relationship between the vessel-wall compliance and permeability and the blood-flow rate in the microvessels in the tumor tissue, in the tumor tissue with the normalized vasculature, as well as in the normal tissue, we developed a transport model to simultaneously predict the IFP, the IFV, and the blood-flow rate in a counter-current microvessel loop, which frequently occurs from anastomosis during tumor-induced angiogenesis [32,33] [see Fig. 1(a)]. Tumor angiogenesis is a very complicated and poorly regulated process including sprout formation from parental vessels, sprout extension and branching, anastomosis, tube formation, and vessel maturation [34]. The resulting vasculature forms a complex network and is structurally and functionally abnormal: tumor vessels are very leaky and tortuous; the vessel–vessel interconnections are quite random and irregular [32,34]. The simplified counter-current microvessel loop shown in Fig. 1(a) represents anastomosis in the formation of tumor vessel network during tumor angiogenesis [33,35]. Figure 1(a) also shows the vessel sprouts from the parent vessels, branches, and four counter-current microvessel loops during anastomosis. For this simplified geometry, three analytical solutions were obtained for the rigid or compliant impermeable vessel, and the rigid permeable vessel. These analytical solutions served to validate the numerical simulation for the compliant and permeable vessel. Open in a separate window Fig. 1 Simplified model geometry. (a) Schematic of counter-current microvessels (shaded regions) in anastomosis during tumor-induced angiogenesis. Three processes are plotted: sprouting from the parental vessels, sprout branching, and anastomosis. (b) Model geometry: enlarged shaded region in (a) comprises counter-current microvessels that are connected at X = L. The Krogh cylinder of radius Rt represents the tumor tissue surrounding the counter-current vessels. Blood flow is driven by the intravascular pressure at X = 0 in the entrance branch P a0 and that in the exit branch P v0. X = 0 represents the outer edge of the tumor tissue, which is the interface between the tumor and normal tissues. The drawing is not to scale.