The effect of interstitial pressure on tumor growth: Coupling with the blood and lymphatic vascular systems
Highlights
► We study interstitial fluid pressure/flow (IFP/IFF) during vascular tumor growth. ► Vessel collapse results in transport barriers and decreased tumor growth rate. ► High interstitial hydraulic conductivity leads to plateau profile in tumor IFP. ► Increasing vascular hydraulic conductivity maintains high IFP in tumor. ► Tumor vascular pathologies promote local invasion and metastasis through IFF.
Introduction
Vascularized tumor growth is a complex process spanning a wide range of spatial and temporal scales, and involves inter-related biophysical, chemical and hemodynamic factors in the interplay between tumor formation, vascular remodeling, and angiogenesis. In the early stages of carcinogenesis, tumor cells are believed to be supported by the pre-existing vasculature sustaining the normal tissue. These factors remodel the surrounding pre-existing vessel network without necessarily generating new vessels (e.g., by cooption and circumferential growth, Holash et al., 1999a, Holash et al., 1999b). The secretion of TAF also leads to tumor-induced angiogenesis as the vasculature becomes unable to support the increasing number of tumor cells, causing new blood vessels to form from the pre-existing vascular network (Raza et al., 2010, Folkman, 1971) through endothelial cell sprouting, proliferation, anastomosis, and remodeling. These processes enable oxygen and cell nutrients circulating in the vasculature to be transported and released closer to the hypoxic tumor cells. However, the interaction between tumor cells and the surrounding vasculature is abnormal due to inadequate signaling from the tumor cells, leading to the creation of new vessels that are inefficient, tortuous and leaky (De Bock et al., 2011, Greene and Cheresh, 2009, Hashizume et al., 2000, Jain, 2001). In order to elucidate these complex processes from a biophysical perspective, modeling of vascularized tumor growth has been an important focus in mathematical oncology, e.g., see the recent reviews (Byrne, 2010, Lowengrub et al., 2010, Frieboes et al., 2011, Roose et al., 2007, Astanin and Preziosi, 2007, Harpold et al., 2007, Anderson and Quaranta, 2008, Deisboeck and Couzin, 2009, Ventura and Jacks, 2009).
Two critical components in tumor growth and vascularization are the interstitial fluid pressure (IFP) and the interstitial fluid flow (IFF) in the tumor and surrounding tissues. Mathematical models of IFP and macromolecule transport were pioneered in Baxter and Jain (1989) under several simplifying assumptions including radial symmetry and spatially uniform blood vessel distributions and intravascular pressures. The models demonstrated that in steady-state, the IFP attains a plateau profile in which the IFP is high and nearly constant in the tumor interior and drops to a lower value near the tumor boundaries and surrounding host tissues. Accordingly, there is little IFF in the tumor interior whereas near the tumor boundary, the IFF is mainly directed outward towards the surrounding tissue. Experimentally, such plateaus of IFP have been observed in tumor samples (Lunt et al., 2008, Milosevic et al., 2008, Boucher et al., 1990). An increase in IFP has been implicated in the development of barriers to the transport of drugs and macromolecules in the tumor microenvironment (Ferretti et al., 2009, Jain, 1987a, Jain, 1987b). This has led to the concept of vascular normalization to reduce IFP and to decrease transport barriers to improve drug penetration into tumors (Jain, 2001, Jain, 2005b, Tong et al., 2004, Jain et al., 2007). Further, other biological factors in the tumor microenvironment, such as TAFs (Phipps and Kohandel, 2011) and ligands (Shields et al., 2007), can be convected by the interstitial fluid flow similar to drug molecules, which indicates that IFP and IFF may also play an important role in biochemical signaling (Shieh and Swartz, 2011). IFF may also promote tumor invasion via autologous chemotaxis up gradients of ligands (Shields et al., 2007). In order to predict tumor progression and response to therapy, it is therefore necessary to model and simulate both IFP and IFF.
Recently, mathematical models have been developed to investigate the role of IFP and IFF on the transport of TAFs and tumor-induced angiogenesis and on the chemotaxis of tumor cells in response to gradients of various ligands. For example, Phipps and Kohandel (2011) assumed that TAFs were convected with the IFF using Darcy's law as the constitutive assumption relating IFP with IFF, and a simplified measure of angiogenic activity (Stoll et al., 2003) was used. It was found that under the conditions of spherical symmetry and a fixed tumor radius, the highest TAF concentrations were located in the tumor interior, angiogenesis was suppressed in the tumor core, and angiogenic activity was greatest near the tumor boundary, consistent with experimental observations (Endrich et al., 1979, Fukumura et al., 2001). In Shields et al. (2007), a Darcy–Stokes (Brinkman) model was used to simulate the velocity field around a single cell to investigate the effect of IFF on gradients of . It was found that IFF could increase the gradient by approximately a factor of 3 compared to pure diffusive transport. Recently, IFP, IFF and vascularized tumor growth were coupled dynamically in a model developed by Cai et al. (2011). Here, we extend this line of research by incorporating a lymphatic system and a pre-existing vasculature.
In recent work (Macklin et al., 2009), we developed a model of vascularized tumor growth following a strategy pioneered by Zheng et al. (2005) and further developed by Bartha and Rieger (2006), Lee et al. (2006a), Welter et al., 2008, Welter et al., 2009, Welter et al., 2010, and Frieboes et al. (2010). In particular, we coupled a continuum model of solid tumor progression (Cristini et al., 2003, Zheng et al., 2005, Macklin and Lowengrub, 2008), which accounts for cell–cell, cell–ECM adhesion, ECM degradation, tumor cell migration, proliferation, and necrosis, together with an angiogenesis model (Anderson and Chaplain, 1998, Pries et al., 1998, Pries et al., 1992, Pries et al., 2009, McDougall et al., 2002, McDougall et al., 2006, Stephanou et al., 2005), which incorporates sprouting, branching and anastomosis, endothelial cell (EC) proliferation and migration, blood flow and vascular network remodeling. The tumor and angiogenesis models were coupled via oxygen extravasated from vessels and TAFs secreted by tumor cells. Oxygen, which represented the total effects of growth-promoting factors, was assumed to affect the phenotype of tumor cells and secretion of TAFs. In particular, hypoxic tumor cells were assumed to secrete TAFs, which initiated sprouting and branching in the vasculature. Once newly formed vessels anastomosed (looped), blood was able to flow through the neo-vascular network, which was modeled using a non-Newtonian Poiseuille law. Stresses induced by the growing tumor and blood flow were assumed to induce remodeling of the vascular network.
In this paper, we extend this previous model to account for (i) IFP and IFF; (ii) lymphatic vessels and drainage; and (iii) transcapillary interstitial fluid flow (e.g., vessel leakage). We model the lymphatic vessels using a continuum approach. We do not model the process of lymphangiogenesis – see Friedman and Lolas (2005) and Pepper and Lolas (2008) for such models – but instead we model the lymphatic drainage capacity, which is affected by the hydrostatic tumor pressure and the degradation of ECM by matrix degrading enzymes (MDE). We investigate how nonlinear interactions among the vascular and lymphatic networks and proliferating tumor cells influence IFP, IFF, transport of oxygen, and tumor progression. We also investigate the consequences of tumor-associated pathologies such as elevated vascular hydraulic conductivities and decreased osmotic pressure differences.
The outline of the paper is as follows. In Section 2 we present the mathematical models, and describe the numerical algorithm and parameter choices in Section 3. Then we present the results in Section 4 and discuss them in Section 5. In the appendices, we present modeling details regarding microenvironmental interactions (Appendix A) and TAFs (Appendix B).
Section snippets
The mathematical model
In this section, we present the coupled systems of equations for tumor growth, IFF and IFP, lymphatic vessels and drainage, and angiogenesis and intravascular flow. We describe each system and the coupling between them.
The coupling of variables in the continuous field
To solve for the oxygen concentration, tumor pressure, IFP and other diffusible chemical factors (MDE and TAF), we discretize the corresponding elliptic/parabolic equations (1), (4), (20), (32), (34) in space using centered finite difference approximations and the backward Euler time-stepping algorithm. The discrete equations are then solved using a nonlinear adaptive Gauss–Seidel iterative method (NAGSI) (Macklin and Lowengrub, 2007, Macklin and Lowengrub, 2008). The ghost cell method
Simulation and parameter studies
We begin by presenting a simulation of vascular tumor growth under the effect of blood/lymphatic vessels fluid extravasation/drainage with the parameters listed in Table 1 for the lymphatics, Table 2 for the discrete vasculature and Table 3 in the Appendix for the tumor model in which the parameters are as in Macklin et al. (2009). At first, oxygen extravasation is not affected by IFP (). Then, we consider the effects of IFP on oxygen extravasation (), and discuss the effects of
Discussion
We have extended previous vascular tumor modeling work by accounting for interstitial fluid pressure (IFP) and flow (IFF) as well as drainage by lymphatic vessels. We have considered blood flow with leaky vessels and have coupled the transcapillary flux with IFP. In contrast with previous work where oxygen extravasation was directly regulated by the tumor hydrostatic pressure, here regulation occurs via IFP and the hydrostatic pressure indirectly regulates extravasation by contributing to the
Acknowledgments
V.C. acknowledges funding by the Cullen Trust for Health Care, NIH/NCI PS-OC Grants U54CA143907 and U54CA143837, NIH-ICBP Grant U54CA149196, and NSF Grant DMS-0818104. J.L. acknowledges funding by the NSF, Division of Mathematical Sciences, and NIH Grant P50GM76516 for a Center of Excellence in Systems Biology at the University of California, Irvine.
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