A Quantitative Theory of Solid Tumor Growth, Metabolic Rate and Vascularization

The relationships between cellular, structural and dynamical properties of tumors have traditionally been studied separately. Here, we construct a quantitative, predictive theory of solid tumor growth, metabolic rate, vascularization and necrosis that integrates the relationships between these properties. To accomplish this, we develop a comprehensive theory that describes the interface and integration of the tumor vascular network and resource supply with the cardiovascular system of the host. Our theory enables a quantitative understanding of how cells, tissues, and vascular networks act together across multiple scales by building on recent theoretical advances in modeling both healthy vasculature and the detailed processes of angiogenesis and tumor growth. The theory explicitly relates tumor vascularization and growth to metabolic rate, and yields extensive predictions for tumor properties, including growth rates, metabolic rates, degree of necrosis, blood flow rates and vessel sizes. Besides these quantitative predictions, we explain how growth rates depend on capillary density and metabolic rate, and why similar tumors grow slower and occur less frequently in larger animals, shedding light on Peto's paradox. Various implications for potential therapeutic strategies and further research are discussed.


Derivation of metabolic scaling
The total volume flow rate of blood to the tumor,Q, is the product of the total number of supply vessels, N 0 , and the volume flow rate in each of them,Q 0 . By fluid conservation we haveQ = N 0Q0 = N capQcap , where N cap is the total number of capillaries, each with flow rateQ cap . The total blood flow rate for the tumor is thus directly proportional to the total number of capillaries,Q ∝ N cap . The total blood volume in the network, summed over all vessels, is V b = N k=0 N k V k = N k=0 πn k r 2 k l k , where k is the generation number, N is the number of generations from feeding vessel to capillary, and V k is the volume of a vessel at level k. A further prediction of energy minimization is that V b scales linearly with (viable) mass, in agreement with data for some tumors (V b ∝ m v ) and for the whole body however, when the opposite condition holds, 2a + b > 1, V b scales super-linearly with capillary number, Total blood flow rate,Q ∝ N cap , is therefore predicted to exhibit power-law scaling with viable tumor size:Q ∝ m β v , consistent with a large amount of data and past conclusions. Finally, since the rate of supply of nutrients and oxygen is directly proportional to blood flow rate, we have for the metabolic rate of the tumor with B 0 (M ) a normalization factor that, in principle, depends on host mass, M .

Network Resistance
The resistance of the network can be found by summing over the resistances of each level in the network.
The viscous resistance to blood undergoing steady, laminar flow is given by the Poiseuille formula R k = 8µl k /πr 4 k , where R k is the resistance of a vessel at level k in the network. Setting δ = r k+1 /r k and λ = l k−1 /l k , and ignoring small effects such as turbulence and non-linearities at junctions, the total resistance for the tumor is given by where µ is the blood viscosity. Now, if nδ 4 /λ < 1 and N 1, a good approximation is , with the smallest vessels in the network dominating the resistance. Since nδ 4 /λ < 1 for the malignant tissues described in this paper, the tumor resistance is inversely proportional to the number of capillaries, and hence the total blood flow , consistent with results from [1], where the resistance to flow in tumors of various weights was measured as the slope at the point where the flow rateQ T became linearly proportional to pressure. Without knowing the actual perfusion pressure and flow rate in vivo, this is a good approximation to the resistance of the tumor vascular network.

Metabolic rate of small tumors
Very small tumors are supplied by the diffusion of nutrients from nearby host capillaries so that the supply to the tumor is proportional to the product of the number of host capillaries within the diffusion distance from the tumor, N cap T , and the flow rate through each host capillary, Q cap . In this initial stage we assume that all host capillaries displaced by the tumor volume, V T , act as supply sources because they are adjacent to the tumor and within a diffusion distance of its surface. The total blood flow rate to the tumor is then given bẏ Thus, for very small tumors, B T is predicted to increase linearly with total tumor mass, m T , but decrease with host mass as M −1/4 .

Recruitment of vessels from host
The host tissue from which the tumor draws blood is effectively a shell surrounding the tumor with thickness determined by the diffusion distance τ , and volume τ S T , where S T is the tumor surface area.
The diffusion distance, τ , which depends on production and consumption rates of angiogenic factors, is assumed to vary inversely with endothelial cell density. The endothelial cell density in the tissue shell is determined by the density of the surface area of host vasculature, k 2πN k r k l k /V . Since capillaries have constant surface area and often dominate this scaling, the endothelial cell density scales linearly with capillary density, ρ cap . For large tumors, this gives This result is consistent with the idea that tumors attach to the host vasculature several levels above the capillaries and that tumors are able to attach to successively larger vessels in larger hosts.
If the host supply vessels are recruited from the k = Lth level of the host hierarchy, then the total number of such supply vessels in the shell of thickness τ is ∼ ρ L S T τ , where ρ L ≈ N L /V is their average density and N L their total number in the whole body. If all these vessels are recruited by the tumor, they contributeQ to its total blood supply rate, whereQ L is the flow rate in an Lth level vessel. Assuming Euclidean Development of necrotic core during angiogenesis: Two regimes ii) The later phase when the tumor has access to more and larger vessels, and tumor vascular inefficiencies become increasingly important. In this case, τ ∝ M 1/4 and B 0 (M ) is approximately independent of M . Moreover, we find that as the tumor grows, β evolves from 1 towards 3/4 implying that γ changes from 2/3 to 8/9 so that ultimately with the tumor becoming increasingly necrotic.

Data Selection and Analysis
We culled the literature for available empirical data to test the many predictions of our theory. We provide references for these data, which also explain the methods used to collect those data. All datasets were fitted using the Matlab curve-fitting toolbox. The data in Figs. 2-3 were fitted using Type 1 linear regression. For tumor growth curves, nonlinear fits were made using a Trust-region algorithm. To estimate the size of the tumor growth transition from exponential to sigmoidal phases, a small sample of early-time growth data was fitted with an exponential, and late-time data points were fitted to the sigmoidal functional form. Remaining intermediate data were added to the exponential regime if they fell within the 95% confidence intervals for the initial exponential fit or were added to the sigmoidal phase if they fell within the 95% confidence intervals for the initial sigmoidal fit. Because a sigmoidal curve provides a reasonable description to the full range of data, empirical data that were within the 95% confidence intervals of both initial fits were included as part of the sigmoidal phase. After assigning the data according to this algorithm, new and final fits were calculated for the exponential and sigmoidal phases, and these fits were smoothly connected at their intersecting point. The coarseness of the data preclude the ability to rigorously determine the growth phases by statistics alone. Therefore the fits displayed are motivated strongly by our theory and should be considered demonstrative of our general approach. These fits often do not differ significantly from fitting the entire range of data with a sigmoidal curve, which represents a more phenomenological approach as in Guiot et al [2].