Effect of tumor shape and size on drug delivery to solid tumors
© Soltani and Chen; licensee BioMed Central Ltd. 2012
Received: 6 January 2012
Accepted: 30 March 2012
Published: 25 April 2012
Tumor shape and size effect on drug delivery to solid tumors are studied, based on the application of the governing equations for fluid flow, i.e., the conservation laws for mass and momentum, to physiological systems containing solid tumors. The discretized form of the governing equations, with appropriate boundary conditions, is developed for predefined tumor geometries. The governing equations are solved using a numerical method, the element-based finite volume method. Interstitial fluid pressure and velocity are used to show the details of drug delivery in a solid tumor, under an assumption that drug particles flow with the interstitial fluid. Drug delivery problems have been most extensively researched in spherical tumors, which have been the simplest to examine with the analytical methods. With our numerical method, however, more complex shapes of the tumor can be studied. The numerical model of fluid flow in solid tumors previously introduced by our group is further developed to incorporate and investigate non-spherical tumors such as prolate and oblate ones. Also the effects of the surface area per unit volume of the tissue, vascular and interstitial hydraulic conductivity on drug delivery are investigated.
KeywordsDrug delivery Solid tumors Computational fluid dynamics Tumor shape and size Interstitial fluid
Cancer causes one in every four deaths in North America and is the second most common cause of death worldwide . Solid tumors account for 85% of human cancers . Although the most important cancer treatment is surgical removal of such tumors, the key to a successful cure often involves efficient delivery of anticancer drugs to the tumor site after the surgery. Many new drugs have been developed, but lack of an efficient means of delivery makes them less effective. Moreover, some of these drugs induce biochemical reactions in the body that produce toxicity. Bioengineers are primarily concerned with both the transport of drugs within the body (which usually involves systemic delivery through the blood supply) and with biochemical reactions or conversions at tumor sites [2, 3]. All of these problems demonstrate that solutions to drug delivery limitations are urgently needed .
Two factors inhibit the effective delivery of drugs within tumors: non-uniform blood supply and non-uniform interstitial pressure distribution [4, 5]. Drugs concentrate most readily in areas with the best blood supply. In solid tumors, these areas are closest to the blood vessels (the vasculature) and a tumor’s peripheral walls; however, 90% of a tumor receives little or no drug, meaning that treated tumors tend to regrow, as only their outer cells have been killed by the drug [6–9]. Variations in the interstitial (i.e., of connective tissue in which cells are embedded) pressure reduce fluid exchange and further inhibit the movement of the drug into the center of the tumor.
Modeling drug delivery involves processes such as drug diffusion, convective transport in extracellular matrices, drug extravasation from blood vessels, tissue elimination by the lymphatic system, and intracellular internalization. In all of these processes, computational fluid dynamics (CFD) can play a crucial role in clarifying the mechanisms of drug delivery from the injection site to absorption by a solid tumor. In-vitro release profiles of systemically administered drugs have been combined with state of the art computational fluid dynamics simulations to predict both temporal and spatial drug delivery in many studies [15–20]. Temporal and spatial changes in blood flow have also been studied with a focus on capillary-network or single-vessels [21, 22]. Before Baxter et al. [2, 23, 24] introduced their innovative model of interstitial pressure as a function of tumor radius, little was known about tumor modeling, except that interstitial pressure is highest at the center of a tumor  and that pressure is directly proportional to tumor size [25, 26]. The main focus of future drug delivery modeling involves the transport of the drug in tissues after drug release by either systemic administration or implantation.
As spherical tumors are the easiest to examine analytically, they have been used in most studies. The effect of tumor shape has not been addressed in the literature except the analytical study of by El-Kareh and Secomb . The numerical method introduced here allows modeling of more complex shapes and promotes a better understanding of the complex mechanisms of interstitial fluid transport that effective drug delivery must depend on. In studying tumor modeling, the numerical method, which introduces more features of drug delivery to solid tumors, is more effective than the analytical method. To design an optimum scheme for drug delivery, the transport mechanisms and obstacles to drug delivery have to be clarified, which is one of the main objectives of this paper.
The proposed CFD model is made for both spherical and non-spherical tumors and their surrounding normal tissues. This model can be further extended to study geometries reconstructed from high resolution images. In this study the tumor and its surrounding tissue are assumed to be rigid porous media. The vasculature as a source term varies spatially. The grid generation divides the whole domain or geometry into finite volumes, called meshes. Interstitial fluid flow equations in porous media are solved using a CFD code (based on the proposed CFD model) that employs unstructured grids (tetrahedral elements) to handle non-spherical tumors.
The tissues most relevant to this discussion are the vasculature (vessels that, with the heart, comprise the circulatory system that carries blood throughout the body), the interstitium (or interstitial space), and the cells. Also relevant is the lymphatic system, which, simply put, is responsible for tissue drainage. The vasculature system involves vessels (essentially tube-like structures) of varying sizes, from the large arteries and veins down to the much smaller arterioles, venules and capillaries . The interstitium is formed of fibers such as collagen, which gives structural stability, glycosaminoglycans (GAG), and other proteins. Together, these fibers make up the gel-like region between blood vessels and cells. The cells, occupying the cellular space, include specific tissue cells (i.e., the cancer cells of a solid tumor) and others such as pericytes, macrophages, and fibroblasts not discussed in this paper [2, 23, 24]. In normal physiology, fluid seeps slowly but constantly from blood vessels into the surrounding tissues. The lymphatic system then reabsorbs this lost fluid and returns it to the blood stream. All the above tissues must be considered in any discussion of drug delivery to tumors. Specifically, to be effective, drugs must cross the blood vessel wall, traverse the interstitial space containing the cancer cells, and bind to and (if the target is intracellular) penetrate the cancer cell membrane. The lymphatic system then removes excess fluid and debris. However, a lack of such lymphatic drainage in solid tumors has been reported in the literature [2, 3]. Computer simulations show that this lack of lymphatic system involvement may result in a build up of interstitial pressure, leading to cessation of the usual blood seepage from vessels. Consequently, large molecules cannot be carried out of vessels to interact with tissue. As some drug particles, including Monoclonal Antibodies (MAbs) used to fight cancer, are large and move very slowly within tissues, they cannot reach the tumor site and are thus ineffective .
Mathematical Model of Interstitial Fluid Transport
where and r cm are the hydraulic conductivity of the interstitium, the interstitial fluid pressure, the interstitial fluid velocity and the radial position, respectively. In the case of anisotropic and heterogeneous porous media, k is a tensor and function of the location in the medium.
The parameters used in these equations are , the volumetric flow rate into the lymphatics; , the surface area per unit volume of the lymphatics; , the hydraulic conductivity of the lymphatic wall; and , the hydrostatic pressure of the lymphatics.
in which the interstitial pressure that yields zero net volume flux out of the vasculature is called the effective pressure, P e , Eq. (16). The steady state pressure and effective pressure in solid tumors with no lymph vessels are the same. If P i = P e , no exchange of fluid occurs between the interstitial space and blood vessels.
Material properties used in numerical simulations, as taken from []
Rippe et al. (1978)
Swabb et al. (1974)
Pappenheimer et al. (1951)
Hilmas and Gilette (1974)
Brace and Guyton (1977)
Brace and Guyton (1977)
Ballard and perl (1978)
The parameters used in Eq. (24) are , the volumetric flow rate out of the vasculature per unit volume of tissue; , the surface area per unit volume for transport in the tumor; , the hydraulic conductivity of the microvascular wall; , the effective pressure; and , the interstitial fluid pressure. All parameters used to solve this equation are listed in Table 1.
Figure 5, shows that in a spherical solid tumor with a certain radius, R, if hydraulic conductivity is increased, the volumetric flow rate first increases to reach an optimum value (maximum) and then decreases. This graph also indicates that an increase in the diameter of spherical solid tumors causes a decrease in the optimum value of the volumetric flow rate. Decreasing one order of magnitude of tumor radius increases the volumetric flow rate approximately two orders of magnitude. This optimum value occurs in response to a small value of hydraulic conductivity in spherical solid tumors with a relatively larger diameter. The values of hydraulic conductivity that result in optimum values of the flux are called optimum values of hydraulic conductivity. The filtration flux (or the volumetric flow rate), shown in Figure 5 as a function of L p S/V, indicates that the usual values for L p or S/V in the literature, listed in Table 1, are much greater than the optimum values shown in this graph. , the optimum value of hydraulic conductivity, can be found easily in a graph such as Figure 5 for different tumor sizes, or one can find the appropriate size for a specific tumor tissue with a known value of L p to have the maximum drug flow rate. In Figure 5, L p changes linearly from its minimum values to its maximum values. On the other hand, the pressure difference (between effective pressure and IFP) changes from P e (when IFP is equal to zero) to zero (when IFP is equal to the effective pressure). When L p is at its minimum, the pressure difference is at its maximum, and vice versa. This circumstance results in a peak in the volumetric flow rate curve in terms of the hydraulic conductivity of the microvascular wall, L p , and surface area per unit volume, S/V, or a combination of these factors through Eq. (24).
The effect of increasing the hydraulic conductivity of microvessels, L p , in terms of drug delivery to solid tumors is significant to this discussion. There are two opposing phenomena: the total fluid filtration flux for the whole solid tumor and local filtration within the solid tumor; therefore, the liquid source term, the filtration flux at the center of the tumor, has its highest value for an optimum value of L p , as shown by El-Kareh and Secomb . As they have shown in their study analytically, and is indicated here numerically, this optimum value of L p is a function of the size and shape of solid tumors. An increase in L p has a direct effect on the total rate and an opposite effect on the local filtration flux. Increased vessel leakiness cannot result in more uniform drug distribution through tumors much larger than 0.25 cm.
Starling’s law shows that, because of low diffusivity, the dominant form of drug transport through vessel walls (transvascular transport) is convection . Based on Eq. (5), both hydraulic conductivity of the vessel wall and pressure difference across the wall are effective in convective transport, again according to Starling’s law. Similar behavior is true for S/V; thus, an increase in L p and S/V does not have a positive effect, and there are optimum values for both of these parameters. However, the analysis of the work done in , as well as this study, shows that, except when the nodule radius is smaller than 0.1 cm, the advantage of decreasing P i is more important than that of decreasing the surface area.
As indicated, an increase in the hydraulic conductivity results in a volumetric flow increase, but after a continued increase, volumetric flow approaches a certain value for all different tumors. This pattern for tumors of all diameters is the same; the only difference is that the smaller the tumor diameter, the sooner the leveling in volumetric flow.
The parameter αdoes not apply exactly in a non-spherical tumor; therefore, the general form of the governing equation, different from the more specific form of that applied in spherical solid tumors, has to be used here. This parameter, α, is a combination of other fundamental variables such as hydraulic conductivity, which turns out to have a clean short definition when solved for a sphere. This study shows that for a distinctly non-spherical tumor (such as a prolate or oblate), everything hinges on the shortest dimension, and the longer dimension is irrelevant, as any fluid or material flows to reach the closest surface.
Because El-Kareh et al.  used an analytical approach to solve the governing equations in prolate and oblate spheroidal shapes, they had to change the coordinate system from a spherical one to a more complicated coordinate system, with oblate and prolate spheroidal coordinates. In this coordinate system, the governing equation, the Helmholtz equation, has a much more complicated form than the original equation. The solution for this new form of equation is a Fourier series. In the work proposed here, the numerical method is applied and, obviously the solution is not limited to any specific shape, offering a freedom that is one of the main advantages of the numerical method over the previous one.
The parameter α/R is independent of tumor geometry. It is a function of hydraulic conductivity (an interstitial property) and vessel permeability. The main problem is defining a characteristic length instead of R in non-spherical tumors. The pressure profile, however, is a function of tumor geometry. Using a solution for IFP that was calculated assuming a spherical profile will give a solution as a function of α, r, and R. Not surprisingly, equations tailored to the different tumor shapes will yield different IFPs, as these pressures are the outcomes of the equations, and thus reflect the discrepancies. What will happen, whether a tumor is a sphere or not, is that IFP will be approximately equal to the vascular pressure, P B , throughout most of the interior. Only near the boundary, whether that boundary is for a sphere or spheroid, will there be a steep pressure gradient as IFP falls to the pressure of the surrounding tissue. For a very small radius, the differences will be significant. Above the critical radius, shape is almost irrelevant.
This study shows that, as a rule, it is not true that the leakier the vessels, the higher the value of convective transport of drugs to solid tumors. The results do show that only in spherical solid tumors with a radius of less than 0.25cm, or in spheroidal ones with the same volume, can drug convection be increased by making vessels more leaky. For spheroidal shapes, the convection of drugs inside is higher than it is in spherical ones, and it seems that for more irregular shapes, which are generally found in the body due to limitations imposed by neighboring tissues and organs such as the brain, this effect is more marked.
For shapes studied in this paper, results show that the dependency of the maximized flux (as a function of L p and S/V or their multiplication) on size is much stronger than its dependency on shape. It should be mentioned that due to very low diffusivity, the high permeability of vessels cannot support homogeneous distribution inside tumors because this high value results in more diffusive transport only in a narrow area around the vessels.
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