To complement experimental work in nanoparticle therapies, mechanistic mathematical modeling and computer simulation can be used to better understand experimental results and provide quantitative guidance for more efficient design of nanotherapeutics. Optimizing carrier and drug penetration into the tumor tissue is critical to maximize the therapeutic effect. Toward this goal, I developed a mathematical model of nanoparticle diffusion and uptake in a spheroid approximation of a solid tumor segment without capillaries. Our model differs from other models that can be found in the literature by taking into account the effect of shape and size on the diffusion constant of nanoparticles. Our model also builds on the previous models by incorporating the rate of endocytosis and how it is affected by size, shape, and surface modification of nanoparticles. As the model nanoparticle, I used the nucleoprotein components of the tobacco mosaic virus . Virus-based plant nanoparticles, such as TMV, provide a unique platform for nanomedical engineering because their dimensions are known and tunable on the molecular level, which cannot readily be accomplished with synthetic nanoparticles. Native TMV particles form a cylindrical structure measuring 300 × 18 nm with a 4 nmwide hollow interior channel. TMV is composed of single-stranded RNA wrapped inside a hollow nanotube formed by 2130 identical coat proteins. TMV offers a programmable scaffold for both genetic engineering and chemical bio-conjugation to impart new functionalities, e.g. therapeutic payloads. TMV virion formation can be initiated by self-assembly of coat proteins from an RNA hairpin forming sequence. This origin of assembly site is the only sequence required to promote a bidirectional coat protein self-assembly along the template RNA. This principle has been exploited to produce TMV nanotubes with diverse shapes such as kinked nanoboomerangs or branched tetrapods.This RNA-templated self-assembly principle has also been used to produce TMV-like nanotubes with distinct longitudinal domains as well as materials of defined aspect ratio.In previous studies,nft channel we have shown that bio-distribution and tumor homing is a function of the carrier’s aspect ratio.
With higher aspect ratio, particles avoid clearance by the mononuclear phagocyte system, resulting in increased tumor homing.Nevertheless, a balance must be established between immune evasion, tumor homing, and tissue penetration. While higher aspect-ratio materials have enhanced tumor homing, the higher molecular weight particles have slower diffusion rates. The TMV platform technology provides a high precision platform with which to specify aspect ratio and surface chemistries that affect tissue penetration in tumorspheroids. I chose to study TMV diffusion in a spheroidal cell-culture system with different sizes and cell densities. This 3D cellular system mimics a small segment of a solid tumor between capillaries and bridges the gap between 2D tissue culture and in vivo mouse models for screening therapeutics. In tumor tissue, the combination of leaky vasculature and deficient lymphatic clearance leads to diffusion as the driving mode of nanoparticle transport and penetration into the tumor tissue.For this study, I focused on ‘stealth’ TMV formulations with reduced cell uptake rates produced by coating the particle surface with polyethylene glycol . Targeted TMV formulations with molecular specificity and increased cellular uptake rates were simulated by displaying the integrin specific peptide ligand RGD on their surface. PEG and RGD are surface modifiers frequently used in nanoparticle engineering to promote immune invasion and targeted endocytosis, respectively. These coatings serve as good model systems whose results can be translated to other nanoparticle formulations.I developed a mathematical model of TMV diffusion and uptake in a spheroid tumor model to evaluate the effect of particle aspect ratio . The input to this model was a bolus injection of a known TMV mass in the medium surrounding the tumor . The rate of diffusion in the interstitial space is much slower than in the surrounding medium so that the distribution of TMV in the surrounding medium is uniform . In addition, the volume of the surrounding medium is much greater than the volume of the spheroid so that the changes in TMV concentration in the surrounding medium are negligible Supporting Equations 6.1.
The tumor cell density within the spheroid segment is uniform and considered as a continuum. The rates of cell proliferation and death are assumed negligible relative to the other TMV rate processes so that the viable cell volume remains constant. From the interstitial space, I assume that TMV is taken up irreversibly by tumor cells at a constant rate that is dependent on the aspect ratio and surface chemistry of TMV nanorods . Furthermore, the TMV does not interact with extracellular matrix components and cannot bind to them. The model parameter values known from direct measurement are the mass of TMV injected, the volume of the surrounding medium , and the radius of the spheroid . The parameters that must be estimated indirectly are the cellular uptake rate coefficient and the diffusion coefficient . For each experiment, these coefficients are constants. This implies that free receptors are always available at the cell surface so that k is constant in any experiment. For different experiments, however, their values change depending on their surface area, shape and cell density within the spheroid . The uptake rate coefficient is directly proportional to the total cell surface area as indicated by the cell density, where k = k0. The diffusion coefficient is a complex function of the cell shape and cell density. Our tumor micro-environment system consists of a spheroidal cancer cell-culture whose diameter can vary between a few hundred micrometers to 1 mm, which corresponds to the heterogeneous spacing of capillary distribution within the tumor . Modeling the diffusion of nanoparticles in the tumor tissue and quantifying the time scales as a function of capillary and cell density could inform dosing and administration schedules. The physiological barriers and diffusion rates of nanoparticles also depend on nanoparticle shape, size and surface chemistry. The simulated effects of spheroid radius, cell density, and aspect ratio on the TMV concentration distributions without cellular uptake are shown in Figures 6.2, 6.3 and 6.4, respectively. The parameters used in each of these figures are summarized in Table 6.1. These results and their significance are discussed in the following sections.The spheroid segment radius represents the distance between capillaries.
The intercapillary distance is highly regulated by a fine balance between angiogenic factors that promote or inhibit vessel growth, as well as the oxygen and nutrient consumption by the surrounding cells.In healthy tissue, particle diffusion from the vessels to the cytoplasmic membrane of surrounding cells does not exceed 100 μm. In the tumor micro-environment, however,hydroponic nft the oxygen consumption is lowered and the tolerance of cancer cells to hypoxic conditions is increased. Tumors with a high rate of oxygen consumption have a higher microvascular density and, therefore, a smaller intercapillary distance. On the other hand, tumors with a low rate of oxygen consumption have a lower microvascular density and, therefore, a higher intercapillary distance.This phenomenon is currently being investigated for nanoparticle-based antiangiogenic tumor therapy.By reducing the oxygen supply to the tumor site, anti-angiogenic tumor therapies aim to prevent the growth and aggressiveness of the tumor. To quantify the effect of different intercapillary distances within the tumor micro-environment, I simulated the diffusion of TMV in a spheroid system without cellular uptake for a range of radii in the absence of cellular uptake . Within the tumor cell spheroid, the simulated concentration distributions at various times of TMV with different spheroid radii are shown in two and three dimensions. This poor tumor penetration correlates with increasing risks of survival of cancer cells and promotes drug resistance.When the tumor cell density increases, the cytotoxicity of chemotherapeutic drugs such as vincristine, bleomycin, and doxorubicin is impaired.Increasing the cancer cell density within the spheroid decreases the void volume through which nanoparticles can diffuse as represented by a smaller diffusion coefficient. With high cell density, the limitation of TMV nanoparticle penetration is a major barrier to chemotherapeutic drug delivery in the deep tissue, which also correlates with increasing risks of survival of cancer cells and promotes drug resistance.While smaller aspect-ratio rod-shaped nanoparticles have higher diffusion and accumulates more easily in the deep tumor tissue, the higher aspect-ratio nanoparticles have enhanced margination toward blood-vessel walls, increased transport across tissue membranes, and reduced clearance by phagocytosis.In other words, a “one-size-fits-all” nanoparticle does not exist and a compromise must be made to optimize the diffusion and accumulation of nanoparticles within the tumor without impairing their ability to extravasate , cross tissue membranes, and evade the immune system. With complementary data, this model can provide a basis for predicting the aspect ratio that promotes optimal accumulation of nanoparticles injected intravenously.Perhaps, a better approach would be to inject intravenously a cocktail of TMV nanoparticles with various aspect ratios. In this scenario, the lowest aspect-ratio TMV are less likely to reach the tumor site, but the fraction that do penetrate the tumor can diffuse more readily than the higher aspect ratio TMV in the deep tumor tissue. In the meantime, the higher aspect ratio nanoparticles can reach the tumor site more readily, but only accumulate in the peripheral tissue of the tumor. The net result would be to improve overall drug distribution and maximize efficacy. The simulations presented above do not include TMV uptake by cells so that the effects on diffusion are not obfuscated. While targeted nanoparticle formulations can increase delivery, endocytotic clearance of targeted nanoparticle can reduce drug distribution and tumor cell access.
To assess the effect of cell uptake on TMV distribution throughout the spheroid, I evaluated the cell uptake rates of TMV in cancer cells experimentally: fluorescently-labeled, RGDtargeted TMV formulations were obtained as described by Pitek et al.310 A fluorescence assay was developed to quantify TMV particle uptake cancer cells over time . I chose triple negative breast cancer cells as our model cell line for their relatively highexpression of v3 integrins.I determined that the targeted TMV formulation exhibits a cell uptake rate of 130 particles/h/cell. With this experimental value, we can extrapolate cell uptake rates of PEGylated and RGD-targeted TL, TM, and TS particles .These data and resulting cell uptake rates are summarized in Table 6.2. While RGD-targeted formulations are readily taken up by the cells, PEGylated formulations show negligible cell interactions. The PEGylated formulations with TS and TM aspect ratios have comparable effects on uptake. The targeted formulations with TL and TM have comparable effects on uptake, but the uptake with TS increases significantly . The experimental data shows that TMV-PEG formulations exhibit low uptake with time. The TMV-RGD formulations, however, display a biphasic behavior: rapid cell uptake within the first 3 h followed by a plateau region with little to no cellular uptake, most likely indicating saturation. This behavior is typical of particle internalization mediated by cell surface receptors.The rate of cellular uptake of TMV reported in this study is much smaller than the rates reported for synthetic nanoparticles. Doiron et al.320 reported that spherical polystyrene nanoparticles with diameters ranging from 20 nm to 500 nm had uptake rates ranging from 6.6×107 particles/h/cell to 12,000 particles/h/cell respectively within the first 3 h of incubation. In addition Huang et al.reported rod-shaped gold nanocrystals displaying RGD peptideson their surface had a rate of internalization in A549 lung carcinoma cells equivalent to 4500 particles/cell/h within the first 2 h of incubation at 37°C. However, the same nanoparticles coated with single-chain variable fragment peptide to target the epidermal growth factor receptor were internalized at a slower rate of 1,250 particles/cell/h. This demonstrates that the rate of cellular uptake is dependent on nanoparticle shape, surface chemistry, as well as the nature of the molecular receptor targeted. Using the evaluated cellular uptake of TMV formulations , I simulated TMV diffusion in a spheroid cell system with different rate coefficients of cell uptake and aspect ratios . This prevents deep tissue penetration because cell uptake occurs at a rate much higher than diffusion. Coating TMV with RGD peptides to target integrin receptors further decreases the TMV concentration within the spheroid. Active targeting of receptors overexpressed on cancer cells is commonly used to promote tissue specificity and accumulation. However, it is counterproductive for tissue penetration.For TM , the TMV concentration reached the center of the tumor within 3 h. At steady state, its concentration at the center was 4% of the concentration in the surrounding medium. For TS , the TMV concentration reached the center of the tumor within 3 h, but its steady-state concentration at the center was 12% of the concentration in the surrounding medium.