Investigation of the Spatiotemporal Responses of Nanoparticles in Tumor Tissues with a Small-Scale Mathematical Model

Abstract

The transport and accumulation of anticancer nanodrugs in tumor tissues are affected by many factors including particle properties, vascular density and leakiness, and interstitial diffusivity. It is important to understand the effects of these factors on the detailed drug distribution in the entire tumor for an effective treatment. In this study, we developed a small-scale mathematical model to systematically study the spatiotemporal responses and accumulative exposures of macromolecular carriers in localized tumor tissues. We chose various dextrans as model carriers and studied the effects of vascular density, permeability, diffusivity, and half-life of dextrans on their spatiotemporal concentration responses and accumulative exposure distribution to tumor cells. The relevant biological parameters were obtained from experimental results previously reported by the Dreher group. The area under concentration-time response curve (AUC) quantified the extent of tissue exposure to a drug and therefore was considered more reliable in assessing the extent of the overall drug exposure than individual concentrations. The results showed that 1) a small macromolecule can penetrate deep into the tumor interstitium and produce a uniform but low spatial distribution of AUC; 2) large macromolecules produce high AUC in the perivascular region, but low AUC in the distal region away from vessels; 3) medium-sized macromolecules produce a relatively uniform and high AUC in the tumor interstitium between two vessels; 4) enhancement of permeability can elevate the level of AUC, but have little effect on its uniformity while enhancement of diffusivity is able to raise the level of AUC and improve its uniformity; 5) a longer half-life can produce a deeper penetration and a higher level of AUC distribution. The numerical results indicate that a long half-life carrier in plasma and a high interstitial diffusivity are the key factors to produce a high and relatively uniform spatial AUC distribution in the interstitium.

Extension of a multiphase tumour growth model to study nanoparticle delivery to solid tumours

“…In our example with a fixed pore radius of r 0 = 150 nm, a permeability coefficient of P v = 3.2 × 10 −5 mm/s as determined by Dreher et al [54] has little influence on the nanoparticles distribution, which rises by only 5% when compared to the baseline simulation with P v = 0. In contrast, the values P v = 3.5 × 10 −4 mm/s by Ho et al [53] and P v = 1.28 × 10 −3 mm/s by Chou et al [55] result in an increase of the mass fraction of nanoparticles in the IF by 41% and 84%, respectively.…”

“…As depicted in Fig 7, the permeability coefficients measured by Dreher et al [54], Chou et al [55] and Ho et al [53] differ by more than one order of magnitude. Dreher et al [54] use a mouse model to measure the apparent permeability in vivo.…”

“…We choose dextran as model carrier to analyse the effect of different vascular permeabilities on the distribution of nanoparticles in the IF. In Fig 7, we summarise permeability coefficients of dextrans with different molecular weights, as determined by Dreher et al [54], Chou et al [55] and Ho et al [53]. The summary in Fig 7 reveals that the measured permeability coefficients differ by more than one order of magnitude.…”

“…This includes an unknown influence of convection as stated by the authors. Chou et al [55] use the Kedem-Katchalsky equation to fit permeability and diffusivity of dextrans to the data of Dreher et al [54]. Ho et al [53], in contrast, reason that a systematic quantitative study of permeabilities utilising animal models is difficult.…”

A novel individual‐cell‐based mathematical model based on multicellular tumour spheroids for evaluating doxorubicin‐related delivery in avascular regions

“…Mathematical models are uniquely advantageous for describing and elucidating complex kinetic profiles (Hicks et al, ; Evans et al, ; Lu et al, ). Although there is a wealth of mathematical modelling of the drug penetration profile of macromolecular drugs in the avascular region of tumours (Goodman et al, ; Thurber and Wittrup, ; Thurber et al, ; Chou et al, ; Vasalou et al, ), only a limited number of studies have focused on low MW drugs. Richard et al quantitatively measured the actual diffusion coefficient of vinblastine in multicellular layers (MCL) according to the Fickian diffusion model but did not consider the effect of cellular uptake (Modok et al, ).…”