Drug delivery in a tumour cord model: a computational simulation

Abstract

The tumour vasculature and microenvironment is complex and heterogeneous, contributing to reduced delivery of cancer drugs to the tumour. We have developed an in silico model of drug transport in a tumour cord to explore the effect of different drug regimes over a 72 h period and how changes in pharmacokinetic parameters affect tumour exposure to the cytotoxic drug doxorubicin. We used the model to describe the radial and axial distribution of drug in the tumour cord as a function of changes in the transport rate across the cell membrane, blood vessel and intercellular permeability, flow rate, and the binding and unbinding ratio of drug within the cancer cells. We explored how changes in these parameters may affect cellular exposure to drug. The model demonstrates the extent to which distance from the supplying vessel influences drug levels and the effect of dosing schedule in relation to saturation of drug-binding sites. It also shows the likely impact on drug distribution of the aberrant vasculature seen within tumours. The model can be adapted for other drugs and extended to include other parameters. The analysis confirms that computational models can play a role in understanding novel cancer therapies to optimize drug administration and delivery.

Keywords: computational modelling; drug delivery; drug transport and binding; mathematical modelling; pharmacokinetic resistance.

Figure 1.

A three-compartment model of drug distribution in tissue. C₁ represents extracellular drug concentration, C₂ is free intracellular drug concentration and C₃ is bound intracellular drug concentration.

Figure 2.

Geometry for a two-dimensional cylindrically symmetric compartment model. The rectangle with the thick outline represents the computational domain.

Figure 3.

Spatialvariation of exposure ($\int \ast \hspace{0.17em}\mathrm{d}t$) to extracellular drug, C₁ (a,c), and bound drug, C₃ (b,d), both at t=72 h. The black circles represent ‘exposure’ in the vessel and the surface plots represent exposure in the tissue. Each surface is coloured according to its height. Parameter values are as in table 1 (standard vasculature) for the top two plots, but modified so that $k_v\to 10\hspace{0.17em}{k}_v$, $\mathrm{\lambda }\to \mathrm{\lambda }/10$ and $l\to l/2$ (narrow, leaky vessels) for the bottom two plots. The single short-infusion pharmacokinetic profile, PK1, was used as input.

Figure 4.

(a–f) Dependence of exposure to bound drug ($\int C_3\hspace{0.17em}\mathrm{d}t$) at t=72 h on model parameters, for the two-dimensional, cylindrically symmetric model: each plot shows the response for all three PK profiles at points ‘near’ to the supply (r=l+10 μm, z=10 μm) and ‘far’ from the supply (r= l+170 μm, z=490 μm). The vertical dashed lines indicate the standard parameter values in table 1.

Figure 5.

Dependence of exposure to bound drug ($\int C_3\hspace{0.17em}\mathrm{d}t$) at t=72 h on binding ratio β for the two-dimensional, cylindrically symmetric model: the plot shows a magnified version of the bottom right corner of figure 4a.

Figure 6.

Dependence of exposure to bound drug ($\int C_3\hspace{0.17em}\mathrm{d}t$) at t=72 h on binding rate k₂, and unbinding rate k₋₂, for the two-dimensional, cylindrically symmetric model with profile PK1: close to the supplying vessel (r=26 μm, z= 10 μm, a); far from the supplying vessel (r=186 μm, z= 490 μm, b). Each surface is coloured according to its height. The vertical dashed lines indicate the standard values of k₂ and k₋₂ in table 1.

Figure 7.

Dependence of exposure to bound drug ($\int C_3\hspace{0.17em}\mathrm{d}t$) at t=72 h on dose, D₀, for the two-dimensional, cylindrically symmetric model: close to the supplying vessel (r=26 μm, z=10 μm; a,c); far from the supplying vessel (r=186 μm, z=490 μm; b,d). In one set of simulations, the binding rate k₂, and unbinding rate k₋₂, are the standard values taken from table 1 (top), in the other, much stronger binding and weaker unbinding are used (k₂=2.95×10⁻² μM⁻¹ s⁻¹ and $k_{-2}=4.38\times {10}^{-7}\hspace{0.17em}{\mathrm{s}}^{-1}$, bottom). The vertical dashed lines indicate the value of D₀ in §3.

Figure 8.

Comparison of bound drug, C₃, profiles for different total doses (${\int }_0^{\mathrm{\infty }}{C}_v(t)\hspace{0.17em}\mathrm{d}t$): single, short infusion (PK1, a) and prolonged subjection to constant concentration (PK3, b).

Figure 9.

Dependence of survival fraction on dosefor the two-dimensional cylindrically symmetric model where a cell is assumed to die if its exposure to bound drug exceeds 165 μM h: standard binding rate k₂, and unbinding rate k₋₂, from table 1 (a); much stronger binding and weaker unbinding (k₂=2.95×10⁻² μM⁻¹ s⁻¹ and $k_{-2}=4.38\times {10}^{-7}\hspace{0.17em}{\mathrm{s}}^{-1}$, b).

Figure 10.

Dependence of the exposure to bound drug ($\int C_3\hspace{0.17em}\mathrm{d}t$) at t=72 h on spheroid radius, for the one-dimensional, spherically symmetric model: at the edge of the spheroid (a) and in its centre (b). Each plot shows the exposure for all three PK profiles.