Theoretical analysis of conjugate localization in two-step cancer chemotherapy

Spatio-temporal tumour model for analysis and mechanism of action of intracellular drug accumulation

We have developed a one-dimensional tumour simulator to describe the biodistribution of chemotherapeutic drugs to a tumoral lesion and the tumour cell's response to therapy. A three-compartment model is used for drug dynamics within the tumour. The first compartment represents the extracellular space in which cells move, the second corresponds to the intracellular fluid space (including cell membrane) which is in direct equilibrium with the extracellular space, and the third is a non-exchangeable compartment that represents sequestered drug which is trapped in the nucleus to damage the cellular DNA,directly triggering cell death. Analytical and numerical techniques (Finite Element Method) are used to describe the tumour's response to therapy and the effect of parameter variation on the drug concentration profiles in the three compartments.

Systems biology and the mathematical modelling of antibody-directed enzyme prodrug therapy (ADEPT)

Antibody-directed enzyme prodrug therapy (ADEPT) can generate highly localised concentrations of cytotoxic agents directly in a tumour, thereby reducing the collateral toxicity associated with normal tissue exposure. ADEPT is a two-component approach. First, a non-toxic antibody-enzyme fusion protein is localised in the tumour matrix by binding a specific antigen expressed only on the surface of a cancer cell. Once the fusion protein is bound, an inert small molecule prodrug is administered which is the substrate for the enzyme bound to the tumour surface. When the prodrug comes into contact with the bound enzyme, an active cytotoxic agent is generated. A multiple length-scale model of ADEPT therapy in solid tumours is presented. A four-compartment pharmacokinetic (PK) model is formulated where the tumour is comprised of interstitial and cell-surface subcompartments. The macroscopic PK model which describes the biodistribution of antibody-enzyme conjugate, prodrug and active drug at the largest length scale is coupled to a reaction-diffusion tumour model. The models are qualitatively validated against current literature and experimental understanding. The relationship between tumour localisation and the affinity of the antibody-enzyme conjugate for its surface antigen is explored by simulation. The influence of pharmacokinetic and biophysical parameters such as renal elimination rate and permeability of the tumour vasculature upon tumour uptake and retention of the fusion protein are also explored. Lastly, a technique for establishing an optimal prodrug dosing schedule is formulated and initial simulation results are presented.

Tumor growth instability and the onset of invasion

Motivated by experimental observations, we develop a mathematical model of chemotactically directed tumor growth. We present an analytical study of the model as well as a numerical one. The mathematical analysis shows that: (i) tumor cell proliferation by itself cannot generate the invasive branching behavior observed experimentally, (ii) heterotype chemotaxis provides an instability mechanism that leads to the onset of tumor invasion, and (iii) homotype chemotaxis does not provide such an instability mechanism but enhances the mean speed of the tumor surface. The numerical results not only support the assumptions needed to perform the mathematical analysis but they also provide evidence of (i), (ii), and (iii). Finally, both the analytical study and the numerical work agree with the experimental phenomena.

Mathematical modelling of the use of macrophages as vehicles for drug delivery to hypoxic tumour sites

Poor drug delivery and low rates of cell proliferation are two factors associated with hypoxia that diminish the efficacy of many chemotherapeutic drugs. Since macrophages are known to migrate specifically towards, and localize within, hypoxic tumour regions, a promising resolution to these problems involves genetically engineering macrophages to perform such anti-tumour functions as inducing cell lysis and inhibiting angiogenesis. In this paper we outline a modelling approach to characterize macrophage infiltration into early avascular solid tumours, and extensions to study the interaction of these cells with macrophages already present within the tumour. We investigate the role of chemotaxis and chemokine production, and the efficacy of macrophages as vehicles for drug delivery to hypoxic tumour sites. The model is based upon a growing avascular tumour spheroid, in which volume is filled by tumour cells, macrophages and extracellular material, and tumour cell proliferation and death is regulated by nutrient diffusion. Crucially, macrophages occupy volume, and hence contribute to the volume balance and hence the size of the tumour. We also include oxygen-dependent production of macrophage chemokines, which can lead to accumulations in the hypoxic region of the tumour. We find that the macrophage chemotactic sensitivity is a key determinant of macrophage infiltration and tumour size. Although increased infiltration should be beneficial from the point of view of macrophage-based therapies, such infiltration in fact leads to increased tumour sizes. Finally, we include terms representing the induced death of tumour cells by hypoxic engineered macrophages. We demonstrate that reductions in tumour size can be achieved, but predict that a combination of therapies would be required for complete eradication. We also highlight some counter-intuitive predictions-for example, absolute and relative measures of tumour burden lead to different conclusions about prognosis. In summary, this paper illustrates how mathematical models may be used to investigate promising macrophage-based therapies.

Intracellular accumulation and mechanism of action of doxorubicin in a spatio-temporal tumor model

A spatio-temporal model of tumor response to sequestered, intracellular doxorubicin is presented and simulated. An important feature of the model is the characterization of different mechanisms by which doxorubicin initiates the cell death cascade. The model predicts that the long-term response of the tumor to repeated rounds of therapy is very sensitive to changes in the threshold level of doxorubicin required to initiate apoptosis at the maximum rate. In fact, perturbations of this parameter mediate the difference between effective tumor regression and minimal growth delays. The model is also used to investigate which parameters are most influential in rendering the tumor drug resistant. Sensitivity analysis shows that decreasing cellular permeability, as opposed to decreasing sequestration rate or increasing cellular efflux, is the most effective way for tumor cells to overcome the growth control afforded by successive rounds of treatment.

A mathematical model to study the effects of drug resistance and vasculature on the response of solid tumors to chemotherapy

A mathematical model is developed that describes the reduction in volume of a vascular tumor in response to specific chemotherapeutic administration strategies. The model consists of a system of partial differential equations governing intratumoral drug concentration and cancer cell density. In the model the tumor is treated as a continuum of two types of cells which differ in their proliferation rates and their responses to the chemotherapeutic agent. The balance between cell proliferation and death within the tumor generates a velocity field which drives expansion or regression of the spheroid. Insight into the tumor's response to therapy is gained by applying a combination of analytical and numerical techniques to the model equations.