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

Abstract

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. r

A Novel, Patient-Specific Mathematical Pathology Approach for Assessment of Surgical Volume: Application to Ductal Carcinomain situof The Breast

“…Corresponding diffusion penetration lengths ranged in value from 77-224 m, which compare well with typical published values of 100 m under ideal conditions (e.g., [37]). The larger values corresponded to low grade cribriform DCIS where the local density of cells is less than it is for higher grade and more dense (solid) DCIS.…”

“…In the population model 2 , τ P is the (constant) duration of the cell cycle; cell death processes (e.g., apoptosis) have time duration τ A . We set τ P = 18 hours to complete a cell cycle and proliferate [37]. We set τ A = 6.6 hours [24,30,33] by applying the population model to (benign) breast epithelium [38] and correcting for the early portion of apoptosis that cannot be detected by TUNEL assay but is detected by cleaved caspase-3 [39]; this estimate is consistent with the experimental literature (e.g., [40,41]).…”

“…where σ H is the threshold concentration value at a distance T from the duct wall (viable-rim thickness), below which cells become necrotic due to lack of oxygen and nutrients (we set σ H = 0.2 [24,30] based upon published models for breast cancer [37,42]), and I n is the n-th order modified Bessel function of the first kind. We use the average duct radius R duct and viable-rim thickness T values from duct morphometric measurements (see below) in place of R duct and T in Eq.…”

Macrophage-tumour interactions: In vivo dynamics

“…The former eigenvector is real-valued, and we may insist with no loss of generality thatn M 1 = 1; the latter eigenvector is complex-valued and is normalised so thatω H = 1. The terms indicated by '· · · ' in (23) represent either those of higher order in δ than needed in our calculation, or non-resonant terms that need not be computed in order to determine the coefficients in (24) and (25). The quantities x(T ) and q(T ) represent the amplitudes of the monotonic and oscillatory disturbances to the two-way coexistence steady state; these amplitudes evolve on the slow time scale T = δ 2 t. The coefficients ω j are determined by substituting (22) in (7)- (10), together with (23) and analogous expansions for n 1 , n 2 , m. The simplest form of the equations governing the evolution of the amplitudes x and q is the normal form…”

“…The next step in the calculation is now, given the biological parameters, to find the corresponding coefficients in (24) and (25). Such a calculation will enable us to determine, for example, the nature of the solutions near the codimension-two point, their existence and stability.…”