Modeling bacteria-based therapy in tumor spheroids

Tumor-targeting bacteria elicit anticancer effects by infiltrating hypoxic regions, releasing toxic agents and inducing immune responses. As the mechanisms of action of bacterial therapies are still to be completely elucidated, mathematical modeling could aid the understanding of the dynamical interactions between tumor cells and bacteria in different cancers. Here we propose a mathematical model for the anti-tumor activity of bacteria in tumor spheroids. We consider constant infusion and time-dependent administration of bacteria in the culture medium, and analyze the effects of bacterial chemotaxis and killing rate. We show that active bacterial migration towards tumor hypoxic regions is necessary for successful spheroid infiltration and that intermediate chemotaxis coefficients provide the smallest spheroid radii at the end of the treatment. We report on the impact of the killing rate on final spheroid composition, and highlight the emergence of spheroid size oscillations due to competing interactions between bacteria and tumor cells.


Introduction
terial growth leads to increased bacterial densities. A cytotoxic protein in 48 Escherichia coli was cloned to investigate its effects on tumors as discussed in modeling the authors showed that bacteria display higher effective diffusivi-In mixture theory velocity fields are determined by considering the me-110 chanical response of the phases to mutual interactions. Neglecting inertial 111 effects, as usually done for growth phenomena (Preziosi, 2003;Byrne, 2012), 112 the balance of linear momentum can be written as: Here σ i is the partial stress tensor of the i-th phase, m ij represent the meaning that we assume that there are no empty spaces within the mix-121 ture (Preziosi, 2003;Byrne, 2012). 122 We conclude the set of governing laws by stating an equation for the 123 normalized nutrient concentration n in the mixture, i.e. the tumor: 124 ∂ t n = D n div (grad n) + S n , in which D n is the nutrient diffusion coefficient and S n represents the 125 nutrient mass exchange with the model phases. In the following we will 126 consider a single nutrient, i.e. oxygen. 127 2.1. Constitutive relationships 128 We close the model by selecting suitable constitutive assumptions. First, 129 we assume that the interaction terms m ij depend linearly on the relative 130 phase velocities (Preziosi, 2003;Byrne, 2012): with the same linearity constant µ for all the phases (i = c, b, f). We 132 consider only a single external force m b acting on bacteria. This term de-133 scribes bacteria chemotaxis following spatial hypoxic gradients and models 134 active cell migration towards waste products from dying cancer cells (Forbes, in which χ b describes the strength of chemoattraction. The partial stress tensors in Equation (2) are defined such that the interfacial 142 pressure of each phase is given by the pressure in the extracellular material 143 plus a correction term, specific to its phase (Boemo and Byrne, 2019): where I is the identity tensor. The ratio π i /µ characterizes the movement 145 of the i-th phase in the mixture and is generally identified as the phase 146 motility coefficient D i (i = c, b) (Boemo and Byrne, 2019). In the following 147 we will also define χ = χ b /µ as the bacterial chemotactic coefficient.

148
To formulate the mass exchange terms in Equations (1) and (4) we assume 149 the following assumptions: 150 A1 TCs proliferate when oxygen is available. As soon as the latter de-

163
The resulting mass exchange terms read: Here γ i and δ i are the proliferation and death rate of the i-th phase re-164 spectively (i = c, b), whereas δ n is the oxygen consumption rate.
in which v i is the radial velocity of the i-th phase (i = c, b, f). Substituting

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Equations (5), (7)-(9) and (13) in Equation (2) we obtain for the radial 179 velocities: after summing over the phases in Equation (2) Note that we do not solve for φ f since it can be obtained as . 185 We model growth of the spheroid as a free-boundary problem, in which 186 the outer tumor radius r = R(t) moves with the same velocity as the TC 187 phase, Finally, we define a set of boundary and initial conditions to close the In the following, we assume a uniform initial tumor volume fraction φ c0 = 194 0.8 across the spheroid (Byrne and Preziosi, 2003) and consider a small value 195 for the bacterial volume fraction at the spheroid outer radius, i.e. φ b0 = 0.01.

196
Regarding the initial conditions, we consider a spheroid devoid of bacteria 197 and displaying a uniform TC volume fraction and nutrient concentration over 198 its radius: Finally, we prescribe an initial spheroid radius, i.e. R(0) = 150 µm.

Parameter estimation 201
The parameters used in the model simulations are reported in Table 1.     Table 1 for the simulation.

231
The model is able to reproduce the two phases of spheroid growth usually where the integral is performed over the spheroid volume V sf (i = c, b, f).

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At early time points, V c is in a phase of fast growth, since nutrient is available