Effect of heterogeneity and spatial correlations on the structure of a tumor invasion front in cellular environments

Youness Azimzade, Abbas Ali Saberi, and Muhammad Sahimi

Phys. Rev. E 100, 062409 – Published 18 December 2019

Abstract

Analysis of invasion front has been widely used to decipher biological properties, as well as the growth dynamics of the corresponding populations. Likewise, the invasion front of tumors has been investigated, from which insights into the biological mechanisms of tumor growth have been gained. We develop a model to study how tumors' invasion front depends on the relevant properties of a cellular environment. To do so, we develop a model based on a nonlinear reaction-diffusion equation, the Fisher-Kolmogorov-Petrovsky-Piskunov equation, to model tumor growth. Our study aims to understand how heterogeneity in the cellular environment's stiffness, as well as spatial correlations in its morphology, the existence of both of which has been demonstrated by experiments, affects the properties of tumor invasion front. It is demonstrated that three important factors affect the properties of the front, namely the spatial distribution of the local diffusion coefficients, the spatial correlations between them, and the ratio of the cells' duplication rate and their average diffusion coefficient. Analyzing the scaling properties of tumor invasion front computed by solving the governing equation, we show that, contrary to several previous claims, the invasion front of tumors and cancerous cell colonies cannot be described by the well-known models of kinetic growth, such as the Kardar-Parisi-Zhang equation.

Received 26 June 2019
Revised 9 November 2019

DOI:https://doi.org/10.1103/PhysRevE.100.062409

Physics Subject Headings (PhySH)

Growth processes

Statistical Physics & ThermodynamicsPhysics of Living Systems

Authors & Affiliations

Youness Azimzade ¹, Abbas Ali Saberi ^1,2,*, and Muhammad Sahimi³

¹Department of Physics, University of Tehran, Tehran 14395-547, Iran
²Institut für Theoretische Physik, Universitat zu Köln, 50937 Köln, Germany
³Mork Family Department of Chemical Engineering Materials Science, University of Southern California, Los Angeles, California 90089-1211, USA

^*Corresponding author: ab.saberi@ut.ac.ir

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Issue

Vol. 100, Iss. 6 — December 2019

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Images

Figure 1
Schematic illustration of the model. (a) The discretized medium with a grid block size of $δ$ . The initial condition in the medium of size of $L$ is $C = C_{0}$ up to the width $n = 3$ . Periodic boundary condition was imposed in the horizontal direction. (b) The tumor's surface is defined as those edges that are around the border units, with each edge above the occupied units being the border edge, and with the border edges generating the straight solid line. For a homogeneous cellular environment, the morphology of the interface remains unchanged and the solid line moves with a fixed velocity. (c) A realization of a heterogeneous cellular environment with no spatial correlations in the values of the diffusivity $D$ and the corresponding cell density with $R / 〈 D 〉 = 10^{7}$ . (d) The distribution of the density $C$ corresponding to (c) after some time. Where density approaches zero is considered as the invasion front. Contrary to homogeneous cellular media, the invasion front deviates from linear geometry, giving rise to and average height $\bar{h}$ . The initially straight line becomes ragged as time passes. Similarly to the conventional approach in interface studies, we analyze the interface and quantify its dependence on the environmental fluctuations, initial conditions, and $R / D$ .
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Figure 2
Analysis of the dependence of (a) the mean height $\bar{h}$ (and, hence, the velocity) and (b) the width $W (L, t)$ of the interface on the threshold $ε$ with $C_{0} = 1$ , $n = 6$ , and $R / D = 5 \times 10^{6} m^{- 2}$ . While for invasion velocity one needs to consider a smaller values of $ε$ , the interface width converges faster.
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Figure 3
(a) Dynamic evolution of the width $W (L, t)$ for $ξ = 1$ , $C_{0} = 0.2$ , and $R / D = 10^{5}$ . Once $W$ saturates, it remains the same for longer times. The roughness exponent is $α = 0$ . (b) Scaling of $W (L, t)$ with time $t$ for $C_{0} = 10^{- 2}, 10^{- 1}$ , and 1. (c) $W_{sat}$ versus the intensity $ξ$ of the local diffusivity fluctuations. $W_{sat}$ increases with increasing $ξ$ and decreases with increasing $R / D$ . (d) Dynamic evolution of the mean interface height $\bar{h}$ versus time in a logarithmic plot for $C_{0} = 0.2$ and $n = 6$ as the initial condition, and $R = 10^{- 5} s^{- 1}$ , $D = 10^{- 13} m^{2}$ /s, and $ε = 10^{- 3}$ . The linear behavior confirms the establishment of the invasion waves. One can determine, respectively, the invasion velocity by $v = d \bar{h} / d t$ (inset), which for the aforementioned parameters in an environment with $ξ = 0$ yields $v_{0} \sim 1.6 \times 10^{- 9}$ m/s. (e) Effect of $ξ$ on the invasion velocity. The velocities are normalized by $v_{0}$ , the velocity in a homogeneous environment. Disorder decreases the invasion velocity in the same manner for various values of $ε$ .
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Figure 4
(a) Realizations for heterogeneous environments with $〈 D 〉$ being the mean local diffusion coefficient and $H$ the Hurst exponent. (b) The corresponding cell density for $R / D = 2 \times 10^{6} m^{- 2}$ , $D_{0} = 0$ , and the strength of the fluctuation of diffusivity, $ξ = 1$ . (c) Scaling of the width $W (L, t)$ with time $t$ in a cellular medium with $H = 0.1$ , $R / D = 10^{8} m^{- 2}$ , the initial concentration $C_{0} = 1$ , and $n = 6$ . The inset shows the dependence of the saturation width $W_{sat} (L)$ on the linear size $L$ that leads to the roughness exponent $α = 0.92 \pm 0.02$ . (d) Dynamic evolution of $W (L, t)$ for several values of $H$ and the same initial condition for $L = 64$ and $R / D = 10^{8} m^{- 2}$ . The inset shows the dependence of $α$ on $H$ . (e) Dependence of $W_{sat} (L)$ on $R / D$ and $H$ for $L = 64$ .
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