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Avascular tumour growth models based on anomalous diffusion

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Abstract

In this study, we model avascular tumour growth in epithelial tissue. This can help us to understand that how an avascular tumour interacts with its microenvironment and what type of physical changes can be observed within the tumour spheroid before angiogenesis. This understanding is likely to assist in the development of better diagnostics, improved therapies, and prognostics. In biological systems, most of the diffusive processes are through cellular membranes which are porous in nature. Due to its porous nature, diffusion in biological systems are heterogeneous. The fractional diffusion equation is well suited to model heterogeneous biological systems, though most of the early studies did not use this fact. They described tumour growth with simple diffusion-based model. We have developed a spherical model based on simple diffusion initially, and then the model is upgraded with fractional diffusion equations to express the anomalous nature of biological system. In this study, two types of fractional models are developed: one of fixed order and the other of variable order. The memory formalism technique is also included in these anomalous diffusion models. These three models are investigated from phenomenological point view by measuring some parameters for characterizing avascular tumour growth over time. Tumour microenvironment is very complex in nature due to several concurrent molecular mechanisms. Diffusion with memory (fixed as well as variable) formation may be an oversimplified technique, and does not reflect the detailed view of the tumour microenvironment. However, it is found that all the models offer realistic and insightful information of the tumour microenvironment at the macroscopic level, and approximate well the physical phenomena. Also, it is observed that the anomalous diffusion based models offer a closer description to clinical facts than the simple model. As the simulation parameters get modified due to different biochemical and biophysical processes, the robustness of the model is determined. It is found that the anomalous diffusion models are moderately sensitive to the parameters.

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Acknowledgements

We would like to thank Dr. Sonya Bahar, Editor-in-Chief of Journal of Biological Physics for being supportive towards the publication of this paper. We show our sincere gratitude to the anonymous reviewer(s) whose valuable comments helped us greatly to improve the presentation. We are sincerely thankful to Dr. Naveen Kumar, Department of Mathematics, Banaras Hindu University, Varanasi 221005, India for his cooperation and help. Special thanks go to Debarpita Santra, PhD scholar, Department of Computer Science and Engineering, University of Kalyani, West Bengal 741245, India for her valuable inputs during the revision of the manuscript. The first author is also thankful to the University Grants Commission, Government of India, for supporting him by a Junior Research Fellowship.

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  1. Department of Computer Science, Institute of Science, Banaras Hindu University, Varanasi, 221005, India

    Sounak Sadhukhan & S. K. Basu

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Appendix

Appendix

1.1 A. Non-dimensionalization of Eq. (2)

Non-dimensionalize Eq. (2) by rescaling the distance l with time τ = l2/DC. Proliferative cell, quiescent cell, necrotic cell, nutrients and EGF concentrations, and fibronectin are rescaled with P0, Q0, N0, C0, E0, and F0 respectively, where P0, Q0, N0, C0, E0, and F0 are the appropriate reference variables. Therefore,

$$ {P}^{\ast }=\frac{P}{P_0},{Q}^{\ast }=\frac{Q}{Q_0},{N}^{\ast }=\frac{N}{N_0},{C}^{\ast }=\frac{C}{C_0},{E}^{\ast }=\frac{E}{E_0},{F}^{\ast }=\frac{F}{F_0},{t}^{\ast }=\frac{t}{\tau } $$

We obtain the new system of Eqs. (3) (by dropping the stars), where,

$$ {\displaystyle \begin{array}{c}{\eta}^{\ast }=\frac{\eta {E}_0}{D_C},\kern0.5em {\rho}^{\ast }=\frac{\rho {F}_0}{D_C},\kern0.5em {\alpha}^{\ast }=\frac{\alpha {l}^2}{D_C},\kern0.5em {\beta}^{\ast }=\frac{\beta {l}^2}{D_C},\kern0.5em {d}_P^{\ast }=\frac{d_P{l}^2}{D_C},\kern0.5em {\omega}^{\ast }=\frac{\beta {l}^2{P}_0}{D_C{Q}_0},\kern0.5em {\gamma}^{\ast }=\frac{\gamma {l}^2}{D_C},\kern0.5em {d}_Q^{\ast }=\frac{d_Q{l}^2}{D_C},\kern0.5em {\lambda}^{\ast }=\frac{\gamma {l}^2{Q}_0}{D_C{N}_0},\\ {}{\mu}_C^{\ast }=\frac{\mu_C{l}^2}{D_C},\kern0.5em {w}_1^{\ast }=\frac{w_1{l}^2}{D_C},\kern0.5em {w}_2^{\ast }=\frac{w_2{l}^2{P}_0}{D_C},\kern0.5em {w}_3^{\ast }=\frac{w_3{l}^2{Q}_0}{D_C},\kern0.5em {\mu}_E^{\ast }=\frac{\mu_E{l}^2}{D_C},\kern0.5em {k}_1^{\ast }=\frac{k_1{l}^2}{D_C},\kern0.5em {k}_2^{\ast }=\frac{k_2{l}^2{P}_0}{D_C},\kern0.5em {k}_3^{\ast }=\frac{k_3{l}^2{Q}_0}{D_C},\\ {}\begin{array}{cccccccc}{\mu}_{FP}^{\ast }=\frac{\mu_{FP}{l}^2{P}_0}{D_C{F}_0},& {\mu}_{FQ}^{\ast }=\frac{\mu_{FQ}{l}^2{Q}_0}{D_C{F}_0},& {z}_1^{\ast }=\frac{z_1{l}^2{P}_0}{D_C},& {z}_2^{\ast }=\frac{z_2{l}^2{P}_0}{D_C},& {D}_{PC}=\frac{D_P}{D_C},& {D}_{QC}=\frac{D_Q}{D_C},& {D}_{CC}=\frac{D_C}{D_C},& \mathrm{and}\end{array}\\ {}{D}_{EC}=\frac{D_E}{D_C}\end{array}} $$

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Sadhukhan, S., Basu, S.K. Avascular tumour growth models based on anomalous diffusion. J Biol Phys 46, 67–94 (2020). https://doi.org/10.1007/s10867-020-09541-w

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