Research Papers

Finite Element Modeling of Avascular Tumor Growth Using a Stress-Driven Model

[+] Author and Article Information
Faezeh Iranmanesh

Department of Mechanical Engineering,
College of Engineering,
University of Tehran,
Tehran 1439955961, Iran
e-mail: f.iranmanesh@ut.ac.ir

Mohammad Ali Nazari

Department of Mechanical Engineering,
College of Engineering,
University of Tehran,
Tehran 1439955961, Iran
e-mail: manazari@ut.ac.ir

1Corresponding author.

Manuscript received February 5, 2017; final manuscript received June 9, 2017; published online June 22, 2017. Assoc. Editor: Thao (Vicky) Nguyen.

J Biomech Eng 139(8), 081009 (Jun 22, 2017) (10 pages) Paper No: BIO-17-1045; doi: 10.1115/1.4037038 History: Received February 05, 2017; Revised June 09, 2017

Tumor growth being a multistage process has been investigated from different aspects. In the present study, an attempt is made to represent a constitutive-structure-based model of avascular tumor growth in which the effects of tensile stresses caused by collagen fibers are considered. Collagen fibers as a source of anisotropy in the structure of tissue are taken into account using a continuous fiber distribution formulation. To this end, a finite element modeling is implemented in which a neo-Hookean hyperelastic material is assigned to the tumor and its surrounding host. The tumor is supplied with a growth term. The growth term includes the effect of parameters such as nutrient concentration on the tumor growth and the tumor's solid phase content in the formulation. Results of the study revealed that decrease of solid phase is indicative of decrease in growth rate and the final steady-state value of tumor's radius. Moreover, fiber distribution affects the final shape of the tumor, and it could be used to control the shape and geometry of the tumor in complex morphologies. Finally, the findings demonstrated that the exerted stresses on the tumor increase as time passes. Compression of tumor cells leads to the reduction of tumor growth rate until it gradually reaches an equilibrium radius. This finding is in accordance with experimental data. Hence, this formulation can be deployed to evaluate both the residual stresses induced by growth and the mechanical interactions with the host tissue.

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Cristini, V. , and Lowengrub, J. , 2010, Multiscale Modeling of Cancer: An Integrated Experimental and Mathematical Modeling Approach, Cambridge University Press, New York.
Drasdo, D. , and Hoehme, S. , 2005, “ A Single-Cell-Based Model of Tumor Growth In Vitro: Monolayers and Spheroids,” Phys. Biol., 2(3), pp. 133–147. [CrossRef] [PubMed]
Ambrosi, D. , and Mollica, F. , 2002, “ On the Mechanics of a Growing Tumor,” Int. J. Eng. Sci., 40(12), pp. 1297–1316. [CrossRef]
Araujo, R. P. , and McElwain, D. , 2004, “ A Linear-Elastic Model of Anisotropic Tumour Growth,” Eur. J. Appl. Math., 15(3), pp. 365–384. [CrossRef]
MacLaurin, J. , Chapman, J. , Jones, G. W. , and Roose, T. , 2012, “ The Buckling of Capillaries in Solid Tumours,” Proc. R. Soc. A, 468(2148), pp. 4123–4145. [CrossRef]
Kim, Y. , Stolarska, M. A. , and Othmer, H. G. , 2007, “ A Hybrid Model for Tumor Spheroid Growth In Vitro I: Theoretical Development and Early Results,” Math. Models Methods Appl. Sci., 17(supp01), pp. 1773–1798. [CrossRef]
Breward, C. , Byrne, H. , and Lewis, C. , 2002, “ The Role of Cell-Cell Interactions in a Two-Phase Model for Avascular Tumour Growth,” J. Math. Biol., 45(2), pp. 125–152. [CrossRef] [PubMed]
Breward, C. J. , Byrne, H. M. , and Lewis, C. E. , 2003, “ A Multiphase Model Describing Vascular Tumour Growth,” Bull. Math. Biol., 65(4), pp. 609–640. [CrossRef] [PubMed]
Roose, T. , Netti, P. A. , Munn, L. L. , Boucher, Y. , and Jain, R. K. , 2003, “ Solid Stress Generated by Spheroid Growth Estimated Using a Linear Poroelasticity Model,” Microvasc. Res., 66(3), pp. 204–212. [CrossRef] [PubMed]
Stylianopoulos, T. , Martin, J. D. , Snuderl, M. , Mpekris, F. , Jain, S. R. , and Jain, R. K. , 2013, “ Coevolution of Solid Stress and Interstitial Fluid Pressure in Tumors During Progression: Implications for Vascular Collapse,” Cancer Res., 73(13), pp. 3833–3841. [CrossRef] [PubMed]
Ambrosi, D. , and Preziosi, L. , 2009, “ Cell Adhesion Mechanisms and Stress Relaxation in the Mechanics of Tumours,” Biomech. Model. Mechanobiol., 8(5), pp. 397–413. [CrossRef] [PubMed]
Jain, R. K. , Martin, J. D. , and Stylianopoulos, T. , 2014, “ The Role of Mechanical Forces in Tumor Growth and Therapy,” Annu. Rev. Biomed. Eng., 16(1), pp. 321–346. [CrossRef] [PubMed]
Voutouri, C. , Mpekris, F. , Papageorgis, P. , Odysseos, A. D. , and Stylianopoulos, T. , 2014, “ Role of Constitutive Behavior and Tumor-Host Mechanical Interactions in the State of Stress and Growth of Solid Tumors,” PLoS One, 9(8), p. e104717. [CrossRef] [PubMed]
Sherratt, J. A. , and Chaplain, M. A. , 2001, “ A New Mathematical Model for Avascular Tumour Growth,” J. Math. Biol., 43(4), pp. 291–312. [CrossRef] [PubMed]
Stylianopoulos, T. , Martin, J. D. , Chauhan, V . P. , Jain, S. R. , Diop-Frimpong, B. , Bardeesy, N. , Smith, B. L., Ferrone, C. R., Hornicek, F. J., Boucher, Y., Munn, L. L., and Jain, R. K., 2012, “ Causes, Consequences, and Remedies for Growth-Induced Solid Stress in Murine and Human Tumors,” Proc. Natl. Acad. Sci., 109(38), pp. 15101–15108. [CrossRef]
Byrne, H. , and Preziosi, L. , 2003, “ Modelling Solid Tumour Growth Using the Theory of Mixtures,” Math. Med. Biol., 20(4), pp. 341–366. [CrossRef] [PubMed]
Nagy, J. A. , Dvorak, A. M. , and Dvorak, H. F. , 2012, “ Vascular Hyperpermeability, Angiogenesis, and Stroma Generation,” Cold Spring Harbor Perspect. Med., 2(2), p. a006544. [CrossRef]
Ronnov-Jessen, L. , Petersen, O. W. , and Bissell, M. J. , 1996, “ Cellular Changes Involved in Conversion of Normal to Malignant Breast: Importance of the Stromal Reaction,” Physiol. Rev., 76(1), pp. 69–125. https://www.ncbi.nlm.nih.gov/pubmed/8592733 [PubMed]
Ateshian, G. A. , Rajan, V. , Chahine, N. O. , Canal, C. E. , and Hung, C. T. , 2009, “ Modeling the Matrix of Articular Cartilage Using a Continuous Fiber Angular Distribution Predicts Many Observed Phenomena,” ASME J. Biomech. Eng., 131(6), p. 061003. [CrossRef]
Jain, R. K. , and Stylianopoulos, T. , 2010, “ Delivering Nanomedicine to Solid Tumors,” Nat. Rev. Clin. Oncol., 7(11), pp. 653–664. [CrossRef] [PubMed]
Pluen, A. , Boucher, Y. , Ramanujan, S. , McKee, T. D. , Gohongi, T. , di Tomaso, E. , Brown, E. B., Izumi, Y., Campbell, R. B., Berk, D. A., and Jain, R. K., 2001, “ Role of Tumor–Host Interactions in Interstitial Diffusion of Macromolecules: Cranial Vs. Subcutaneous Tumors,” Proc. Natl. Acad. Sci., 98(8), pp. 4628–4633. [CrossRef]
Holzapfel, G. A. , Gasser, T. C. , and Ogden, R. W. , 2000, “ A New Constitutive Framework for Arterial Wall Mechanics and a Comparative Study of Material Models,” J. Elasticity Phys. Sci. Solids, 61(1–3), pp. 1–48.
Stylianopoulos, T. , and Barocas, V . H. , 2007, “ Multiscale, Structure-Based Modeling for the Elastic Mechanical Behavior of Arterial Walls,” ASME J. Biomech. Eng., 129(4), pp. 611–618. [CrossRef]
Driessen, N. J. , Bouten, C. V. , and Baaijens, F. P. , 2005, “ A Structural Constitutive Model for Collagenous Cardiovascular Tissues Incorporating the Angular Fiber Distribution,” ASME J. Biomech. Eng., 127(3), pp. 494–503. [CrossRef]
Billiar, K. L. , and Sacks, M. S. , 2000, “ Biaxial Mechanical Properties of the Native and Glutaraldehyde-Treated Aortic Valve Cusp: Part II—A Structural Constitutive Model,” ASME J. Biomech. Eng., 122(4), pp. 327–335. [CrossRef]
Wilson, W. , Van Donkelaar, C. , Van Rietbergen, B. , and Huiskes, R. , 2005, “ A Fibril-Reinforced Poroviscoelastic Swelling Model for Articular Cartilage,” J. Biomech., 38(6), pp. 1195–1204. [CrossRef] [PubMed]
Wilson, W. , Huyghe, J. , and Van Donkelaar, C. , 2007, “ Depth-Dependent Compressive Equilibrium Properties of Articular Cartilage Explained by Its Composition,” Biomech. Model. Mechanobiol., 6(1–2), pp. 43–53. [CrossRef] [PubMed]
Soulhat, J. , Buschmann, M. , and Shirazi-Adl, A. , 1999, “ A Fibril-Network-Reinforced Biphasic Model of Cartilage in Unconfined Compression,” ASME J. Biomech. Eng., 121(3), pp. 340–347. [CrossRef]
Hagendoorn, J. , Tong, R. , Fukumura, D. , Lin, Q. , Lobo, J. , Padera, T. P. , Xu, L., Kucherlapati, R., and Jain, R. K., 2006, “ Onset of Abnormal Blood and Lymphatic Vessel Function and Interstitial Hypertension in Early Stages of Carcinogenesis,” Cancer Res., 66(7), pp. 3360–3364. [CrossRef] [PubMed]
Padera, T. P. , Stoll, B. R. , Tooredman, J. B. , Capen, D. , di Tomaso, E. , and Jain, R. K. , 2004, “ Pathology: Cancer Cells Compress Intratumour Vessels,” Nature, 427(6976), p. 695. [CrossRef] [PubMed]
Helmlinger, G. , Netti, P. A. , Lichtenbeld, H. C. , Melder, R. J. , and Jain, R. K. , 1997, “ Solid Stress Inhibits the Growth of Multicellular Tumor Spheroids,” Nat. Biotechnol., 15(8), pp. 778–783. [CrossRef] [PubMed]
Cheng, G. , Tse, J. , Jain, R. K. , and Munn, L. L. , 2009, “ Micro-Environmental Mechanical Stress Controls Tumor Spheroid Size and Morphology by Suppressing Proliferation and Inducing Apoptosis in Cancer Cells,” PLoS One, 4(2), p. e4632. [CrossRef] [PubMed]
Kaufman, L. J. , Brangwynne, C. P. , Kasza, K. E. , Filippidi, E. , Gordon, V . D. , Deisboeck, T. S. , and Weitz, D. A., 2005, “ Glioma Expansion in Collagen I Matrices: Analyzing Collagen Concentration-Dependent Growth and Motility Patterns,” Biophys. J., 89(1), pp. 635–650. [CrossRef] [PubMed]
Stylianopoulos, T. , and Jain, R. K. , 2013, “ Combining Two Strategies to Improve Perfusion and Drug Delivery in Solid Tumors,” Proc. Natl. Acad. Sci., 110(46), pp. 18632–18637. [CrossRef]
Holzapfel, G. A. , 2000, Nonlinear Solid Mechanics, Vol. 24, Wiley, Chichester, UK.
Roose, T. , Chapman, S. J. , and Maini, P. K. , 2007, “ Mathematical Models of Avascular Tumor Growth,” SIAM Rev., 49(2), pp. 179–208. [CrossRef]
Nicholson, C. , and Phillips, J. , 1981, “ Ion Diffusion Modified by Tortuosity and Volume Fraction in the Extracellular Microenvironment of the Rat Cerebellum,” J. Physiol., 321(1), pp. 225–257. [CrossRef] [PubMed]
Freyer, J. , 1981, “ Heterogeneity in Multicell Spheroids Induced by Alterations in the External Oxygen and Glucose Concentration,” Department of Radiation Biology and Biophysics, Rochester University, Rochester, NY, Report No. DOE/EV/03490-2101. https://www.osti.gov/scitech/biblio/6583865
Casciari, J. , Sotirchos, S. , and Sutherland, R. , 1992, “ Mathematical Modelling of Microenvironment and Growth in EMT6/Ro Multicellular Tumour Spheroids,” Cell Proliferation, 25(1), pp. 1–22. [CrossRef] [PubMed]
Shirinifard, A. , Gens, J. S. , Zaitlen, B. L. , Popławski, N. J. , Swat, M. , and Glazier, J. A. , 2009, “ 3D Multi-Cell Simulation of Tumor Growth and Angiogenesis,” PLoS One, 4(10), p. e7190. [CrossRef] [PubMed]
Harko, T. , and Mak, M. K. , 2015, “ Travelling Wave Solutions of the Reaction-Diffusion Mathematical Model of Glioblastoma Growth: An Abel Equation Based Approach,” Math. Biosci. Eng., 12(1), pp. 41–69. [CrossRef] [PubMed]
Lanir, Y. , 1983, “ Constitutive Equations for Fibrous Connective Tissues,” J. Biomech., 16(1), pp. 1–12. [CrossRef] [PubMed]
Ateshian, G. A. , 2007, “ Anisotropy of Fibrous Tissues in Relation to the Distribution of Tensed and Buckled Fibers,” ASME J. Biomech. Eng., 129(2), pp. 240–249. [CrossRef]
Chahine, N. O. , Wang, C. C. , Hung, C. T. , and Ateshian, G. A. , 2004, “ Anisotropic Strain-Dependent Material Properties of Bovine Articular Cartilage in the Transitional Range From Tension to Compression,” J. Biomech., 37(8), pp. 1251–1261. [CrossRef] [PubMed]
Clauss, M. , and Breier, G. , 2004, Mechanisms of Angiogenesis, Vol. 94, Springer Science & Business Media, Basel, Switzerland.
Nishida, N. , Yano, H. , Nishida, T. , Kamura, T. , and Kojiro, M. , 2006, “ Angiogenesis in Cancer,” Vasc. Health Risk Manage., 2(3), pp. 213–219. [CrossRef]
Maas, S. A. , Ellis, B. J. , Ateshian, G. A. , and Weiss, J. A. , 2012, “ FEBio: Finite Elements for Biomechanics,” ASME J. Biomech. Eng., 134(1), p. 011005. [CrossRef]
Janet, M. T. , Cheng, G. , Tyrrell, J. A. , Wilcox-Adelman, S. A. , Boucher, Y. , Jain, R. K. , and Munn, L. L., 2012, “ Mechanical Compression Drives Cancer Cells Toward Invasive Phenotype,” Proc. Natl. Acad. Sci., 109(3), pp. 911–916. [CrossRef]
Roeder, B. A. , Kokini, K. , and Voytik-Harbin, S. L. , 2009, “ Fibril Microstructure Affects Strain Transmission Within Collagen Extracellular Matrices,” ASME J. Biomech. Eng., 131(3), p. 031004. [CrossRef]
Kokini, K. , Sturgis, J. E. , Robinson, J. P. , and Voytik-Harbin, S. L. , 2002, “ Tensile Mechanical Properties of Three-Dimensional Type I Collagen Extracellular Matrices With Varied Microstructure,” ASME J. Biomech. Eng., 124(2), pp. 214–222. [CrossRef]
Jain, R. K. , 1987, “ Transport of Molecules in the Tumor Interstitium: A Review,” Cancer Res., 47(12), pp. 3039–3051. http://cancerres.aacrjournals.org/content/47/12/3039 [PubMed]
Netti, P. A. , Berk, D. A. , Swartz, M. A. , Grodzinsky, A. J. , and Jain, R. K. , 2000, “ Role of Extracellular Matrix Assembly in Interstitial Transport in Solid Tumors,” Cancer Res., 60(9), pp. 2497–2503. http://cancerres.aacrjournals.org/content/60/9/2497 [PubMed]
Choi, J. , Credit, K. , Henderson, K. , Deverkadra, R. , He, Z. , Wiig, H. , Vanpelt, H., and Flessner, M. F., 2006, “ Intraperitoneal Immunotherapy for Metastatic Ovarian Carcinoma: Resistance of Intratumoral Collagen to Antibody Penetration,” Clin. Cancer Res., 12(6), pp. 1906–1912. [CrossRef] [PubMed]
Pouysségur, J. , Dayan, F. , and Mazure, N. M. , 2006, “ Hypoxia Signalling in Cancer and Approaches to Enforce Tumour Regression,” Nature, 441(7092), pp. 437–443. [CrossRef] [PubMed]
Kaur, B. , Khwaja, F. W. , Severson, E. A. , Matheny, S. L. , Brat, D. J. , and Van Meir, E. G. , 2005, “ Hypoxia and the Hypoxia-Inducible-Factor Pathway in Glioma Growth and Angiogenesis,” Neuro-Oncology, 7(2), pp. 134–153. [CrossRef] [PubMed]
Chauhan, V . P. , Martin, J. D. , Liu, H. , Lacorre, D. A. , Jain, S. R. , Kozin, S. V. , Stylianopoulos, T., Mousa, A. S., Han, X., Adstamongkonkul, P., Popovic, Z., Huang, P., Bawendi, M. G., Boucher, Y., and Jain, R. K., 2013, “ Angiotensin Inhibition Enhances Drug Delivery and Potentiates Chemotherapy by Decompressing Tumour Blood Vessels,” Nat. Commun., 4, p. 2516. [CrossRef] [PubMed]


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Fig. 1

Finite element representation of tumor tissue and its surrounding host

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Fig. 2

Experimental and model predicted radius of MU89 line tumor spheroid versus time

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Fig. 3

Radial stresses generated in tumor and its surrounding host

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Fig. 4

Circumferential stresses generated in tumor and its surrounding host

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Fig. 5

(a) First and (b) third principal stress variation as a function of radius in tumor and host tissue at day 20

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Fig. 6

Radius of tumor versus time for five different values of ϕs

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Fig. 7

Radial stress inside tumor and through the surrounding host for five different values of ϕs

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Fig. 8

Circumferential stress inside tumor and through the surrounding host for five different values of ϕs

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Fig. 9

Variation of radius of tumor spheroid versus time for the three sets of βi′s

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Fig. 10

Ellipsoidal tumor at the last time point



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