zbMATH — the first resource for mathematics

On the mechanics of a growing tumor. (English) Zbl 1211.74161
Summary: We study tumor growth within the framework of Continuum Mechanics, considering a tumor as a specific case of a growing soft tissue. Using the notion of multiple natural configurations we introduce a mechanical description that splits volumetric growth and mechanical response into two separate contributions. Growth is described as an increase of the mass of the particles of the body and not as an increase of their number. As tumor growth strongly depends upon the availability of nutrients and on the presence of chemical signals, such as growth factors, their diffusion through the growing material is introduced in the description. The model is then applied to describe the homogeneous growth inside a rigid cylinder, a model mimicking the growth of a ductal carcinoma, and to the growth of a multicell spheroid fed by a non-homogeneous diffusion of nutrients. In the latter case residual stresses are generated because the non-uniform distribution of nutrients leads to inhomogeneous growth.

74L15 Biomechanical solid mechanics
92C50 Medical applications (general)
Full Text: DOI
[1] D. Ambrosi, L. Preziosi, On the closure of mass balance models of tumor growth, Math. Models Meth. Appl. Sci., to appear · Zbl 1016.92016
[2] Ambrosi, D., Infiltration through deformable porous media, Zamm, 82, 2, 115-124, (2000) · Zbl 1006.76091
[3] Blatz, P.J.; Ko, W.L., Application of finite elasticity theory to the deformation of rubbery materials, Trans. soc. rheol., 6, 223-251, (1962)
[4] H.M. Byrne, L. Preziosi, Modelling solid tumor growth using the theory of mixtures, IMA J. Math. Appl. Med. Biol., submitted · Zbl 1046.92023
[5] Fung, Y.C., Biomechanics: mechanical properties of living tissues, (1993), Springer Berlin
[6] L. Graziano, D. Morale, Continuum modelling of biological systems based on mass balance law: the mass-flux obtained by microdynamics and parallelism with non-equilibrium thermodynamics, in preparation
[7] Gurtin, M.E., An introduction to continuum mechanics, (1981), Academic Press New York · Zbl 0559.73001
[8] Helmlinger, G.; Netti, P.A.; Lichtenbeld, H.C.; Melder, R.J.; Jain, R.K., Solid stress inhibits the growth of multicellular tumour spheroids, Nature biotech., 15, 778-783, (1997)
[9] J.D. Humphrey, K.R. Rajagopal, A constrained mixture model for growth and remodeling of soft tissues, preprint, 2001
[10] Mollica, F.; Rajagopal, K.R.; Srinivasa, A.R., The inelastic behavior of metals subject to loading reversal, Int. J. plast., 17, 1119-1146, (2001) · Zbl 1044.74505
[11] Murray, J.D., Mathematical biology, (1993), Springer Berlin · Zbl 0779.92001
[12] Netti, P.A.; Baxter, L.T.; Boucher, Y.; Skalak, R.; Jain, R.K., Macro- and microscopic fluid transport in living tissues: application to solid tumours, Aiche j., 43, 818-834, (1997)
[13] Rajagopal, K.R.; Tao, L., Mechanics of mixtures, (1995), World Scientific Singapore · Zbl 0941.74500
[14] Rajagopal, K.R.; Srinivasa, A.R., On the inelastic behavior of solids. part I: twinning, Int. J. plast., 11, 653-678, (1995) · Zbl 0858.73006
[15] Rajagopal, K.R., Multiple natural configurations in continuum mechanics, Rep. inst. comput. appl. mech., 6, (1995) · Zbl 0871.76005
[16] Rajagopal, K.R.; Srinivasa, A.R., Mechanics of the inelastic behavior of materials. part I: theoretical underpinnings, Int. J. plast., 14, 945-967, (1998) · Zbl 0978.74013
[17] Rajagopal, K.R.; Srinivasa, A.R., Mechanics of the inelastic behavior of materials. part II: inelastic response, Int. J. plast., 14, 969-995, (1998) · Zbl 0978.74014
[18] Rajagopal, K.R.; Srinivasa, A.R., On the thermodynamics of shape memory wires, Zamp, 50, 459-496, (1999) · Zbl 0951.74005
[19] Rajagopal, K.R.; Srinivasa, A.R., A thermodynamic frame work for rate type fluid models, J. non-Newtonian fluid mech., 88, 207-227, (2000) · Zbl 0960.76005
[20] Rajagopal, K.R.; Wineman, A.S., A constitutive equation for nonlinear solids which undergo deformation induced microstructural changes, Int. J. plast., 8, 385-395, (1992) · Zbl 0765.73005
[21] Rao, I.J.; Rajagopal, K.R., Phenomenological modeling of polymer crystallization using the notion of multiple natural configurations, Interf. free bound., 2, 73-94, (2000) · Zbl 0948.35123
[22] Rodriguez, E.K.; Hoger, A.; McCulloch, A., Stress dependent finite growth in soft elastic tissues, J. biomech., 27, 455-467, (1994)
[23] Skalak, R.; Dasgupta, G.; Moss, M., Analytical description of growth, J. theor. biol., 94, 555-577, (1982)
[24] Skalak, R.; Zargaryan, S.; Jain, R.K.; Netti, P.A.; Hoger, A., Compatibility and genesis of residual stress by volumetric growth, J. math. biol., 34, 889-914, (1996) · Zbl 0858.92005
[25] Taber, L., Biomechanics of growth, remodeling and morphogenesis, Appl. mech. rev., 48, 487-545, (1995)
[26] Truesdell, C.A.; Noll, W., The non-linear field theories of mechanics, (1992), Springer Berlin · Zbl 0779.73004
[27] Wineman, A.S.; Rajagopal, K.R., On a constitutive theory for materials undergoing microstructural changes, Arch. mech., 42, 53-75, (1990) · Zbl 0733.73001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.