Delsanto, P.P.

Guiot, C.

Pugno, N.

English

Guiot, Caterina

Institutional Research Information System University of Turin

Elastomechanical model of tumor invasionCaterina GuiotDepartment of Neuroscience, Università di Torino, Torino, Italy and CNISM, Torino, 10125 ItalyNicola PugnoaDepartment of Structural Engineering and Geotechnics, Politecnico di Torino, Torino, 10129 ItalyPier Paolo DelsantoDepartment of Physics, Politecnico di Torino, Torino, 10010 Italy and Bioinformatics High PerformanceComputing Laboratories, Bioindustry Park of Canavese, Colleretto Giacosa, 10010 ItalyReceived 2 March 2006; accepted 19 October 2006; published online 4 December 2006Tumor invasion concerns the tumor capability of colonizing the host by means of several complexbiochemical processes. Since certain aspects of the problem present a striking resemblance with wellknown physical mechanisms, the authors propose here an analogy between tumoral invasivebranching in a tissue and the mechanical insertion of a solid inclusion in an elastic materialspecimen. The model, which is an extension of a previous one, based on the universal growth lawof West et al. Nature London 413, 628 2001, is discussed in the case of multicellular tumorspheroids and cords, but it may be adapted to understand invasion also in real tumors. © 2006American Institute of Physics. DOI: 10.1063/1.2398910The urgent need for a “system biology” approach to bio-medical sciences, in particular oncology, has been widelyrecognized.1 It calls for a new theoretical framework of in-tegrative cancer biology, bringing together the contributionsof researchers from biology to mathematics, physics, infor-mation technology, and computational sciences.2In the present contribution we wish to propose, as anexample of cross-fertilization between different fields,3–5 amechanistic description of tumor invasion as an extension ofa previous cancer growth model5 based on the law of West etal.3 The term “invasion” accounts for the capability of thetumor of colonizing its host by means of several biochemicalprocesses, which can induce tissue degradation.Our model is not intended as an alternative to previousmodels, but rather as being complementary to them, captur-ing some aspects of the invasion process, which present astriking resemblance with well known mechanisms of frac-ture mechanics.6 Incidentally, some concepts from fracturemechanics can account also for chemical diffusion.7 In fact,if we compare the growth of a tumor in a tissue with thephysical insertion of a solid inclusion in an elastic fracturablematerial specimen, we find that from a purely mechanisticpoint of view the dynamics of the two processes is similar.To the pressure or stress exerted by the tumor or invasionon the host, the latter reacts with a compression and/or com-paction of the former.8 As a consequence, the external stressis released allowing a new phase of tumor growth invasion.The whole process shows the transition from the initial con-dition through a noninvasive growth to an invasive phasewhen the matrix strength tolerance is reached, at the end ofwhich the fracture-regrowth cycle FRC may start anew.In the present contribution we limit for simplicity ourinvestigation to the analysis of multicellular tumor spheroidsMTSs and cords. In particular, MTSs are sphericalaggregates of malignant cells,9,10 which can be grown invitro under strictly controlled nutritional and mechanicalconditions.We call R, M, and V the radius, mass, and volume of theMTS, m and v the average mass and volume of a single cell,and N=M /m=V /v the total number of cells in the MTS. Weuse the indices i and f to denote the initial and final values,i.e., at the beginning and at the end of the FRC. We assumethat m is constant throughout the whole cycle, while vchanges, due to the external pressure , but with v f =vi, sincewe assume zero pressure at both ends of the FRC.Let B be the MTS input power from the surroundingmatrix a continuously replenished medium, such as agarosegel. Then, due to energy conservation, B=N+dN /dt,where the two terms on the right hand side represent theenergy rate of consumption due to the metabolism andgrowth, respectively.  is the metabolic rate for a single celland  the energy required to create a new cell. In Ref. 3,West et al. write B=Mp with p=3/4 see also Ref. 11.In order to include the mechanistic effect of the interfa-cial pressure, we now add to the right hand side of the energyconservation law the mechanical term dV /dt. It follows thatdNdt+ dvNdt− Np + N = 0, 1where =mp. At any time during the FRC,  may be com-puted as the mechanical stress resulting from the compatibil-ity between the elastic displacements in the MTS and in theannular region around it: =DN − DN0ctDN + cmDN0, 2where D=3 and cm and ct are the compliances of the matrixand tumor materials given as functions of their elasticmodulus and Poisson’s ratio by Eqs. 7 and 9; for cylin-drical symmetry, i.e., tumor cords, cm and ct are given byEq. 14 and D=2. Finally, to complete the system in thethree unknowns N, v, and , we write the constitutivestress-strain equation:aElectronic mail: nicola.pugno@polito.itAPPLIED PHYSICS LETTERS 89, 233901 20060003-6951/2006/8923/233901/3/$23.00 © 2006 American Institute of Physics89, 233901-1Downloaded 05 Dec 2006 to 165.124.161.25. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jspdvd= − 3ctv . 3Since v cannot change too much during the FRC, it may beconvenient to linearize Eq. 3 by replacing v with v0. Theconstant 3ct may be obtained using for a single cell the sameprocedure, leading to the same equation for the MTS i.e.,comparing the linearized equation 3 with Eq. 12.It may be interesting to note that the “first-order”asymptotic solution of Eq. 1 is identical to the one pre-dicted by West et al.,3 if the energy required to create a newcell  is incremented by the mechanical factor v0.Returning to the mechanistic analogy, in which crackscorrespond to the infiltration channels, failure is usually as-sumed to arise, for un-notched structures, when  reaches thematerial strength tolerance C. In notched specimens, not thestress  but the stress-intensity factor K must reach a criticalvalue KC for fracture propagation;6 thus, Kl=KC,i.e., the stress-intensity factor at the tip of a crack of length lloaded by a stress  must be equal to the fracture toughnessKC of the material;  is a geometrical factor, e.g., for a crackat the edge of a large medium 1.12.Recently a more powerful criterion valid both for smalland large values of l for predicting the strength of solids hasbeen derived,12 by simply removing the assumption of con-tinuous crack propagation. Accordingly, the failure stress isestimated as f =C1 + 2C2 /KIC2 l, 4which represents an asymptotic matching between the twopreviously discussed solutions.As a first example of application of our formalism, wenow use Eqs. 1–3 to predict the time evolution of thetumor volume V=Nv for noninvasive MTS of human colonadenocarcinoma LS174T for different agarose concentra-tions, see Fig. 1. The parameters used for the calculations arereported in the left column of Table I. The agreement withthe corresponding experimental data13 is excellent.Next, to treat the invasive phase, we apply Eq. 4. Weanalyze data from U87MGmEGFR glioma spheroids, rang-ing between 500 and 700 m in diameter, which were cul-tured for six consecutive days in a growth-factor-reducedMatrigel–OPTI-MEM medium mixture 3:1.14 In 40% ofthem there is an amplification of the epidermal growth factorreceptor gene, producing mutant cells able to invadeMatrigel.14 The parameter values are given in the right col-umn of Table I. The observational data empty dots in Fig. 2correspond to an intermediate stage between growth in anunstressed medium solid line and in an elastic mediumdotted line, thus confirming the occurrence of invasion. Thedashed line has been obtained by assuming that a criticalstress occurs at a value  f given by Eq. 4. Also in this casethe agreement with the experimental data14 is very good.Mathematical details. Let us consider a linear elastic iso-tropic spherical shell of inner and outer radii a and b sub-jected to inner and outer pressures p and q, respectively. TheFIG. 1. Comparison between theoretical and observed time evolutions of theLS174T MTS volume. The theoretical lines were obtained using the param-eter values listed in Table I some of which resulted from a numericalfitting. They refer to MTS cultured in a gel with agarose concentrations of0% –, 0.5% ¯, 0.8% –·–, and 0.9% ---. The corresponding experi-mental data Ref. 13 are represented by the symbols 	, , , and *,respectively.TABLE I. Values of the parameters in noninvasive left column and inva-sive right column cases and corresponding references. The parameterswithout reference are the result of a numerical fitting from experimentaldata. AC is the agarose concentration.Parameter LS174T MTS U87MGmEGFR MTS1/ct kPa 3.8 21/cm kPa unstressed 0 0.05a1 /cm kPa at 0.5% AC 2.41b ¯1/cm kPa at 0.8% AC 9.17b ¯1/cm kPa at 0.9% AC 12.6b ¯p 34c 34c /mc1−p g0.25/day 0.042 0.185 / day−1 0.26 0.82aReference 14.bReference 13.cReference 3.FIG. 2. Comparison between theoretical and observed rates of volumegrowth vs volumes of U87MGmEGFR glioma spheroids Ref. 14. Theempty dots represent the observational data and are fitted by the dashed lineusing the parameter values given in Table I. The solid and dotted linescorrespond to the simulation results for growth in an unstressed and anelastic medium, respectively.233901-2 Guiot, Pugno, and Delsanto Appl. Phys. Lett. 89, 233901 2006Downloaded 05 Dec 2006 to 165.124.161.25. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jspradial and circumferential stress components at a generic ra-dius arb are15r =1b3 − a3− b3q + a3p + a3b3r3q − p	 ,5 =1b3 − a3− b3q + a3p − a3b32r3q − p	 .Linear isotropic elasticity and spherical symmetry yieldthe following for the radial component of the displacement uand circumferential component of the strain :ur = r =rE1 − v − r , 6where E and v are Young’s modulus and Poisson’s ratio ofthe material in the shell.Let us now first apply Eqs. 5 and 6 for a=R0 and b→ hence q→0. It follows for rR0, i.e., in the matrixsurrounding the MTS, and p=,r = −R03r3 ,7 = −12r,ur = cmR03r2 ,cm = 1 + vm/2Em .We use the indices m and t for the matrix and tumor,respectively. The radius of the deformed matrix surroundingthe MTS at the pressure  is consequentlyA = R0 + uR0 = R01 + cm . 8Next we apply Eqs. 5 and 6 again, but with a→0,b=Rf, and q=. It follows thatr =  = −  ,uRf = − ctRf , 9ct =1 − 2vtEt.Consequently the radius of the deformed MTS at thepressure , computed starting from the relaxed state at theend of the cycle, becomesB = Rf + uRf = Rf1 − ct . 10The compatibility equation requires A=B contact at theinterface between matrix and MTS, from which we derive =Rf − R0cmR0 + ctRf. 11We now observe that the number of cells, N, increases,due to the term dN /dt, only in the deformation between R0and A, and does not change sensibly during the quick relax-ation from B to Rf. Hence Nf =N. Also, since R03 and Rf3 are,respectively, proportional to V0=v0N0 and Vf =v fN, and v0=v f, from Eq. 11 one obtains immediately Eq. 2.For cylindrical symmetry tumor channels the extensionof our formalism is straightforward. Eqs. 8–11 remainvalid but with different values of ct,m; in fact Eqs. 5–7become, respectively,r =1b2 − a2− b2q + a2p + a2b2r2q − p	 ,12 =1b2 − a2− b2q + a2p − a2b2r2q − p	 ,ur = r =rE − vr , 13with E=E, v=v for plane stress or E=E /1−v2,v=v /1−v for plane strain, and thus,r = −R02r2 , = + r,u = cmR02r , 14cm =1 + vmEm,ct =1 − vtEt.This work has been partly supported by the Centre forthe Development of a Virtual Tumor. Helpful discussionswith T. S. Deisboeck and M. Griffa are gratefullyacknowledged.1NIH Guide to Integrative Cancer Biology Programs 2003.2G. B. West and J. H. Brown, Phys. Today 57, 36 2004.3G. B. West, J. H. Brown, and B. J. Enquist, Nature London 413, 6282001.4P. P. Delsanto, M. Griffa, C. A. Condat, S. Delsanto, and L. Morra,Phys. Rev. Lett. 94, 148105 2005.5C. Guiot, P. G. Degiorgis, P. P. Delsanto, P. Gabriele, and T. S. Deisboeck,J. Theor. Biol. 225, 147 2003.6A. A. Griffith, Philos. Trans. R. Soc. London, Ser. A 221, 163 1920.7D. J. Unger, W. W. Gerberich, and E. C. Aifantis, Scr. Metall. 16, 10591982.8T. S. Deisboeck, Y. Mansury, C. Guiot, P. G. Degiorgis, and P. P. Delsanto,Med. Hypotheses 65, 785 2005.9R. Chignola, A. Schenetti, G. Andrighetto, E. Chiesa, R. Foroni,S. Sartoris, G. Tridente, and D. Liberati, Cell Prolif 33, 219 2000.10M. T. Santini, G. Rainaldi, R. Romano, A. Ferrante, S. Clemente,A. Motta, and P. L. Indovina, FEBS Lett. 557, 148 2004.11A. Carpinteri and N. Pugno, Nat. Mater. 4, 421 2005.12N. Pugno and R. Ruoff, Philos. Mag. 84, 2829 2004.13G. Helminger, P. A. Netti, H. C. Lichtenbeld, R. J. Melder, and R. K. Jain,Nat. Biotechnol. 15, 778 1997.14V. D. Gordon, M. T. Valentine, M. L. Gardel, D. Andor-Ardo,S. Dennison, A. Bogdanov, D. A. Weitz, and T. S. Deisboeck, Exp. CellRes. 289, 58 2003.15J. Prescott, Applied Elasticity Dover, New York, 1946.233901-3 Guiot, Pugno, and Delsanto Appl. Phys. Lett. 89, 233901 2006Downloaded 05 Dec 2006 to 165.124.161.25. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp

Elastomechanical model of tumor invasion

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Elastomechanical model of tumor invasion

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