Most organisms grow according to simple laws, which in principle can be
derived from energy conservation and scaling arguments, critically dependent on
the relation between the metabolic rate B of energy flow and the organism mass
m. Although this relation is generally recognized to be of the form B(m) = mp,
the specific value of the exponent p is the object of an ongoing debate, with
many mechanisms being postulated to support different predictions. We propose
that multicellular tumor spheroids provide an ideal experimental model system
for testing these allometric growth theories, especially under controlled
conditions of malnourishment and applied mechanical stress

Condat, Carlos A.

Degiorgis, Piero Giorgio

Deisboeck, Thomas S.

Delsanto, Pier Paolo

Guiot, Caterina

Mansury, Yuri

English

arXiv

DELSANTO, PIER PAOLO

C. GUIOT

P. G. DEGIORGIS

C. A. CONDAT

Y. MANSURY

T. S. DEISBOECK

PORTO@iris (Publications Open Repository TOrino - Politecnico di Torino)

A Growth Model for Multicellular Tumor Spheroids

Guiot, C.

Degiorgis, P.G.

Condat, C.A.

Mansury, Y.

Deisboeck, T.S.

PORTO Publications Open Repository TOrino

Pier Paolo Delsanto

Caterina Guiot

Piero Giorgio Degiorgis

Carlos A. Condat

Yuri Mansury

Thomas S. Deisboeck

Crossref

Growth model for multicellular tumor spheroids

Institutional Research Information System University of Turin

08 January 2022AperTO - Archivio Istituzionale Open Access dell'Università di TorinoOriginal Citation:A growth model for multicellular Tumor SpheroidsTerms of use:Open Access(Article begins on next page)Anyone can freely access the full text of works made available as "Open Access". Works made available under aCreative Commons license can be used according to the terms and conditions of said license. Use of all other worksrequires consent of the right holder (author or publisher) if not exempted from copyright protection by the applicable law.Availability:This is a pre print version of the following article:This version is available http://hdl.handle.net/2318/36884 since  PROOF COPY 072444APL    PROOF COPY 072444APL  Growth model for multicellular tumor spheroidsPier Paolo Delsantoa)Dip Fisica, Politecnico di Torino and Bioindustry Park Canavese, Colleretto Giacosa, Torino, ItalyCaterina Guiot and Piero Giorgio DegiorgisDip. Neuroscience, Università di Torino, 10125 Torino, Italy,and INFM, sezioni di Torino Università e Politecnico, ItalyCarlos A. CondatDepartment of Physics, University of Puerto Rico, Mayagüez, PR 00681,and CONICET and FaMAF, Universidad Nacional de Córdoba, 5000-Córdoba, ArgentinaYuri MansuryComplex Biosystems Modeling Laboratory, Harvard-MIT (HST) Athinoula A. Martinos Centerfor Biomedical Imaging, HST-Biomedical Engineering Center, Massachusetts Institute of Technology,Cambridge, Massachusetts 02139Thomas S. DeisboeckComplex Biosystems Modeling Laboratory, Harvard-MIT (HST) Athinoula A. Martinos Centerfor Biomedical Imaging, HST-Biomedical Engineering Center, Massachusetts Institute of Technology,Cambridge, Massachusetts 02139 and Molecular Neuro-Oncology Laboratory, Harvard Medical School,Massachusetts General Hospital, Charlestown, Massachusetts 02129(Received 5 April 2004; accepted 2 September 2004)Most organisms grow according to simple laws, which in principle can be derived from energyconservation and scaling arguments, critically dependent on the relation between the metabolic rateB of energy flow and the organism mass m. Although this relation is generally recognized to be ofthe form Bm=mp, the specific value of the exponent p is the object of an ongoing debate, withmany mechanisms being postulated to support different predictions. We propose that multicellulartumor spheroids provide an ideal experimental model system for testing these allometric growththeories, especially under controlled conditions of malnourishment and applied mechanicalstress. © 2004 American Institute of Physics. [DOI: 10.1063/1.1812842]It was recently proposed1 that tumors may follow thegeneral model of ontogenetic growth for all living organisms(from mammals to mollusks to plants) developed by West,Brown and Enquist (WBE).2 The beauty of the WBE modelis that it is entirely based on energy conservation and othergeneral physical arguments, i.e., scaling. The authors startwith the assumption that the energy intake, supplied by in-gested nutrients, is spent partly to support the metabolicfunctions of the organism’s existing cells and partly for cellreplication, i.e., to reproduce new cells. Based on this keyassumption, a universal growth curve is derived, which theauthors conjectured to be applicable to all living organisms.Their conjecture, based on a fractal-like distribution model,is supported by data encompassing many different species.2Although they assumed Bmp with p=3/4, their value of pis not universally accepted. Indeed there is strong evidencethat growth may be consistent with a 2/3 law in the case ofbirds and small mammals.3 Due to its implications for tumormetastasis, cell turnover rates, angiogenesis, and invasion,the proposal of Ref. 1 has immediately contributed to anongoing debate.4,5Many different explanations have been put forward forthe scaling laws, ranging from four-dimensional biology.6and quantum statistics7 to long-bone allometry combinedwith muscular development,8 but, as yet, the debate is farfrom being settled. In this letter we suggest that multicellulartumor spheroids (MTS) are excellent experimental modelsystems to test the validity of the proposed mechanisms.MTS are spherical aggregations of malignant cells,9,10 whichcan be grown in vitro under strictly controlled nutritional andmechanical conditions. Although cells appear to preserve theevolutionary self-organization rules governing the growth ofthe original tumor line, they also evolve according to theenvironmental conditions of the particular experimental set-ting. Here we show that MTS indeed grow following a scal-ing law and obtain their expected properties under conditionsof malnourishment and rising mechanical stress—conditionsthat often apply to tumors in vivo.Following WBE, we assume, for simplicity, that metabo-lism and growth are the same for all MTS cells. Let B be theresting metabolic energy expenditure. Then, at any time t andfor any discrete short time interval t, the law of energyconservation yieldsBt = Nt + N , 1where N is the total number of cells,  is the metabolic ratefor a single cell, and  is the energy required to create a newcell. Since in general volumetric, rather than mass, measure-ments are performed on MTS, we rewrite Eq. (1) in terms ofvolumes (assuming constant cell density). Let  be the vol-ume of a single cell and v=N the volume of the MTS.Writing B=B0vp, for a general choice of the exponent p, Eq.(1) becomesa)Author to whom correspondence should be addressed; electronic mail:caterina.guiot@unito.itAPPLIED PHYSICS LETTERS VOLUME 85, NUMBER 18 1 NOVEMBER 20040003-6951/2004/85(18)/1/0/$22.00 © 2004 American Institute of Physics1  PROOF COPY 072444APL    PROOF COPY 072444APL    PROOF COPY 072444APL  dvdt= avp − bv , 2where a=B0 / and b= /. It follows that, starting from avolume v0 at birth, v increases up to a maximum value V= a /b1/1−p, corresponding to the zero growth point dv /dt=0.Note that Eq. (2) follows von Bertalanffy’s allometriclaw, which has been used to describe growth processes.11 Anumber of alternative specifications, such as the logistic andGompertzian laws, have been previously used to fit spheroidgrowth data.11.Introducing the functionyt = 1 − vtV1−p, 3Eq. (2) becomesdydt= − y , 4with =b1− p. If we now define2 the dimensionless vari-able = 1− pbt−ln1− v0 /V1−p, Eq. (4) has the solutiony=e−, which does not depend on the choice of p and,therefore, is universally valid.Since effective nutrient absorption cannot start immedi-ately after spheroid implantation into the experimental sur-rounding; we model this start-up phase or “accommodationtime” by replacing the parameter a in Eq. (2) with the time-dependent function a1t=a1−e−t/T, where T is the effectiveaccommodation time.Let us now turn our attention to the impact of nutrientsreplenishment on MTS growth. Data from Freyer andSutherland.12 show that when the nutrient content of the tis-sue culture medium is depleted (in particular of glucose andoxygen), the growth curve of the spheroids is modified, i.e.,it reaches a volume lower than the asymptotic value V. Thecondition of insufficient nutrient availability can be repro-duced in our model by the subtraction of a term in Eq. (2)representing the missing power. It is plausible to write such aterm as fvp, with the fraction f  0,1 and f =0 in condi-tions of full nutrient availability. Then Eq. (2) becomesdydt+ y = f + 1 − fe−t/T. 5Equation (5) has a simple exact analytical solution whoseasymptotic value y= f corresponds to the asymptotic vol-umeV = V1 − f1/1−p 0 	 f  1 . 6Figure 1 shows experimental data from Ref. 12 for differentdegrees of undernourishment. Specifically, these authors cul-tured EMT6/Ro mouse mammary carcinoma cell spheroidsin suspension with various oxygen and glucose concentra-tions and measured spheroid growth saturation sizes as wellas viable rim thickness. Both correlated positively with theoxygen and glucose concentrations in the medium. Since theavailable data do not allow us to select an optimal value forp, we choose p=2/3 (which would correspond to diffusion-controlled nutrient inflow) as our working assumption for thefits (solid lines). The agreement is excellent, except for thecase of most severe undernourishment f =0.8. This is notsurprising, because a strong nutrient deficiency should alsoresult in a substantial reduction of the metabolic rate.It has also been suggested that tumor growth and me-tastasis may be influenced by mechanical stress. In particu-lar, Helmlinger et al. have investigated the volumetricgrowth of MTS cultured in agarose gels, showing that it isinhibited when the gel concentration is increased.13 This re-sult can be explained with increasing local stress, which inturn leads to a higher cell (rather then mass) density . Cor-respondingly, Freyer and Schor report that cells in the inter-nal MTS layers experience a size reduction by up to a factorof 3.14 In our model we assume that MTS exhibit the abilityto be compacted, if not compressed, under mechanical stress,so that  varies (e.g., exponentially) from an initial value 0to an asymptotic value R.Defining the auxiliary variables: st= vt /V1−p andgt= t /R1−p, their product “sg” corresponds now to the“mass related” variable “1−y” in Eq. (5). By time differen-tiation and making explicit use of Eq. (5) we obtain−ddtgs = − gdsdt− sdgdt= f − 1 − sg 5bis .After rearranging the terms and inserting the variable yone getsdydt= −dsdt=1 − ygdgdt+ 1 − y +g1 − f5ter .Finally, by noting that 1 /g dg /dt can be rewritten asdn g /dt, Eq. (5ter) becomesdydt+  + d ln gdty − 1 = − 1 − fg1 − e−t/T . 7Here the parameter f accounts for the effect of both the“missing energy input” (as before) and the mechanical stress.Growth data for various gel concentrations from Ref. 13are shown together with model fits in Fig. 2. Note the sub-stantial increase in the accommodation time as the gel con-centration (and thus the mechanical stress) is increased.These results suggest that it takes a relatively long time forFIG. 1. Time variation of the undernourished spheroid mass [see Eq. (3),data from Ref. 12]. Solid curves are model fits for a choice of p=2/3. They intercept corresponds to v0=210−6 cm3. Final volumes V are, startingfrom the lowest curve: 4.410−3, 3.710−3, 1.9510−3, and 3.5610−5 cm3. The accommodation time is, in all cases, T=10 h. The values ofthe parameter f (fitting results) are given next to the corresponding curves.2 Appl. Phys. Lett., Vol. 85, No. 18, 1 November 2004 Delsanto et al.  PROOF COPY 072444APL    PROOF COPY 072444APL    PROOF COPY 072444APL  the tumor cells to develop a suitable nutrient uptake within astressed matrix. Finally, the results show that the effectiveundernourishment f is increased with the gel concentration,while the terminal tumor volume is sharply reduced.In here we suggest that multicellular tumor spheroidscan be used as an experimental tool to test allometric growthlaws under controlled environmental conditions of insuffi-cient nutrient supply and rising mechanical stress. The expo-nent in Eq. (6) indicates that growth is less affected by mal-nourishment when p is farther away from unity. Although wederived Eq. (5) for MTS, one could speculate that the finalsize of tumors that grow according to the 3/4 law will bemore severely modified by nutrient scarcity than in thosefollowing the 2/3 law. In two-dimensional (2D) in vitrodiffusion-controlled growth, p=2/3 should be replaced byp= 12 (if nutrient enters only through the rim), while for alinear cell array (i.e., growing in a tube and fed through itsends) we should assume p=0. This indicates that sensitivityto nutrient deprivation is likely to be decreased in lower-dimensionality tumor cell aggregates (e.g., for single inva-sive tumor cells forming chain-like patterns along preferredanatomically pathways in vivo). We note that our results donot apply to a 2D culture submerged in nutrients, for whichEq. (2) needs to be modified. Finally, our results for growthunder stress indicate that normal nutrient uptake patterns areseverely impaired by increasing mechanical stress. Thisshould have implications for in vivo cancer growth in theneighborhood of robust tissues, such as bone.We note that the current setup does not yet explicitlyconsider space nor does it account for necrosis and apopto-sis, i.e., cell death in the tumor’s center. In vivo, tumors oftendevelop such a necrotic core once they reach a certain vol-ume. We have not included necrosis at this stage because (i)it would make the data interpretation less transparent as newparameters must be added, and (ii) other than in vivo tumors,microscopic tumor spheroids often develop much smaller ne-crotic cores, if at all. However, introducing spatial heteroge-neity may open new possibilities aiming at predictions ofMTS’ spatio-temporal evolution, both from macro- and me-soscopic reference scales.Our conclusions should stimulate more such experimen-tal studies in order to shed light not only on various aspectsof the dynamics of cancer growth, but also on the currentallometric growth law controversy.Y.M. is the recipient of a NCI-Training Grant Fellowshipfrom the National Institutes of Health (CA09502).1C. Guiot, P. G. Degiorgis, P. P. Delsanto, P. Gabriele, and T. S. Deisboeck,J. Theor. Biol. 225,147 (2003).2G. B. West, J. H. Brown, and B. J. Enquist, Nature (London) 413, 628(2001).3P. S. Dodds, D. H. Rothman, and J. S. Weitz, J. Theor. Biol. 209, 9 (2001).4M. Martin, J. Natl. Cancer Inst. 95, 704 (2003).5P. Cohen, New Sci. (2003).6J. J. Blum, J. Theor. Biol. 64, 599 (1977).7L. Demetrius, Physica A 322, 477 (2003).8V. B. Kokshenev, Physica A 322, 491 (2003).9O. Oudar, Crit. Rev. Oncol. Hematol. 36, 99 (2000).10R. Chignola, A. Schenetti, E. Chiesa, R. Foroni, S. Sartoris, A. Brendolan,G. Tridente, G. Andrighetto, and D. Liberati, Cell Prolif 33, 219 (2000).11M. Marusic, Z. Bajzer, S. Vuk-Pavlovic, and J. P. Freyer, Bull. Math. Biol.56, 617 (1994); M. Marusic, Z. Bajzer, J. P. Freyer, and S. Vuc-Pavlovic,Cell Prolif 27, 73 (1994).12J. P. Freyer, and R. M. Sutherland, Cancer Res. 46, 3504 (1986).13G. Helmlinger, P. A. Netti, H. C. Lichtenbeld, R. J. Melder, and R. K. Jain,Nat. Biotechnol. 15, 778 (1997).14J. P. Freyer and P. L. Schor, J. Cell Physiol. 138, 384 (1989).FIG. 2. Time variation of the spheroid volume growing under differentmechanical stress conditions due to various gel concentrations [see Eqs. (6)and (7), data from Ref. 13]. Solid curves are model fits. The y interceptcorresponds to v0=410−9 cm3. Final volumes and accommodation timesV ,T are, starting from the lowest curve: (610−4 cm3, 30 h), (3.810−5 cm3, 100 h), (2.6510−5 cm3, T=110 h), (4.8810−6 cm3, 120 h).The values of f and R (fitting results) are given next to the correspondingcurves.Appl. Phys. Lett., Vol. 85, No. 18, 1 November 2004 Delsanto et al. 3  PROOF COPY 072444APL  

A growth model for multicellular Tumor Spheroids

A Growth Model for Multicellular Tumor Spheroids

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