Thermodynamically Constrained Averaging Theory for Cancer Growth

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IFAC-PapersOnLine 49-26 (2016) 289–294 Thermodynamically constrained averaging Thermodynamically constrained averaging theory for cancer growth modelling theory for cancer growth modelling Marco Albrecht ∗ Giuseppe Sciumè ∗∗ Philippe Lucarelli ∗ Marco Albrecht ∗ Giuseppe Sciumè ∗∗ Philippe Lucarelli ∗ Marco Albrecht ∗ GiuseppeThomas Sciumè Sauter∗∗∗ Philippe Lucarelli ∗ Thomas Sauter ∗ Thomas Sauter ∗ ∗ University of Luxembourg, Belvaux, 4367 Luxembourg ∗ University of Luxembourg, Belvaux, 4367 Luxembourg (e-mail:∗ University [email protected] of Luxembourg, Belvaux, or [email protected]). 4367 Luxembourg (e-mail:∗ University [email protected] of Luxembourg, Belvaux, or [email protected]). 4367 Luxembourg ∗∗ University(e-mail: of [email protected] Bordeaux I2M-TREFLE, or [email protected]). Talence Cedex, 33405 France ∗∗ University of Bordeaux I2M-TREFLE, Talence Cedex, 33405 France ∗∗ University of(e-mail: Bordeaux [email protected]) I2M-TREFLE, Talence Cedex, 33405 France ∗∗ University of(e-mail: Bordeaux [email protected]) I2M-TREFLE, Talence Cedex, 33405 France (e-mail: [email protected]) Abstract: In Systems Biology, network models are often used to describe intracellular mechanismsAbstract: In at Systems the cellular Biology, level. network The obtained models results are often are difficult used to to describe translate intracellular into three mechanismsAbstract: In at Systems the cellular Biology, level. network The obtained models results are often are difficult used to to describe translate intracellular into three dimensionalmechanisms atbiological the cellular systems level. of The higher obtained order. results The multiplicity are difficult and to time translate dependency into three of cellulardimensional system biological boundaries, systems mechanical of higher phenomena order. The and multiplicity spatial concentration and time gradients dependency affect of cellulardimensional system biological boundaries, systems mechanical of higher phenomena order. The and multiplicity spatial concentration and time gradients dependency affect of thecellular intercellular system boundaries, relations and mechanical communication phenomena of biochemical and spatial networks. concentration These gradients environmental affect effectsthe intercellular can be integrated relations with and ourcommunication promising cancer of biochemical modelling networks. environment, These that environmental is based on effectsthe intercellular can be integrated relations with and ourcommunication promising cancer of biochemical modelling networks. environment, These that environmental is based on thermodynamicallyeffects can be integrated constrained with our averaging promising theory cancer (TCAT). modelling Especially, environment, the TCAT that is parameter based on viscositythermodynamically can be used constrained as critical averaging player in theory tumour (TCAT). evolution. Especially, Strong cell-cell the TCAT contacts parameter and a viscositythermodynamically can be used constrained as critical averaging player in theory tumour (TCAT). evolution. Especially, Strong cell-cell the TCAT contacts parameter and a highviscosity degree can of be differentiation used as critical make player cancer in cells tumour viscous evolution. and support Strong compact cell-cell tumourcontacts growth and a withhigh degreehigh tumour of differentiation cell density make and accompanied cancer cells viscous displacement and support of the extracellular compact tumour material. growth In withhigh degreehigh tumour of differentiation cell density make and accompanied cancer cells viscous displacement and support of the extracellular compact tumour material. growth In contrast,with high dedifferentiation tumour cell density and losing and accompanied of cell-cell contacts displacement make cancer of the cells extracellular more fluid material. and lead Into ancontrast, infiltrating dedifferentiation tumour growth and behaviour losing of cell-cell without contacts resistance make due cancer to the cells ECM. more The fluid fast and expanding lead to ancontrast, infiltrating dedifferentiation tumour growth and behaviour losing of cell-cell without contacts resistance make due cancer to the cells ECM. more The fluid fast and expanding lead to tumouran infiltrating front of tumour the invasive growth type behaviour consumes without oxygen resistance and the due limited to the oxygen ECM. The availability fast expanding behind thetumour invasive front front of the results invasive automatically type consumes in a oxygen much smaller and the average limited tumour oxygen availability cell density behind in the thetumour invasive front front of the results invasive automatically type consumes in a oxygen much smaller and the average limited tumour oxygen availability cell density behind in the tumourthe invasive core. front The proposed results automatically modelling technique in a much is most smaller suitable average for tumour tumour growth cell density phenomena in the intumour stiff tissues core. The like proposed skin or bone modelling with high technique content is of most extracellular suitable for matrix. tumour growth phenomena intumour stiff tissues core. The like proposed skin or bone modelling with high technique content is of most extracellular suitable for matrix. tumour growth phenomena in stiff tissues like skin or bone with high content of extracellular matrix. © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: TCAT, Systems Biology, cancer growth, multi phase systems, porous media, tissues Keywords: TCAT, Systems Biology, cancer growth, multi phase systems, porous media, tissues Keywords: TCAT, Systems Biology, cancer growth, multi phase systems, porous media, tissues 1. INTRODUCTION to the softness of several cancer cell lines, especially of 1. INTRODUCTION metastaticto the softness cells. of This several soft cancer conformation cell lines, might especially help cells of 1. INTRODUCTION metastaticto the softness cells. of This several soft cancer conformation cell lines, might especially help cells of The biological field experiences an enormous boost of metastaticsqueezing through cells. This the soft tight conformation and small mightchannels help within cells The biological field experiences an enormous boost of squeezing through the tight and small channels within Themathematical biological and field quantitative experiences methods. an enormous The field boost of Sys- of squeezingthe ECM. through In melanoma the tight it seems, and small that channels the stiffness within of mathematicalThe biological and field quantitative experiences methods. an enormous The field boost of Sys- of the ECM. In melanoma it seems, that the stiffness of mathematicaltems Biology has and emerged quantitative in the methods. interface The of Molecular field of Sys- Bi- the cancer ECM. cells In melanoma changes over it seems, the different that the stages, stiffness always of temsmathematical Biology has and emerged quantitative in the methods. interface The of Molecular field of Sys- Bi- the cancer cells changes over the different stages, always temsology, Biology Mathematics, has emerged Informatics in the interfaceand Engineering. of Molecular Besides Bi- theusing cancer the most cells appropriate changes over mechanical the different invasion stages, strategy always to ology,tems Biology Mathematics, has emerged Informatics in the interfaceand Engineering. of Molecular Besides Bi- using the most appropriate mechanical invasion strategy to ology,this movement, Mathematics, a cooperation Informatics between and Engineering. mainly clinicians, Besides usingcircumvent the most anatomic appropriate obstacles mechanical (Weder invasion et al. (2014)). strategy This to thisology, movement, Mathematics, a cooperation Informatics between and Engineering. mainly clinicians, Besides circumvent anatomic obstacles (Weder et al. (2014)). This thismathematicians movement, a and cooperation civil engineers between pushed mainly forward clinicians, the circumventtumour evolutionary anatomic pattern obstacles has (Weder its equivalent et al. (2014)). phenotype This mathematiciansthis movement, a and cooperation civil engineers between pushed mainly forward clinicians, the tumour evolutionary pattern has its equivalent phenotype mathematiciansfield of Physical Oncology and civil (Frieboes engineers et pushed al. (2011)). forward During the tumouron the transcriptomic evolutionary pattern pattern. has Hoek its et equivalent al. (2008) phenotype state that fieldmathematicians of Physical Oncology and civil (Frieboes engineers et pushed al. (2011)). forward During the on the transcriptomic pattern. Hoek et al. (2008) state that fieldthe last of Physical five years, Oncology those interdisciplinary(Frieboes et al. (2011)). fields coalesce During onmelanoma the transcriptomic cells oscillate pattern. between Hoek a etproliferative al. (2008) state but non- that thefield last of Physical five years, Oncology those interdisciplinary(Frieboes et al. (2011)). fields coalesce During melanoma cells oscillate between a proliferative but non- theand last mechanics five years, of cancer those interdisciplinary is more and more fields related coalesce to melanomainvasive and cells a non-proliferative oscillate between and a proliferative invasive phenotype, but non- andthe last mechanics five years, of cancer those interdisciplinary is more and more fields related coalesce to invasive and a non-proliferative and invasive phenotype, andthe biochemical mechanics of counterparts cancer is more (Hatzikirou and more et al. related (2012)). to invasivecontrolled and byMicrophthalmia-associated a non-proliferative and invasive transcription phenotype, fac- theand biochemical mechanics of counterparts cancer is more (Hatzikirou and more et al. related (2012)). to controlled by Microphthalmia-associated transcription fac- theTensile biochemical and compressive counterparts stress within (Hatzikirou the tumour et al. arise (2012)). due controlledtor (MITF). by MITF Microphthalmia-associated is a differentiation factor transcription of melanoma. fac- Tensilethe biochemical and compressive counterparts stress within (Hatzikirou the tumour et al. arise (2012)). due tor (MITF). MITF is a differentiation factor of melanoma. Tensileto the pressingand compressive proliferating stress tumour within the mass tumour (Stylianopou- arise due torThe (MITF). understanding MITF ofis athe differentiation optimal mechanical factor of environment melanoma. toTensile the pressingand compressive proliferating stress tumour within the mass tumour (Stylianopou- arise due The understanding of the optimal mechanical environment tolos the et al. pressing (2012)) proliferating and cells sense tumour these mass mechanical (Stylianopou- cues. Theand itsunderstanding modulation of might the optimal be helpful mechanical to improve environment the ther- losto the et al. pressing (2012)) proliferating and cells sense tumour these mass mechanical (Stylianopou- cues. and its modulation might be helpful to improve the ther- losSome etcancer al. (2012)) cells and react cells to senseinduced these extracellular mechanical matrix cues. andapeutic its modulation regimen. Greaves might be and helpful Maley to (2012) improve emphasize, the ther- Somelos etcancer al. (2012)) cells and react cells to senseinduced these extracellular mechanical matrix cues. apeutic regimen. Greaves and Maley (2012) emphasize, Somestiffening cancer with cells the react reinforcement to induced of extracellular the own cytoscele- matrix apeuticthat chemotherapy regimen. Greaves removes and the Maley poorly (2012) adjusted emphasize, cancer stiffeningSome cancer with cells the react reinforcement to induced of extracellular the own cytoscele- matrix that chemotherapy removes the poorly adjusted cancer stiffenington and increased with the actin-myosin reinforcement contractility. of the own This cytoscele- sup- thatcells first, chemotherapy but resistant removes cancer the clones poorly can adjusted remain. cancer These tonstiffening and increased with the actin-myosin reinforcement contractility. of the own This cytoscele- sup- cells first, but resistant cancer clones can remain. These tonports and the increased squeezing actin-myosin through the contractility. endothelium This of blood sup- cellsevolutionary first, but selected resistant cells cancer use the clones cleared can space remain. around These to portston and the increased squeezing actin-myosin through the contractility. endothelium This of blood sup- evolutionary selected cells use the cleared space around to portsvessels the - a squeezing step toward through the metastatic the endothelium aggressive of cancer blood evolutionarygrow in an aggressive selected cells manner. use the Some cleared scientists space try around to limit to vesselsports the - a squeezing step toward through the metastatic the endothelium aggressive of cancer blood grow in an aggressive manner. Some scientists try to limit vesselsform (Butcher - a step et toward al. (2009)). the metastatic The internal aggressive cell stiffness cancer growthe available in an aggressive space for manner. tumour Some growth scientists with cytostatica. try to limit formvessels (Butcher - a step et toward al. (2009)). the metastatic The internal aggressive cell stiffness cancer the available space for tumour growth with cytostatica. formfollows (Butcher the stiffness et al. of (2009)). the environment, The internale.g. cellvia stiffness protein theCancer available as chronic space disease for tumour might growth be better with for cytostatica. the patient followsform (Butcher the stiffness et al. of (2009)). the environment, The internale.g. cellvia stiffness protein Cancer as chronic disease might be better for the patient followstyrosine the kinase stiffness 2 (PTK2 of the (FAK)) environment, mediatede.g. mechanovia protein signal Cancerin some as cases, chronic than disease trying might to remove be better cancer for completely.the patient tyrosinefollows the kinase stiffness 2 (PTK2 of the (FAK)) environment, mediatede.g. mechanovia protein signal in some cases, than trying to remove cancer completely. tyrosinetransduction. kinase The 2 (PTK2 mechanical (FAK)) homeostasis mediated mechano is therby signal regu- inTo some better cases, analyse than these trying findings, to remove a mathematical cancer completely. model transduction.tyrosine kinase The 2 (PTK2 mechanical (FAK)) homeostasis mediated mechano is therby signal regu- To better analyse these findings, a mathematical model transduction.lated by positive The and mechanical negative feedbackshomeostasis (Humphrey is therby regu-et al. Tois helpful better to analyse understand these the findings, mechanical a mathematical feedback and model the latedtransduction. by positive The and mechanical negative feedbackshomeostasis (Humphrey is therby regu-et al. is helpful to understand the mechanical feedback and the lated(2014)). by positive Guck et and al. negative (2005) established feedbacks (Humphrey a fluidics device et al. isconcentration helpful to understand gradients in the order mechanical to minimize feedback the risk and and the (2014)).lated by positive Guck et and al. negative (2005) established feedbacks (Humphrey a fluidics device et al. concentration gradients in order to minimize the risk and (2014)).that stretches Guck cells et al. with (2005) the established help of lasers a fluidics and he devicepoints concentrationto maximize the gradients therapy in success. order to minimize the risk and that stretches cells with the help of lasers and he points Into maximize this paper the we therapy would like success. to demonstrate the benefits thatHorizon stretches 2020 MSCA cells withgrantagreement the help Noof lasers642295 andwww.melplex.eu he points Into maximize this paper the we therapy would like success. to demonstrate the benefits Horizon 2020 MSCA grant agreement No 642295 www.melplex.eu In this paper we would like to demonstrate the benefits Horizon 2020 MSCA grant agreement No 642295 www.melplex.eu Copyright2405-8963 © 2016,2016 IFAC (International Federation of Automatic Control)1 Hosting by Elsevier Ltd. All rights reserved. Copyright © 2016 IFAC 1 CopyrightPeer review © under 2016 responsibilityIFAC of International Federation of Automatic1 Control. 10.1016/j.ifacol.2016.12.141 2016 IFAC FOSBE 290October 9-12, 2016. Magdeburg, GermanyMarco Albrecht et al. / IFAC-PapersOnLine 49-26 (2016) 289–294 of using thermodynamically constrained averaging the- gaps, whose volume taken together is called porosity . The ory (TCAT) as base for cancer modelling. Following the saturation degree Sα states how these pores are saturated overview of the modelling framework TCAT, we briefly with three immiscible liquids α, representing interstitial explain the model characteristics. We demonstrate how fluid (IF,l), host cell population (HC,h) and tumour cell tumour cell viscosity can be used as switcher between the population (TC,t). Favourable growth conditions for the invasive and the non-invasive tumour type and finally we tumour cells lead to TC mass expansion by converting IF will show why our modelling approach is ideally suited to TC. The expanding TC liquid presses against the other to be coupled with the network models in Systems Bi- fluids and the ECM. The ECM has a general permeability ology. Thereby, we focus on the description of biological k for liquids, depending on the tissue characteristic. A α compartments and the cellular sensors, which react to the relative permeability krel represents the migration of a mechanical state of a tissue. Moreover, we discuss cellular cell phase in relation to the others, while the IF moves last functions modifying the mechanical state of the tissue. because it’s more attached to the ECM (high wettability or hydrophilicity). From the TCAT momentum equation we 2. THERMODYNAMICALLY CONSTRAINED obtain a relationship between the fluid velocity vα relative AVERAGING THEORY to the ECM scaffold velocity vs kα k Several phenomena in geology, hydrology, petroleum engi- vα vs = rel ( pα)(α = h, t, l). (1) neering and technology deal with porous media in which − µαSα · −∇ multiphase flows occur. Gases and immiscible liquids flow faster than the solid bulk material or stone. Usually The velocity difference depends on the pressure gradient ( pα). The dynamic viscosity µα of the cell population Darcy’s law is applied. But this is a formulation on the −∇ macroscale only. Several microscale phenomena like wet- represents their ability to be moved. This movement is tability have a significant impact on the macroscale but inhibited by cell-cell contacts modelled with high viscosity are not mathematically considered (Sciumèet al. (2013)). up to a pressure gradient threshold. After exceeding this These inconsistencies are frequently bridged with pure pressure gradient threshold, the cell-cell contacts unbound mathematical constructs without physical relations. The and the viscosity is assumed to be reduced smoothly by a thermodynamically constrained averaging theory (TCAT), factor of ten. as developed by Gray and Miller (2014), is a rigorous The tissue model can be formulated at the macro-scale and methodology to interconnect conservation equations across is derived from TCAT mass conservation and momentum different length scales. Conservation equations of phases, conservation equations. The five primary variables are the interphases, common curves and common points are rep- oxygen concentration ωol, the pressure of the IF phase pl, resented in material derivative form and integrated in the pressure difference between IF and HC phase phl, and the entropy inequality equation. The entropy inequality is the pressure between TC and HC phase pth - while the then maximized in a Langrangian framework, constrained sum of these pressures corresponds to the tumour pressure by the conservation equations to obtain a constrained pt = pl + phl + pth. The displacement of the extracellular entropy inequality. Variational methods are used to find matrix us is the fifth primary variable. closure relations near equilibrium. Force-flux pairs in the The momentum balance equation of the extracellular constrained entropy inequality can be used to reduce the matrix is system, because each pair must equal zero. Averaging theorems are used to transfer microscale equations to ∂ts ∂ps macroscale equivalents and evolution equations account eff =0. (2) ∇· ∂t − ∂t for the maintenance of geometric properties. ˙s s The effective stress rate is defined as teff = Ds : e˙ el = Ds : 3. TCAT TISSUE MODEL DEFORMED BY TUMOUR s GROWTH (e˙ s e˙ s ), where the strain rate e˙ s = S ∂u is of visco- − vp ∂t plastic (vp) and elastic (el) origin. The tangent matrix Ds The use of TCAT eases the formulation of a model de- represents the mechanical properties of the ECM and links scribing tumour growth. The framework allows a straight the stress with the strain, while tensor S links the strain forward model extension using partial differential equa- with the displacement. The solid pressure is coupled with tions with at most second order derivatives in time and the proportionately considered liquid pressures ps = pl + space. Other methods in the field of continuous modelling (1 Sl)phl + Stpth, using pore saturation Sα as Bishop − like mixture theory are confronted with complicated fourth parameter. The result is the pressure-strain relationship order Cahn Hilliard equations and are difficult to extent α˜ ∂ps s K ∂t = 1 : dsp. Each liquid and solid component is not (Sciumèet al. (2013)). TCAT was applied by Giuseppe compressible but the deformation of the ECM scaffold is Sciumè et. al. to investigate the impact of mechanical taken into account by bulk modulus K. The final governing stress on diabetic foot ulceration (Sciumèet al. (2014a)) ECM displacement equation, based on Equation 2, is and to explain tumour growth within healthy tissues. This section summarizes the latest model of melanoma growth ∂us (D :(S )) (D : e˙ s )= (K(1 : e˙ s )). (3) in skin from Sciumèet al. (2014b) with some minor mod- ∇· s ∂t −∇· s vp ∇· sp ifications as detailed below. The solid porous medium represents the ECM, a protein The mass balance equation of the solid phase is integrated network secreted by the cells to establish a stable cell in the following mass balance equations of three immiscible environment. This ECM contains tortuous interconnected liquids: The interstitial fluid phase

2 2016 IFAC FOSBE 2016 IFAC FOSBE October 9-12, 2016. Magdeburg, Germany October 9-12, 2016. Magdeburg, GermanyMarco Albrecht et al. / IFAC-PapersOnLine 49-26 (2016) 289–294 291

t th of using thermodynamically constrained averaging the- gaps, whose volume taken together is called porosity . The 1 ∂S ∂p 0 for 0 ωol ωol α (St + pth ) ≤ ≤ crit ory (TCAT) as base for cancer modelling. Following the saturation degree S states how these pores are saturated th 1 1 ωol ωol K ∂p ∂t ol − crit ol ol ol G1(ω )= + cos π 1+ for ωcrit ω ωenv overview of the modelling framework TCAT, we briefly with three immiscible liquids α, representing interstitial  2 2 ol ol ≤ ≤ l hl l ωenv ω 1 l hl ∂S ∂p 1 ∂p  − crit explain the model characteristics. We demonstrate how fluid (IF,l), host cell population (HC,h) and tumour cell + (1 S p ) + 1 for ωol ωol tumour cell viscosity can be used as switcher between the population (TC,t). Favourable growth conditions for the K − − ∂phl ∂t K ∂t ≥ env (9) invasive and the non-invasive tumour type and finally we tumour cells lead to TC mass expansion by converting IF t t h  t t krelk th krelk krelk hl 1 for 0 p pcr1 = p + + p t t ≤ ≤ will show why our modelling approach is ideally suited to TC. The expanding TC liquid presses against the other t t h t 1 1 p pcr1 t t t ∇· µ ·∇ ∇· µ µ ·∇ G2(p )= + cos π 1+ − for p p p 2 2 pt pt cr1 ≤ ≤ cr2 to be coupled with the network models in Systems Bi- fluids and the ECM. The ECM has a general permeability t h l  cr2 − cr1 k k k k k k 0 for pt pt ology. Thereby, we focus on the description of biological k for liquids, depending on the tissue characteristic. A + rel + rel + rel pl  ≥ cr2 compartments and the cellular sensors, which react to the relative permeability kα represents the migration of a ∇· µt µh µl ·∇ (10) rel l t  1 1 ωol mechanical state of a tissue. Moreover, we discuss cellular cell phase in relation to the others, while the IF moves last ρ ρ l t + cos π 1+ for 0 ωol ωol s s → ol ≤ ≤ crit functions modifying the mechanical state of the tissue. because it’s more attached to the ECM (high wettability or 1 :(e˙ e˙ sp) + − M , R(ω )= 2 2 ωol (11) − − ρtρl  crit hydrophilicity). From the TCAT momentum equation we 1 for ωol ωol  ≥ crit α (4) 2. THERMODYNAMICALLY CONSTRAINED obtain a relationship between the fluid velocity v relative 1 1 ωol s the host cell phase + cos π for 0 ωol ωol to the ECM scaffold velocity v ol  ≤ ≤ crit AVERAGING THEORY N1(ω )= 2 2 ωol Sh ∂St ∂St ∂pth  crit α t th 0 for ωol ωol k k (S + p th ) th  ≥ crit Several phenomena in geology, hydrology, petroleum engi- α s rel α K ∂p − ∂p ∂t (12) v v = α α ( p )(α = h, t, l). (1) neering and technology deal with porous media in which − µ S · −∇ Sh Sl ∂Sl ∂phl Sh ∂pl  multiphase flows occur. Gases and immiscible liquids flow + (1 Sl phl ) + K − − phl − ∂phl ∂t K ∂t faster than the solid bulk material or stone. Usually The velocity difference depends on the pressure gradient α α h ol ( p ). The dynamic viscosity µ of the cell population k k Below the oxygen threshold ωcrit, the tumour cells have a Darcy’s law is applied. But this is a formulation on the −∇ = rel (pl + phl) Sh 1 :(e˙ s e˙ s ) vs Sh, macroscale only. Several microscale phenomena like wet- represents their ability to be moved. This movement is ∇· µh ·∇ − − sp − ∇ reduced basic respiratory or metabolic turnover rate with ol tability have a significant impact on the macroscale but inhibited by cell-cell contacts modelled with high viscosity (5) basic oxygen consumption rate R(ω ) and start to die, are not mathematically considered (Sciumèet al. (2013)). up to a pressure gradient threshold. After exceeding this while above this threshold, tumour cells start growing until pressure gradient threshold, the cell-cell contacts unbound and the tumour cell phase they reach the maximum proliferation rate at the envi- These inconsistencies are frequently bridged with pure t t t th and the viscosity is assumed to be reduced smoothly by a S t th ∂S ∂S ∂p ol mathematical constructs without physical relations. The (S + p )+ ronmental oxygen concentration ωenv. The proliferation thermodynamically constrained averaging theory (TCAT), factor of ten. K ∂pth ∂pth ∂t t rate is maximal if tumour pressure is below pcr1. Above as developed by Gray and Miller (2014), is a rigorous The tissue model can be formulated at the macro-scale and t l hl t l this threshold, cells have a reduced growth rate to mimic S l hl ∂S ∂p S ∂p methodology to interconnect conservation equations across is derived from TCAT mass conservation and momentum + (1 S p ) + K − − ∂phl ∂t K ∂t contact inhibition. different length scales. Conservation equations of phases, conservation equations. The five primary variables are the Necrosis is a process, that shuts down the work of ion ol l kt k interphases, common curves and common points are rep- oxygen concentration ω , the pressure of the IF phase p , = rel (pl + phl + pth) St 1 :(e˙ s e˙ s ) pumps due to energy shortage. Osmotic water influx lets resented in material derivative form and integrated in the pressure difference between IF and HC phase phl, and ∇· µt ·∇ − − sp the necrotic tumour cells swell within 3 hours to the th the entropy inequality equation. The entropy inequality is the pressure between TC and HC phase p - while the l t s t 1 → doubled size, before the cells lose water through a stretched then maximized in a Langrangian framework, constrained sum of these pressures corresponds to the tumour pressure v S + t M . and leaky cell membrane. The concentrated cytoplasm t l hl th − ∇ ρ by the conservation equations to obtain a constrained p = p + p + p . The displacement of the extracellular (6) builds calcified crystals, accumulating in the necrotic entropy inequality. Variational methods are used to find matrix us is the fifth primary variable. core of tumours (Macklin et al. (2013)). Accordingly, the closure relations near equilibrium. Force-flux pairs in the The momentum balance equation of the extracellular The tumour cell phase can be partitioned in two miscible swelling can be modelled with a factor of two in the constrained entropy inequality can be used to reduce the matrix is liquids accounting for the living tumour cell fraction (1 necrosis rate equation Nt − system, because each pair must equal zero. Averaging ω ) and the necrotic tumour cell fraction t StrN =2 [γt N (ωol)](1 ωNt)St γt ωNtSt. theorems are used to transfer microscale equations to s s necrosis 1 clear ∂t ∂p ∂ωNt 1 l t · − − macroscale equivalents and evolution equations account eff =0. (2) t Nt Nt → t t t Nt (13) = t t S r ω M S ρ v ω , (7) ∇· ∂t − ∂t ∂t S ρ − growth − ·∇ t for the maintenance of geometric properties. Equation 13 contains the maximal necrosis rate γnecrosis which arises with a low oxygen mass fraction ωol, whose and an additional term describing cell degradation with ˙s s t The effective stress rate is defined as teff = Ds : e˙ el = Ds : distribution is described by species mass conservation clearance rate γ . Because this is only a transfer within 3. TCAT TISSUE MODEL DEFORMED BY TUMOUR s clear GROWTH (e˙ s e˙ s ), where the strain rate e˙ s = S ∂u is of visco- equation the tumour phase, the tumour growth rate has to be − vp ∂t plastic (vp) and elastic (el) origin. The tangent matrix Ds ∂ωol 1 l t ol t adjusted, so that the necrotic swelling arises from the Sl = (SlDol(Sl)σ ωol)+ (ωol M→ M)→ interstitial fluid and not from the living tumour mass The use of TCAT eases the formulation of a model de- represents the mechanical properties of the ECM and links ∂t ∇· 0 ∇ ρl − scribing tumour growth. The framework allows a straight the stress with the strain, while tensor S links the strain (1 ωNt)St. The growth rate (8) − forward model extension using partial differential equa- with the displacement. The solid pressure is coupled with ol l t s l with the oxygen diffusion coefficient D0 . The mass trans- → t ol t Nt t tions with at most second order derivatives in time and the proportionately considered liquid pressures p = p + M =[γgrowthG1(ω )G2(p )](1 ω )S l hl t th α l t growth − space. Other methods in the field of continuous modelling (1 S )p + S p , using pore saturation S as Bishop fer from the IF phase to the TC phase M → is related − to tumour growth using free IF from the environment. t ol Nt t (14) like mixture theory are confronted with complicated fourth parameter. The result is the pressure-strain relationship +[γnecrosisN1(ω )](1 ω )S s ol t − order Cahn Hilliard equations and are difficult to extent α˜ ∂p s The related oxygen species transfer M → account for the t Nt t K ∂t = 1 : dsp. Each liquid and solid component is not γ ω S (Sciumèet al. (2013)). TCAT was applied by Giuseppe compressible but the deformation of the ECM scaffold is oxygen consumption. Cell death due to oxygen limitation − clear is modelled as species transfer rNt between two miscible t Sciumè et. al. to investigate the impact of mechanical taken into account by bulk modulus K. The final governing with maximal growth rate γgrowth can be increased by the stress on diabetic foot ulceration (Sciumèet al. (2014a)) ECM displacement equation, based on Equation 2, is liquids. The living tumour mass is transferred to the frac- swelling of the necrotic fraction and can be reduced due and to explain tumour growth within healthy tissues. This tion of dead tumour mass within the immiscible tumour to the shrinkage and degradation of the necrotic mate- s cell phase. These three transfer terms are composed of section summarizes the latest model of melanoma growth ∂u s s rial. The swelling and degradation of necrotic material (Ds :(S )) (Ds : e˙ )= (K(1 : e˙ )). (3) cosine functions 9 to 12 containing threshold parameters in skin from Sciumèet al. (2014b) with some minor mod- ∇· ∂t −∇· vp ∇· sp are extensions for the model from Sciumèet al. (2014b) ifications as detailed below. that describe the transition zones between minimum and while Equations 9 - 12 replace former used Heaviside step The solid porous medium represents the ECM, a protein The mass balance equation of the solid phase is integrated maximum tumour growth, oxygen consumption, and tu- functions for a better numeric stability. network secreted by the cells to establish a stable cell in the following mass balance equations of three immiscible mour cell death rates. The model equations are discretized in space with Galerkin environment. This ECM contains tortuous interconnected liquids: The interstitial fluid phase and in time with the approach of Crank-Nicolson using the

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A initial condition A oxygen mass fraction 1 5 living tumor mass interstitial fluid 4 0.5 [-] 3 6 proportion solid ECM −

0 10 2 0 200 400 600 800 1000 high viscous 1 medium viscous B high viscous tumour [ 6300 Pa · s] 1 low viscous 0 0 200 400 600 800 1000 distance from the center [µm] 0.5 necrotic mass B radial displacement 20 proportion high viscous 0 medium viscous 0 200 400 600 800 1000 m] 15 µ low viscous C medium viscous tumour [ 63 Pa · s] 1 10

0.5 5 displacement [ proportion 0 0 0 200 400 600 800 1000 0 200 400 600 800 1000 distance from the center [µm] D low viscous tumour [ 0.63 Pa · s] 1 Fig. 2. Spatial proﬁle along the radius after 31 days 0.5 simulated. A: Oxygen mass fraction (unitless). B: ECM displacement. Positive displacement means a

proportion displacement in direction of the spheroid surface while 0 a negative sign indicates a compression in direction 0 200 400 600 800 1000 tumour center. distance from the center [µm] high viscous tumour (µt = 6300 Pa s), the medium vis- Fig. 1. Volume proportions of solid ECM (dark green), in- cous tumour (µt = 63 Pa s) and the· low viscous tumour terstitial fluid (white), living tumour mass (cyan) and (µt =0.63 Pa s). · the necrotic core (black) changes along the spheroid The starting· condition is illustrated in Figure 1 A. An radius. The left side is the micro tumour spheroid initial tumour mass (radius: 50 µm) in a three dimensional core. A: Initial condition. B-D: Model simulation collagen gel spheroid (radius: 1000 µm) has been defined. of tumour growth with high (B), medium (C) and After the simulation of 31 days, three different viscosities low (D) viscosity after 31 days. The change of the yield three clearly different tumour phenotypes. Exact viscosity is comparable with a switch between benign values are listed in Table 1. and malignant tumour growth. The high viscous tumour grows compact and slow (Figure Wilsons Theta method. The equations are implemented 1 B) and 33.5 % of the cells are alive. in Cast3M (www-cast3m.cea.fr) and solved with a finite The medium viscous tumour infiltrates a 46% larger radius element method. The mathematical model contains two and a 3 times bigger volume. The absolute tumour cell blocks: One describes pressure and diffusion with Equa- tions 4, 5, 6 and 8; and one takes into account the solid Table 1. Final results after 31 days of simula- mechanics with Equation 3. Both blocks are solved iter- tion. The space infiltrated by the tumour con- atively, until convergence is reached at each single time tains, besides the tumour cells, interstitial fluid step. and ECM. The cytoplasm volume of all dead and living tumour cells (TC) or of all living 4. SIMULATION RESULTS tumour cells (LTC) is lower. The displacement of the ECM decreases with lower viscosity.

A physical model of cancer growth should explain bio- high medium low logical phenomena observed experimentally. One impor- viscous viscous viscous tant experimental observation is the change from a com- tumour tumour tumour pact growing tumour to an invasive tumour with reduced Radius infiltrated [µm] 199 291 695 growth. In this paper, the tumour evolution is investi- Volume infiltrated [mm3] 33 103 1408 3 3 gated when cell-cell contacts are weakened and intracel- Volume of TC [10− mm ] 8.55 8.73 33.32 3 3 lular stiffness is decreased. Accordingly, three different Volume of LTC [10− mm ] 2.86 5.31 23.67 viscosities are simulated for the tumour cell phase: The Max displacement [µm] 18.8 2.68 0.49

4 2016 IFAC FOSBE October 9-12, 2016. Magdeburg, Germany 2016 IFAC FOSBE October 9-12, 2016. Magdeburg, GermanyMarco Albrecht et al. / IFAC-PapersOnLine 49-26 (2016) 289–294 293

A initial condition A oxygen mass fraction volume remains almost constant, while the fraction of considered for the model in subsequent papers. Our model 1 5 interstitial fluid living cells is with 60.8% distinctly larger. The tumour predicts a much lower cancer cell density in accordance living tumor mass cell density is clearly smaller (Figure 1 C). with the related oxygen profile for the invasive tumour, 4 The low viscous tumour infiltrates a 3.5 times larger radius although the molecular link between dedifferentiation and 0.5 [-] 3 than the high viscous tumour and it infiltrates a 42.7 larger growth rate has been neglected. Hoek et al. (2008) dis- 6 proportion solid ECM − volume. The absolute tumour mass increases by a factor covered in the flanks of mice, that the invasive tumour

0 10 2 of 3.9. Figure 1 D indicates the very small tumour cell grows slower than the proliferative phenotype. But their 0 200 400 600 800 1000 high viscous density. The low viscous tumour has the largest necrotic result might be underestimated as dedifferentiated cancer 1 medium viscous core with a radius of around 400 µm and has the thickest cells cannot be stained specifically and because the tumour B high viscous tumour [ 6300 Pa · s] 1 low viscous living tumour front with 295 µm. 71% of the low viscous size measurement using vernier calipers might be limited 0 tumour mass belongs to the living tumour cell mass. to a small visible subpart of the infiltrated area. Especially, 0 200 400 600 800 1000 The correlation between viscosity and tumour cell density if the cancer cell density of the originated invasive cancer distance from the center [µm] 0.5 necrotic can be explained with the oxygen mass fraction. The oxy- cells is too small to be detected, the injected initial tumour mass B radial displacement 20 gen profiles of all three cases are shown in Figure 2 A and mass might be the only part of the tumour, that can be proportion high viscous the profile of each case can be separated in three sections. measured with vernier calipers. 0 medium viscous The first section is located in the necrotic core zone in The proposed model is easy to extend. For instance, cells 0 200 400 600 800 1000 m] 15 µ low viscous which the oxygen mass fraction is constant at minimum produce ECM reducing metallo-matrix proteins or they C medium viscous tumour [ 63 Pa · s] 1 level. The second section is in the zone of the living tumour change the metabolism to an oxygen independent but 10 cell front. Here, the biggest change of the oxygen mass acidic lactate producing modus. Both weaken the mechan- fraction is observed. The third section is the zone of the ical properties of the ECM. The first reduces the amount of 0.5 5 pure gel. The profile of the oxygen mass fraction in the solid material, the second changes the material properties

displacement [ third zone depends on diffusion. The oxygen mass fraction of the solid phase due to an acidic environment (Achilli and proportion 0 increases from the invasive front to the fixed value of Mantovani (2010)). Parameters can be modified based on 0 6 0 200 400 600 800 1000 4.2 10− at the boundary of the collagen spheroid. The environmental concentrations, and mass exchange terms 0 200 400 600 800 1000 · distance from the center [µm] more viscous the tumour is, the more time the tumour has can easily be added to take into account the degrada- D low viscous tumour [ 0.63 Pa · s] 1 to grow locally. In the low viscous case, the living tumour tion of solid material. Besides the cells’ impact on the front changes the position quickly. At the new position, the environment, the environmental feedback on the cells can Fig. 2. Spatial profile along the radius after 31 days living tumour front consumes oxygen and interrupts the be considered. The model calculates the interstitial fluid 0.5 simulated. A: Oxygen mass fraction (unitless). B: supply of oxygen for the old position behind. The tumour pressure as well as the oxygen concentration to any discrete ECM displacement. Positive displacement means a become faster necrotic than in the viscous case and the time step. The increasing interstitial fluid pressure does proportion displacement in direction of the spheroid surface while cell density remains small. not have only an effect on the drug efficiency (Heldin 0 a negative sign indicates a compression in direction Figure 2 B shows the displacement of the ECM due to the et al. (2004)) but is also connected to the pressure induced 0 200 400 600 800 1000 tumour center. distance from the center [µm] growing tumour. The high viscous tumour displaces the crosslinking of ECM molecules. This crosslinking activates surrounding ECM by almost 20 µm. This could explain the mechano signal transduction, which activates proto- high viscous tumour (µt = 6300 Pa s), the medium vis- the lower oxygen level in the necrotic core in comparison oncogene tyrosine-protein kinase SRC (Levental et al. Fig. 1. Volume proportions of solid ECM (dark green), in- cous tumour (µt = 63 Pa s) and the· low viscous tumour to the less viscous cases. The compressed matrix reduces (2009)), a protein that is also triggered by limited oxygen terstitial fluid (white), living tumour mass (cyan) and (µt =0.63 Pa s). · the pore space for diffusion and the sharp living tumour concentration (Hanna et al. (2013)). This will be modelled the necrotic core (black) changes along the spheroid The starting· condition is illustrated in Figure 1 A. An front takes more from the remaining oxygen of the tumour in subsequent papers as concentration of active SRCa radius. The left side is the micro tumour spheroid initial tumour mass (radius: 50 µm) in a three dimensional zone instead of the tumour environment. The displacement depending on ECM displacement [SRCa] us. SRC core. A: Initial condition. B-D: Model simulation collagen gel spheroid (radius: 1000 µm) has been defined. decreases from the high viscous to the medium viscous is directly responsible for the loose of cell-cell∝ contacts of tumour growth with high (B), medium (C) and After the simulation of 31 days, three different viscosities tumour by a factor of 7 and from the medium viscous and migration (Avizienyte and Frame (2005)), which we low (D) viscosity after 31 days. The change of the yield three clearly different tumour phenotypes. Exact to the low viscous tumour by a factor of 5.5, according can in turn consider as viscosity change. SRC activates viscosity is comparable with a switch between benign values are listed in Table 1. Table 1. Within medium and low viscous tumours, cells extracellular signal-regulated kinase (ERK), which is, with and malignant tumour growth. The high viscous tumour grows compact and slow (Figure can squeeze easier through the ECM and the solid ECM its interacting partners, a central player in cancer biology 1 B) and 33.5 % of the cells are alive. Wilsons Theta method. The equations are implemented has a lower resistance to tumour growth. While in the and a target for pharmaceuticals. Moreover, our model The medium viscous tumour infiltrates a 46% larger radius in Cast3M (www-cast3m.cea.fr) and solved with a finite solid tumour, ECM has an impact, the displacement of the provides at any time the relation between interstitial fluid, and a 3 times bigger volume. The absolute tumour cell element method. The mathematical model contains two ECM for the medium and low viscous growth is negligible tumour cytoplasm and host cell cytoplasm, which can be blocks: One describes pressure and diffusion with Equa- and no limitation of oxygen delivery is expected due to used to relate quantitatively the intracellular pathways to tions 4, 5, 6 and 8; and one takes into account the solid Table 1. Final results after 31 days of simulation. The space infiltrated by the tumour con- compressed ECM. each other, e.g. based on the input-output behaviour of mechanics with Equation 3. Both blocks are solved iter- molecular network models. The per cell production rate atively, until convergence is reached at each single time tains, besides the tumour cells, interstitial fluid 5. DISCUSSION and ECM. The cytoplasm volume of all dead of any molecule can be directly translated to the actual step. concentration in the limited interstitial volume. and living tumour cells (TC) or of all living The change of the viscosity is like the change between be- 4. SIMULATION RESULTS tumour cells (LTC) is lower. The displacement nign and malign tumour growth. The question is, whether 6. CONCLUSION of the ECM decreases with lower viscosity. it is easier for the tumour to displace or to flow through the A physical model of cancer growth should explain bio- high medium low ECM in order to grow. The observed smoothness in cancer The thermodynamically constrained averaging theory has logical phenomena observed experimentally. One impor- viscous viscous viscous cell lines by Guck et al. (2005) has also in our simulation a been used to model cancer growth in three dimensional tant experimental observation is the change from a com- tumour tumour tumour profound benefit for the tumour evolution. The low viscous collagen gels or tissues. The benefit of our modelling ap- pact growing tumour to an invasive tumour with reduced Radius infiltrated [µm] 199 291 695 tumour has infiltrated a much larger space than the high proach is, that the ECM and the host cells can be actuated growth. In this paper, the tumour evolution is investi- Volume infiltrated [mm3] 33 103 1408 viscous tumour, representing stiff and highly differentiated separately by the cancer cells, while most other continuous 3 3 gated when cell-cell contacts are weakened and intracel- Volume of TC [10− mm ] 8.55 8.73 33.32 cancer cells with strong cell-cell contacts. But the dediffer- modelling approaches treat the environment as a mixture 3 3 lular stiffness is decreased. Accordingly, three different Volume of LTC [10− mm ] 2.86 5.31 23.67 entiation of melanoma cells - that supports this invasive with averaged material properties. This allows us to use viscosities are simulated for the tumour cell phase: The Max displacement [µm] 18.8 2.68 0.49 growth - is also connected at the molecular level with a the advantages of continuous and bio mechanical mod- reduced growth rate (Hoek et al. (2008)), which will be elling while reaching additional advantages of agent-based 4 5 2016 IFAC FOSBE 294October 9-12, 2016. Magdeburg, GermanyMarco Albrecht et al. / IFAC-PapersOnLine 49-26 (2016) 289–294 modelling. While the agent-based models define clearly Hanna, S.C., Krishnan, B., Bailey, S.T., Moschos, S.J., the reaction volume of each cell, we define clearly the Kuan, P.F., Shimamura, T., Osborne, L.D., Siegel, reaction volumes of the cytoplasm within a representative M.B., Duncan, L.M., O’Brien, E.T., et al. (2013). HIF1α volume element. This can be justified if neither single and HIF2α independently activate SRC to promote cell information is available nor cellular memory must be melanoma metastases. The Journal of clinical inves- tracked. While in agent-based systems the number of cell tigation, 123(5), 2078–2093. types are nearly unlimited, our described cell liquids can be Hatzikirou, H., Chauviere, A., Bauer, A.L., Leier, A., subdivided in several miscible and cell type specific liquids. Lewis, M.T., Macklin, P., Marquez-Lago, T.T., Bearer, Our approach shifts the focus on inter cell communication E.L., and Cristini, V. (2012). Integrative physical to a more holistic view with cell-ECM interplay. This oncology. Wiley Interdisciplinary Reviews: Systems improves the modelling for cancer growth in comparison Biology and Medicine, 4(1), 1–14. to agent-based models with its accumulation of mechanical Heldin, C.H., Rubin, K., Pietras, K., and Ostman,¨ A. deformable beads and in comparison to other continuous (2004). High interstitial fluid pressure- an obstacle in models without clear definition of reaction spaces, molec- cancer therapy. Nature Reviews Cancer, 4(10), 806–813. ular networks relate to. The reduction of cell-cell contacts Hoek, K.S., Eichhoff, O.M., Schlegel, N.C., Döbbeling, and the reduction of the intracellular stiffness reduce the U., Kobert, N., Schaerer, L., Hemmi, S., and Dummer, overall viscosity of the tumour cell population. The TCAT R. (2008). In vivo switching of human melanoma parameter viscosity is a central player for the tumour cells between proliferative and invasive states. Cancer growth shape and the overall tumour volume. A low viscos- research, 68(3), 650–656. ity reduces the displacement of the ECM and the tumour Humphrey, J.D., Dufresne, E.R., and Schwartz, M.A. cells infiltrate a much larger volume. Due to a changed (2014). Mechanotransduction and extracellular matrix oxygen profile in the low viscous tumour, the cancer cell homeostasis. Nature Reviews Molecular Cell Biology, density within the tumour remains small. The parameter 15(12), 802–812. viscosity can be linked with several biologically meaningful Levental, K.R., Yu, H., Kass, L., Lakins, J.N., Egeblad, sub-systems or molecules and the proposed model has the M., Erler, J.T., Fong, S.F., Csiszar, K., Giaccia, A., potential to extend network orientated models, formulated Weninger, W., et al. (2009). Matrix crosslinking forces in the field of Systems Biology. tumor progression by enhancing integrin signaling. Cell, 139(5), 891–906. ACKNOWLEDGEMENTS Macklin, P., Mumenthaler, S., and Lowengrub, J. (2013). Modeling multiscale necrotic and calcified tissue biome- This work has been financed by the European Union. chanics in cancer patients: application to ductal carci- We acknowledge the Horizon 2020 MSCA grant agree- noma in situ (DCIS). In Multiscale computer modeling ment, No 642295, www.melplex.eu. Moreover, we thank in biomechanics and biomedical engineering, 349–380. the Fonds National de la Recherche, Luxembourg (IN- Springer. 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