*Corresponding author: Mattenstrasse 26, CH-4058 Basel,

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1 Simulaing Organogenesis in COMSOL: Image-based Modeling Z. Karimaddini 1,, E. Unal 1,,3, D. Menshykau 1, and D. Iber* 1, 1 Deparemen for Biosysems Science and Engineering, ETH Zurich, Swizerland Swiss Insiue of Bioinformaics (SIB), Swizerland 3 Developmenal Geneics, Deparmen Biomedicine, Universiy of Basel, Swizerland *Corresponding auhor: Maensrasse 6, CH-4058 Basel, dagmar.iber@bsse.ehz.ch Absrac: Mahemaical Modelling has a long hisory in developmenal biology. Advances in experimenal echniques and compuaional algorihms now permi he developmen of increasingly more realisic models of organogenesis. In paricular, 3D geomeries of developing organs have recenly become available. In his paper, we show how o use image-based daa for simulaions of organogenesis in COMSOL Muliphysics. As an example, we use limb bud developmen, a classical model sysem in mouse developmenal biology. We discuss how embryonic geomeries wih several subdomains can be read ino COMSOL using he Malab LiveLink, and how hese can be used o simulae models on growing embryonic domains. The ALE mehod is used o solve signaling models even on srongly deforming domains. Keywords: in silico organogenesis, image-based modeling, limb developmen, compuaional biology, numerical simulaion, COMSOL 1. Inroducion Organogenesis is a highly dynamic process ha is ighly regulaed during embryogenesis. Many of he individual regulaory componens, e.g. signaling molecules and heir recepors, as well as heir regulaory ineracions have been idenified in experimens. However, an inegraive mechanisic undersanding of he regulaory nework is missing [1]. Mahemaical modeling has a long hisory in developmenal biology [,3]. Limb developmen, in paricular, has araced much aenion from modellers [4]. Early models were raher simplisic, and o his dae mos models are sill solved on idealized domains ha a mos qualiaively resemble he physiological domains. However, he geomery can grealy impac he paerning process [5], and i is herefore imporan o solve hese models on physiological domains. COMSOL Muliphysics is a versaile package ha provides finie elemen mehod (FEM)-based solvers o solve a wide range of parial differenial equaion (PDE)-based problems on complex domains. We have used COMSOL o solve models of limb developmen [5,6], bone developmen [7], ovarian follicle developmen [8], and branching morphogenesis [9,10,11]. In a series of papers on simulaing organogenesis in COMSOL [1,13,14,15], we have discussed mehods o efficienly solve models for organogenesis on complex saic and growing domains as well as models, which consider cells explicily. Iniially, hese models were formulaed on idealized geomeries. Recenly, we have sared o ake advanage of advancemens in imaging echniques, which now provide us wih deailed imaging daa of organogenesis [15]. This now allows us o simulae our models on realisically growing embryonic domains in COMSOL Muliphysics [16]. In his paper, we show how o use image-based daa for simulaions of organogenesis in COMSOL Muliphysics. As an example, we use limb bud developmen, a classical model sysem in mouse developmenal biology. In he firs sep, compuer readable geomeries mus be exraced from he 3D images and mus hen be impored ino COMSOL. Many issues conain clearly defined subdomains wih differen properies. These can be idenified wih suiable saining proocols for marker proeins or marker proein expression. We show how complex domains wih subdomains can be impored. In a second sep, he displacemen fields beween wo consecuive image frames mus be calculaed and impored ino COMSOL. Finally, he impored displacemen fields can be used o simulae he domain shape evoluion in COMSOL. Given he large number of sages ha we use, we implemen our models using Malab LiveLink. We use he ALE mehod o solve our PDE-based Excerp from he Proceedings of he 014 COMSOL Conference in Cambridge

2 A) B) domain Lengh uni Figure 1. An idealized D limb bud domain a wo differen ime poins. The enire domain is divided ino hree subdomains ( green, domain - blue, - red). The domains and subdomains deform during developmen. (A) Domain a ime, and (B) a +1. signaling models even on srongly deforming domains. The simulaion resuls can be compared o experimenal daa, and parameer values can be opimized o obain an opimal mach of model predicions and experimenal resuls [14]. We conclude ha he image-based modeling approach allows us o build realisic models of highly dynamic developmenal processes, and allows us o sudy he combined impacs of paerning and growh.. Mehod.1 Model Formulaion Our models are defined as a se of n reaciondiffusion equaions in he form of: C i + u C i advecion! + C! i u =! D iδc + R i i(c 1,...,C n )! #" ## $ diluion diffusion reacion where C i denoes he concenraion of componen i (n oal componens), D i is diffusion consan, and Δ refers o he Laplace operaor such ha D i Δ C i describes he diffusion flux of C i. In case of growing domains, advecion and diluion erms have o be added o he reacion-diffusion equaions; u represens he velociy field of he domain, for more deails refer o [15]. The reacion erm, R i (C 1,,C n ), is very ofen nonlinear and describes all reacions of componen i, i.e. is producion, degradaion, and complex formaion. For more deails refer o [1]. The presence of species can be resriced o pars of he domain, and in ha case also some reacions can become spaially resriced.. Regulaory Nework To illusrae our approach, we consider a concree example. Consider a domain wih 3 subdomains as shown in Figure 1, and a regulaory nework ha involves hree componens, A, B and C (Figure ). All componens are assumed o diffuse in he enire domains and o be degraded everywhere. Moreover, we assume ha he componens A and C are produced only in and, respecively, whereas componen B is produced in all domains. We formulae he sub domains using he uni funcion, #% Ι j = $ &% 1 if (x, y) domain j 0 oherwise. The spaio-emporal dynamics of he aforesaid nework can be described by he following reacion erms: R A (A, B,C) = ρ A R B (A, B,C) = ρ B ( K BA K BA A K AB R C (A, B,C) = ρ C A K AC + B Ι d A A K CB + A + C ) d B B K CB + A Ι d C C, Figure. Regulaory Nework. The nework consiss of hree componens, A, B and C. These componens are produced and regulaed in specific subdomains: A is produced and is inhibied only in (green), C is acivaed only in (red), and B is acivaed and repressed in he enire domain. All componens diffuse in all domains. Excerp from he Proceedings of he 014 COMSOL Conference in Cambridge

3 where K C j C i C j + C j and K C j C i K C j C i + C j describe he acivaing and inhibiory acions of C j, respecively. Iniial and Boundary Condiions: The iniial condiion of A is 1.Ι domain3 ; he iniial values of B and C are se o zero. Zero flux boundary condiions, n. C i, are used for all componens on he ouer boundary, as he ouer layer, he ecoderm, can be considered impermeable..3 Boundaries and Displacemen Fields Using sandard echniques for image segmenaion, exernal and inernal boundaries can be exraced [16]. This process can be repeaed a differen developmenal ime poins o obain a developmenal sequence of shapes [15]. In his sudy, we consider he wo geomeries in Figure 1 as our exraced D geomeries a wo subsequen developmenal ime seps, and +1. As can be seen, he enire domain as well as he subdomains deform from o +1. To describe he growing domains, we need o calculae he displacemen fields beween he wo shapes a and +1. A range of algorihms can be employed, which have heir advanages and disadvanages dependen on he deails of he geomeries and heir deformaions (Schwaninger e al., submied). Here, we use he uniform displacemen field algorihm proposed by (Schwaninger e al., submied): consider a curve a ime, γ, ha is deformed o γ +1 wihin he nex ime sep. This algorihm inerpolaes N poins on boh curves: { } γ = (x 1, y 1 ),...,(x N, y N ) γ +1 = {(x +1 1, y +1 1 ),...,(x +1 N, y +1 N )} such ha (x i T, y i T ),(x j T, y j T ) is equal for all i, j and T {, +1}. The displacemen filed marix, D, is defined as D i = " # x i, y i,(x +1 i x i ),(y +1 i y i ) $ %. For Figure 3. Displacemen fields. Blue arrows show he displacemen fields beween wo curves a wo ime seps, (orange) and +1 (red). inner boundaries, i is imporan o use he COMSOL buil-in surface-boundary parameer, S: every poin (x i, y i ) on he curve γ maps o S i and he displacemen marix is defined as D = " # S,(X +1 X ),(Y +1 Y ) $ %. Figure 3 shows he displacemen fields beween wo curves a wo subsequen ime seps and +1. The displacemen fields are impored ino COMSOL as Inerpolaion funcion and are laer employed in he Moving Mesh (ale) module o describe he domain deformaion due o he growh..4 Displacemen of inersecing Curves The inroducion of subdomains resuls in inersecing boundary curves (Figure 1). The funcion Inerpolaion Curve, ha we used o generae he boundaries, does no discriminae beween inersecion poins and oher poins on he curve. All poins are inerpolaed in he same way. Given he inerpolaion, here is no guaranee ha he inersecion poin of wo curves a ime will be accuraely displaced o heir prescribed inersecion poin a ime +1. This issue can cause disored meshes, invered meshes, and numerical problems close o he inersecion poins. In case of spaially resriced variables, his inaccuracy can resul in leakage of variables ou of heir resriced domains. From here on we will refer o he model wih hese problems as Model 1. To deal wih his problem, we propose he following algorihm. Assume curve 1 a ime, γ 1, inersecs wih γ a poin P=(X,Y) (Figure 4A) such ha γ 1 = {(x 1, y 1 ),...,(X,Y ),...,(x n, y n )} γ = (X,Y ),...,(x { m, y m )}. Excerp from he Proceedings of he 014 COMSOL Conference in Cambridge

4 A) B) C) P P Figure 4. Framework o deal wih inersecing boundary curves. (A) Curve 1 (green) and curve (red) of Figure inersec a poin P. (B) To map he inersecion poin (blue poin) a ime o he inersecion poin a +1, he curves are divided ino segmens, such ha he poin P becomes he sar/end poin of he inersecing curves. (C) This algorihm is applied o all curves of Figure. Curve 1 is divided ino wo segmens; Curve 3 is divided ino hree segmens, whereas Curve has only one segmen. Since we wan o preserve he inersecion poin, he curves γ 1 and γ have o be divided ino segmens such ha poin P is he sar/end poin of segmens. We divide γ 1 ino wo segmens (Figure 4B) such ha: γ 1 segmen1 γ 1 segmen { } { } = (x 1, y 1 ),...,(X,Y ) = (X,Y ),...,(x n, y n ) We implemened his algorihm for all inersecing curves (Figure 4C) and deermined A) B) he displacemen fields for each segmen individually. The coordinaes of each segmen and heir corresponding displacemen fields were hen impored ino COMSOL separaely. Using he above algorihm, we obain an accurae mapping of all domains and inersecion poins. From here on, we will refer o his model as Model. 3. Resuls Model 1 and Model were implemened in COMSOL wih he parameer values as given in C) domain 1 D) E) F) domain 1 0 Figure 5. The deformed domain a he final ime poin. All boundary poins a ime are mapped o heir corresponding poins a ime +1 using he displacemen field marix D. (A-C) Final deformed domain a ime +1 using Model 1. In his case, no only he inersecion poins are no displaced correcly, bu also heir adjacen poins are displaced improperly. (D-F) Final deformed domain a ime +1 using Model. In his case, all inersecing curves are divided ino segmens a heir inersecion poins (Figure 4C). The inersecion poins and heir neighbours are displaced perfecly. (B,E-op) Focus on he inersecion of Curve 1, Curve and Curve 3. (C,F-op) Focus on he inersecion of Curve 1 and Curve. (B,C-boom) Inaccuracies in he mapping of he inersecion poins resul in invered meshes close o hese poins in Model 1. (E,F-boom) High qualiy meshes close o he inersecion poins in Model. The color bar indicaes he qualiy of mesh elemens; negaive values indicae invered meshes. Excerp from he Proceedings of he 014 COMSOL Conference in Cambridge

5 Area Model 1 Model 1 Model Model area in Model 1 differs from he real size of he domain a +1. The invered meshes also lead o numerical problems and consequenly inaccurae soluions. Thus, Figure 7 shows ha he expression paerns, i.e. he effecive spaial producion raes, of A and C in Model 1 and Model differ a ime +1. The componens A and C have lower expression in Model 1 han in Model ime Figure 6. The domain areas over simulaion ime in Model 1 and Model. The real areas (red circles) of he domains a ime and +1 are provided for comparison. Table 1 in he Appendix. Figure 5A-C shows ha in Model 1 he inersecion poins are no displaced correcly. This problem leads o he invered meshes close o he inersecion poins (Figure 5B,C boom panels), and he subdomains are disored (Figure 5B,C op panels) as compared o Figure 1B. Segmening he curves a he inersecion poins before deformaion (Model ) leads o a higher qualiy of mesh elemens and herefore accurae numerical soluion and correc domain deformaion (Figure 5D-F). The invered meshes ha resul from inaccuracies in he mappings also lead o differences in he domain areas. Thus, Figure 6 repors he area of and domain a differen ime seps. As can be seen, he final A) 0.18 B) 4. Conclusion In his paper, a framework is presened o simulae PDE models on growing, embryonic domains wih subdomains, using COMSOL Muliphysics. Inersecing boundaries are no mapped accuraely using he sandard COMSOL Inerpolaion funcion. We addressed his problem by inroducing segmened boundaries. Using his algorihm and COMSOL s Malab LiveLink inerface, one can implemen also complicaed domains and large ses of PDEs. This permis he simulaion of large, complex regulaory neworks on physiological domains. We expec ha his will furher increase he predicive value of he models, and will allow o beer es, improve, and validae he models wih experimenal daa. 5. References 1. Iber, D., Zeller, R., Making sense - daabased simulaions of verebrae limb developmen, Curren Opinion in Geneics & Developmen,, (01) C) 0.19 D) Figure 7. Expression paerns. Effecive producion rae as prediced by (A,B) Model 1 for (A) species A, and (B) species C, and by (C,D) Model for (C) species A, and (D) species C. Excerp from he Proceedings of he 014 COMSOL Conference in Cambridge

6 . Wolper, L.,Posiional informaion and he spaial paern of cellular differeniaion, Journal of heoreical biology, 5, 1-47 (1969) 3. Turing, A. M., The chemical basis of morphogenesis. Philosophical Transacions of he Royal Sociey of London. Series B, Biological Sciences, 73, 37-7 (195) 4. Iber, D., Germann, P., How do digis emerge? - Mahemaical Models of Limb Developmen, Embryo Today, Special Issue on Musculoskeleal Developmen, 10, 1-1 (014) 5. Badugu, A., Conradin, K., Germann, P., Menshykau, D., Iber, D., Digi paerning during limb developmen as a resul of he BMP-recepor ineracion, Scienific Repors,, 99 (01) 6. Lopez-Rios, J., Duchesne, A., Speziale, D., Andrey, G., Peerson, K., Germann, P., Unal, E., Liu, J., Florio, S., Barbey, S., e al., Aenuaed sensing of SHH by Pch1 underlies evoluion of bovine limbs, Naure, 511, (014) 7. Tanaka, S., Iber, D., Iner-dependen issue growh and Turing paerning in a model for long bone developmen, Physical Biology, 10, (013) 8. Bächler, M., Menshykau, D., De Geyer, C., Iber, D., Species-specific differences in follicular anral sizes resul from diffusionbased limiaions on he hickness of he granulosa cell layer, Molecular Human Reproducion, 0, 08- (014) 9. Menshykau, D., Iber, D., Kidney branching morphogenesis under he conrol of a ligand- -recepor-based Turing mechanism, Physical biology, 10, (013) 10. Menshykau, D., Kraemer, C., Iber, D., Branch mode selecion during early lung developmen, PLoS compuaional biology, 8, e (01) 11. Cellière, G., Menshykau, D., Iber, D., Simulaions demonsrae a simple nework o be sufficien o conrol branch poin selecion, smooh muscle and vasculaure formaion during lung branching morphogenesis, Biology open, 1, (01) 1. Germann, P., Menshykau, D., Tanaka, S., Iber, D., Simulaing Organogenesis in COMSOL, Proceedings of COMSOL Conference (01) 13. Menshykau, D., Iber, D., Simulaion Organogenesis in COMSOL: Deforming and Ineracing Domains, Proceedings of COMSOL Conference (01) 14. Menshykau, D., Adivarahan, S., Germann, P., Lermuzeaux, L., Iber, D., Simulaing Organogenesis in COMSOL: Parameer Opimizaion for PDE-based models, Proceedings of COMSOL Conference (013) 15. Iber, D., Tanaka, S., Fried, P., Germann, P., Menshykau, D.: Simulaing Tissue Morphogenesis and Signaling. In Nelson, C., ed., Tissue Morphogenesis: Mehods and Proocols, Springer Book Series: Mehods in Molecular Biology (013) 16. Adivarahan, S., Menshykau, D., Michos, O., Iber, D., Dynamic Image-Based Modelling of Kidney Branching Morphogenesis. In Henzinger, A., ed., Compuaional Mehods in Sysems Biology (CMSB), 8130, (013) 6. Acknowledgemens We hank he COMSOL suppor eam, specially Sven Friedel for heir suppor and insighful discussions. The auhors acknowledge funding from he SysemsX RTD NeurosemX, a SysemsX iphd, and a SNF Sinergia gran. 7. Appendix Table 1: Non-dimensionalized Model Parameers wih characerisic ime T = 3600 sec and characerisic lengh L = 150 µm Parameer Value Descripion D 1*T Diffusion consan ρ A 1E-4*T Producion rae ρ B 50* ρ A Producion rae ρ c 00* ρ A Producion rae d A, d B, d C 1E-6*T Degradaion rae K BA 0. Hill consan K AB 0.15 Hill consan K CB 0.5 Hill consan K AC 0.05 Hill consan Excerp from he Proceedings of he 014 COMSOL Conference in Cambridge