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In-silico analysis on biofabricating vascular networks using kinetic Monte Carlo simulations

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2014 Biofabrication 6 015008

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Biofabrication

Biofabrication 6 (2014) 015008 (15pp) doi:10.1088/1758-5082/6/1/015008

In-silico analysis on biofabricatingvascular networks using kineticMonte Carlo simulations

Yi Sun1,4, Xiaofeng Yang1 and Qi Wang1,2,3

1 Department of Mathematics and Interdisciplinary Mathematics Institute, University of South Carolina,Columbia, SC 29208, USA2 School of Mathematics, Nankai University, 300071, Tianjin, People’s Republic of China3 Beijing Computational Science Research Center, 100084, Beijing, People’s Republic of China

E-mail: yisun@math.sc.edu, xfyang@math.sc.edu and qwang@math.sc.edu

Received 30 May 2013, revised 4 December 2013Accepted for publication 4 December 2013Published 15 January 2014

AbstractWe present a computational modeling approach to study the fusion of multicellularaggregate systems in a novel scaffold-less biofabrication process, known as ‘bioprinting’.In this novel technology, live multicellular aggregates are used as fundamental building blocksto make tissues or organs (collectively known as the bio-constructs,) via the layer-by-layerdeposition technique or other methods; the printed bio-constructs embedded in maturogens,consisting of nutrient-rich bio-compatible hydrogels, are then placed in bioreactorsto undergo the cellular aggregate fusion process to form the desired functional bio-structures.Our approach reported here is an agent-based modeling method, which uses the kinetic MonteCarlo (KMC) algorithm to evolve the cellular system on a lattice. In this method, the cells andthe hydrogel media, in which cells are embedded, are coarse-grained to material’s points on athree-dimensional (3D) lattice, where the cell–cell and cell–medium interactions are quantifiedby adhesion and cohesion energies. In a multicellular aggregate system with a fixed number ofcells and fixed amount of hydrogel media, where the effect of cell differentiation, proliferationand death are tactically neglected, the interaction energy is primarily dictated by the interfacialenergy between cell and cell as well as between cell and medium particles on the lattice,respectively, based on the differential adhesion hypothesis. By using the transition state theoryto track the time evolution of the multicellular system while minimizing the interfacial energy,KMC is shown to be an efficient time-dependent simulation tool to study the evolution of themulticellular aggregate system. In this study, numerical experiments are presented to simulatefusion and cell sorting during the biofabrication process of vascular networks, in whichthe bio-constructs are fabricated via engineering designs. The results predict the feasibilityof fabricating the vascular structures via the bioprinting technology and demonstrate themorphological development process during cellular aggregate fusion in various engineeringdesigned structures. The study also reveals that cell sorting will perhaps not significantly impactthe final fabricated products, should the maturation process be well-controlled in bioprinting.

Keywords: bioprinting, tissue fusion, multicellular aggregates, vascular networks, differentialadhesion hypothesis, kinetic Monte Carlo

(Some figures may appear in colour only in the online journal)

4 Author to whom any correspondence should be addressed.

1758-5082/14/015008+15$33.00 1 © 2014 IOP Publishing Ltd Printed in the UK

Biofabrication 6 (2014) 015008 Y Sun et al

1. Introduction

Tissue fusion is a ubiquitous phenomenon during embryonicdevelopment and morphogenesis, in which tissues and organsare formed by living cells through cell–cell and cell–extracellular matrix (ECM) interactions [1]. In the past,bioengineering processes have been devised to fabricatetissues under controlled conditions using cell self-assemblyon well-designed scaffold, in which one seeds cells intothe biodegradable polymer scaffolds or gels, which are thencultured in bioreactors for several weeks and finally implantedinto the recipient organism, where the maturation of thenew organ continues [2–5]. In a novel biomimetic scaffold-less or solid scaffold-free biofabrication process, known as‘bioprinting’, multicellular tissue spheroids or other shapesof aggregates are used as the fundamental building blocksto construct the 3D tissue or organ [6–16]. The multicellularaggregates are first prepared in the form of tissue spheroids,rods or other stable configurations that consist of cellsblended with bio-compatible hydrogels. They are then directlydeposited or laid by computer-aided design tools [17, 18] intothe desired 3D tissue or organ constructs via the layer-by-layerdeposition technique or other plausible deposition methods.The printed bio-constructs immersed in a hydrogel are thenplaced in bioreactors for further maturation, during which themulticellular aggregates undergo the fusion process to formthe desired functional tissue or organ products by followingthe natural rule of histogenesis and organogenesis [8–10, 14,19, 20]. Tissue fusion driven biofabrication is a fundamentalbiophysical and biochemical process in the emerging organbioprinting technology, whose main mechanisms dictatingfinal fused bio-constructs warrant a thorough experimental aswell as theoretical investigation.

In bioprinting, the bio-constructs ranging from the onescomprised of tissue spheroids or other stable cellular constructsto functioning tissues or organs all exhibit fluid behaviorduring tissue fusion processes (in the time and lengthscale of engineering interest). Numerical simulations of themorphogenesis phenomenon in bioprinting using the MonteCarlo method and experimental evidence [8, 9, 19, 21, 22]reveal a strong influence of surface tension on the multicellularaggregate fusion in the biofabrication process when thebiomaterials (tissue spheroids and hydrogel matrices) areregarded as viscous fluids. A separate study using experimentalmethods and numerical simulations with the cellular Pottsmodel [23, 24] also explored the effect of surface tensionand hydrodynamics on the rounding of cellular aggregates[25]. Recently Yang et al developed a coarse-grain phasefield model for studying the fusion of cellular aggregates ata coarse-grain length and time scales featuring a long rangemulticellular aggregate–aggregate interaction in addition tothe surface tension impact [26, 27].

In this paper, we present a multicellular lattice modelwhich describes the interactions between cells as well asbetween cells and media based on the differential adhesionhypothesis (DAH) proposed by Steinberg [28] to studycellular aggregate fusion in the process of fabricating thevascular constructs via 3D bioprinting. DAH implies that

early morphogenesis is a self-assembly process, wherebymobile and interacting cells spontaneously give rise to themorphological structure [29–31]. This hypothesis states that(i) cell adhesion in the multicellular systems depends onthe energy differences between different types of cells and(ii) an aggregate of cells is motile enough to reach theconfiguration which minimizes the interfacial energy of thesystem. In light of DAH, embryonic tissues mimic the behaviorof highly viscous, incompressible liquids and their liquid-like phenomena can then be interpreted using the theory ofordinary viscous liquids, consistent with the multiphase fluidtheory adopted by Yang et al [26, 27]. Many embryonic tissueshave been characterized in terms of effective tissue surfacetension, which were measured and found to be consistent withthe mutual sorting behavior of these tissues [32]. As one ofthe most important principles of developmental biology, DAHprovides a connection between the tissue surface tension andthe strength of adhesion between the cells constituting thetissue. The implications of DAH have been confirmed notonly in vitro and in silico [23, 24], but also in vivo [33, 34].

DAH provides a useful mechanism for predictions onequilibrium tissue configurations using Metropolis MonteCarlo (MMC) methods [35], but this direct Monte Carloimplementation cannot address the question of how theseconfigurations develop in time since the time used in the MMCis not the real time. Instead, in our numerical experiments,we employ the kinetic Monte Carlo (KMC) algorithm [36]to simulate the stochastic dynamics of the cells based onthe Arrhenius law [37, 38]. The reason to choose the KMCinstead of the MMC method is that the KMC method canprovide the transition rates which are associated with possibleconfigurational changes of the multicellular system, andthen the corresponding time evolution of the system can beexpressed in terms of these rates while the traditional MMCcannot. Our goal of developing the KMC simulation tool isto describe the self-assembly process of cells within tissueconstructs during the cellular aggregate fusion. The KMCmethod treats each cell as a point on the lattice and likewisea comparable sized medium chunk as a particle point on thelattice as well. It can thus deal with various types of particles(i.e., cells, including different cells, or hydrogel medium)present on each lattice site rather than following cell shapechanges.

Our research reported here is a part of the South CarolinaProject for Organ Biofabrication [16], in which scientificprogress includes computational modeling of vascular treesand experimental testing of natural and engineered constructs.As the mathematical/computational thrust, we focus on thedevelopment of the necessary predicative tools, includingmathematical models, to support testing of the vascularconstructs and prefabrication engineering design. Our researchgoals are to develop computer-aided design or simulation toolsfor naturally branched vascular trees and predict structureformation results to aid bioengineering design and fabrication.Our KMC simulation tool is thus developed against thisbackdrop and tested against experiments and benchmarkingsimulations, for which some preliminary simulation resultscan be found in [38]. In this paper, we extend our simulations

2

Biofabrication 6 (2014) 015008 Y Sun et al

(a) (b)

n = 6im n = 6jcn = 6 n = 4jc

(c)

im

(e)

n = 7im n = 7jc

( f )

n = 6im n = 8jc

mediumparticle jcell i

swap

(d)

n = 6im n = 7jc

im n = 6jcn = 5

Figure 1. Schematic representation of swapping a cell i and a medium particle j (for 2D, nn = 8). Six examples are shown. nim is the numberof bonds connecting cell i with neighboring medium particles and njc is the number of bonds connecting medium particle j to neighboringcells. Therefore, the average number of cell–medium bonds connected to cell i and medium particle j given in (5) are (a) ni, j

cm = 5;(b) ni, j

cm = 112 ; (c) ni, j

cm = 6; (d) ni, jcm = 13

2 ; (e) ni, jcm = 7 and ( f ) ni, j

cm = 7.

to engineering designed vascular networks in 3D bioprintingto show the morphological development of vascular veinsand networks when they are fabricated using solid as well asuniluminal multicellular spheroids. These simulations explorenot only the feasibility of bioprinting 3D vascular networkconstructs using multicellular spheroids, but also demonstratethe potential to aid engineering design in biofabrication, whichwill be an integral part of the 3D biofabrication technology.

We organize the rest of the paper into three sections.In section 2, we describe the lattice model and discussthe KMC algorithm and its implementation for multicellularsystems. In section 3, we present numerical results involvingcellular aggregate fusion in bioprinting of the bifurcating andnetworked vascular constructs and discuss their biologicalimplication in terms of developmental biology. We concludethe study in section 4.

2. Kinetic Monte Carlo algorithms for multicellularsystem

We first describe the KMC method. In the lattice model, wediscretize the space and present the multicellular system ona 3D cubic lattice, in which each site is occupied either by acell or a similar-sized volume element of hydrogel medium orECM [8, 21, 22]. The configuration at each site r = (x, y, z) isdefined by an integer type index σr ∈ {0, 1, 2, . . . , Ntype − 1},where Ntype is the number of particle types in the system.For instance, in a binary particle system (Ntype = 2) formedby cells of type-1 embedded in a hydrogel medium, we takeσr = 0 if the medium occupies the site or σr = 1 if a cell isat the site. The total pair interaction energy of the system isgiven by:

E =∑

〈r,r′〉J(σr, σr′ ), (1)

where J(σr, σr′ ) = −Ei j is the contact interaction energybetween two particles, located at neighboring sites r andr′, of type σr = i and σr′ = j, respectively, and Ei j is apositive quantity to represent the mechanical work neededto break the bond between them. Each pair is counted onlyonce in the summation 〈r, r′〉 over the neighboring sites.In the binary particle system, each term J(σr, σr′ ) can takethe values J(0, 0) = −Emm, J(1, 1) = −Ecc, J(0, 1) =J(1, 0) = −Ecm, where Emm, Ecc and Ecm represent theinteraction energy strengths between medium–medium, cell–cell and cell–medium pairs, respectively [39].

By separating interfacial and bulk terms in the sum in (1),the total interaction energy can be rewritten into [21]

E =Ntype−1∑

i, j=0, i< j

γi jNbi j − 1

2nn

Ntype−1∑

i=0

NiEii, (2)

where γi j = Eii+Ej j

2 −Ei j are the interfacial tension parametersbetween different particle types i and j and Nb

i j are thetotal numbers of bonds between them [8, 39]. Clearly,there are Ntype(Ntype −1)/2 independent γi j parameters and thecorresponding numbers Nb

i j of bonds between different particletypes. The parameter nn denotes the number of significantneighbors. In our 3D simulations, we consider interactionsbetween first, second and third nearest neighbors, for whichnn = 26, and we assume that the interaction strengthsbetween a particle and all of its nn neighbors are the same.During simulations that do not include cell proliferation,differentiation and death, the numbers of particles of eachtype, Ni, are constant so that the second term in (2) is constantas well; therefore, it can be omitted. The conformationalevolution of the multicellular system is energetically driven tothe configuration which minimizes the interfacial free energy(2) of the system, i.e., the first term is minimized after thesecond one is omitted.

3

Biofabrication 6 (2014) 015008 Y Sun et al

(a) t = 0 step (s e) t = 0 steps

(b) t = 2 × 107 step (s f) t = 2 × 107 steps

(c) t = 2 × 108 step (s g) t = 2 × 108 steps

(d) t = 6 × 108 step (s h) t = 6 × 108 steps

Figure 2. Time evolution of tube formation via fusion of 480 identical cellular aggregates in a KMC simulation. Initially, each aggregatecontains 280 cells of the same type (in red) and the aggregates are surrounded by agarose rods (in gray). The snapshots are taken at t = 0,2 × 107 steps (≈30τ0 ≈1.4 h), 2 × 108 steps (≈422τ0 ≈20 h), and 6 × 108 steps (≈2250τ0 ≈105 h), respectively. The cross-sectional viewsin (a)–(d) show that the cellular aggregates are embedded in the structure of agarose rods. (e)–(h) depict only the tube construct, which isshrinking. We set the interaction energy strength Ecc = 1.0.

We have applied the KMC simulations to study thefusion of the cellular clusters when they are embedded ina host hydrogel medium, i.e., the time evolution of thewhole multicellular system arranged in a designer’s patternfor tissue and organ generation [38]. We note that theMMC simulations [35] using (2) yielded results in qualitativeagreement with experiments on living tissue self-assembly[8, 19, 21]. However, in the MMC method, trial steps aresometimes rejected because the acceptance probability issmall, in particular when a system approaches equilibrium,or is in a metastable state. The KMC method that we adopthere is related to the method proposed by Bortz, Kalos andLebowitz as a speed-up to the MMC method for simulatingthe evolution of the Ising model [36]. A main feature of theKMC algorithm is that it is ‘rejection-free’. In each step,the transition rates for all possible changes from the current

configuration are calculated and then a new configurationis chosen with a probability proportional to the rate of thecorresponding transition. Since the interaction is short-range,there is only a small number of local states that need to bechanged due to the previous transition. The other featureof KMC is its capability of simulating the dynamics of thesystem in real time. Since the corresponding time evolutionis expressed in terms of these rates, the KMC method shouldprovide a more accurate description of the time evolution of amulticellular system than the MMC method.

The dynamics of the cells depends on the transition rates ofswapping cells with adjacent cells of different type and/or withmedium particles [37, 38]. In the spin-exchange Arrheniusdynamics, the simulation is driven based on the energy barriertwo particles have to overcome exchanging with each other.We perform a swap only if the potential energy of the cells is

4

Biofabrication 6 (2014) 015008 Y Sun et al

0 1 2 3 4 5 6

x 108

0

1

2

3

4

5

6x 10

5

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Ene

rgy

0 1 2 3 4 5 6

x 108

100

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120

130

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150

160

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Tub

e le

ngth

0 1 2 3 4 5 6

x 108

0

500

1000

1500

2000

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KM

C ti

me

in τ

0

1.5 2 2.5

x 107

20

25

30

35

40

1.9 2 2.1

x 108

400

410

420

430

440

450

Figure 3. Time evolution of the total interfacial energy E (top-left) and the tube length (bottom-left) calculated for tube formation in theKMC simulation shown in figure 2. Right: the relationship between the time (in the unit of τ0) and the number of KMC steps. The insets inthe right panel show that the times at 2 × 107 and 2 × 108 KMC steps are about 30τ0 ≈ 1.4 h (square) and 422τ0 ≈20 h (diamond),respectively. All panels show the results up to t = 6 × 108 steps (≈2250τ0 ≈105 h). In the long time regime (the number of steps> 4 × 108), the actual KMC time scales linearly with the number of steps.

higher than a given threshold. The swap is taken with its rategiven by the Arrhenius relation:

r = w0 exp (−Eb), (3)

where the prefactor w0 = 1/τ0 corresponds to the cellswapping frequency with τ0 the characteristic or relaxationtime unit. The energy barrier Eb is assumed to depend only onthe local state of the swapping cells/particles. For instance, inthe binary particle system, the energy barrier of swapping acell i and a medium particle j is given by

Ei, jb = Es + 2γcm

(nn − ni, j

cm

), (4)

where Es is the energy associated with the site binding of theparticle, which could vary in both space and time to accountfor the spatial and temporal flow situations (in this study we setEs = 0). The parameter γcm is the interfacial tension parameterand nn = 26 is the number of significant neighbors. As shownin figure 1, ni, j

cm is the average number of cell–medium bondsconnected to cell i and medium particle j, which is

ni, jcm = 1

2 (nim + n jc). (5)

Here nim is the number of bonds connecting cell i with theneighboring medium particles and n jc is the number of bondsconnecting medium particle j to the neighboring cells. Thenall swapping events can be classified into at most 2nn foldsaccording to the value of ni, j

cm. To summarize, the followingparameters need to be given for KMC simulations: (i) thecharacteristic or relaxation time unit τ0; (ii) the interfacialtension parameter γi j which depends on the interaction energystrengths Eii, Ej j and Ei j.

The KMC algorithm is built on the assumptionthat the model features N independent Poisson processes(corresponding to N swapping events) with transition ratesri that sum to give the total rate R = ∑N

i=1 ri. The generalKMC algorithm at each step is given as follows.

Algorithm 2.1. (The KMC algorithm)Step 1: Generate a uniform random number, ξ1 ∈ (0, 1)

and decide which process will take place by using a binarysearch to choose the event index s such that

s−1∑

i=1

ri

R< ξ1 �

s∑

i=1

ri

R. (6)

Step 2: Perform the selected event leading to a newconfiguration.

Step 3: Use R and another random number ξ2 ∈ (0, 1) todecide the time it takes for that event to occur (the transitiontime), i.e., the nonuniform time step �t = − log(ξ2)/R.

Step 4: Update the total rate R and any rate ri that mayhave changed due to that event.

In simulations with a finite number of distinct processesit is more efficient to consider the groups of events accordingto their rates [40–42]. This can be done by forming groupsof the same kinds of events according to the number of cell–medium bonds ni, j

cm in (5). A faster algorithm at each step ofthe KMC simulation based on the grouping of events can befound in [38]. The KMC simulations are implemented withfixed boundary conditions so that the cells are constrained tomove within the hydrogel medium and the medium boundaryis treated as a fixed physical boundary of the system. It isonly when a significant portion of the cells migrates awayfrom the cell ensemble in the model construct and reachesthe system boundary that finite-size effects may interferewith the energetically-driven rearrangement of cells andmedium particles. However, this is usually not the case inour simulations since only a few cells can detach from theconstruct formed by the ensemble.

5

Biofabrication 6 (2014) 015008 Y Sun et al

(a) t = 0 step (s e) t = 0 steps

(b) t = 5 × 105 step (s f) t = 5 × 105 steps

(c) t = 1 × 106 step (s g) t = 1 × 106 steps

(d) t = 2 × 106 step (s h) t = 2 × 106 steps

Figure 4. Time evolution of T-shaped branched tube formation via fusion of nine uniluminal cellular spheroids in a KMC simulation.Initially, each spheroid contains 982 cells, which include 680 SMCs (red) in the outer layer and 302 ECs (green) in the inner layer thatengulfs 436 lumen particles (blue). The four snapshots of external views (a)–(d) are taken at t = 0, 5 × 105 steps (≈126τ0 ≈6 h), 1 × 106

steps (≈244τ0 ≈11 h) and 2 × 106 steps (≈500τ0 ≈23 h), respectively. For better visualization of cell dynamics (e)–(h) show thecorresponding cross-sectional views by cutting the spheroids through the plane normal to direction [0 1 0]. We set the interfacial tensionparameter γi j as follows: γ01 = 0.25, γ02 = 0.6, γ03 = 0.9, γ12 = 0.2, γ13 = 0.6, and γ23 = 0.25, where the subscripts correspond to the fourphases (see text).

3. Results and discussions

We demonstrate the numerical methods described in theprevious section by investigating the cell self-assembly andmulticellular aggregate fusion in several cases relevant tobiofabrication with different and complex geometries, namelydirected self-assembly of cells into complex bio-constructsof controlled shape by engineering design. We first study anexample of fusion of single type cellular aggregates. Then wefocus on the cellular aggregates formed by multiple types ofcells with distinct adhesivities.

3.1. Tube construct formation via fusion of single cell typecellular spheroids

Here, we present an example to simulate a tubular constructformation. As shown in figures 2(a) and (e), initially, spherical

aggregates (red) are deposited layer-by-layer in the axialdirection on the circumference of a hexagon supported byagarose rods (gray), which are formed in situ and depositedby the bioprinter automatically, rapidly and accurately. Sinceagarose is an inert and bio-compatible hydrogel that cellsneither invade nor rearrange, the rods can keep their integrityduring post-printing fusion, and can be removed to releasethe fused multicellular construct [14]. There are in total 480identical spheroidal aggregates stacked into 20 layers in theaxial direction. Each aggregate contains 280 cells and itsdiameter is about 8 cell diameters. Thus, the length of initialconstruct is 160 cell diameters.

Since cohesion between medium–medium Emm andadhesion between cell–medium Ecm are considered to benegligibly small compared with Ecc, we set them to zero,which makes γcm = Ecc

2 . The interaction energy strength is

6

Biofabrication 6 (2014) 015008 Y Sun et al

(a) t = 0 step (s e) t = 0 steps

(b) t = 5 × 105 step (s f) t = 5 × 105 steps

(c) t = 1 × 106 step (s g) t = 1 × 106 steps

(d) t = 2 × 106 step (s h) t = 2 × 106 steps

Figure 5. Time evolution of the T-shaped branched tube formation via fusion of nine three-component spheroids in a KMC simulation.Initially, each spheroid contains a total of 1418 randomly mixed cells and internal lumen particles, including about 50% SMCs (red), 20%ECs (green) and 30% lumen particles (blue). The four snapshots of external views (a)–(d) and the corresponding cross-sectional views(e)–(h) are taken at t = 0, 5 × 105 steps (≈112τ0 ≈5 h), 1 × 106 steps (≈234τ0 ≈11 h) and 2 × 106 steps (≈500τ0 ≈23 h), respectively. Wetake the same values for the interfacial tension parameter γi j as the ones in the previous example.

0 0.5 1 1.5 2

x 105

2

3

4

5

6

7x 10

4

Steps

Ene

rgy

UnmixedMixed

0 0.5 1 1.5 2

x 106

2

3

4

5

6

7x 10

4

Steps

Ene

rgy

UnmixedMixed

(a) t = 2 × 105 step (s b) t = 2 × 106 steps

Figure 6. Time evolution of the total interfacial energy E calculated for the T-shaped branched tube formation via fusion of nine cellularspheroids in the KMC simulations shown in figures 4 and 5. The left and right panels show the results up to t = 2 × 105 and t = 2 × 106

steps, respectively.

7

Biofabrication 6 (2014) 015008 Y Sun et al

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 106

0

50

100

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350

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500

Steps

KM

C ti

me

in τ

0

4.5 5 5.5

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x 106

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MC

tim

e in

τ0

4.5 5 5.5

x 105

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x 106

225

230

235

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245

Figure 7. The relationship between the time (in units of τ0) and the number of KMC steps in the simulations of T-shaped branched tubeformation. Left: the initially uniluminal case shown in figure 4. The insets show that the times at 5 × 105 and 1 × 106 KMC steps are about126τ0 ≈ 6 h (square) and 244τ0 ≈ 11 h (diamond), respectively. Right: the initially mixed case shown in figure 5. The insets show that thetimes at 5 × 105 and 1 × 106 KMC steps are about 112τ0 ≈ 5 h (square) and 234τ0 ≈ 11 h (diamond), respectively. All panels show theresults up to t = 2 × 106 steps (≈500τ0 ≈23 h).

set to Ecc = 1.0. We remark that we nondimensionalize theinteraction energy by a characteristic energy value E0, which isthe average biological fluctuation energy of a cell, the analogueof the energy of thermal fluctuations kBT (kB is Boltzmann’sconstant and T is the absolute temperature) [44, 45]. Inthe simulations, only the dimensionless combinations Ei j/E0

appear. In [38] we have calibrated the parameter τ0 = 168s andthe other model parameters in the KMC code by simulatingfusion of two identical cellular aggregates and compared withthe existing experiments and published Monte Carlo results.The relationship between the KMC time (the accumulatedtransition time �t of each KMC step in the unit of τ0) and thenumber of KMC steps is shown in figure 3 (right panel).

Once the aggregates start fusing, the tubular wall thinsfirst upon fusion shown in figures 2(b) and ( f ) at t = 2 × 107

KMC (nonuniform) steps (≈30τ0 ≈1.4 h, see the inset offigure 3 (right panel)). As fusion goes on, the tube substantiallyshrinks its length in the axial direction to maintain the totalvolume of the aggregates. Meanwhile, the thickness of thetubular wall recovers and the cells fill the gap between theinternal and external layers of agarose rods (figures 2(c), (d)and (g), (h)). As the free energy of the multicellular system isminimized after 5×108 steps (≈1600τ0 ≈75 h), the tube lengthalso approaches its steady state value at about 107–109 celldiameters (figure 3 (left panels)). The shrinkage rate duringfusion is approximately 107/160 ≈ 2/3. This is consistentwith the height shrinkage rate with which a sphere with thevolume 4

3πR3 (radius R) is changed into a cylinder with thesame volume. If the sectional area of the latter is πR2, itsheight is h = 4R

3 . Thus, the shrinkage rate is h2R = 2

3 . This isa potential way to fabricate a vascular tube if stability of the

initial packing can be achieved. The simulation gives a fairlyaccurate description of the final fused tube product.

3.2. Branched tube formation via fusion of uniluminal cellularspheroids

Next, we study the fusion process of the uniluminal cellularspheroids which have outer layers of vascular smooth musclecells (SMCs) and a contiguous inner layer of endothelial cells(ECs) lining a central lumen [43]. This is perceived as apotential means to fabricate vascular veins via engineeringdesigns. To simulate the fusion process with the 3D latticemodel and the KMC method, we take the type index σ toaccount for all four experimental materials in the system: theexternal hydrogel medium (σ = 0), SMCs (σ = 1), ECs(σ = 2) and the internal hydrogel medium/lumen (σ = 3).The summation in the first term in (2) includes six terms thataccount for the interaction energies along the six interfacesbetween the four phases. After many tests for tuning theinterfacial tension parameter γi j, we choose the followingvalues that are related to the ones used in [43]: γ01 = 0.25,γ02 = 0.6, γ03 = 0.9, γ12 = 0.2, γ13 = 0.6 and γ23 = 0.25.We simulate the fusion process using the 3D lattice model andthe general KMC algorithm 2.1.

Figure 4 shows the KMC simulation results of fabricatinga branched tube with interior and exterior views of thespheroids, respectively. Initially, six uniluminal cellularspheroids are positioned side by side along the x-axis andanother three spheroids are positioned along the z-axis to forma T-shaped joint (figures 4(a) and (e)). Each aggregate has aradius of about 7 cell diameters and contains 982 cells, which

8

Biofabrication 6 (2014) 015008 Y Sun et al

(a) t = 0 step (s e) t = 0 steps

(b) t = 5 × 105 step (s f) t = 5 × 105 steps

(c) t = 1 × 106 step (s g) t = 1 × 106 steps

(d) t = 2 × 106 step (s h) t = 2 × 106 steps

Figure 8. Time evolution of the Y-shaped branched tube formation via fusion of ten uniluminal cellular spheroids in a KMC simulation. Thecomponents in each spheroid at initial time and the values of interfacial tension parameter γi j are the same as the ones in figure 4. The foursnapshots of external views (a)–(d) and the corresponding cross-sectional views (e)–(h) are taken at t = 0, 5 × 105 steps (≈124τ0 ≈6 h),1 × 106 steps (≈243τ0 ≈11 h) and 2 × 106 steps (≈495τ0 ≈23 h), respectively.

includes 680 SMCs (red) in the outer layer and 302 ECs (green)in the inner layer that engulfs 436 lumen particles (blue). Oncethe spheroids make contact, partial fusion of the SMCs in theouter layers ensues. At t = 5 × 105 steps (≈126τ0 ≈6 h), theECs from several spheroids begin merging (figures 4(b) and( f )). At t = 1 × 106 steps (≈244τ0 ≈11 h), the lumens insideseveral spheroids have begun merging (figures 4(c) and (g)). Atthe final t = 2 × 106 steps (≈500τ0 ≈23 h), the lumens insideall spheroids have joined together and the fused branched tubehas a fairly smooth outer surface (figures 4(d) and (h)). Thebiological analogues of such structures are branched bloodvessels, which are contractile due to the SMCs and ECslining their interior wall. To arrive at lumens, the internalhydrogel medium would have to be removed in the final tubularstructure.

Figure 5 depicts another set of KMC simulation results.Initially, each of the nine spheroids has a radius of about 7 cell

diameters and contains a total of 1418 randomly intermixedSMCs, ECs and internal lumen particles. It consists of 50%SMCs (red), 20% ECs (green), and 30% lumen particles (blue)shown in figures 5(a) and (e). Cell sorting in these three-component aggregates takes a major role at the beginningof the fusion process, which sorts the cells and the lumenparticles such that each spheroid has outer layers of SMCs anda contiguous inner layer of ECs lining a central lumen, justlike the case with initially uniluminal spheroids simulated inthe previous example. At t = 5×105 steps (≈112τ0 ≈5 h), theECs from all spheroids have joined and the lumens inside mostspheroids have begun merging (figures 5(b) and ( f )). Then,at t = 1 × 106 steps (≈234τ0 ≈11 h), the lumens continuemerging and there are still some SMCs that have not beensorted out (figures 5(c) and (g)). Finally, at t = 2 × 106 steps(≈500τ0 ≈23 h), the lumens inside all spheroids join togetherand the fused tube has a fairly smooth outer surface made up

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(a) t = 0 step (s e) t = 0 steps

(b) t = 1 × 106 step (s f) t = 1 × 106 steps

(c) t = 2 × 106 step (s g) t = 2 × 106 steps

(d) t = 5 × 106 step (s h) t = 5 × 106 steps

Figure 9. Time evolution of ‘trident’-shaped branched tube formation via fusion of 14 uniluminal cellular spheroids in a KMC simulation.The components in each spheroid at initial time and the values of interfacial tension parameter γi j are the same as the ones in figure 4. Thefour snapshots of external views (a)–(d) and the corresponding cross-sectional views (e)–(h) by cutting through the planes normal to thedirections [0 1 0] and [0 0 1] are taken at t = 0, 1 × 106 steps (≈246τ0 ≈11 h), 2 × 106 steps (≈494τ0 ≈23 h) and 5 × 106 steps(≈1280τ0 ≈60 h), respectively.

of SMCs and inner surface made up of ECs (figures 5(d) and(h)). This simulation is hypothetical. However, if we can findthe right hydrogel, biodegradable to be removed from the tubeinterior after the formation of the well-separated tube structure,this would be a viable way to fabricate a bifurcating vascularvein.

Figure 6 shows the time evolution of the total interfacialenergy during the fusion process in equation (2) for bothinitially uniluminal and mixed cases above. As shown infigure 6(a) of the simulation up to t = 2×105 steps, cell sortingin the initially mixed aggregates dominates at the beginningof the fusion process, which results in a very quick drop ofthe energy until t = 8 × 104 steps. Then the energy decreasesmuch slower in both mixed and uniluminal cases as shownin figure 6(b) when lumenized multicellular spheroids startfusing. At the time the simulation terminates, the energy is stilldecaying, though slowly. This indicates that the branched tube

formation structure is in a quasi steady state. The relationshipbetween the KMC time (the accumulated transition time �t ofeach KMC step in the unit of τ0) and the number of KMC stepsis shown in the left and right panels in figure 7 for the initiallyuniluminal and mixed cases, respectively, in which they scalelinearly.

Figure 8 depicts the KMC simulation result of fabricatinga Y-shaped branched tube with interior and exterior views ofthe spheroids, respectively. Initially, six uniluminal cellularspheroids are positioned side by side along the x-axis andanother four spheroids are positioned slantingly in the xz-plane(figures 8(a) and (e)). The components in each spheroid atinitial time and the values of interfacial tension parameters γi j

are the same as the ones used in figure 4. Once the spheroidsmake contact with each other, partial fusion of the SMCs in theouter layers ensues. At t = 5 × 105 steps (≈124τ0 ≈6 h), theECs from most spheroids have begun merging and the lumens

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(a) t = 0 step (s e) t = 0 steps

(b) t = 4 × 105 step (s f) t = 4 × 105 steps

(c) t = 1.2 × 106 step (s g) t = 1.2 × 106 steps

(d) t = 2.4 × 106 step (s h) t = 2.4 × 106 steps

Figure 10. Time evolution of large branched tube formation via fusion of uniluminal and mixed cellular spheroids in a KMC simulation.Initially, each two-component spheroid of the main vessel tube contains a total of 908 randomly intermixed cells, including about 80%SMCs (red) and 20% ECs (green). The three-component uniluminal spheroids of the branching vessels include 35% SMCs in the outer layerand 20% ECs in the inner layer that engulfs lumen particles (45% in blue). The values of interfacial tension parameter γi j are the same as theones in figure 4. The four snapshots of external views (a)–(d) are taken at t = 0, 4 × 105 steps (≈106τ0 ≈5 h), 1.2 × 106 steps(≈351τ0 ≈16 h) and 2.4 × 106 steps (≈855τ0 ≈40 h), respectively. For better visualization of cell dynamics (e)–(h) show the correspondingcross-sectional views by cutting the spheroids through the plane normal to direction [0 0 1].

in the bottom two spheroids have joined together (figures 8(b)and ( f )). At t = 1 × 106 steps (≈243τ0 ≈11 h), the lumensinside the three spheroids on the left in the horizontal line haveconnected with those from the slanting branch (figures 8(c)and (g)). At the final t = 2 × 106 steps (≈495τ0 ≈23 h), thelumens inside all spheroids have joined together and the fusedbranched tube has a fairly smooth outer surface (figures 8(d)and (h)). We also performed a similar KMC simulation withten spheroids formed by randomly mixed cells such as theones used in producing the result shown in figure 5 and thefinal result at t = 2 × 106 steps is similar to the one shown infigures 8(d) and (h).

Figure 9 shows the KMC simulation result of fabricatinga ‘trident’-shaped branched tube with interior and exteriorviews of the spheroids, respectively. Initially, six uniluminalcellular spheroids are positioned side by side along the x-axis and another two branches, each of which contains four

spheroids, are positioned slantingly in the xz- and xy-plane,respectively (figures 9(a) and (e)). The components in eachspheroid at the initial time and the values of interfacial tensionparameters γi j are the same as the ones used in producingthe result shown in figure 4. Once the spheroids make contactwith each other, partial fusion of the SMCs in the outer layersensues. At t = 1 × 106 steps (≈246τ0 ≈11 h), the ECs frommost spheroids have begun merging and the lumens in eachbranch have joined together, respectively (figures 9(b) and ( f )).At t = 2 × 106 steps (≈494τ0 ≈23 h), the lumens inside theleft part of the horizonal line have connected with those fromthe lower branch in the xz-plane, (figures 9(c) and (g)). At thefinal t = 5×106 steps (≈1280τ0 ≈60 h), the lumens inside allspheroids have joined together and the fused, branched tubehas a fairly smooth outer surface (figures 9(d) and (h)). Wealso performed a similar KMC simulation with 14 spheroids

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(a) t = 0 step (s e) t = 0 steps

(b) t = 8 × 105 step (s f) t = 8 × 105 steps

(c) t = 2 × 106 step (s g) t = 2 × 106 steps

(d) t = 3.2 × 106 step (s h) t = 3.2 × 106 steps

Figure 11. Time evolution of large branched tube formation via fusion of uniluminal and mixed cellular spheroids in a KMC simulation.Initially, the components of uniluminal or mixed spheroids are the same as the ones used in previous example shown in figure 10, but themain vessel tube is longer and has four branching vessels. The values of interfacial tension parameter γi j are the same as the ones in figure 4.The four snapshots of external views (a)–(d) and the corresponding cross-sectional views are taken at t = 0, 8 × 105 steps (≈207τ0 ≈10 h),2 × 106 steps (≈568τ0 ≈26 h) and 3.2 × 106 steps (≈1026τ0 ≈48 h), respectively.

formed by randomly mixed cells such as the ones used infigure 5 and the final result at t = 5 × 106 steps is similar tothe one shown in figures 9(d) and (h) although the time takento reach the final state is longer.

3.3. Large branched tube/network formation via fusion ofuniluminal and mixed cellular spheroids

Here, we simulate large branched tube formation by using thesame four-phase materials as the ones reported in the previoussubsection, but with more complex engineering designs in theconstruction. To form the main vessel tube, instead of usingthree-component spheroids, we take two-component spheroidsmade up of mixtures of SMCs and ECs, and pack them alonga column of internal hydrogel medium which gives rise to

the lumen upon removal. But the smaller branching vesselsstill consist of three-component uniluminal cellular spheroidsshown in the previous subsection. Then the whole constructis immersed in the external hydrogel medium for maturation.Again, we simulate the fusion process with the 3D latticemodel and the general KMC algorithm 2.1. The values ofinterfacial tension parameters are the same as we used in theprevious subsection.

Figure 10 depicts the KMC simulation results with interiorand exterior views of the construct. As shown in figures 10(a)and (e), the main vessel tube initially consists of three rings ofeight spheroids across the rings. Each two-component spheroidis made up of randomly mixed SMCs (80% in red) and ECs(20% in green). Another two smaller vessels branch out fromthe main vessel, each of which contains five three-component

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(a) t = 0 step (s e) t = 0 steps

(b) t = 1 × 106 step (s f) t = 1 × 106 steps

(c) t = 5 × 106 step (s g) t = 5 × 106 steps

(d) t = 1 × 107 step (s h) t = 1 × 107 steps

Figure 12. Time evolution of tube network formation via fusion of uniluminal and mixed cellular spheroids in a KMC simulation. Initially,the components of uniluminal or mixed spheroids are the same as the ones used in the previous example shown in figure 10, but the twomain vessel tubes are longer and have six branching vessels. Moreover, the two main vessel tubes are connected through branching vesselsbetween them, which forms a network. The values of the interfacial tension parameter γi j are the same as the ones in figure 4. The foursnapshots of external views (a)–(d) and the corresponding cross-sectional views (e)–(h) are taken at t = 0, 1 × 106 steps (≈242τ0 ≈11 h),5 × 106 steps (≈1388τ0 ≈65 h) and 1 × 107 steps (≈3289τ0 ≈153 h), respectively.

uniluminal cellular spheroids positioned side by side along thex-axis, respectively. Each three-component spheroid includesSMCs (35%) in the outer layer and ECs (20%) in the innerlayer that engulfs lumen particles (45% in blue). Each spheroidhas a radius of about 6 cell diameters and contains a total of908 cells/particles. The inside column contains 10 800 internalhydrogel particles (blue). We conducted many tests to adjustthe percentages of different cells so that there are sufficient ECslining the central lumen in the final fused tube structure. In thetwo-component spheroids of the main vessel, cell sorting takesa major role at the beginning of the fusion process such thateach spheroid has outer layers of SMCs which engulf the ECs.Meanwhile, the lumens inside three-component uniluminalcellular spheroids of the branching vessels have begun mergingand joining together at t = 4 × 105 steps (≈106τ0 ≈5 h)(figures 10(b) and ( f )). At t = 1.2 × 106 steps (≈351τ0 ≈16h), while the SMCs in the outer layers start fusing, the SMCsand the ECs also begin sorting such that the ECs move towardthe internal hydrogel column (figures 10(c) and (g)). At the

final t = 2.4 × 106 steps (≈855τ0 ≈40 h), the ECs arecompletely sorted out forming a thin layer separating the SMCsfrom the internal hydrogel in both main and branched vessels.Eventually, the construct shrinks to a solid fused tube whichhas a fairly smooth outer surface (figures 10(d) and (h)). Toarrive at lumens in the tube, the internal hydrogel medium hasto be removed from the final structure afterward.

Figure 11 shows the KMC simulation result with interiorand exterior views of a longer branched tube construct. Asshown in figures 11(a) and (e), the main vessel tube initiallyconsists of five rings of eight spheroids across the ringsand the inside column contains 18 000 internal hydrogelparticles. Another four smaller vessels branch out from themain vessel, each of which contains five three-componentuniluminal spheroids positioned side by side along thex-axis, respectively. The components of uniluminal or mixedspheroids are the same as the ones used in the previousexample. Here, we observe a similar result. At the beginning,cell sorting makes the SMCs engulf the ECs in the mixed

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two-component spheroids of the main vessel while the lumensinside three-component uniluminal spheroids of the branchingvessels have begun merging and joining together with theinternal hydrogel column at t = 8×105 steps (≈207τ0 ≈10 h)(figures 11(b) and ( f )). At t = 2 × 106 steps (≈568τ0 ≈26 h),the SMCs and the ECs sort such that the ECs move toward theinternal hydrogel column (figures 11(c) and (g)). At the finalt = 3.2 × 106 steps (≈1026τ0 ≈48 h), the SMCs and ECs arecompletely sorted out forming a thin EC layer separating theSMCs from the internal hydrogel in both main and branchedvessels, where the final construct has a fairly smooth outersurface (figures 11(d) and (h)).

Finally, figure 12 shows another set of KMC simulationresults of tube network formation via fusion of uniluminaland mixed cellular spheroids. This time, each of the two mainvessel tubes initially consists of seven rings of eight spheroidsacross the rings and the inside column contains 25 200 internalhydrogel particles. Another six smaller vessels branch outfrom the main vessel. Moreover, the two main vessel tubes areconnected through the branching vessels between them, whichform a network. Like the results in the previous examples, cellsorting takes a major role at the beginning of the fusion processsuch that each spheroid has outer layers of SMCs which engulfthe ECs. Then the lumens inside three-component uniluminalspheroids of the branching vessels begin merging and joiningtogether with the internal hydrogel column at t = 1×106 steps(≈242τ0 ≈11 h) (figures 12(b) and ( f )). At t = 5 × 106 steps(≈1388τ0 ≈65 h), the SMCs and the ECs sort such that the ECsmove toward the internal hydrogel column (figures 12(c) and(g)). At the final t = 1×107 steps (≈3289τ0 ≈153 h), the ECsare completely sorted out forming a thin layer separating theSMCs from the internal hydrogel and the networked constructhas a fairly smooth outer surface (figures 12(d) and (h)).

4. Conclusion

We have used the kinetic Monte Carlo (KMC) method witha three-dimensional (3D) lattice model developed in [38] tostudy fusion of cellular aggregates during their morphogenesisprocess where cell sorting, mixing and biomechanicalrelaxation takes place. Our work is motivated by the growingneed to understand morphological changes and biomechanicalproperties of the engineered tissue constructs during theirfabrication processes that have practical applications inbioengineering and regenerative medicine, in particular, theemergent field of 3D bioprinting [9, 14, 15].

The 3D lattice model using the KMC method is based onthe differential adhesion hypothesis (DAH) on the liquid-likenature of embryonic tissues consisting of engineered cellularaggregates proposed by Steinberg [28], which provides usefulpredictions on quasi-equilibrium tissue configurations. AsDAH cannot address the question of how these configurationsdevelop in time, we use the KMC method to simulate thetime evolution of the multicellular system, in which the self-assembly of cells in the systems is described in terms ofthe transition rates corresponding to possible configurationalchanges of the system and then the corresponding timeevolution of the systems can be expressed in terms of these

rates. After a careful calibration on the time scales and otherenergetic model parameters on experiments and benchmarkingcomputations [38], we use the KMC simulations to describeand predict the time evolution of the morphology anddynamics of the simulated multicellular tissue systems. Thesesimulations explore the feasibility of the 3D bioprinting ofvascular networks using engineering designed multicellularspheroids either in solid spheroids or uniluminal spheroids,providing a powerful simulation tool for the novel 3Dbioprinting technology.

The time for fusion of the cellular systems depends onthe initial packing of the cellular spheroids, their structuralstability and bio-compatibility with the supporting ambientmaterials. Tight packing will certainly result in shorter fusiontime. However, the time-saving in tight packing is negligiblecompared to the system fusion to quasi-steady state, which weexplored in our simulations but did not show any figures inthe text. The KMC simulation tool used here can certainly beemployed to test various packing options which may providean optimal strategy for biofabrication using 3D bioprintingtechnology. The bio-compatibility issue is largely ignored inthis study, which we will look into in future research. Thissimulation tool developed for cellular aggregate fusion can beextended to aid design of tissue constructs via tissue fusiontechnology. A computer-aided design and monitoring tool istherefore plausible in the near future to virtually simulatethe entire biofabrication process and provide guidance to thebioprinting process in precision.

Acknowledgments

The research of YS is partially supported by National ScienceFoundation (NSF) grant DMS-1318866 and an USC startupfund. The research of XY is supported in part by NSF grantDMS-1200487. The research of QW is partly supported byNSF grants DMS-0908330 and DMS-1200487. The researchof all three authors is also partially supported by a SCEPSCoR GEAR award. The authors thank Drs Scott Argraves,Christopher J Drake, Roger R Markwald, Vladimir Mironov,and Michael J Yost for their sharing of the experimentalfindings, vision, encouragement and guidance for thisproject.

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