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Research Papers

A Phase Field Approach for Multicellular Aggregate Fusion in Biofabrication

[+] Author and Article Information
Xiaofeng Yang

e-mail: xfyang@math.sc.edu

Yi Sun

e-mail: yisun@math.sc.edu
Department of Mathematics and Interdisciplinary Mathematics Institute,
University of South Carolina,
Columbia, SC 29208

Qi Wang

Department of Mathematics,
Interdisciplinary Mathematics Institute, and NanoCenter at USC,
University of South Carolina,
Columbia, SC 29208;
School of Mathematics,
Nankai University,
Tianjin, 300071, China
e-mail: qwang@math.sc.edu

1Corresponding author.

Contributed by the Bioengineering Division of ASME for publication in the JOURNAL OF BIOMECHANICAL ENGINEERING. Manuscript received February 3, 2013; final manuscript received March 20, 2013; accepted manuscript posted April 4, 2013; published online June 11, 2013. Assoc. Editor: Keith Gooch.

J Biomech Eng 135(7), 071005 (Jun 11, 2013) (9 pages) Paper No: BIO-13-1062; doi: 10.1115/1.4024139 History: Received February 03, 2013; Revised March 20, 2013; Accepted April 04, 2013

We present a modeling and computational approach to study fusion of multicellular aggregates during tissue and organ fabrication, which forms the foundation for the scaffold-less biofabrication of tissues and organs known as bioprinting. It is known as the phase field method, where multicellular aggregates are modeled as mixtures of multiphase complex fluids whose phase mixing or separation is governed by interphase force interactions, mimicking the cell-cell interaction in the multicellular aggregates, and intermediate range interaction mediated by the surrounding hydrogel. The material transport in the mixture is dictated by hydrodynamics as well as forces due to the interphase interactions. In a multicellular aggregate system with fixed number of cells and fixed amount of the hydrogel medium, the effect of cell differentiation, proliferation, and death are neglected in the current model, which can be readily included in the model, and the interaction between different components is dictated by the interaction energy between cell and cell as well as between cell and medium particles, respectively. The modeling approach is applicable to transient simulations of fusion of cellular aggregate systems at the time and length scale appropriate to biofabrication. Numerical experiments are presented to demonstrate fusion and cell sorting during tissue and organ maturation processes in biofabrication.

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Figures

Grahic Jump Location
Fig. 7

Fusion of two blood vessels involving four material phases. Initially, the SMCs and ECs are mixed up in the layer, where they occupy 50% each. As time evolves, they separate completely into two separate phases with the ECs lining the wall of the blood vessel with ξ1 = 0, ξ2 = 0, ξ3 = 0.005.

Grahic Jump Location
Fig. 6

Fusion of three blood vessels in close proximity laterally. Inside: blood; outside: SMC; layer: EC. The snapshots are shown at t = 0, 0.2, 0.4, 0.6, 0.8, 1, 1.6, 2 with ξ1 = 0, ξ2 = 0.005.

Grahic Jump Location
Fig. 5

Fusion of seven blood vessels in close proximity laterally. Inside: blood; outside: SMC; layer: EC. The snapshots are taken at t = 0, 2, 2.4, 2.8, 4, 6, 8, 14, 18 with ξ1 = 0, ξ2 = 0.005.

Grahic Jump Location
Fig. 4

Fusion of three blood vessels in close proximity laterally. Inside: blood; outside: SMC; layer: EC. The snapshots are shown at t = 0, 2, 4, 6, 8, 10 with ξ1 = 0, ξ2 = 0.005.

Grahic Jump Location
Fig. 3

Fusion of three blood vessels in close proximity linearly. Inside: blood; outside: SMC; layer (or wall): EC. The snapshots are shown at t = 0, 2, 4, 6, 8 with ξ1 = 0, ξ2 = 0.005.

Grahic Jump Location
Fig. 2

Fusion of two blood vessels in close proximity. Inside: blood; outside: SMC; layer or wall: EC. The simulation is shown at t = 0, 2, 4, 6, 8, 10, 12 with ξ1 = 0, ξ2 = 0.005.

Grahic Jump Location
Fig. 1

Cell sorting in a multicellular aggregate that engulfs a blood vessel. Inside: blood; outside: the mixture of SMC: 80% and EC: 20%. The plots are snapshots at t = 0, 5, 10, 13, 15, 20 with ξ1 = 0, ξ2 = 0.005. Initially, the SMCs and ECs are well mixed in the exterior. As fusion starts, ECs migrate toward the interior where the blood vessel is located leaving behind the purer SMC aggregate.

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