Phase-field modeling and simulation of two- and three-dimensional curvature-dependent tissue growth on surfaces

Authors: Zecheng Qiu, Yutong Wu, Junxiang Yang

This paper presents a novel numerical framework for modeling curvature-dependent tissue growth on complex scaffolds. The manuscript is currently Under Review at Physica D: Nonlinear Phenomena.

Abstract: Curvature-dependent tissue accretion in porous scaffolds can be modeled by phase-field formulations, but explicit discretizations impose severe stability restrictions, particularly for three-dimensional simulations. We develop a second-order stabilized semi-implicit ADI operator-splitting solver for a curvature-modulated phase-field tissue growth model. The stiff interfacial relaxation operators are advanced by a stabilized ADI update on Cartesian grids, while the nonlinear curvature-dependent growth term is treated by a frozen-coefficient strategy with a pointwise closed-form update. Second-order temporal accuracy is achieved via step-doubling (Richardson extrapolation). Convergence and efficiency are verified by numerical tests, and the method enables robust 3D simulations of tissue infilling in complex scaffold geometries, including TPMS and orthogonal log-pile architectures. The simulation code corresponding to Section 4.1 (Table 1), as well as the 2D visualization of Section 4.2.1 (Figure 2, left) and 4.2.2 (Figure 3, left) in this paper can be accessed at https://github.com/aaron-z-chiu/phase-field-tissue-growth.

Keywords: Phase-field model; Curvature-dependent tissue growth; Stabilized semi-implicit ADI; Operator splitting

Main Contributions:

• Developed and open-sourced a C++ simulation framework implementing a stabilized semi-implicit ADI operator-splitting solver from the ground up, addressing the stability constraints of traditional explicit time-stepping.
• Achieved second-order temporal accuracy, enabling reliable long-time simulations with improved accuracy–cost balance.
• Extended the baseline 2D formulation to 3D volumetric geometries and implemented the corresponding solver pipeline, supporting simulations in realistic porous/scaffold-like structures.

Recommanded BibTEX:

@article{qiu2026phase,
  title={Phase-Field Modeling and Simulation of Two-and Three-Dimensional Curvature-Dependent Tissue Growth on Surfaces},
  author={Qiu, Zecheng and Wu, Yutong and Yang, Junxiang},
  journal={Available at SSRN 6663678},
  year={2026}
}

Synchronized evolution of curvature-driven growth mechanism in 2D and 3D perspectives.
Three-dimensional tissue infilling process within a Diamond (D) scaffold.	Three-dimensional tissue infilling process within a Gyroid (G) scaffold.
Three-dimensional tissue infilling process within a Primitive (P) scaffold.	Evolution of mean curvature heatmap during tissue growth in an orthogonal log-pile scaffold.

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