Hele-Shaw flow in multi-connected regions

Physical Review Fluids · 2026-01-20

An Efficient Constant-Coefficient MSAV Scheme for Computing Vesicle Growth and Shrinkage

ArXiv.org · 2026-01-15

We present a fast, unconditionally energy-stable numerical scheme for simulating vesicle deformation under osmotic pressure using a phase-field approach. The model couples an Allen-Cahn equation for the biomembrane interface with a variable-mobility Cahn-Hilliard equation governing mass exchange across the membrane. Classical approaches, including nonlinear multigrid and Multiple Scalar Auxiliary Variable (MSAV) methods, require iterative solution of variable-coefficient systems at each time step, resulting in substantial computational cost. We introduce a constant-coefficient MSAV (CC-MSAV) scheme that incorporates stabilization directly into the Cahn-Hilliard evolution equation rather than the chemical potential. This reformulation yields fully decoupled constant-coefficient elliptic problems solvable via fast discrete cosine transform (DCT), eliminating iterative solvers entirely. The method achieves O(N^2 log N) complexity per time step while preserving unconditional energy stability and discrete mass conservation. Numerical experiments verify second-order temporal and spatial accuracy, mass conservation to relative errors below 5 x 10^-11, and close agreement with nonlinear multigrid benchmarks. On grids with N >= 2048, CC-MSAV achieves 6-15x overall speedup compared to classical MSAV with optimized preconditioning, while the dominant Cahn-Hilliard subsystem is accelerated by up to two orders of magnitude. These efficiency gains, achieved without sacrificing accuracy, make CC-MSAV particularly well suited for large-scale simulations of vesicle dynamics.

An Efficient Constant-Coefficient MSAV Scheme for Computing Vesicle Growth and Shrinkage

arXiv (Cornell University) · 2026-01-15

We present a fast, unconditionally energy-stable numerical scheme for simulating vesicle deformation under osmotic pressure using a phase-field approach. The model couples an Allen-Cahn equation for the biomembrane interface with a variable-mobility Cahn-Hilliard equation governing mass exchange across the membrane. Classical approaches, including nonlinear multigrid and Multiple Scalar Auxiliary Variable (MSAV) methods, require iterative solution of variable-coefficient systems at each time step, resulting in substantial computational cost. We introduce a constant-coefficient MSAV (CC-MSAV) scheme that incorporates stabilization directly into the Cahn-Hilliard evolution equation rather than the chemical potential. This reformulation yields fully decoupled constant-coefficient elliptic problems solvable via fast discrete cosine transform (DCT), eliminating iterative solvers entirely. The method achieves O(N^2 log N) complexity per time step while preserving unconditional energy stability and discrete mass conservation. Numerical experiments verify second-order temporal and spatial accuracy, mass conservation to relative errors below 5 x 10^-11, and close agreement with nonlinear multigrid benchmarks. On grids with N >= 2048, CC-MSAV achieves 6-15x overall speedup compared to classical MSAV with optimized preconditioning, while the dominant Cahn-Hilliard subsystem is accelerated by up to two orders of magnitude. These efficiency gains, achieved without sacrificing accuracy, make CC-MSAV particularly well suited for large-scale simulations of vesicle dynamics.

BiLO: Bilevel Local Operator Learning for PDE Inverse Problems

Journal of Computational Physics · 2026-01-11

An Efficient Constant-Coefficient MSAV Scheme for Computing Vesicle Growth and Shrinkage

bioRxiv (Cold Spring Harbor Laboratory) · 2026-01-23

Abstract We present a fast, unconditionally energy-stable numerical scheme for simulating vesicle deformation under osmotic pressure using a phase-field approach. The model couples an Allen–Cahn equation for the biomembrane interface with a variable-mobility Cahn–Hilliard equation governing mass exchange across the membrane. Classical approaches, including nonlinear multigrid and Multiple Scalar Auxiliary Variable (MSAV) methods, require iterative solution of variable-coefficient systems at each time step, resulting in substantial computational cost. We introduce a constant-coefficient MSAV (CC-MSAV) scheme that incorporates stabilization directly into the Cahn–Hilliard evolution equation rather than the chemical potential. This reformulation yields fully decoupled constant-coefficient elliptic problems solvable via fast discrete cosine transform (DCT), eliminating iterative solvers entirely. The method achieves O ( N 2 log N ) complexity per time step while preserving unconditional energy stability and discrete mass conservation. Numerical experiments verify second-order temporal and spatial accuracy, mass conservation to relative errors below 5 × 10 −11 , and close agreement with nonlinear multigrid benchmarks. On grids with N ≥ 2048, CC-MSAV achieves 6–15× overall speedup compared to classical MSAV with optimized preconditioning, while the dominant Cahn–Hilliard subsystem is accelerated by up to two orders of magnitude. These efficiency gains, achieved without sacrificing accuracy, make CC-MSAV particularly well-suited for large-scale simulations of vesicle dynamics.

BiLO: Bilevel Local Operator Learning for PDE Inverse Problems

bioRxiv (Cold Spring Harbor Laboratory) · 2026-01-12

Abstract We propose a new neural network based method for solving inverse problems for partial differential equations (PDEs) by formulating the PDE inverse problem as a bilevel optimization problem. At the upper level, we minimize the data loss with respect to the PDE parameters. At the lower level, we train a neural network to locally approximate the PDE solution operator in the neighborhood of a given set of PDE parameters, which enables an accurate approximation of the descent direction for the upper level optimization problem. The lower level loss function includes the least-square penalty of both the residual and its derivative with respect to the PDE parameters. We apply gradient descent simultaneously on both the upper and lower level optimization problems, leading to an effective and fast algorithm. The method, which we refer to as BiLO (Bilevel Local Operator learning), is also able to efficiently infer unknown functions in the PDEs through the introduction of an auxiliary variable. We provide a theoretical analysis that justifies our approach. Through extensive experiments over multiple PDE systems, we demonstrate that our method enforces strong PDE constraints, is robust to sparse and noisy data, and eliminates the need to balance the residual and the data loss, which is inherent to the soft PDE constraints in many existing methods.

Membrane heterogeneity-driven dynamics of multicomponent vesicles in shear flow

Journal of Fluid Mechanics · 2026-03-05 · 1 citations

Despite its significance in biology and materials science, the dynamics of multicomponent vesicles under shear flow remains poorly understood because of its nonlinear and strongly coupled nature, especially regarding the role of membrane heterogeneity in driving non-equilibrium behaviour. Here we present a thermodynamically consistent phase-field model, which is validated against experiments, for the quantitative investigation of the dynamics. While prior research has primarily focused on viscosity or bending rigidity contrasts, we demonstrate that surface tension heterogeneity can also trigger swinging and tumbling in vesicles under shear. Additionally, our systematic phase diagram reveals three previously unreported dynamical regimes arising from the interplay between bending rigidity heterogeneity and shear flow. Overall, our model provides a robust framework for understanding multicomponent vesicle dynamics, with findings offering new physical insights and design principles for tuneable vesicle-based carriers.

Chemomechanical regulation of growing tissues from a thermodynamically-consistent framework and its application to tumor spheroid growth

Journal of Mathematical Biology · 2025-09-01 · 1 citations

Cahn-Hilliard dynamical models for condensed biomolecular systems

bioRxiv (Cold Spring Harbor Laboratory) · 2025-07-17

Biomolecular condensates create dynamic subcellular compartments that alter systems-level properties of the networks surrounding them. One model combining soluble and condensed states is the Cahn-Hilliard equation, which specifies a diffuse interface between the two phases. Customized approaches required to solve this equation are largely inaccessible. Using two complementary numerical strategies, we built stable, self-consistent Cahn-Hilliard solvers in Python, MATLAB, and Julia. The algorithms simulated the complete time evolution of condensed droplets as they dissolved or persisted, relating critical droplet size to a coefficient for the diffuse interface in the Cahn-Hilliard equation. We applied this universal relationship to the chromosomal passenger complex, a multi-protein assembly that reportedly condenses on mitotic chromosomes. The fully constrained Cahn-Hilliard simulations yielded dewetting and coarsening behaviors that closely mirrored experiments in different cell types. The Cahn-Hilliard equation tests whether condensate dynamics behave as a phase-separated liquid, and its numerical solutions advance generalized modeling of biomolecular systems.

Integrating Mathematical and Mouse Models Identifies T Regulatory Cell Influx as A Key Determinant of Acquired Resistance to PD-1 Immunotherapy

bioRxiv (Cold Spring Harbor Laboratory) · 2025-11-03

ABSTRACT The immune system can eradicate cancer, but various immunosuppressive mechanisms active within a tumor curb this beneficial response. However, unraveling the effects of multimodal interactions between tumor and immune cells and their contributions to tumor control using an experimental approach alone is time- and resource-intensive. To identify the critical immunological features associated with tumor control and escape, we built a mechanistic mathematical model of the interactions between CD8 + T cells, Tregs, DCs, and tumor cells deeply rooted in current biological concepts. A distinguishing feature of our model is that it captures Treg accrual occurring after checkpoint blockade immunotherapy. After successfully fitting the model to experimental data of a mouse model of immunogenic melanoma, we generated hundreds of parameter sets, each representing a unique ‘virtual mouse’, that fit the data equally as well to capture variability across individuals. Our model indicates that the tumor and immune states before therapy are a key limiting factor of the immune response. Increasing the initial number of tumor-killing CD8 + T cells alone doesn’t always result in a better outcome; instead, the model implies that there exist optimal initial ratios of immune cells that will result in improved tumor control. The model further predicts that the Treg influx into the tumor is a key determinant of resistance to PD-1 immunotherapy. We validated this predictions experimentally. Overall, this integrated approach of modeling and experimental validation identified crucial determinants of resistance to immunotherapy and can be used to guide the development of more effective therapeutic strategies.