Kei Fong Lam

A Cahn–Hilliard–Darcy model for tumour growth with chemotaxis and active transport

We propose and investigate a model for lipid raft formation and dynamics in biological membranes. The model describes the lipid composition of the membrane and an interaction with cholesterol. To account for cholesterol exchange between cytosol and cell membrane we couple a bulk-diffusion to an evolution equation on the membrane. The latter describes a relaxation dynamics for an energy taking lipid-phase separation and lipid-cholesterol interaction energy into account. It takes the form of an (extended) Cahn--Hilliard equation. Different laws for the exchange term represent equilibrium and non-equilibrium models. We present a thermodynamic justification, analyze the respective qualitative behavior and derive asymptotic reductions of the model. In particular we present a formal asymptotic expansion near the sharp interface limit, where the membrane is separated into two pure phases of saturated and unsaturated lipids, respectively. Finally we perform numerical simulations and investigate the long-time behavior of the model and its parameter dependence. Both the mathematical analysis and the numerical simulations show the emergence of raft-like structures in the non-equilibrium case whereas in the equilibrium case only macrodomains survive in the long-time evolution.Using basic thermodynamic principles we derive a Cahn–Hilliard–Darcy model for tumour growth including nutrient diffusion, chemotaxis, active transport, adhesion, apoptosis and proliferation. In contrast to earlier works, the model is based on a volume-averaged velocity and in particular includes active transport mechanisms which ensure thermodynamic consistency. We perform a formally matched asymptotic expansion and develop several sharp interface models. Some of them are classical and some are new which for example include a jump in the nutrient density at the interface. A linear stability analysis for a growing nucleus is performed and in particular the role of the new active transport term is analysed. Numerical computations are performed to study the influence of the active transport term for specific growth scenarios.

Well-posedness of a Cahn--Hilliard system modelling tumour growth with chemotaxis and active transport

We consider a diffuse interface model for tumour growth consisting of a Cahn--Hilliard equation with source terms coupled to a reaction-diffusion equation. The coupled system of partial differential equations models a tumour growing in the presence of a nutrient species and surrounded by healthy tissue. The model also takes into account transport mechanisms such as chemotaxis and active transport. We establish well-posedness results for the tumour model and a variant with a quasi-static nutrient. It will turn out that the presence of the source terms in the Cahn--Hilliard equation leads to new difficulties when one aims to derive a priori estimates. However, we are able to prove continuous dependence on initial and boundary data for the chemical potential and for the order parameter in strong norms.

Analysis of the Diffuse Domain Approach for a Bulk-Surface Coupled PDE System

We analyze a diffuse interface type approximation, known as the diffuse domain approach, of a linear coupled bulk-surface elliptic PDE system. The well-posedness of the diffuse domain approximation is shown using weighted Sobolev spaces and we prove that the solution to the diffuse domain approximation converges weakly to the solution of the coupled bulk-surface elliptic system as the approximation parameter tends to zero. Moreover, we can show strong convergence for the bulk quantity, while for the surface quantity, we can show norm convergence and strong convergence in a weighted Sobolev space.

A multiphase Cahn–Hilliard–Darcy model for tumour growth with necrosis

We derive a Cahn–Hilliard–Darcy model to describe multiphase tumour growth taking interactions with multiple chemical species into account as well as the simultaneous occurrence of proliferating, quiescent and necrotic regions. A multitude of phenomena such as nutrient diffusion and consumption, angiogenesis, hypoxia, blood vessel growth, and inhibition by toxic agents, which are released for example by the necrotic cells, are included. A new feature of the modelling approach is that a volume-averaged velocity is used, which dramatically simplifies the resulting equations. With the help of formally matched asymptotic analysis we develop new sharp interface models. Finite element numerical computations are performed and in particular the effects of necrosis on tumour growth are investigated numerically. In particular, for certain modelling choices, we obtain some form of focal and patchy necrotic growth that have been observed in experiments.

Optimal Control of Treatment Time in a Diffuse Interface Model of Tumor Growth

We consider an optimal control problem for a diffuse interface model of tumor growth. The state equations couples a Cahn–Hilliard equation and a reaction-diffusion equation, which models the growth of a tumor in the presence of a nutrient and surrounded by host tissue. The introduction of cytotoxic drugs into the system serves to eliminate the tumor cells and in this setting the concentration of the cytotoxic drugs will act as the control variable. Furthermore, we allow the objective functional to depend on a free time variable, which represents the unknown treatment time to be optimized. As a result, we obtain first order necessary optimality conditions for both the cytotoxic concentration and the treatment time.

Two-Phase Flow with Surfactants: Diffuse Interface Models and Their Analysis

New diffuse interface and sharp interface models for soluble and insoluble surfactants fulfilling energy inequalities are introduced. We discuss their relation with the help of asymptotic analysis and present an existence result for a particular diffuse interface model.

On a Cahn-Hilliard-Darcy system for tumour growth with solution dependent source terms

We study the existence of weak solutions to a mixture model for tumour growth that consists of a Cahn–Hilliard–Darcy system coupled with an elliptic reaction-diffusion equation. The Darcy law gives rise to an elliptic equation for the pressure that is coupled to the convective Cahn–Hilliard equation through convective and source terms. Both Dirichlet and Robin boundary conditions are considered for the pressure variable, which allow for the source terms to be dependent on the solution variables.

A Hele–Shaw–Cahn–Hilliard Model for Incompressible Two-Phase Flows with Different Densities

Topology changes in multi-phase fluid flows are difficult to model within a traditional sharp interface theory. Diffuse interface models turn out to be an attractive alternative to model two-phase flows. Based on a Cahn–Hilliard–Navier–Stokes model introduced by Abels et al. (Math Models Methods Appl Sci 22(3):1150013, 2012), which uses a volume-averaged velocity, we derive a diffuse interface model in a Hele–Shaw geometry, which in the case of non-matched densities, simplifies an earlier model of Lee et al. (Phys Fluids 14(2):514–545, 2002). We recover the classical Hele–Shaw model as a sharp interface limit of the diffuse interface model. Furthermore, we show the existence of weak solutions and present several numerical computations including situations with rising bubbles and fingering instabilities.

Shape optimization for surface functionals in Navier--Stokes flow using a phase field approach

We consider shape and topology optimization for uids which are governed by the Navier{Stokes equations. Shapes are modelled with the help of a phase eld approach and the solid body is relaxed to be a porous medium. The phase eld method uses a Ginzburg{Landau functional in order to approximate a perimeter penalization. We focus on surface functionals and carefully introduce a new modelling variant, show existence of minimizers and derive rst order necessary conditions. These conditions are related to classical shape derivatives by identifying the sharp interface limit with the help of formally matched asymptotic expansions. Finally, we present numerical computations based on a Cahn{Hilliard type gradient descent which demonstrate that the method can be used to solve shape optimization problems for uids with the help of the new approach.

Cahn--Hilliard Inpainting with the Double Obstacle Potential

The inpainting of damaged images has a wide range of applications and many different mathematical methods have been proposed to solve this problem. Inpainting witht the help of Cahn--Hilliard models has been particularly successful, and it turns out that Cahn--Hilliard inpainting with the double obstacle potential can lead to better results compared to inpainting with a smooth double well potential. However, a mathematical analysis of this approach is missing so far. In this paper we give first analytical results for a Cahn--Hilliard double obstacle model and in particular we can show existence of stationary solutions without constraints on the parameters involved. With the help of numerical results we show the effectiveness of the approach for binary and grayscale images.