Solving the advection-diffusion equations in biological contexts using the cellular Potts model

The cellular Potts model (CPM) is a robust, cell-level methodology for simulation of biological tissues and morphogenesis. Both tissue physiology and morphogenesis depend on diffusion of chemical morphogens in the extra-cellular fluid or matrix (ECM). Standard diffusion solvers applied to the cellular potts model use finite difference methods on the underlying CPM lattice. However, these methods produce a diffusing field tied to the underlying lattice, which is inaccurate in many biological situations in which cell or ECM movement causes advection rapid compared to diffusion. Finite difference schemes suffer numerical instabilities solving the resulting advection-diffusion equations. To circumvent these problems we simulate advection diffusion within the framework of the CPM using off-lattice finite-difference methods. We define a set of generalized fluid particles which detach advection and diffusion from the lattice. Diffusion occurs between neighboring fluid particles by local averaging rules which approximate the Laplacian. Directed spin flips in the CPM handle the advective movement of the fluid particles. A constraint on relative velocities in the fluid explicitly accounts for fluid viscosity. We use the CPM to solve various diffusion examples including multiple instantaneous sources, continuous sources, moving sources, and different boundary geometries and conditions to validate our approximation against analytical and established numerical solutions. We also verify the CPM results for Poiseuille flow and Taylor-Aris dispersion.

Figure 1

A typical cell configuration in CPM. The bold lines denote cell boundary.

Figure 2

(Color online) (a) Velocity profile in a cylindrical flow under gravity and its fit to the analytical solution. (b) Velocity profile across a diameter of the tube with error bars, the bold segment along the $\widehat{x}$ direction denotes the width of a fluid particle.

Figure 3

(a) The absolute error between the analytical solution $V_{\text{analytical}}$ and the CPM simulation $V_{\text{simulation}}$ at $x=150$. (b) The rms error of over 50 ensembles.

Figure 4

(Color online) Symbols denote chemical profile projected on the $x$ axis (summing in the $Y$ and $Z$ directions) at $t=50$, 100, and 200 MCS for reflecting barriers at $x=0$ and 100 denoted by symbols. The solid line is a fit to the exact solution using the fitting parameter $D$. The inset shows variations in $D$ with time, due to coarse graining and the initial sharp distribution.

Figure 5

(Color online) Chemical profile projected on the $x$ axis from diffusion of two instantaneous point sources for an absorbing barrier at $x=0$ and $x=100$ for $t=70$, 100, and 200 MCS. The inset shows the variation of $D$ with fluid particle volume on a log scale.

Figure 6

(Color online) Relative error along the $x$ lattice direction for an absorbing barrier.

Figure 7

(Color online) Chemical profile cross section along $z=40$, 50, and 60 in the presence of a cylindrical reflecting barrier with axis along the $\widehat{x}$ direction and rectangular cross section. The sources with initial concentration 5 and 10 are placed at 50, 40, 40 and 50, 60, 60. The inset shows 2D projection of the profile. The lower figure shows a two-dimensional projection obtained from finite element calculations. The concentration decreases as we go away from the center contours (yellow lines).

Figure 8

(Color online) CPM simulation of the chemical profile from a moving source (the other two coordinates integrated out) at $t=300$ MCS. The solid line denotes the fit to Eq. (10). The inset shows a two-dimensional projection of the simulation, where red denotes the highest chemical concentration and black the lowest.

Figure 9

(Color online) Effective diffusion coefficient for mean flow velocities $\overline{\upsilon }=0.056$, 0.028, and 0.016, respectively from top to the bottom curve. The solid lines show the analytical results corresponding to above mean velocities.

Figure 10

(Color online) Cross-sectional profile ($\widehat{x}\text{−}\widehat{z}$ plane, $y=20$) of chemical concentration at $t=500$ MCS starting from an initial set of particles at $x=75$ with uniform concentration; (a) with no diffusion, (b) with diffusion. Particles which move away from the plane in the $\widehat{y}$ direction cannot be seen in the figures.