English

Português

日本語

Diffusion Limited Growth Diffusion Equation The growth of the new vasculature depends on the diffusion of the angiogenic factor, VEGF. The diffusion equation that describes this type of process is: $$\frac{\partial\phi}{\partial t}({\textbf{x}},t)=D\nabla^{2}\phi({\textbf{x}},t)$$ (1.1) where $\phi({\textbf{x}},t)$ represents the concentration of particles in solution at point ${\textbf{x}}$ and time $t$. If the particle concentration is sufficiently low, we can treat each particle independently. Mathematically, this corresponds to replacing the diffusion equation by the Laplacian equation: $$\nabla^{2}\phi({\textbf{x}})=0$$ (1.2) with appropriate boundary conditions, since the time dependence is no longer relevant. In other words, the Laplace equation corresponds to the steady state solution of the diffusion equation. Since the particles are absorbed on contact, the value of the field at the interface is identically 0 and in order to maintain a gradient, the value of the frilled at infinity is set to $1$. This results in a velocity for the growth of the interface proportional to the gradient of the field. $${\bf v}({\bf x})\propto{\bf\nabla}\phi({\bf x})$$ (1.3) Two different approaches can be taken in the solution of this equation. We can solve the equation numerically, using the methods available to us from the study of Numerical Partial Differential Equations, or, alternatively, we can perform a numerical simulation using random walkers. We explore the second approach in the remainder of this text. Computer Simulations In this section we describe the algorithms used to simulate the most basic aspects of Laplacian Growth and Choroidal Neovascularization. These algorithms were implemented using C++. The graphical aspects were implemented using the C++ wrapper for the widely used OPENGL graphics library. We made extensive used of the polymorphic and inheritance capabilities of the programming language that we chose. The most important C++ classes and header files that resulted of this programming effort are transcribed in the Appendices at the end of this text. Elastic Breakdown Model We can simulate the damage that Bruch's membranes suers throughout the life of an individual in random locations by using a set of random walkers and keeping track of how many random walkers have been trough a specific location on our grid. If that number passes a previously specified threshold $T$ we consider that the membrane has been permanently damage on that location and we can use that location as the origin for the growth of a cluster. We implemented this algorithm on a spherical surface and represent the result in Fig. 1.1. Where the colors go from Blue to Red as the level of damage to the membrane increases. This was implemented in the files ``OPENGL.h'', ``randomgl.h'' and ``RW.h''. Figure 1.1: Simulated damage to the Bruch Membrane in a spherical surface. Laplacian Growth In the limit of low concentrations, we can take each particle to be a random walker undergoing Brownian motion in 2D independently of each other. Suppose further that in the origin of our coordinate system there exists a fixed particle of a different kind and that when a diffusing particle collides with a particle at rest, the first one is absorbed and a new fixed particle of the second kind is created in it's place. Figure 1.2: Illustration of the diffusion and growth process. We represent this process in Fig. 1.2 where each of the green squares represents a diffusing particle, the circle represents our concept of infinity1.1 and the red square is the fixed particle at the origin. The blue square is a newly created particle after the absorption process. As we can see, particles start at infinity and move randomly in all directions. If they leave the circle after entering it, we consider that they diffused to infinity and dispose of them. If we repeat this process for each of the diffusing particles until they diffuse to infinity or are absorbed by the fixed particles, we generate a cluster similar to the one represented in Fig. 1.3. The branching properties of this cluster stem from the fact that the diffusive nature of the process implies that different branches of the same cluster will shadow each other. In turn this will lead to the concentrating the growth on the edges of each branch as demonstrated by the color coding of the figure where the color of the cluster varies from blue to red from the oldest particles to the youngest. If we study the volume occupied by the cluster as a function of the cluster radius, we obtain a power law of the form: $$V(r)\propto r^D$$ (1.4) with $D\approx1.7$. The fractional value of the dimension $D$ reveals the intrinsically fractal nature of the clusters generated by the process. Our implementation of these algorithms is available in files ``CLdefs.h'', ``DLA.h'' and ``planegl.h''. Figure 1.3: Cluster grown by the process illustrated in Fig. 1.2 with N=10000 particles. We can see the parallel between this process and the angiogenic processes described in the previous chapter if we think of the diffusing particles as being the growth factors and of the fixed particles as being the vasculature. The creation of a new fixed particle after the absorption of a diffusing particle corresponds to the growth of the blood vessel after it absorbed a molecule of VEGF. From this analogy, we expect that Choroidal Neovascularization would have a similar structure to DLA. This was shown to be true by Family et al in [1] for the retinal vascularization. Figure 1.4: Multiple clusters grown simultaneously. a) Initial particles are far apart. b) Initial particles are closer together. Generalizations of the model We later extended the Laplacian growth model to include multiple clusters as shown in Fig. 1.4. As in the previous image, the color denotes the ``age'' of the particle. As we can see in Fig. 1.4b), the different branches or clusters shadow each other, which causes them to, in effect, repel one another. We define a probability $p<1$ that the diffusing particle is absorbed or not. This will reduce the shadowing effect and result in fatter branches by allowing the particles to get closer the the center of the cluster. We expect that the resulting clusters will be increasingly fatter as $p\to 0$. We demonstrate this in Fig. 1.5. We also expect that as $p$ decreases the fractal dimension defined in 1.4 will increase toward $2$ as the cluster becomes closer and closer to covering the plane completely. This allows us to use $p$ to fine tune the fractal dimension in the range $1.67<p< 2$ and obtain a solution that satisfies some condition that we specify. Figure 1.5: Variation in the results of the simulation as a function of the sticking probability $p$. The left image corresponds to $p=0.2$ and the right one to $p=0.01$. These results were also obtained using the code in files ``CLdefs.h'', ``DLA.h'' and ``planegl.h'' with a slightly different set of parameters. This represents the care that was taken in producing code that can be reused for multiple applications and also the potential that this simple model has to represent a great array of different physical systems. Bibliography 1 F. Family, B. R. Masters, and D. E. Platt. Fractal pattern formation in human retinal vessels. Physica D, 38:98-103, 1989. Footnotes ... infinity1.1 We consider everything outside the circle to be at infinity.   © Copyright 2004 Bruno Goncalves - All rights reserved