Why is the oxygen concentration modeled using the Laplacian?

Question

I am trying to understand the following equation from (1):

The C, D and M members are quite self-explanatory and, as I understand it from this explanation the laplacian term simply tells how "different" the current position is from the average value of its neighbours.

I've tried a graphical representation, I'm not sure how accurate this is:

The brown surface represents the Laplacian component. In the image I'm assuming that the concentration at our point is greater than that at the neighbours, so the oxygen molecules (Orange spheres -collectively C) would flow away. The diffusivity D indicates how many can escape at any one point, and it's represented by the barrier D. So obviously the bigger D and laplacian value the quicker concentratio would increase/decrease.

What I don't understand is why this is equal to M?

(1) Hsu, R., & Secomb, T. W. (1989). A Green’s function method for analysis of oxygen delivery to tissue by microvascular networks. Mathematical Biosciences, 96(1), 61–78. http://doi.org/10.1016/0025-5564(89)90083-7

Answer 1

Here is a slightly lengthier explanation.

The standard assumptions that some quantity $C$ experiences the brownian motion lead to the standard diffusion equation $$ C_t=\nabla\cdot D\nabla C+M(x), $$ where $D$ is the diffusion coefficient and $M$ is the function that describes the appearance or disappearance of $C$ withing the feasible domain $\Omega$. Assuming that the quantity $C$ is in equilibrium state yields exactly the equation in the referenced paper, since in this case $C_t=0$.

Answer 2

Laplacian is the divergence of the gradient. Let's rewrite the equation as $$\operatorname{div}(-D\nabla C) = -M$$ Here, $-D\nabla C$ is the vector field describing the flow of oxygen (it's opposite to concentration gradient, and proportional to diffusivity $D$).

The divergence of a vector field within some region describes the amount of substance generated within that region. Here, the substance is being consumed at rate $M$, rather than generated, hence the rate of generation is $-M$.

By the way, the partial differential equation $(1)$ is known as the Poisson's equation.