# Context for the question

I am a mathematics researcher working on the mathematics of imaging and parameter estimation (inverse problems). I found an interesting diffusion equation that models the growth of biofilms. A question I am interested in: Given some measurements of the biofilm, and assuming the mathematical model is accurate, can we deduce the coefficients in the model? This, however, is not the question I want to ask here.

Please pardon nonstandard use and misuse of biological terminology and gaps in knowledge; this is by no means my field. I have never done biological experiments in a laboratory, for example, or studied any university-level biology.

# The model

Let us assume a simple scenario where we have a biofilm and some substrate it consumes to grow. We consider a surface where the model should tell how fast and to where the biofilm grows. The parameters in the model are:

• $d_1$ substrate diffusion coefficient;
• $d_2$ biomass diffusion coefficient;
• $K_1$ maximum specific consumption rate;
• $K_2$ biomass decay rate;
• $K_3$ maximum specific growth rate;
• $K_4$ Monod half saturation constant (relative to initial data $S_0$);
• $a$, $b$ biomass spreading parameters.

The quantities that are solved from the equations are the following, both of them scaled so that $0 \le M, S \le 1$. Both of them are functions of space $x$ and time $t$, which I omit from the notation.

• $S$ is the concentration of the substrate that the biofilm feeds on
• $M$ is the biomass density

The governing equations are the following, with some boundary conditions I will not specify here:

$$\begin{cases} \partial_t S = d_1 \Delta_x S - K_1 \frac{SM}{K_4+S} \\ \partial_t M = d_2 \nabla_x \cdot \left( \frac{M^b}{(1-M)^a} \nabla_x M \right) - K_2 M + K_3 \frac{SM}{K_4+S} \end{cases}$$

# The questions

Can both $M$ and $S$ be measured with fair accuracy in laboratory conditions? How is it done and how reliable is it? Can the experiments continue afterwards, or does the measurement significantly alter the experimental set-up in some way (by for example destroying the biofilm or mixing the entire thing etc.)?

Of the parameters in the model, which are well-known?

I suppose the answer to last question depends on whether the biofilm is of known or unknown nature, and in the case of an unknown substance, only $d_1$ might be known. What about a known kind of biofilm?