# Mathematical Modelling of the Effects of Plasma Treatment on the Diffusivity of Biofilm
## 2 Materials and Methods
### Development of a Mathematical Model of CHX Diffusion in a Biofilm
Equation (7) then becomes,  
\[\frac{d\ ln\ \overline{X}}{dt}=-k_{d}C(x,t) \tag{8}\]  
Integrating Equation (8) from the initial cell concentration \(\left(ln\overline{X}_{0}\right)\) at \(t=0\) to a final cell concentration \(\left(ln\overline{X}\right)\) at time \(t\).x  
\[\overline{X}(x,t)=\overline{X}_{0}e^{-k_{d}\int_{0}^{t}C(x,t)dt} \tag{9}\]  
Equation (9) shows how the cell concentration at any given position or value of \(x\) changes with time. Now, let \(\phi=\) total number of cells in the biofilm at time \(t\). Then, the total number of cells measured in the biofilm at time \(t\) is given by,  
\[\phi(t)=S\ \int_{0}^{\delta}\overline{X}(x,t)dx \tag{10}\]  
At \(t=0\), the initial number of cell concentration is \(\phi_{0}=\overline{X}_{0}\ S\). \[\frac{\phi_{0}}{\delta}=\overline{X}_{0}\ S \tag{11}\]  
Therefore, inserting Equation (9) into (10),  
\[\phi(t)=S\ \overline{X}_{0}\int_{0}^{\delta}e^{-k_{d}\int_{0}^{t}C(x,t)dt}dx \tag{12}\]  
From Equations (11) and (12),  
\[\frac{\phi(t)}{\phi_{0}}=\frac{1}{\delta}\int_{0}^{\delta}e^{-k_{d}f(x,t)}dx \tag{13}\]  
where  
\[f(x,t)=\int_{0}^{t}C(x,t)dt \tag{14}\]  
Substituting the value of \(C(x,t)\) from Equation (4) into (14) and interchanging, we get \(f(x,t)\) as,  
\[f(x,t)=C_{b}\Bigg{\{}t-\frac{16\delta^{2}}{\pi^{3}D_{e}}\ \sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2n+1)^{3}}.\Bigg{[}1-exp\Bigg{[}\frac{-D_{e}(2n+1)^{2}\pi^{2 }t}{4\delta^{2}}\Bigg{]}\Bigg{]}\cos\bigg{[}\frac{(2n+1)\pi x}{2\delta}\Bigg{]} \Bigg{\}} \tag{15}\]  
As Equation (15) is very complex and we expect the summation to converge rapidly as \(t\) increases, we only use the term for \(n=0\). Therefore,  
\[\frac{d\overline{X}}{dt}=\mu_{d}\ \overline{X}\quad\text{and},\mu_{d}=k_{d}\ C(x,t) \tag{7}\]0  
For steady state (SS) or long treatment times,  
\[\frac{d\overline{X}}{dt}=\mu_{d}\ \overline{X}\quad\text{and},\mu_{d}=k_{d}\ C(x,t) \tag{7}\]1and at \(x=0\) (at the coupon surface),  
\[\frac{d\overline{X}}{dt}=\mu_{d}\ \overline{X}\quad\text{and},\mu_{d}=k_{d}\ C(x,t) \tag{7}\]2  
Further, at \(x=\delta\),  
\[\frac{d\overline{X}}{dt}=\mu_{d}\ \overline{X}\quad\text{and},\mu_{d}=k_{d}\ C(x,t) \tag{7}\]3  
Now, because the biofilm is very thin, we use the average value of the function \(f\) across the biofilm, which can be written as,  
\[\frac{d\overline{X}}{dt}=\mu_{d}\ \overline{X}\quad\text{and},\mu_{d}=k_{d}\ C(x,t) \tag{7}\]4  
From Equations (16) and (20), we obtain,  
\[\frac{d\overline{X}}{dt}=\mu_{d}\ \overline{X}\quad\text{and},\mu_{d}=k_{d}\ C(x,t) \tag{7}\]5  
Substituting the result into Equation (13), we obtain,  
\[\frac{d\overline{X}}{dt}=\mu_{d}\ \overline{X}\quad\text{and},\mu_{d}=k_{d}\ C(x,t) \tag{7}\]6  
From Equation (21), for \(t>\tau=\frac{4\delta^{2}}{\pi^{4}D_{e}}\), Equation (21) becomes,  
\[\frac{d\overline{X}}{dt}=\mu_{d}\ \overline{X}\quad\text{and},\mu_{d}=k_{d}\ C(x,t) \tag{7}\]7  
Then, using this result in Equation (22),  
\[\frac{d\overline{X}}{dt}=\mu_{d}\ \overline{X}\quad\text{and},\mu_{d}=k_{d}\ C(x,t) \tag{7}\]8  
where \(\phi\) is the final number of cells after plasma treatment; \(\phi_{0}\) is the initial number of cells before plasma treatment; and \(t_{lag}\) is the x-intercept of the linear plot of \(log_{10}\frac{\phi}{\phi_{0}}\) versus CHX treatment time, which is given by the equation,  
\[\frac{d\overline{X}}{dt}=\mu_{d}\ \overline{X}\quad\text{and},\mu_{d}=k_{d}\ C(x,t) \tag{7}\]9  
Now, taking the log of Equation (24), we get,  
\[\frac{d\ ln\ \overline{X}}{dt}=-k_{d}C(x,t) \tag{8}\]0  
Equation (26) says a plot of \(log_{10}\frac{\phi}{\phi_{0}}\) versus \(t\) should be linear. The value of \(t_{lag}\) can be found at that time where the regression line crosses the time axis. From the calculated \(t_{lag}\), we can then find the effective diffusivity of CHX in the biofilm from Equation (25). The calculated value of \(D_{e}\) should be less than the previously calculated value of the effective diffusivity of CHX in water at \(20\ ^{\circ}\)C (\(D_{e}=3.77\times 10^{-6}\)cm\({}^{2}\)/ sec).