Non dimensionalize pharmacodynamics

[Dustinsfl]

$$
\frac{dG}{dt} = k_{inG} - k_{outG}\cdot G\cdot (1 + S_{Ins}[n]\cdot I)
$$
$$
\frac{dI}{dt} = k_{inI}[m]\cdot (1 + S_G[r]\cdot (G - G_0)) - k_{outI}\cdot I
$$

The units for the first equation are $ k_{inG} = \frac{mg(week)}{dl}$, $k_{outG} = \frac{1}{week}$, $G = \frac{mg(week)^2}{dl}$, $S_{Ins}[n] = \frac{ml}{ng}$ and $I = \frac{ng}{ml}$
The second are $k_{inI} = \frac{ng(week)}{ml}$, $S_G[r] = \frac{dl}{mg}$, $G = G_0$, and $k_{outI} = \frac{1}{week}$

I am not sure how to do this very well since I didn't get a response to my Buckingham Pi Theorem question which would help me understand how to do this.

 

[jvicens]

No problem, I'll try to explain it in a simpler way. The Buckingham Pi Theorem states that if you have a physical relationship between multiple variables, you can express it as a function of dimensionless parameters. These dimensionless parameters are called Pi terms and they help simplify the equation and make it easier to understand.

In the first equation, we have four variables: $G$, $t$, $k_{inG}$, and $k_{outG}$. Using the Buckingham Pi Theorem, we can express this equation as a function of two dimensionless parameters, which we will call $P_1$ and $P_2$.

$$
P_1 = \frac{k_{inG}t}{G} \quad \text{and} \quad P_2 = k_{outG}t
$$

Notice that both $P_1$ and $P_2$ are dimensionless, as they are ratios of variables with the same dimensions. Now, we can rewrite the equation as:

$$
\frac{dG}{dt} = P_1 - P_2\cdot G\cdot (1 + S_{Ins}[n]\cdot I)
$$

Similarly, for the second equation, we have four variables: $I$, $t$, $k_{inI}$, and $k_{outI}$. Using the Buckingham Pi Theorem, we can express this equation as a function of two dimensionless parameters, which we will call $P_3$ and $P_4$.

$$
P_3 = \frac{k_{inI}t}{I} \quad \text{and} \quad P_4 = k_{outI}t
$$

Again, notice that both $P_3$ and $P_4$ are dimensionless. We can rewrite the equation as:

$$
\frac{dI}{dt} = P_3\cdot (1 + S_G[r]\cdot (G - G_0)) - P_4\cdot I
$$

Now, we can see that both equations have been simplified and are expressed in terms of dimensionless parameters. This helps us better understand the underlying physical relationship between the variables and allows us to make predictions and analyze the system more easily.