Homework Help: Help with equation format

1. Aug 21, 2015

cbrons

1. The problem statement, all variables and given/known data
Trying to create an equation to calculate the expected bicarbonate level given a respiratory acidosis or alkalosis (this is purely a math question, though).

I am not sure if my equations are in the right format in terms of the order of operations and use of brackets, etc.

2. Relevant equations

The laws are:

The [HCO3-] will increase by 4 mmol/l for every 10 mmHg increase in PaCO2 above 40mmHg.
The [HCO3-] will decrease by 5 mmol/l for every 10 mmHg decrease in pCO2 below 40 mmHg.

By the way "24" comes from the midpoint of the normal value of HCO3-.

3. The attempt at a solution

Do these equations look correct to you given the rules outlined? Particularly the subtraction for alkalosis (24- and 40 - PaCO2 as opposed to PaCO2 - 40 in the above example).

2. Aug 21, 2015

haruspex

Yes, that's fine. Alternatively, you could make the signs in the second equation the same as in the first, i.e. $24+\left[5\times\frac{(PaCO_2-40)}{10}\right]$, so that the only difference is the 5. It comes to the same thing, but you would need to be careful with signs when using it to calculate.

3. Aug 21, 2015

Ray Vickson

Yes, your expressions look correct, as written. However, it would be better to put the restrictions right in the equation(s) themselves:
$$\text{Expected HCO}_3^{-} = 24 + \begin{cases} \;\;(4/10) (\text{PaCO}_2 - 40) &, \text{PaCO}_2 > 40\\ - (5/10) ( 40 - \text{PaCO}_2) &, \text{PaCO}_2 < 40 \end{cases}$$

It would also work if you wrote
$$\text{Expected HCO}_3^{-} = 24 + \begin{cases} (4/10) (\text{PaCO}_2 - 40) &, \text{PaCO}_2 > 40\\ (5/10) ( \text{PaCO}_2 - 40) &, \text{PaCO}_2 < 40 \end{cases}$$
The reason the latter works is that when $\text{PaCO}_2 > 40$ the quantity $\text{PaCO}_2 - 40$ is $>0$ and so we are adding, but when $\text{PaCO}_2 < 40$ the quantity $\text{PaCO}_2 - 40$ is $< 0$, and so we are subtracting. This is exactly what you want to do.

Note: the equations above were written using "LaTeX" and employing the " \ begin{cases} ... \ end{cases}" construction. (Remove the space between the '\' and 'begin'; I put it in just to prevent LaTeX from trying to process the explanation.)