How Do You Calculate Bicarbonate Levels in Respiratory Disorders?

[cbrons]

Homework Statement

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.

Homework 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-.

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).

 

[haruspex]

Homework Statement

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.

Homework 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-.

The Attempt at a Solution

View attachment 87656Do 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).

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.

 

[Ray Vickson]

Homework Statement

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.

Homework 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-.

The Attempt at a Solution

View attachment 87656Do 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).

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.)