# 10.3: The Variables

• • Kerry Brandis
• Gold Coast Hospital
$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

The equation for calculating [H+] developed by Stewart contains 3 independent variables and 6 dependent ones. The nature of the independent variables will seem strange at first but the purpose of this section is to introduce them and briefly discuss what they are and why they are independent.

### The Three Independent Variables

These are:

• pCO2 -the partial pressure of CO2 in the solution under examination
• SID -this stands for the 'strong ion difference' in the solution
• [ATot] -the total concentration of weak acid in the solution.

(These 3 variables are explained further in the subsections below)

## The first independent variable : pCO2

The pCO2 is the easiest to understand. Some facts:

• Carbon dioxide is produced by all cells in the body
• It crosses all cell membranes easily, traverses the ISF and enters the blood
• It is excreted from the body by the lungs
• The arterial pCO2 is under sensitive and powerful feedback control via the peripheral and central chemoreceptors

These receptors respond to an increase in arterial pCO2 by increasing ventilation and this returns arterial pCO2 to normal. Arterial pCO2 is frequently said to be determined by the ratio of CO2 production to alveolar ventilation (See Section 2.3). This is quite correct but does not indicate the effect of the control system which is very effective at maintaining normal arterial pCO2. A consideration of the equation would suggest that a doubling of CO2 production would result in a doubling of arterial pCO2 but this does not occur in the intact person (unless ventilation is fixed eg as in an anaesthetised ventilated patient).

Any rise in arterial pCO2 is detected by the sensors (ie the chemoreceptors) and activates the control system resulting in increased alveolar ventilation. This returns the arterial pCO2 towards normal. In abnormal situations, the control system is disturbed or otherwise ineffective at keeping arterial pCO2 constant.

The gist is that the value of pCO2 in arterial blood and all body fluids is effectively set by mechanisms other than the chemical equilibria occurring in the fluids. The value is determined and controlled by factors external to the chemical system in the body fluids. It is therefore an independent variable.

## The second independent variable: SID

This abbreviation stands for Strong Ion Difference. It is defined as:

SID = (the sum of all the strong cation concentrations in the solution) minus (the sum of all the strong anion concentrations in the solution).

For example: if a solution contained Na+, K+ and Cl- as the only strong ions present, then:

$$SID= [Na^{+}] + [K^{+}] - [Cl^{-}]$$

SID = [Na+] + [K+] - [Cl-]

If these strong ions were the only charged species present, then the powerful requirement for electrical neutrality would mean that SID would be zero. Most biological fluids contain weak electrolytes (mostly weak acids). If the SID is not zero, then it means that the solution must contain other charged species ie weak electrolytes. The SID represents the net charge which must be balanced by charges on the weak acids in the solution for electrical neutrality to be maintained.

In plasma, the formula for SID is approximately:

$SID = [Na^{+}] + [K^{+}] + [Ca^{2+}] + [Mg^{2+}] - [Cl^{-}] - [\text{Other strong anions}^{-}]$

### Why is SID considered an 'independent variable'?

The components (ie the strong ions) which are used to calculate the SID are not altered by any of the reactions in the system. None of these ions are produced or consumed. The concentrations are imposed on the solution from outside and are controlled by outside mechanisms. The kidney is the most important regulator of most of these ion concentrations.

Inorganic strong ions (eg Na+, Cl-) are mostly absorbed from the gut and control is mostly by variations in renal excretion due to various control systems in the body.

Organic strong ions (eg lactate, keto-anions) are produced by metabolism and may be metabolised in the tissues or excreted in the urine. However, their concentrations in most body fluids are not dependent on the reactions within the solution but are regulated by mechanisms external to the system.

The derived value SID is used because it is a term which arises in the equation for electrical neutrality and allows us to lump together all the independent concentrations in the form in which the strong ions are involved in affecting acid-base balance (ie by their overall net charge). The SID is that part of the charge on the strong ions which has to be balanced (because of the electroneutrality requirement) by the net opposite charges of the total weak ions present. Unlike the strong ions, the amount of these weak ions varies because of varying amounts of dissociation. The amount of dissociation of these weak ions varies such that the net amount of charge of them all considered together, is equal and opposite to the charge due to the strong ions. This is just a chemical fact due to the requirement for electroneutrality that is imposed on the system by physical laws.

If only the strong ions which are typically present in health are considered, the apparent SID (SIDa) can be calculated as:

$SID_{a} = [Na^{+}] + [K^{+}] + [Ca^{2+}] + [Mg^{2+}] - [Cl^{-}] - [\text{lactate}^{-}]$

SIDa has a normal value of 40 to 42 mEg/l.

This is a useful simplification but it is possible to go further. Only [Na+] and [Cl-] are present in high concentrations so the SID can be roughly approximated as ( [Na+] - [Cl-] ). Now if we remember that [Na+] is tightly controlled by the body because it controls tonicity, then the major way that the ECF pH can be altered is by changes in [Cl-] relative to a constant [Na+].

## The third independent variable: [ATot]

The abbreviation represents the total amount of non-volatile weak acid present in the system.

All the weak acids in the system are represented collectively as HA. The anion for each acid will be different but because they all behave similarly all the weak acids are represented as though they were a single acid (for which the symbol HA is used) which has a single apparent dissociation constant. This is a useful simplifying assumption which is basically an averaging process. The dissociation reaction is:

$$HA \Leftrightarrow H^{+} + A^{-}$$

The law of conservation of mass means that the total amount of A (symbol: [ATot]) in the system must be constant. None of the reactions in the system produce or consume A. Conservation of A can be represented as:

$$[A_{Tot}] = [HA] + [A^{-}]$$

In plasma, the major non-volatile weak acids present are:

• Proteins $$([Pr_{Tot}] = [Pr^{-}] + [HPr])$$
• Phosphates $$([Pi_{Tot}] = [PO_{4}^{3-}] + [HPO_{4}^{2-}] + [H_{2}PO_{4}^{-}] + [H_{3}PO_{4}])$$

Albumin is the most important protein present that acts as a weak acid so the total amount of protein is approximated by the albumin concentration ([Alb]). Globulins do not contribute significantly to the total negative charge due to plasma protins. The level of albumin in body fluids is imposed upon the acid-base system and is not regulated by it. The colloid osmotic pressure & osmolality of the extravascular liver space is the primary factor which controls the rate of production of albumin. (Pietrangelo et al, 1992).

Phosphates are present in several forms but the total amount is normally fairly constant. Its level in plasma is controlled as part of the system for regulating calcium levels. Phosphates normally contribute only about 1mM of ATot. Phosphates represent only 5% of ATot at normal phosphate levels. If phosphate levels are elevated then its contribution becomes more important.

### The point of all this is that the [Albumin] alone can be used as an estimate of ATot in plasma.

As an overview of these independent factors, consider the following generalisations that have been made:

• The first independent variable is pCO2 which is controlled by a respiratory control system.
• The 2nd independent variable is SID and this can be roughly estimated as ([Na+] - [Cl-]) and this is controlled by the kidney.
• The 3rd independent variable is ATot and this is estimated as [Alb] which is controlled by the liver.

10.3: The Variables is shared under a CC BY-NC-SA 2.0 license and was authored, remixed, and/or curated by Kerry Brandis via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.