Henderson–Hasselbalch and Stewart Approaches

Acid–base frameworks organize the same chemical system using different variables. The bicarbonate–carbon dioxide relationship describes an equilibrium, while strong-ion analysis makes electrolyte and weak-acid accounting explicit. Both must satisfy chemical equilibria, conservation of matter, and electrical neutrality. Their value lies in explaining the patient's physiology without mistaking a calculation for the process that produced it.

53.1 Different Coordinates Describe the Same Chemical System

The pH expresses hydrogen-ion activity on a negative logarithmic scale. Activity reflects effective chemical behavior rather than simply the counted concentration of free hydrogen ions. Concentration-based clinical equations use approximations appropriate to the solution and measurement conditions; their constants incorporate assumptions about activities, temperature, and composition.

A blood sample contains interacting carbonate species, proteins, phosphate, strong electrolytes, and other ions. Changing one component can alter several others as equilibria re-establish. A change in bicarbonate concentration, for example, may accompany carbon dioxide retention, acid buffering, altered renal handling, or a change in extracellular water.

The Henderson–Hasselbalch equation expresses the carbonate equilibrium in terms of pH, bicarbonate, and carbon dioxide tension. Stewart's formulation combines carbonate equilibrium with electroneutrality, water dissociation, and conservation of weak-acid species. It selects a different set of variables for solving the system, rather than introducing a separate chemistry.

Charge accounting is indispensable in either description. In a macroscopic fluid compartment, positive and negative charge concentrations balance to a very close approximation. An anion gap or strong-ion gap identifies substances omitted from a particular calculation; it is not evidence that plasma has acquired a large unbalanced electrical charge.

Mass conservation is equally important. Reversible proton binding redistributes hydrogen among chemical forms but does not itself excrete an acid load. Ventilation, renal excretion, metabolism, and exchange between compartments change the available inventories or boundary conditions. Those processes must be specified when explaining a patient's trajectory.

Wooten derived relationships among titratable base, strong-ion difference, bicarbonate, and related quantities within a common equilibrium formalism. The relationships depended on buffer concentrations and reference states. This mathematical connection supports comparing complete descriptions, rather than comparing one isolated equation with a more fully specified model. 1

53.2 The Bicarbonate–Carbon Dioxide Relation Requires Consistent Constants

For plasma under specified conditions, let [HCO₃⁻] be bicarbonate concentration in mmol/L and PCO₂ carbon dioxide partial pressure in mmHg. The solubility coefficient αCO₂, in mmol/(L·mmHg), converts that pressure into dissolved carbon dioxide concentration; pK′ is the negative logarithm of the apparent acid dissociation constant. The bicarbonate form of the Henderson–Hasselbalch equation is:

The ratio inside the logarithm must be dimensionless, so numerator and denominator must use compatible concentration units.

For the following illustrations at 37°C, use rounded values pK′ = 6.10 and αCO₂ = 0.030 mmol/(L·mmHg). These are teaching approximations. Instrument algorithms may employ more detailed constants, so small numerical differences need not indicate a physiological discrepancy.

With bicarbonate 24 mmol/L and PCO₂ 40 mmHg, dissolved CO₂ is 0.030 × 40 = 1.2 mmol/L. The concentration ratio is 24/1.2 = 20, and pH is 6.10 + log₁₀(20) = 7.401, approximately 7.40. This calculation describes the equilibrium compatible with those selected values.

Pressure units cannot be changed without changing the solubility coefficient. Because 1 kPa is approximately 7.5006 mmHg, the same rounded αCO₂ becomes 0.030 × 7.5006 ≈ 0.225 mmol/(L·kPa). The corresponding PCO₂ is approximately 5.33 kPa. Using a coefficient intended for mmHg with a pressure in kPa would create a major calculation error.

Temperature changes gas solubility, dissociation behavior, and the measured pH. Blood-gas instruments commonly analyze at 37°C and may additionally report temperature-adjusted estimates. The equation must use mutually consistent values and constants; combining a temperature-adjusted pH with an incompatible PCO₂ or coefficient undermines the calculation.

Carbonic acid and dissolved molecular CO₂ are related species, but the conventional apparent constant groups their behavior for clinical use. The denominator should therefore be interpreted according to that convention rather than replaced with an independently guessed carbonic-acid concentration.

53.3 An Equilibrium Identity Is Not a Complete Causal Explanation

If bicarbonate falls at unchanged PCO₂, the Henderson–Hasselbalch relation predicts a lower pH. If PCO₂ rises at unchanged bicarbonate, it also predicts a lower pH. Those mathematical comparisons are useful, but whether the supposedly unchanged variable actually remains fixed is a physiological question.

When CO₂ accumulates, nonbicarbonate buffers bind some of the hydrogen generated as carbonate equilibria shift. Bicarbonate consequently changes along with pH. Treating it as fixed may illustrate one algebraic dependency without reproducing the response of whole blood. Chronic renal adaptation introduces additional changes in bicarbonate and other ion balances.

Likewise, buffering an added nonvolatile acid alters bicarbonate and produces carbon dioxide that can exchange with the lungs. The final state depends on ventilation and other buffers, not simply on subtraction from a static bicarbonate pool. A low bicarbonate may also accompany an appropriate response to sustained hypocapnia rather than an isolated primary metabolic disturbance.

Stewart analysis does not make bicarbonate chemically inactive. It remains a real buffer species, participates in transport, and carries carbon. Calling it a dependent variable means that its equilibrium concentration is solved from a selected set of model inputs. It does not mean that bicarbonate transport or administration has no physiological effects.

Conversely, the bicarbonate relationship can be used within a model that explicitly includes chloride, albumin, phosphate, and charge balance. A clinician using conventional terminology is not required to ignore these components. The adequacy of an analysis depends on what it actually measures and conserves, rather than the name attached to it.

An intervention also changes more than a word in an equation. Adding a sodium bicarbonate solution introduces sodium, carbonate species, and water. Removing gastric fluid removes several species together. These perturbations can be described through changes in bicarbonate and carbon dioxide or through strong-ion, weak-acid, and carbon balances, provided the same transfers and boundaries are included.

53.4 Calculated Bicarbonate and Measured Total CO₂ Are Related Measurements

Blood-gas analyzers usually measure pH and PCO₂, then calculate plasma bicarbonate from the carbonate equilibrium. The displayed bicarbonate is therefore not an independent third measurement confirming the first two. A systematic error in either measured input can propagate into the calculated result.

Routine chemistry commonly measures total carbon dioxide in serum or plasma. Most is bicarbonate, but dissolved CO₂ and smaller additional forms contribute. With the preceding illustrative bicarbonate of 24 mmol/L and dissolved CO₂ of 1.2 mmol/L, a simplified total is approximately 24 + 1.2 = 25.2 mmol/L. Exact sample chemistry and analytical methods add qualifications.

A modest difference between calculated bicarbonate and measured total CO₂ is therefore expected under compatible conditions. A substantial disagreement deserves investigation, but it does not by itself establish which instrument or physiological model is wrong. Arterial whole blood and venous serum are different specimens, and samples obtained at different times may reflect real change.

Collection and handling also matter. Exposure of a chemistry specimen to air can permit CO₂ loss. Excess liquid anticoagulant can dilute a blood-gas sample; delayed analysis permits cellular metabolism to continue. Lipemia, unusual proteins, and assay-specific interference can affect selected chemistry methods. These effects require method-specific assessment rather than one universal correction.

Story and colleagues compared bicarbonate estimates in critically ill patients using blood-gas, enzymatic, and a strong-ion-derived method. Agreement depended on the method, with particularly poor agreement for their strong-ion-derived estimate. This laboratory comparison cautions against treating values obtained through different analytical and modeling routes as interchangeable; it does not invalidate charge conservation or the carbonate equilibrium. 2

Before interpreting a disagreement, establish specimen type, timing, temperature convention, analyzer method, and whether the bicarbonate is measured indirectly as total CO₂ or calculated. Repeating a derived number from the same inputs provides less independent information than a compatible measurement obtained through a different physical method.

53.5 Stewart Selects Strong Ions, Weak-Acid Totals, and PCO₂

Stewart's original quantitative formulation selected PCO₂, strong-ion difference, and total nonvolatile weak acid as inputs to an equilibrium solution. Hydrogen ions, bicarbonate, and the dissociated weak-acid fraction were then calculated by satisfying the system's simultaneous equations. This is a statement about a specified mathematical model and its boundary conditions. 3

Strong ions are treated as effectively fully dissociated over the relevant physiological pH range. Sodium, potassium, and chloride are major examples. Lactate is generally included as a strong anion in this accounting because its conjugate acid is extensively dissociated at blood pH. That classification does not by itself explain lactate production or equate its concentration with the rate of proton generation.

The true strong-ion difference is the sum of strong-cation charge concentrations minus strong-anion charge concentrations. It is expressed in mEq/L. This is not the same as subtracting numbers of molecules without regard to valence: freely dissolved calcium and magnesium contribute two equivalents per millimole.

Positive SID is balanced mainly by bicarbonate and negatively charged weak-acid species. Hydrogen and hydroxyl concentrations remain essential to equilibrium but are too small to account for tens of mEq/L of bulk charge. The charge constraint and dissociation relationships must be solved together; electroneutrality alone does not provide a complete pH equation.

The term independent requires care. In an open CO₂ model, PCO₂ can be specified by a gas reservoir or respiratory boundary. In a closed sample, total carbon and gas volume may instead constrain the eventual PCO₂. Physiological variables can also be correlated because transport and regulation change them together.

A membrane proton flux is not made irrelevant by this variable choice. It occurs with accompanying charge-balancing transport or reactions and changes compartment balances. Likewise, an isolated chloride addition without a counterion or compensating flux is not a physically complete intervention. Model coordinates organize the consequences of real transfers; they do not replace those transfers as mechanisms.

53.6 Weak-Acid Amount, Concentration, and Charge Must Be Distinguished

A simple weak acid can be represented as HA ⇌ H⁺ + A⁻. Its total concentration, Atot, is [HA] + [A⁻], whereas its negative charge depends on the dissociated fraction. Changing pH can change that fraction without adding or removing the weak-acid molecules themselves.

Albumin is more complex because it contains many ionizable groups. Its net charge varies with pH, binding, and molecular properties. Phosphate also exists in several protonation states. Neither a gram of albumin nor a millimole of phosphate carries one fixed number of negative charges under every condition.

Figge, Rossing, and Fencl measured and modeled manipulated albumin-containing solutions and human serum, establishing a quantitative contribution from plasma proteins. A subsequent model incorporated additional information about albumin histidine ionization. These studies concern chemical behavior in defined preparations, rather than the clinical benefit of correcting a protein concentration. 4, 5

In human blood studied in vitro, Rossing and colleagues found that reducing protein concentration under controlled gas conditions produced an alkalinizing change, whereas increasing albumin produced the opposite effect. This demonstrates a weak-acid contribution while preserving the importance of the experimental conditions. 6

Useful linear approximations express the magnitudes of albumin and phosphate negative charge. Let albumin be in g/L, inorganic phosphate Pi in mmol/L, and each calculated charge magnitude in mEq/L:

These approximations were used in Fencl and colleagues' clinical analysis and apply near the modeled physiological conditions. 7

At pH 7.40, albumin 40 g/L contributes 40 × 0.2792 = 11.17 mEq/L, and phosphate 1 mmol/L contributes 1 × 1.8176 = 1.82 mEq/L. These are positive numerical magnitudes of negative charge. Atot and this dissociated charge must not be substituted for one another.

53.7 Apparent SID, Effective SID, and the Strong-Ion Gap Answer Different Questions

The true SID includes all relevant strong ions. In practice, apparent SID, or SIDa, is calculated from a selected measured subset. For the following convention, sodium, potassium, chloride, lactate, ionized calcium, and ionized magnesium are in mmol/L, and SIDa is in mEq/L:

The ion list and chemical fractions must be stated. Total calcium cannot automatically replace freely ionized calcium without considering binding and complex charge. Some published conventions use different components or approximations; their resulting gaps and reference distributions are consequently different.

Effective SID, or SIDe, estimates the charge balanced by measured bicarbonate and modeled weak acids. Using bicarbonate in mmol/L, numerically equal to mEq/L for this monovalent ion, and the preceding weak-acid charge magnitudes:

SIG is the strong-ion gap. A positive value suggests net omitted anionic charge relative to the selected accounting, while a negative value can reflect omitted cationic charge, measurement error, or model mismatch. It does not identify a chemical species or directly measure a total acid amount.

Consider sodium 140, potassium 4, ionized calcium 1.2, ionized magnesium 0.6, chloride 108, and lactate 1 mmol/L. SIDa is 140 + 4 + 2.4 + 1.2 − 108 − 1 = 38.6 mEq/L. With pH 7.40, bicarbonate 24 mmol/L, albumin 40 g/L, and Pi 1 mmol/L, SIDe is 24 + 11.17 + 1.82 = 36.99 mEq/L. SIG is approximately 38.6 − 36.99 = 1.61 mEq/L.

This illustrative residual is not a universal normal value. Minor species, binding-related charge, assay bias, and rounding remain in the omitted balance. Lactate is already included here; a gap calculated without it would contain a different set of unaccounted anions.

53.8 Hypoalbuminemia Can Conceal Opposing Effects

Return to the illustrative sample with pH about 7.40, PCO₂ 40 mmHg, bicarbonate 24 mmol/L, and SIDa 38.6 mEq/L. Now consider another sample with the same gas values and measured strong ions but albumin 20 rather than 40 g/L. Keep phosphate at 1 mmol/L.

The carbonate relationship still gives the same pH because its bicarbonate-to-dissolved-CO₂ ratio is unchanged. A report limited to those gas values therefore does not reveal the altered protein environment. This is not a failure of the equilibrium equation; the equation was never a complete inventory of plasma constituents.

At pH 7.40, albumin charge magnitude becomes 20 × 0.2792 = 5.58 mEq/L. SIDe is now 24 + 5.58 + 1.82 = 31.40 mEq/L, and SIG is 38.6 − 31.40 = 7.20 mEq/L. Compared with the previous sample, the residual has increased by approximately 5.58 mEq/L as the modeled albumin charge has fallen by the same amount.

The interpretation is conditional. If all relevant strong ions and PCO₂ were truly fixed and albumin alone fell, the equilibrium pH would tend to rise. Holding the gas values unchanged in this constructed comparison therefore requires another balancing difference, represented here by greater net unmeasured anionic charge or an equivalent omitted/model contribution.

This does not prove that a particular organic acid accumulated, nor does it establish two independent disease processes from arithmetic alone. The example shows how the same pH and bicarbonate can coexist with different underlying composition. Conventional analysis can also detect this problem when it includes albumin and an appropriately interpreted anion gap.

Albumin concentration is additionally not the total albumin amount. Dilution, redistribution, loss, and synthesis can change it through different mechanisms. A low value identifies a changed sampled concentration; it does not establish which transfer occurred or imply that replacing albumin would improve acid–base outcomes.

53.9 Base Excess Standardizes a Titration Question

Base excess describes departure from a standardized acid–base reference using a modeled or measured titration. Conceptually, it asks how much strong acid or base would be required per liter to return the specified sample system to pH 7.40 at PCO₂ 40 mmHg and 37°C, with a defined oxygenation convention.

A positive base excess means strong acid would be required to reach that endpoint; a negative value means strong base would be required. For illustration, a base excess of −6 mmol/L represents a modeled base requirement of 6 mmol per liter of the specified sample system under the standardization. It is not an administered dose or a measurement of whole-body acid inventory.

Siggaard-Andersen's Van Slyke equation linked bicarbonate, pH, hemoglobin, and base excess, providing an arithmetic algorithm alongside Henderson–Hasselbalch. Its inclusion of nonbicarbonate buffering is precisely why base excess is more than the bicarbonate concentration alone. 8

Whole-blood base excess and standard base excess are related but distinct. Whole blood contains substantial hemoglobin buffering. Standard or extracellular-fluid base excess uses a standardized representation intended to approximate the effective buffering of blood plus surrounding extracellular fluid. Analyzer labels and algorithms should be identified before comparing numbers.

In controlled human-blood experiments, Morgan and colleagues found that calculated base excess remained nearly constant during large changes in PCO₂ and quantified added nonrespiratory acid with useful accuracy. The experiment supports its intended separation of a metabolic component under those conditions, rather than making it an infallible whole-patient measurement. 9

Buffer concentrations, oxygenation, compartment assumptions, and the chosen reference affect interpretation. An apparently modest net base excess can coexist with substantial opposing acidifying and alkalinizing influences. It cannot identify whether chloride, altered proteins, phosphate, or unmeasured ions account for the result. Those questions require additional chemical measurements and the clinical history.

53.10 Chloride Must Be Interpreted Relative to Other Ions and Water

Chloride concentration cannot be interpreted independently of sodium and the rest of the charge balance. The same chloride value can accompany different strong-ion differences. Conversely, water redistribution can change several concentrations together without equivalent gains or losses of their total body amounts.

A sodium–chloride difference is a useful partial descriptor, not the complete SID. For sodium 140 and chloride 108 mmol/L, the difference is 32 mmol/L. At the same chloride but sodium 154 mmol/L, it is 46 mmol/L. Potassium, divalent ions, lactate, and omitted strong ions still determine the full charge difference.

An explicit mixing model separates amount from concentration. Choose an extracellular compartment of 14 L containing sodium 140 and chloride 104 mmol/L. Its sodium amount is 14 × 140 = 1,960 mmol and chloride amount 14 × 104 = 1,456 mmol. Mathematically add 1 L containing 154 mmol each of sodium and chloride, with no other exchange.

The new sodium concentration is (1,960 + 154)/15 = 140.93 mmol/L. Chloride is (1,456 + 154)/15 = 107.33 mmol/L. The sodium–chloride difference falls from 36 to 33.60 mmol/L even though equal amounts of the two ions were added. Their initial concentration difference has been diluted by a solution whose own sodium–chloride difference is zero.

That calculation does not alone predict final pH. Weak acids are also diluted, CO₂ exchanges with the lungs, other ions and buffers respond, and the kidney changes balance. A bicarbonate-based description must include those same processes rather than assuming a permanently fixed bicarbonate amount.

Scheingraber and colleagues randomized surgical patients to saline or lactated Ringer's solution and observed different acid–base trajectories, including a hyperchloremic, lower-SID pattern with saline. This controlled physiological comparison supports the importance of composition; it does not prove that one explanatory framework or biochemical change independently determines clinical outcomes. 10

53.11 Compartment and Measurement Assumptions Limit Numerical Precision

Plasma is not the whole organism. Red cells contain hemoglobin and exchange bicarbonate, chloride, carbon dioxide, and water with plasma. Interstitial and intracellular spaces have different buffers and transport properties. A model of separated plasma can describe that sample accurately while incompletely predicting a whole-blood or in-vivo response.

Wooten extended equilibrium calculations to multicompartment systems, explicitly accounting for differences between plasma and whole blood. This work illustrates why moving between system boundaries requires additional relationships rather than merely inserting the same numbers into a plasma equation. 11

Equilibrium models also do not fully specify rates. Ventilation can change PCO₂ quickly, whereas renal balance, organic-acid metabolism, protein redistribution, and tissue exchange evolve differently. Two successive samples may represent different stages of a moving system. Their difference is not necessarily the response to one isolated intervention.

Calculated residuals accumulate uncertainty from their components. Because sodium and chloride are large concentrations whose difference is much smaller, modest assay biases can materially change SIDa. In the stated formula, a 2-mmol/L positive sodium bias increases SIDa and SIG by 2 mEq/L if all other inputs remain fixed. That is arithmetic propagation, not evidence that the patient gained a new cation.

Albumin and phosphate charge estimates contribute additional modeling uncertainty. Abnormal proteins, binding, unusual ionic strength, or pH outside the approximation's useful range can reduce fidelity. Different sodium measurement methods or calcium fractions can create apparent differences between serial calculations even without equivalent biological change.

Use contemporaneous, compatible measurements and state the adopted formula. Retaining many decimal places cannot overcome uncertain inputs or omitted chemistry. Trends become more interpretable when the analytical method and system boundary remain consistent, while a surprising residual should prompt reconciliation with direct measurements and plausible physiological transfers.

53.12 Compare Explanations Against the Full Sample and Its Trajectory

Consider a final illustrative sample with bicarbonate 18 mmol/L and PCO₂ 40 mmHg, using the same rounded 37°C constants. Dissolved CO₂ is 1.2 mmol/L, so pH is 6.10 + log₁₀(18/1.2) = 7.276, approximately 7.28. The bicarbonate view identifies a reduced carbonate ratio; the unchanged selected PCO₂ is an example condition, not an expected compensation rule.

Now specify sodium 140, potassium 4, ionized calcium 1.2, ionized magnesium 0.6, chloride 114, and lactate 1 mmol/L, with albumin 40 g/L and phosphate 1 mmol/L. SIDa is 140 + 4 + 2.4 + 1.2 − 114 − 1 = 32.6 mEq/L.

At calculated pH 7.276, modeled albumin charge is approximately 10.56 mEq/L and phosphate charge 1.78 mEq/L. SIDe is 18 + 10.56 + 1.78 ≈ 30.34 mEq/L, leaving SIG about 2.26 mEq/L. The strong-ion description makes the chloride-rich measured composition explicit while preserving a residual for omitted charge and model effects.

These calculations agree because they use compatible equilibria and balances. They do not establish why chloride rose, whether the disturbance is acute, or which intervention would improve the patient. Those questions require history, serial measurements, external balance, metabolism, and respiratory and renal assessment.

Fencl and colleagues demonstrated that net gas and base-excess findings could conceal opposing compositional abnormalities in ICU samples. Later, Dubin and colleagues compared approaches in a large observational ICU cohort and found that adding albumin-corrected anion-gap information substantially narrowed diagnostic differences. Neither study randomized patients to an interpretation framework or demonstrated that choosing one improves outcomes. 7, 12

A useful analysis therefore explains the measured pH and PCO₂, checks the bicarbonate method, and accounts for strong ions, weak acids, and unexplained charge. The subsequent disorder-specific chapters develop causes, compensatory responses, and intervention evidence. The framework should make those processes easier to see without substituting terminology for physiological explanation.

References

1. Wooten EW. Analytic calculation of physiological acid-base parameters in plasma. Journal of Applied Physiology. 1999;86:326–334. doi:10.1152/jappl.1999.86.1.326

2. Story DA, Poustie S, Bellomo R. Comparison of three methods to estimate plasma bicarbonate in critically ill patients: Henderson-Hasselbalch, enzymatic, and strong-ion-gap. Anaesthesia and Intensive Care. 2001;29:585–590. doi:10.1177/0310057X0102900603

3. Stewart PA. Independent and dependent variables of acid-base control. Respiration Physiology. 1978;33:9–26. doi:10.1016/0034-5687(78)90079-8

4. Figge J, Rossing TH, Fencl V. The role of serum proteins in acid-base equilibria. Journal of Laboratory and Clinical Medicine. 1991;117:453–467. PMID:2045713

5. Figge J, Mydosh T, Fencl V. Serum proteins and acid-base equilibria: a follow-up. Journal of Laboratory and Clinical Medicine. 1992;120:713–719. PMID:1431499

6. Rossing TH, Maffeo N, Fencl V. Acid-base effects of altering plasma protein concentration in human blood in vitro. Journal of Applied Physiology. 1986;61:2260–2265. doi:10.1152/jappl.1986.61.6.2260

7. Fencl V, Jabor A, Kazda A, Figge J. Diagnosis of metabolic acid-base disturbances in critically ill patients. American Journal of Respiratory and Critical Care Medicine. 2000;162:2246–2251. doi:10.1164/ajrccm.162.6.9904099

8. Siggaard-Andersen O. The Van Slyke equation. Scandinavian Journal of Clinical and Laboratory Investigation Supplement. 1977;146:15–20. doi:10.3109/00365517709098927

9. Morgan TJ, Clark C, Endre ZH. Accuracy of base excess—an in vitro evaluation of the Van Slyke equation. Critical Care Medicine. 2000;28:2932–2936. doi:10.1097/00003246-200008000-00041

10. Scheingraber S, Rehm M, Sehmisch C, Finsterer U. Rapid saline infusion produces hyperchloremic acidosis in patients undergoing gynecologic surgery. Anesthesiology. 1999;90:1265–1270. doi:10.1097/00000542-199905000-00007

11. Wooten EW. Calculation of physiological acid-base parameters in multicompartment systems with application to human blood. Journal of Applied Physiology. 2003;95:2333–2344. doi:10.1152/japplphysiol.00560.2003

12. Dubin A, Menises MM, Masevicius FD, et al. Comparison of three different methods of evaluation of metabolic acid-base disorders. Critical Care Medicine. 2007;35:1264–1270. doi:10.1097/01.CCM.0000259536.11943.90