If you’re like me, you learned and then promptly forgot the Henderson Hasselbalch equation (HH eq) in medical school . After all, in clinical rotations it was never invoked, and our patients seemed to have fared well without it. So why bring up the topic now?
Medicine is changing. The ubiquitous nature of computing allows a level of sophistication exponentially greater than before. To a large extent we’re freed from much of the onerous work of rote memorization. In the ideal, that should free us to be more thoughtful about the way we approach our work and to have a deeper understanding of health and disease. Going forward, medicine will become increasingly computational. With that in mind, I’ll make three points about the HH eq.
3 Important Points About the Henderson Hasselbalch Equation
- The HH eq is a classic example of science at its best. There aren’t that many situations in medicine where understanding is achieved in a mathematically elegant and robust form. It should be astonishing to us that pH, pCO₂, and bicarbonate concentration are so intricately related and that we can perfectly quantitate that relationship.
- We are capable of both understanding and manipulating the HH Eq. The logarithms are easy if you know just a few simple declarations. Log(A × B) is equal to logA + logB, and similarly log(A ÷ B) is the same as logA – logB. Finally, if logA = 4, then A = 10⁴ . That’s it. That’s all you need to know to derive the equation!
- We can and should use it in the emergency department. One example of its use is for rapid estimation of pH and pCO₂ in patients with metabolic acidosis when the only data point available is a serum bicarbonate from a Chem 7. This requires the assumption of appropriate respiratory compensation.
Derivation of the Henderson Hasselbalch Equation
All acids in solution dissociate in a mathematically quantifiable way. Kₐ is the acid dissociation constant for that relationship.
In medicine it is the convention to refer to acidity using pH instead of [H⁺], so the logarithm of both sides of the equation is taken.
Remember that log(A × B) is equal to logA + logB, so we can rewrite the equation as follows:
Since we want the equation to solve for pH, we’ll move the log[H⁺] to the left hand side of the equation and the log Kₐ to the right.
By definition, pH means the negative log (base 10) of the hydrogen ion concentration. Similarly, pKₐ means the negative log of Kₐ, so we can rewrite the equation as follows:
We will be using the equation to scrutinize the bicarbonate buffer system, so carbonic acid and bicarbonate are substituted for HA and A⁻ respectively.
Now we get to the really interesting part of the derivation. Carbon dioxide in solution forms carbonic acid, and there is a solubility constant described by Henry’s law that quantifies the relationship.
The solubility constant is 0.03, so we can substitute that into the equation to reach our final state. We’ll also substitute in the constant for pKₐ which is 6.1:
Figure 8. The Henderson Hasselbalch Equation as used in medicine
Now we’re ready to take this show on the road!
The Equation In Clinical Practice
Consider a patient with a bicarbonate value of 12 mEq/L on a chemistry panel. A bicarbonate value of 24 is considered normal, with a range of 22 – 26. In most laboratories an autoanalyzer is used that measures total CO₂ content as an approximation of bicarbonate. This estimation will exceed the actual bicarbonate value by 1-2 mEq/L , so we may need to adjust for that. In this example we will use the actual value.
The low bicarbonate value indicates that an acidosis is present.
From our history and physical examination we will have a good sense whether our patient has some degree of respiratory compromise resulting in a respiratory acidosis. If it appears the patient has no respiratory compromise — and this could be confirmed by end tidal CO₂ monitoring — then we conclude there is a metabolic acidosis.
To estimate pH, we need one more value, the pCO₂. Assuming normal respiratory compensation, the pCO₂ is estimated using Winter’s equation , as follows:
Figure 9. Winter’s equation
So our patient’s pCO₂ is 12 × 1.5 + 8 ± 2, which is 26 ± 2.
Entering those values into the HH eq gives a pH estimate of 7.27 (7.26 – 7.31). So what we’ve just accomplished is rapid estimation of pH starting with a bicarb value from a Chem 7. Or from a mathematical perspective, we used 2 equations to solve a three variable problem.
These estimations can be prepared in advance to create a quick reference (Table 1).
Table 1. Blood gas values derived from Winter’s equation and the Henderson Hasselbalch Equation.
The Decimal Digits Rule (aka “last 2 digits” rule)
As an alternative to the Winter’s equation, there’s another great way to estimate pCO₂ in metabolic acidosis using the “decimal digits” rule, sometimes referred to as the “last 2 digits” rule . The rule states that in patients with metabolic acidosis and appropriate respiratory compensation, the pCO₂ equals the decimal digits of the pH. The best way to understand this is with an example.
Imagine you are caring for a patient with DKA and you obtain a blood gas to estimate the severity of the illness. The results come back pH 7.27, pCO₂ 27, and bicarb 12.0. (The pO₂ is not relevant for the acid base analysis.) It’s helpful to know that the pH is 7.27, but equally important is whether there is adequate respiratory compensation for the acidosis. The “decimal digits” rule says that if the pH is 7.27, then the predicted pCO₂ in a patient with adequate respiratory compensation is 27. If your measured pCO₂ is close to predicted, then compensation is adequate. A pCO₂ much higher indicates inadequate respiratory compensation. This could be the case if your patient is extremely ill with respiratory muscle fatigue. A pCO₂ much lower indicates a primary respiratory alkalosis. Your patient is hyperventilating, and you should ask yourself why.
For the purposes of our work as emergency physicians, I’m arguing that we should work backwards using the “decimal digits” rule. Let’s start from scratch to create a lookup table that gives pH, pCO₂, and bicarb values that are compatible with the rule. How is that done? You guessed it. Using the Henderson Hasselbalch equation.
Start with pH 7.37 and a pCO₂ of 37 and calculate the predicted bicarb using HH eq. Then do the same thing for pH 7.36 and a pCO₂ of 36. Continue until you’ve reached a pCO₂ of 12, which is about the lower limit of respiratory compensation.
This technique works best if you actually use a pH with 3 decimal digits (i.e, 7.350 instead of 7.35) and a pCO₂ with 1 decimal digit (35.0 instead of 35). That way you can get bicarb results with three significant figures. OK, too much information. Take a look at the resulting table:
Table 2. Blood gas values derived from the “decimal digits” rule and the Henderson Hasselbalch Equation.
How do we use this table? The idea here is that when a Chem 7 result comes back with a significantly low bicarb, we know there is a metabolic acidosis. We use the table to predict pH and pCO₂ under the assumption that our patient has adequate respiratory compensation for the acidosis. So if my patient’s bicarb is 10, I expect to see a pCO₂ of 24 and a pH of 7.24.
25 year old man, brought in for evaluation of a suspected suicide attempt by medication overdose. Chem 7 shows a bicarb of 13 and an anion gap of 20. If he has a pure metabolic acidosis with an appropriate degree of respiratory compensation, what would you expect his pCO₂ and pH to be?
Starting with the bicarb of 13, we can estimate the pCO₂ using Winter’s equation, so the expected pCO₂ is 13 × 1.5 + 8 ± 2 = 27.5 ± 2. Plugging those numbers into the Henderson Hasselbalch equation gives an expected pH of 7.28 (7.27 – 7.33). We would get the same results with the table generated by the “Decimal Digits” rule.
Here’s his actual blood gas: pH 7.50, pCO₂ 17. The pCO₂ is much lower than predicted indicating a respiratory alkalosis superimposed on the high anion gap metabolic acidosis. This is a classic finding in early aspirin overdose.
19 year old man found down unconscious, bicarb of 5, anion gap of 35. What are his expected pH and pCO₂?
Using the lookup table derived from the “Decimal Digits” rule, we expect a pH of 7.15 and a pCO₂ of 15. The actual values were pH of 7.00 and pCO₂ of 20. As you can see, the higher pCO₂ significantly lowers the pH. Any further compromise in his respiratory drive will push the pH into the dangerous sub 7 realm.
To App or Not to App?
Should I use a look up table in a Paucis Verbis card, a mobile app, a web app, or something built into my EHR? How much time do you have? There probably are many situations in our practice where we just want an answer. That may seem shallow, but we don’t want to see or know the derivation if there’s not enough time or if we’re lacking the mental stamina at that particular moment. No problem. I’m like that. A lot. Use the tool that’s available and fast. But we can and should have the opportunity to take a deep dive from time to time, and that’s where a well designed app can make a difference if it has the ability to educate as well as spit out a result. Showing formulas and providing references is a minimum requirement. Particularly valuable is the use of animation. For now, be aware that you can accomplish the same end goal through a variety of means.
Take Home Points
If you have a patient with a significantly low bicarb on a Chem 7, use these quick look-up tables to rapidly estimate pH and pCO₂ based on that single bicarbonate value. Alternatively, run the calculations from scratch, which is pretty awesome when you’re teaching residents and medical students at the bedside. Or use an app. Regardless of what method you choose, if you go on to get a blood gas, compare actual to predicted pCO₂.
If the pCO₂ agrees with predicted values, your patient has an appropriate respiratory compensation attenuating the acidosis. If the pCO₂ is too high, your patient also has a primary respiratory acidosis. The latter is a potentially dangerous scenario that may have an impact on how you manage the airway. On the other hand, if the pCO₂ is lower than expected, your patient has a primary respiratory alkalosis in addition to a metabolic acidosis. Think about aspirin overdose or other reasons why your patient might be hyperventilating.
And always remember to pay tribute to Henderson and Hasselbalch every time you interpret a blood gas.
- Story DA. Bench-to-bedside review: a brief history of clinical acid-base. Crit Care. 2004;8(4):253-8. PMID: 15312207
- Centor RM. Serum Total Carbon Dioxide. In: Walker HK, Hall WD. Clinical methods, the history, physical, and laboratory examinations. Butterworth-Heinemann; 1990. Accessed September 21, 2014.
- Albert MS, Dell RB, Winters RW. Quantitative displacement of acid-base equilibrium in metabolic acidosis. Ann Intern Med. 1967;66(2):312-22. PMID: 6016545
- Fulop M. A guide for predicting arterial CO2 tension in metabolic acidosis. Am J Nephrol. 1997;17(5):421-4. PMID: 9382159
Disclosures: Dr. Ruiz is the creator of the “Acid Base” app for the iPhone and “Likelihood” app for the iPad.
ALiEM Copyedit 1
October 15, 2014
Editor reviewing for educational merit:
I really liked this piece – it breaks down a complicated concept, but most notably, brings it back to the clinical setting.
I think for me one of the biggest jumps was in the last bit – from the “last 2 digits” rule to the Take Home Points. I think you have to highlight the concept a bit more… and highlight the predicted vs. actual measured bicarb. Linking this to the idea of the delta-gap (http://fitsweb.uchc.edu/student/selectives/TimurGraham/Delta_Ratio.html, http://www.ncbi.nlm.nih.gov/pubmed/2240729) or double gap… These might be important because the learners are often learning all these concepts in tandem.
Alternatively, you could write up these other two concepts too… but I think they need to addressed in relation to this piece in order for learners to get the most out of it…
Love the equation diagrams – so clear!
[AUTHOR RESPONSE: I revised the last 2 digits rule to try to make sense out of it. As for the delta gap… if this piece is well received I would like to actually have a three part series, the second of which would be “acid base without the blood gas” which would include the delta gap, double gap, and some other cool stuff. Part three would be a gentle introduction to the physicochemical (Stewart) approach to acid base.
I wanted to add a comment about the intent of the post. It’s really a bit schizophrenic in that it’s a combination of a review of the HH eq, but it is presenting my own concept of quick look up tables derived using the HH eq to achieve rapid prediction of pH and pCO2. So it’s a mashup of a review and a trick of the trade piece. As far as I know, I’m the only one who applies HH eq in this manner.]
Teresa Chan, MD, ALiEM Associate Editor; Assistant Professor, Division of Emergency Medicine, McMaster University
ALiEM Copyedit 2
October 19, 2014
Hi Frank – nice post! Made the esoteric very clear. Some notes:
- Can you make descriptive labels for Tables 1 and 2.
- At the risk of sounding redundant, perhaps you can end with a short clinical scenario/ case to illustrate your take home point? I think readers who are pretty new to this concept would appreciate it.
- Agree with Teresa about the slight leap in concepts moving from Winters to the “last 2 digits” concept. Another alternative is to bridge this gap through your take-home example.
Michelle Lin, MD, ALiEM Editor in Chief; Associate Professor of Emergency Medicine, UC San Francisco
ALiEM Copyedit 3
October 21, 2014
Good morning Dr. Ruiz–really great post. As a learner, this was easy to follow and provided relevant depth to a topic that is often ineffectively covered.
Just a few additional copyedits I made, pending your review:
- Table 2 was re-aligned for better centering.
- Case studies have been placed under droppable menus.
- Take home points were bolded and separated for easy recognition and emphasis.
Overall great post, with a very clear derivation and impact.
Scott Kobner, Medical Student @ NYU, ALiEM-EMRA Social Media and Digital Scholarship Fellow, Founder, EdintheED.com
October 20, 2014
Good morning Frank–
Thank you for this post. The easy to understand derivation of the HH eq was a great thing for me to review and is well done.
Under “The Equation in Clinical Practice”, I might add a reference range for normal bicarb values after 12 mEq/L as this blog attracts learners of all levels.
I really like the last 2 digits rule but wish it had a different name, like FTDAD– (first two digits after decimal, obvi) because the technique works best if you use pH with three decimal digits and the last two digits in that setting do not equal the predicted pCO2. Is there a way to explain this eloquently?
I agree with bringing it all home with a take home example that makes the leaps from concept to concept. An example where being able to calculate pH and pCO2 from a bicarb level and comparing it to VBG really gives you a clue as to how to manage a critically ill patient quickly and appropriately.
I love that you explain how all of this works so that we aren’t just mindlessly typing numbers into an app and not really understanding why/how things work. However, going back to the charts might not be as ‘quick reference’ as we’d like. Do you have any calculators or alternatives that you recommend once we get the underlying concepts down?
As an aside, I have recently joined a community practice and can see how it could take a lot of effort it can take to keep up with the latest and greatest when you don’t have residents bring in new info to shifts and students challenging your assumptions. I think the way you have really embraced technology, kept up with the changes in medicine, and are are now imparting your wisdom to help others understand some of the more difficult topics in medicine is pretty awesome and inspiring.
[AUTHOR RESPONSE: Thank you so much for your kind words! I’m like Mark Twain. I could live for a month on a single complement. Your suggestions were great. I added the bicarb normal range. The “last 2 digits” rule is not a legitimate name – everyone calls it something different. So I changed it to the decimal digits rule. Your point about looking things up in a table or punching data into a black box and getting an answer that lacks insight is well taken. I added a section about that: To app or not to app. The drawback is that I am getting self conscious about writing about apps. See what you think.]
Meghan Schott, MD, Clinical Fellow, Emergency Medicine, UCSF School of Medicine
Expert Peer Review
October 30, 2014
To go along with it, here is the reference list:
- Albert MS, Dell RB, Winters RW. Quantitative Displacement of Acid-Base Equilibrium in Metabolic Acidosis. Ann Intern Med 1967;66(2):312.
- Story DA. Bench-to-bedside review: A brief history of clinical acid-base. Crit Care 2004;8(4):253.
- Kishen R et al. Facing acid-base disorders in the third milleniu – the Stewart approach revisited. Int J Nephro Renovac Dis 2014;7:209.
and the formula:
[AUTHOR RESPONSE: Listening to Dr Weingart’s comments, I was reminded of a patient I took care of who told me that she was allergic to “chloride”. I resisted the temptation to tell her that chloride ions were present in every cell of her body. I cannot, however, resist telling Dr Weingart that the Henderson Hasselbalch equation is an essential component of the Stewart physicochemical approach of acid base analysis, which Dr Weingart espouses.
Let’s take a look:
Peter Stewart took the Henderson Hasselbalch equation and combined it with 5 other equations to derive his theory of acid base. If you accept Stewart’s approach, you must accept the scientific validity and clinical importance of the Henderson Hasselbalch equation. The bottom line is that the Henderson Hasselbalch equation is a fundamental component of both classical and modern (physicochemical) acid base analysis.
The pragmatic ED physician:
In the emergency department, as opposed to Dr Weingart’s ICU, we are on the front lines. We are the first to see the patient, and we initially have a limited data set. Most of our patients with acid base disorders are not sick enough to require hospitalization, and most will not require blood gas analysis. To do a precise physicochemical analysis requires, at a minimum, basic electrolytes, a serum albumin, and a blood gas. But a preliminary acid base assessment is possible without a blood gas analysis, and if you choose to do so, you can incorporate the Henderson Hasselbalch equation into that preliminary assessment. It’s fast, it’s easy, and it’s practical.
The best of both worlds:
It looks like there’s a real need for physicians who care about acid base to have a debate on what we should be teaching our students, residents, and colleagues. We need to resolve the conflict between classical and physicochemical approaches in a way that benefits our specialty. My perspective is that the classical approach is a subset of the physicochemical approach. This can be shown mathematically. For most purposes in the ED, the classical approach yields a fast, actionable assessment. The physicochemical approach refines one's initial assessment and is essential in many situations, for example, where there are derangements of albumin or with infusions of low SID fluids. I think the intellectual ED doc will want the best of both worlds.]
Scott Weingart, MD, ED Intensivist at Stony Brook University, Host of the EMCrit Podcast, Twitter: @emcrit