The Calculus of Emergency Medicine


Have you ever dealt with a patient or specialist who is confused about the scope of emergency medicine? Here’s a quasi-mathematical way to think about it.  

As a relative newcomer to the pantheon of medicine, emergency medicine sometimes requires more of an explanation than other specialties. Patients ask emergency medicine residents, “So what type of doctor do you want to become?” Older attending physicians of other services often seem unfamiliar with the full scope of emergency medicine practice, which now may be quite different than during their years of training. In all fairness, defining emergency medicine does require a different approach than defining other realms of medicine.

Unlike most medical and surgical specialties, emergency medicine does not address a specific organ system or disease process, as in cardiology or oncology. Nor does it limit itself to any specific diagnostic or treatment modality, as in radiology or anesthesiology. We also work in a liminal space between inpatient and outpatient medicine. And while it’s a good start to simply state that emergency medicine deals with emergent pathology, treating truly emergent pathology constitutes a small percentage of how our time is spent in a modern emergency department.

I have developed my own approach to explaining to medical students how to conceptualize the practice of emergency medicine. I explain it using the concept of derivatives. Students are closer to the study of calculus than I am, so I lay it out in the simplest terms possible, without using any numbers, to avoid embarrassing myself. It goes something like this:

Imagine that we can predict the full course of a disease process and model it as a mathematical function. The x-axis is time, the y-axis is health. As an example, consider the hypothetical course of a patient with coronary artery disease presented in Graph 1 below. At the start, the gradual process of atherosclerosis advances slowly and persistently. Then one day, with little warning, a plaque ruptures and causes an acute myocardial infarction. The patient promptly seeks medical care and the coronary artery is appropriately reopened, reperfusing the myocardium and stabilizing the patient. As a result of the MI, the patient develops congestive heart failure. For years they intermittently suffer and recover from CHF exacerbations. Throughout this time, their ejection fraction gradually deteriorates and they eventually die.

Now let’s look at this model in terms of derivatives. The first derivative (the slope at any given point) of the mathematical function of a patient’s disease course is the rate at which a patient is moving from wellness to death. At Point A in our example patient, with the slow progression of atherosclerosis, the patient moves towards death at a constant but slow rate. At Point C, in the midst of a myocardial infarction, the patient moves towards death at constant but fast rate. Patients moving at a high rate are “sick,” patients moving at a low rate are “not sick”. When emergency physicians identify a sick patient, moving towards death at a high and constant rate, we are tasked with decreasing that rate–we resuscitate the patient. When we identify that our patient is at point C on their illness curve, our role is to bend that curve to a lower rate.

But resuscitation is a small part of our job. The rest of our job can be better understood with the concept of a second derivative (the rate of change of the slope of the function at a given point). On a patient’s illness course, the second derivative is essentially the rate at which a patient is becoming more or less sick. It represents a change in the trajectory of their disease course. As emergency physicians, we search for points on a patient’s illness trajectory when the value of the second derivative is high. At these inflection points, a patient may not yet be moving rapidly towards death, but the rate which they are moving towards death is increasing; they are becoming sick. On our example chart, this is point B. The patient’s ruptured plaque is growing and be-coming a major ischemic event, and while the rate at which the patient is moving toward death may be currently slow, that rate is increasing.

Most of our time as emergency physicians is spent estimating this rate of change at which a patient is becoming more or less sick. This estimation drives the other key roles of our specialty: identifying dangerous conditions and determining appropriate disposition. It requires us to obsess over risk stratification and prognosis even more than other specialties, trying to determine where a patient lies on their illness curve. Is this patient with chest pain at Point A, moving to-wards death at a slow and constant rate, or point B, at a currently slow but rapidly increasing rate? Will this patient’s pneumonia improve or worsen if I treat them as an outpatient? Did I just appropriately treat this patient’s migraine or did I mask the sentinel bleed of their sub-arachnoid hemorrhage? Answering these questions is no easy task, in fact often harder than decreasing the rate of change when we resuscitate a patient. This role as skilled decision-makers in patient disposition places us in a critical position in the continuum of care.

Usually, at this point in my overly pedantic explanation, the medical student’s eyes have glazed over. But if they have continued to pay attention, I think they progress beyond a glamorized version of emergency medicine towards a more realistic understanding of our very difficult specialty.


Dr. McVane is an Emergency Medicine Physician at Elmhurst Hospital in New York City


  1. Interesting article Ben. I had to dig deep to remember my calculus, but It’s a surprisingly effective way to explain your point.

  2. \sum_{i=1}^{n} i^3 = 1^3 + 2^3 + 3^3 + … + n^3 = { n^2(n+1)^2 \over 4 } } $ is a much simpler mathematical description of Emergency Medicine, where i is the diagnosis, n the number of patients, $ is the amount of money you take home at the end of the shift. Over 4 is the number of lawyers you have to circumvent to take $ home.

  3. This is simply beautiful in its concept – and I can just see the medical student’s eyes glaze – but that happens with any other explanation, anyway!

  4. I think William Osler said it best. “…,,,……….. it is the science of uncertainty and the art of probability “. He was talking about medicine in general but I think it depicts the ER better.

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