If you find yourself heading down a rabbit hole pursuing a diagnosis, consider using a Markov chain to get back out. No, it’s not a physical chain. It’s a metaphorical chain.
In mathematics, statistics, and research, a Markov chain is a series of branching events where the options of what comes next at each branch point are not limited by what happened at a previous branch point. That is, each step is not constrained by the results of prior steps; you make a decision based on the present state, allowing past results to inform, but not limit, current options.
For example, if a patient has an equivocal CT result, you might either repeat the CT for better images or proceed to an MRI. If radiology denies a repeat CT scan (to minimize dye load or radiation), then the options are limited by having done the prior CT, and it is not a Markov chain. In contrast, if you have the option to repeat the CT, then the options are not limited, and this qualifies as a Markov chain.
A Markov chain has flexibility and opportunities for a “re-think” or “re-do.”
A non-Markov chain is similar to Robert Frost’s observation in “The Road Not Traveled:”… knowing how the way leads on to way, I doubted if I should ever come back. There are no “re-dos.”
This phenomenon of re-traceable or non-retraceable decision points is familiar to any practitioner. It was one consulting physician’s belief she was trapped in a non-Markov chain that led our ED patient to a craniotomy, despite no intracranial pathology.
This patient was referred to our ED by a neurologist for sudden onset of a holocephalic headache without other neurologic symptoms that began within 6 hours of his ED arrival. The neurologist wanted to rule out a subarachnoid hemorrhage (SAH) and requested CT angiography of the brain and neck, which we ordered.
The radiologist reported no SAH or aneurysms; there was one hyperdensity that was most likely a volume-averaging artifact.
Because the sensitivity, specificity, and positive and negative predictive values for subarachnoid hemorrhage of (even) a non-contrast brain CT performed within 6 hours of headache onset is 100 percent, and angiograms are excellent (98.5 percent sensitivity, similar to lumbar punctures) for identifying (even-ruptured) aneurysms, there was essentially no possibility he’d had a SAH. We controlled the patient’s pain and discharged him.
The patient was an employee of our hospital, and, a few months later, I encountered him and noticed a craniotomy scar. He informed me how the neurologist, concerned that the radiology reading was not an artifact but, indeed, represented a SAH, referred him to neurosurgery, who performed a craniotomy looking for aneurysmal structures. Finding none, no intervention was performed.
The neurologist was caught (in her mind) in a non-Markov chain trap, inflexible to the possibility of re-ordering the CT angiogram (i.e., going backward). Her anchoring to the SAH diagnosis trapped her in this non-Markov chain, from which the only possible option was a neurosurgery referral.
We clinicians face similar decisions all the time of how best to pursue a diagnosis. I was recently presented a case by a resident of a patient with BPH symptoms and cystitis. The resident suggested we send a PSA level to evaluate for prostate cancer. Since both BPH and cystitis cause elevated PSA levels, the patient’s result would likely have been high, which would have led to a urology referral, which would (very likely) have led to a transrectal prostate biopsy, which would have potentially led to urosepsis (~3 percent incidence). We might have sent the patient down a non-Markov chain in which he got urosepsis. Instead, we decided it was better not to send the PSA until the cystitis resolved. And, we discussed that, if a PSA sent during an episode of cystitis is elevated, one can repeat it after cystitis resolves (that is, it’s fine to create a Markov chain, where the prior positive PSA result does not limit the option of repeating it).
In many cases (as in the craniotomy example), the feeling of being trapped in a diagnostic inertia or of not being able to repeat a study (non-Markov situation) is in our minds. We can re-think our pre-test probabilities, re-order our differential diagnoses, and convert a non-Markov chain (options-limited) to a Markov chain (options not limited) by repeating an exam.
The possibility of entering a non-Markov chain of events is one of the many reasons it’s important to have a smart, caring physician, with time to prioritize the differential diagnosis in terms of likelihood and severity, and who is familiar with test characteristics and up-to-date on recent literature advances.
In the “many-worlds (multiverse) interpretation” of quantum mechanics, every time a decision or observation occurs, at the same time, its alternative(s) branch off in different universes and also occur. That’s definitely a Markov chain (unlimited options) situation! I like to think that, somewhere, in a different universe, our headache patient is doing fine without his craniotomy.
The author is an anonymous physician.