Tensors are a conceptual framework
Medicine could really use some higher-dimensional thinking…
Updated: 2022-01-17
I just had the best day. Today was the first official day of my clinical clerkships. I’m starting with internal medicine at Grady Memorial Hospital.
I had a relatively tough time with the preclinical curriculum. This is when you sit in a classroom and learn from the firehose of medical knowledge. It was tough and frustrating because you know there’s a pattern to what is being presented. That pattern is the latent system that we call ‘physiology’. We’re still far from really understanding physiology but we see data that is measured from physiologic processes and this data helps us estimate those processes. Those estimates are all thrown at us and we’re expected to learn each one in isolation and/or with heuristics (like initialisms/acronyms).
I reiterate: it was very frustrating.
Today was a cool day, however. Today I really hit my stride in trying to frame the clinical process that will be my life for at least one year as a fundamentally engineering process. This is much easier today than it was 5 years ago, when my classmates went through the M3/M4 curriculum.
It’s easier because concepts like pre-test probability, likelihood, problem solving, differential diagnoses, etc. are all embedded in the medical milieu. Phrased another way: the clinicians have finally embraced Bayesian Inference. They don’t use the math yet but, given the advances in machine learning, they probably won’t have to directly understand the nitty-gritty internals of $P(x | M_{i}) \cdot P(M_{i})$ as an integral. They’ll just use tools that others have developed and apply the rule without having to break things out. It’s actually beautiful and I’m very excited at my training timeline given these massive changes in medical+engineering worlds. I’ll talk about this in more detail later (and maybe even try to do a bigger project around this).
But what I wanted to just comment on today was: I finally grasp the basics of tensor calculus. The reason I do is weird: I was trying to frame physiology, history of present illness, SOAP notes, and clinical assessment as explicit operators and/or a control theoretic chain of processes. In doing so, I had to wrestle with the fact that the space of lab measurements is not fully accessible by the body: real-world measurements exist on a manifold of that space.
I’m a bit embarrased it took me this long, but I’m also very happy. I finally realized the tensor’s beauty in splitting out the process of the covector (functional) and the vector in order to address something that is abstract. In other words, I finally realized that the tensor in $g_{\mu \nu} = \lambda^\rho_\mu \cdot \lambda^\sigma_\nu \cdot g_{\rho \sigma}$ is actually the $g$ and not directly the $x$ and the $y$. I kept watching youtube videos that explicitly said as much, and even wrestled directly with the metric tensor to try to understand it. But it took a combination of framing it as a clinical problem (and, importantly, a few months of not thinking about it) for me to really get it.
It may also have helped that I spent a few months grappling with things like differential forms, Lie algebras, and more generalized concepts, but it’s hard to ascribe any particular understanding to what ended up being a frenetic shotgun approach to trying to understand differential geometry…
Anyways, I’m very excited for my clinical curriculum; moreso now knowing that I’ll actually be improving my math understanding as well!