Mathematical Medicine: Pulse Oximetry
Pulse oximetry (PulseOx) is fundamental to telemetry in modern hospitals. Having a real-time reading of SpO2 (oxygen saturation) allows physicians to get a rough understanding of respiratory function, as well as heart rate. As a bonus, PulseOx is quite cheap (you can buy one on Amazon for around $20) and can be done through a non-invasive clip-on device on either the fingertip or ear lobe.
PulseOx works by shining visible and infrared light through tissue and detecting the amounts of light that make it all the way through the tissue to the detector. This technique is known as absorption spectroscopy, and you may have very well done similar experiments in your college introductory chemistry classes. If you haven’t, here’s a quick refresher:
Absorption spectroscopy is the process by light is shone through a material, and the spectrum of absorbed light is calculated by subtracting a baseline calibration. By using laws of absorption like the Beer-Lambert Law, the concentration of material can be calculated.
In the case of PulseOx, we’re calculating the concentration of oxidized-hemoglobin (HbO2) and reduced hemoglobin (Hb). HbO2 and Hb have slightly different absorption spectra, allowing PulseOx machines to calculate relative concentration by measuring both spectra against a baseline. Tissue itself doesn’t interfere too much with the hemoglobin absorption spectra, as tissue naturally absorbs wavelengths other than red, which is why a light shone through a fingertip often appears red (wavelengths except red are being absorbed).
Thus the setup of a PulseOx is as follows: A light source emitting primarily red to infrared light, a detector to measure the spectra of light absorbed, and a microprocessor to take the signal data and convert it into meaningful values of SpO2. PulseOx is quite simple in principle, but creating a cheap, portable, accurate setup is much harder in practice.
The Detector
First, let’s take a look at how such a detector is created. In general, visible light is first shone onto a diffraction grating, which “splits” the visible light into separate beams of varying wavelength. The mechanisms and mathematics behind diffraction gratings are fascinating, and I highly recommend looking into the physics behind them, as university introductory physics (which most doctors have at some point taken) does not do it justice. Next, the spectra of light are reflected by a mirror through the sample and towards a detector array, similar to what is found in a digital camera. There are many different arrangements of lights, mirrors, gratings, and sensors that accomplish this task with different pros and cons, but below is shown the most simple diagram I could find:
The Beer-Lambert Law
The Beer-Lambert law is the driving force behind the calculations of SpO2. Let’s take a look at what it says about absorbance. We have:
Where the absorbance A is the sum of the individual absorbances Ai of the materials in the sample. This is equivalent to a molar absorptivity constant multiplied by the concentration, c(z), integrated along a path length. The derivation comes from the following first-order differential equation, which is trivial to solve:
Where µ(z) is related to how easily light penetrates the material, and Φ is the flux of light through the material.
For those who have studied first-order ODE’s, it’s clear that this leads to a logarithmic relation between absorption and concentration.
Empirically, the Beer-Lampert law can’t be directly applied to PulseOx, since other complicated effects (molecule-molecule interaction, etc.) skews the data a little bit, so instead empirical data on HbO2 and Hb concentration is gathered beforehand, and a lookup table (a few of the readers here may remember using lookup tables back before calculators existed, but I sure don’t) is used to compute the concentrations.
Wrap-Up
With the setup in place, the PulseOx emits a short pulse of red light, followed by a short pulse of infrared light, followed by no pulse of light, and calculates Hb concentration and HbO2 concentration, using the baseline (gathered from the period of no pulse). This is done almost 30 times a second! The pulse is simply measured by observing the periodic changes in absorbance intensity caused by the inflow of blood. In practice, a lot of cool signal processing goes on behind the scenes, which is much too complicated to delve into in this short Medium piece. To the readers more signal-processing-inclined, I strongly recommend reading this article, of which much of this piece is based off.
