math formatting

pulseox.md

 ## Pulse Oximetry
 Pulse oximetry devices use several LEDs to measure pulse rate and blood oxygen content. The LEDs are tuned to specific wavelengths corresponding to the absorbance bands of oxygenated and reduced hemoglobin; by cycling through the LEDs rapidly the device compensates for skin differences and ambient light, returning saturation and pulse rate.
 
-### Background
+### References
 - overview of pulse oximetry physics and engineering challenges from 1989: Tremper, Kevin K., and Steven J. Barker. "Pulse oximetry." Anesthesiology: The Journal of the American Society of Anesthesiologists 70.1 (1989): 98-108.
     - engineering challenges identified
         - LED center wavelength consistency
@@ -15,6 +15,7 @@ Pulse oximetry devices use several LEDs to measure pulse rate and blood oxygen c
 - earlier overview: Yelderman, Mark, and William New. "Evaluation of pulse oximetry." Anesthesiology: The Journal of the American Society of Anesthesiologists 59.4 (1983): 349-351.
 - changing LED wavelengths with temp: ~0.1 nm/C: Reynolds, K. J., et al. "Temperature dependence of LED and its theoretical effect on pulse oximetry." British journal of anaesthesia 67.5 (1991): 638-643.
     - "... equation (2) is only an approximation and pulse oximeters are usually calibrated empirically using data obtained by inducing hypoxia in healthy volunteers."
+- detailed discussion of pulse-ox machine design: Pologe, Jonas A. "Pulse oximetry: technical aspects of machine design." International anesthesiology clinics 25.3 (1987): 137-153.
 
 ### Commercial Example
 A quick teardown of a ~$20 500BL from Walgreens revealed no [integrated photonics package](https://www.maximintegrated.com/en/products/interface/sensor-interface/MAX30101.html) or [signal processing ASIC](https://www.maximintegrated.com/en/products/interface/sensor-interface/MAX32664.html); instead, the device uses a bi-color IR/red LED on one side of a spring-loaded plastic clam-shell and a PCB with a decent sized photodiode on the other, paired with an [SGM8634](www.sg-micro.com/uploads/soft/20190626/1561538475.pdf) op-amp and an [STM32F100](https://www.st.com/en/microcontrollers-microprocessors/stm32f100-value-line.html)-series 32-bit Arm Cortex M3 microcontroller. The display is a custom multi-segment LED device, but the PCB labels suggest an OLED is used for an alternate model. TX/RX test points were spotted that could be investigated further; with any luck, these could be used to pull live data out of the instrument.
@@ -27,10 +28,18 @@ A quick teardown of a ~$20 500BL from Walgreens revealed no [integrated photonic
 
 ### Operational Theory
 Pulse oximetry is based on the [Beer-Lambert law](https://en.wikipedia.org/wiki/Beer%E2%80%93Lambert_law), a principle that relates the concentration of a species to the attenuation of light through a sample:
+
 ```math
 I=I_{in}e^{-(DC\epsilon)}
 ```
-where $`I`$ is the intensity of light transmitted through the sample; $`I_{in}`$ is the intensity of the light prior to absorption by the sample; $`D`$ is the optical path length; $`C`$ is the solute concentration; and $`\epsilon`$ is the extinction coefficient, the sample's absorption at a given wavelength of light. Typical commercial pulse oximeters use a red LED (660 nm) and an IR LED (940 nm) to quantify the relative concentration of reduced and oxygen-rich hemoglobin in a person's bloodstream based on the following absorbance curves:
+
+where $`I`$ is the intensity of light transmitted through the sample; $`I_{in}`$ is the intensity of the light prior to absorption by the sample; $`D`$ is the optical path length; $`C`$ is the solute concentration; and $`\epsilon`$ is the extinction coefficient, the sample's absorption at a given wavelength of light. For a multi-species compound, the three terms $`D`$, $`C`$, and $`\epsilon`$ for each individual species are combined:
+
+```math
+I=I_{in}e^{-(D_1C_1\epsilon_1+D_2C_2\epsilon_2+\dots+D_nC_n\epsilon_n)}
+```
+
+Typical commercial pulse oximeters use a red LED (660 nm) and an IR LED (940 nm) to quantify the relative concentration of reduced and oxygen-rich hemoglobin in a person's bloodstream based on the following absorbance curves:
 
 ![hemoglobin_curve](img/hemoglobin_curve.png)
 
@@ -42,6 +51,8 @@ In order to differentiate the slight intensity change caused by varying blood ox
 R=\frac{A_{AC_{660}}/A_{DC_{660}}}{A_{AC_{940}}/A_{DC_{940}}}
 ```
 
-As the photodiode sensor does not differentiate by wavelength, the device rapidly cycles between red, IR, and no LED, allowing the system to compensate for ambient light as well. The cycling speed must be substantially faster than the heart rate, since the ratio $`R`$ assumes absorption at all wavelengths is carried out simultaneously in order to cancel out path length.
+As the photodiode sensor does not differentiate by wavelength, the device rapidly cycles between red, IR, and no LED, allowing the system to compensate for ambient light as well. The cycling speed must be substantially faster than the heart rate, since the ratio $`R`$ assumes absorption at all wavelengths is carried out simultaneously in order to cancel out path length. $`R`$ is then related to SpO2 using an empirically determined chart:
+
+
 
 Note that methemoglobin (MetHb) and carboxyhemoglobin (CoHb) are not factored in with this method and will thus cause systematic errors; the above calculation assumes these two compounds are minimally present. Additional wavelengths are needed to quantify all four hemoglobin species.