math formatting

pulseox.md

-## Pulse Oxygenation
-Pulse oxygenation devices use several LEDs to measure pulse rate and blood oxygen content. The LEDs are tuned to specific wavelengths corresponding to the absorbance band 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.
+## 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
-- overview of pulse oxymetry 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.
+- 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
         - the other two hemoglobins (MetHb and COHb)
@@ -36,4 +36,12 @@ where $`I`$ is the intensity of light transmitted through the sample; $`I_{in}`$
 
 _Figure source: Bülbül, Ali & Küçük, Serdar. (2016). Pulse Oximeter Manufacturing & Wireless Telemetry for Ventilation Oxygen Support. International Journal of Applied Mathematics, Electronics and Computers. 211-211. 10.18100/ijamec.270309._
 
-In order to differentiate the slight intensity change caused by varying oxygen concentration from errors related to skin absorbance, the signal processing algorithm only looks at the AC portion of the signal, since within a reasonable range (~0.5 - 3 Hz) this corresponds to blood rushing through arteries with each heartbeat, increasing the optical path length and causing periodic oscillations in the absorption signal. 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. Using two wavelengths to quantify two species results in a system of two equations that can be continuously evaluated. Note that, as mentioned above, methemoglobin (MetHb) and carboxyhemoglobin (CoHb) are not factored in with this method and can thus cause systematic errors; additional wavelengths are needed to quantify all four hemoglobin species.
+In order to differentiate the slight intensity change caused by varying blood oxygen concentration from errors related to skin absorbance and venous blood (whose oxygen has already been taken up by cells), the signal processing algorithm isolates the AC portion of the signal, since within a reasonable range (~0.5 - 3 Hz) this corresponds to blood rushing through arteries with each heartbeat. This _pulsatile arterial blood_ increases the optical path length of the measurement as blood pressure swells the arteries, producing periodic oscillations in the absorption signal. The other contributors to absorption, such as tissue and venous/capillary blood, are effectively constant in this frequency regime. By calculating the ratio of the AC and DC signals at each wavelength, then taking the ratio of these two absorption ratios, a value $`R`$ can be determined which is only related to the relative concentration of oxyhemoglobin (O2Hb) and reduced hemoglobin (Hb):
+
+```math
+R=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.
+
+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.