A \"pulse oximeter\" operates by using light and a photocell to measure oxygen s

Question

A "pulse oximeter" operates by using light and a photocell to measure oxygen saturation in arterial blood. The transmission of light energy as it passes through a solution of light-absorbing molecules is described by the Beer-Lambert law I = I010CL or log10 I I0 = CL, which gives the decrease in intensity I in terms of the distance L the light has traveled through a fluid with a concentration C of the light-absorbing molecule. The quantity is called the extinction coefficient, and its value depends on the frequency of the light. (It has units of m2/mol. ) Assume the extinction coefficient for 660-nm light passing through a solution of oxygenated hemoglobin is identical to the coefficient for 940-nm light passing through deoxygenated hemoglobin. Also assume that 940-nm light has zero absorption ( = 0) in oxygenated hemoglobin and 660-nm light has zero absorption in deoxygenated hemoglobin. If 37.0% of the energy of the red source and 67.8% of the infrared energy is transmitted through the blood, what is the fraction of hemoglobin that is oxygenated?

Explanation / Answer

Lets follow a notation for easier understanding and solving

Call oxygenated hemoglobin 'OH'.
Call deoxygenated hemoglobin 'DOH'.
Concentration of OH = CO.
Concentration of DOH = CD.
Abbreviate 'log to base 10' to 'log'

A transmission of 37% means I/I0 = 37/100 = 0.37
A transmission of 67.8% means I/I0 = 67.8/100 = 0.678

For the red light, the DOH has no affect ( = 0). This means absorption is due only to OH:
log(0.37) = -COL
-0.431= -COL

For the infrared light, the OH has no affect ( = 0). This means absorption is due only to DOH:
log(0.678) = -CDL
-0.168 = -CDL

CD/CO = -0.168 / -0.431 = 0.3897

If the total volume V then:
amount of OH = COV
amount of DOH = CDV

Fraction of hemoglobin that is oxygenated
= amount of OH / total amount of hemoglobin)
= COV / (COV + CDV)
= CO / (CO + CD)
= 1 / (1 + (CD/CO)) (divided top and bottom by CO)
= 1 / (1 + 0.3897)
= 0.714
= 71.4%

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