A pulse oximeter operates by using light and a photocell to

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 29.4% of the energy of the red source and 85.3% of the infrared energy is transmitted through the blood, what is the fraction of hemoglobin that is oxygenated? %

Solution

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 30% means I/I0 = 29.4/100 = 0.294
A transmission of 75% means I/I0 = 85.3/100 = 0.85

For the red light, the DOH has no affect ( = 0). This means absorption is due only to OH:
log(0.3) = -COL
-0.532 = -COL  

For the infrared light, the OH has no affect ( = 0). This means absorption is due only to DOH:
log(0.85) = -CDL
-0.07 = -CDL

CD/CO = -0.07 / -0.532 = 0.132 (this value is needed below)

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 + 0.132 )
= 0.88
= 88%