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 = I010−εCL or log10II0= −εCL,

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 ε 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 (ε = 0) in oxygenated hemoglobin and 660-nm light has zero absorption in deoxygenated hemoglobin. If 32.8% of the energy of the red source and 76.5% of the infrared energy is transmitted through the blood, what is the fraction of hemoglobin that is oxygenated?

= %

Answer 1

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 32.8% means I/I0 = 32.8 /100 = 0.328
A transmission of 76.5% means I/I0 = 76.5/100 = 0.765

For the red light, the DOH has no affect (ε = 0). This means absorption is due only to OH:
log(0.328) = -εCOL
-0.484 = -εCOL

For the infrared light, the OH has no affect (ε = 0). This means absorption is due only to DOH:
log(0.765) = -εCDL
-0.116 = -εCDL

CD/CO = -0.116 / -0.484 = 0.239 (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 + 0.239)
= 0.81
= 81%

the fraction of hemoglobin that is oxygenated is 81%