Operation of the pulse oximeter (see previous problem). The...

Operation of the pulse oximeter (see previous problem). The transmission of light energy as it passes through a solution of light - absorbing molecules is described by the Beer–Lambert law $$I=I_{0} 10^{-e C L} \quad$ or $\quad \log _{10}\left(\frac{I}{I_{0}}\right)=-\epsilon C L$$ 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 $\epsilon$ is called the extinction coefficient, and its value depends on the frequency of the light. (It has units of $\mathrm{m}^{2} / \mathrm{mol} . )$ 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 940 - nm light has zero absorption $(\epsilon=0)$ in oxygenated hemoglobin and 660 - nm light has zero absorption in deoxygenated hemoglobin. If 33% of the energy of the red source and 76% of the infrared energy is transmitted through the blood, what is the fraction of hemoglobin that is oxygenated?

Operation of the pulse oximeter (see previous problem). The transmission of light energy as it passes through a solution of light - absorbing molecules is described by the Beer–Lambert law
$$I=I_{0} 10^{-e C L} \quad$ or $\quad \log _{10}\left(\frac{I}{I_{0}}\right)=-\epsilon C L$$
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 $\epsilon$ is called the
extinction coefficient, and its value depends on the frequency of the light. (It has units of $\mathrm{m}^{2} / \mathrm{mol} . )$ 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 940 - nm light has zero absorption $(\epsilon=0)$ in oxygenated hemoglobin and 660 - nm light has zero absorption in deoxygenated hemoglobin. If 33% of the energy of the red source and 76% of the infrared energy is transmitted
through the blood, what is the fraction of hemoglobin that is oxygenated?

Key Concepts

Beer–Lambert Law

The Beer–Lambert law quantitatively describes how the intensity of light decreases as it passes through a medium containing absorbing species. This law establishes an exponential relationship between the transmitted light intensity and the product of the absorption coefficient, the concentration of the absorbing molecules, and the path length. It forms the fundamental basis for understanding light attenuation in various applications, including spectrophotometry and optical diagnostics.

Extinction Coefficient

The extinction coefficient is a measure of how strongly a substance absorbs light at a particular wavelength. It quantifies the amount of attenuation per unit concentration and path length. This concept is crucial in determining the degree of light absorption in a sample and is a key parameter in applying the Beer–Lambert law to real-world scenarios.

Pulse Oximetry

Pulse oximetry is a non-invasive method used to measure the oxygen saturation of arterial blood. It operates by emitting light at different wavelengths and detecting the relative amounts of light that pass through tissue. By exploiting the distinct absorption characteristics of oxygenated and deoxygenated hemoglobin, pulse oximeters can estimate blood oxygen levels, providing essential information about a patient's respiratory and circulatory health.

Differential Absorption of Hemoglobin

Oxygenated and deoxygenated hemoglobin exhibit different light absorption properties at specific wavelengths, which is the principle behind differential absorption methods. In pulse oximetry, light of one wavelength is preferentially absorbed by oxygenated hemoglobin while another wavelength is absorbed by deoxygenated hemoglobin. This differential absorption allows for the calculation of the fraction of hemoglobin that is oxygenated by comparing the transmitted intensities at these wavelengths.

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