spo2_measurements.md

Pulseoximetery

Physiology

Oxygen and Glucose enable aerobic glycolysis which produces ATP. Oxygen is transported to the cells through perfusion and diffusion. Its taken up from the inhaled air and bound to hemoglobin in red blood cells. The oxygenation is usually efficient and almost all hemoglobin molecules pick up 4 oxygens in the lung's alveoli.

The Oxygen Saturation is determined by the ratio of oxy-hemoglobin to the total hemoglobin:

$HbO_2 Sat = \frac{c_{HbO_2}}{c_{Hb}+c_{HbO_2}}$

This is expressed as per cent.

However, when partial pressure of oxygen is low such as at high altitude or when there is obstruction of ventilation, oxygen saturation can fall below 90%. A lack of oxygen supply results in inability of muscles to exert force and cognition and analytic thinking is impaired. Unfortunately there is no biologic sensory system detecting such condition.

Luckily we can measure oxygenation of hemoglobin non-invasively because it changes its optical properties when it binds oxygen. For example in the red, oxygenated hemoglobin absorbs less then deoxygenated hemoglobin. This is inverted in the near infrared making those wavelengths ideal for a sensitive measurement approach.

As we are less interested in the venous oxygenation as compared to the arterial oxygenation, we need to derive a technique to extract the arterial contribution to the optical signal. This is accomplished by measuring the component changing with each heartbeat as the pressure waves extend the arteries and increase the optical path length through arterial blood while the venous blood flow does not results in vessel extension with each heart beat and its contribution the absorption remains constant.

Blood pressure in artierial vessels is illustrated over distance from the heart:

Optical Properties of Hemoglobin

The color of tissue and skin can be explained with the optical properties of its main constituents. Here the main interest is Hemoglobin.

Absorption in the blue would be much stronger, however optical penetration is also affected by increased scattering. To maintain a strong signal on the sensor, measurements occur in the wavelength range of the "optical window" of tissue which is in the red to near infrared.

To simply detect the heart rate, measurements in the green create stronger signal fluctuation as compared to the red, however the largest differences between oxy and deoxy hemoglobin are in the red and near infrared.

Molar attenunation coeficient $a(\lambda)$ with units [ $\frac{1}{cm M}$ ] expresses the attenuation properties as function of concentration and pathlength. Relevant examples at two wavelengths are given below:

Wavelength [nm]	Deoxyhemoglobin	Oxyhemoglobin	Water [$\frac{1}{cm}$]
660 nm (red)	3226.56	319.6	0.0036
940 nm (nir)	693.44	1214	0.29

$M = moles/Liter$

Absorbance $A = a \cdot c \cdot L$ with $c$ the concentration and $L$ the pathlength.

Absorbance is depending on wavelength $\lambda$ and the optical properties of the absorbers. Here we focus on $a_{Hb}$ and $a_{HbO_2}$ as we need to know the relative concentration of those two constituents.

Absorbance through a path length $L$ is

$A(\lambda) = \left[ a_{Hb}(\lambda) \cdot c_{Hb} + a_{HbO_2}(\lambda) \cdot c_{HbO_2} \right] \cdot L$

with $a$ the optical property and $c$ te concentration.

We choose two wavelengths $\lambda_1$ and $\lambda_2$ with $\lambda$ in the red (660nm) and near infrared (940nm) resulting in $A_{\lambda_1}$ and $A_{\lambda_2}$.

Assuming that the pathlength $L$ is the same for both wavelengths we can show that:

$HbO_2 Sat = \frac{\frac{A_{\lambda_1}}{A_{\lambda_2}} * a_{Hb,\lambda_2} - a_{Hb,\lambda_1} } { \frac{A_{\lambda_1}}{A_{\lambda_2}} \left[ a_{Hb,\lambda_2} - a_{HbO_2,\lambda_2}\right] + \left[ a_{HbO_2,\lambda_1} - a_{Hb,\lambda_1}\right] }$

Light passing through a piece of tissue is attenuated with:

$\frac{I}{I_0} = 10^{-A}$

With $I_0$ the light incident to the tissue, $I$ the light leaving the tissue and $A$ being the sum of all absorbing components in the tissue such as

$A_{skin} + A_{bone}+ A_{muscle} + A_{fat} + A_{water} + L_{venous} \cdot \left[ a_{Hb} \cdot c_{Hb_{venous}} + a_{HbO_2} \cdot c_{HbO_{2_{venous}}} \right] + L_{aterial} \cdot \left[ a_{Hb} \cdot c_{Hb_{arterial}} + a_{HbO_2} \cdot c_{HbO_{2_{arterial}}} \right]$

Assuming that within a short time frame only $L_{arterial}$ changes one can lump all the other components into a constant:

$A = const + L_{arterial} \cdot \left[ a_{Hb} \cdot c_{Hb_{arterial}} + a_{HbO_2} \cdot c_{HbO_{2_{arterial}}} \right]$

Figure: Illustration of PPG with actual measurement.

Lets call the maximum signal $PPG_{relaxed}$ and the minimum signal $PPG_{extended}$ corresponding to diastole (relaxation of blood vessels due to decreased pressure) and systole. This signal as expected follows closely the arterial blood pressure.

$PPG_{relaxed} = I_0 \cdot 10^{-const + L_{aterial_{relaxed}} \cdot \left[ a_{Hb} \cdot c_{Hb_{arterial}} + a_{HbO_2} \cdot c_{HbO_{2_{arterial}}} \right]}$

with $I_0$ being the initial power of the light when entering the tissue and $PPG$ when it leaves.

When forming a ratio between extended and relaxed:

$log \frac{PPG_{extended}}{PPG_{relaxed}} = -\Delta L \cdot \left[ a_{Hb} \cdot c_{Hb_{arterial}} + a_{HbO_2} \cdot c_{HbO_{2_{arterial}}} \right]$ where $\Delta L$ is the difference between $L_{extended}$ and $L_{relaxed}$

We can extract this number from our measurement and call it $R_{lambda}$

The fraction of two of those numbers at different wavelengths will eliminate $\Delta L$.

$R = \frac{R_{660}}{R_{940}} = \frac {a_{Hb_{660}} \cdot c_{Hb_{arterial}} + a_{HbO_{2_{660}}} \cdot c_{HbO_{2_{arterial}}} } {a_{Hb_{940}} \cdot c_{Hb_{arterial}} + a_{HbO_{2_{940}}} \cdot c_{HbO_{2_{arterial}}} } = \frac {a_{Hb_{660}} \cdot (1-HbO_2 Sat) + a_{HbO_{2_{660}}} \cdot HbO_2 Sat } {a_{Hb_{940}} \cdot (1-HbO_2 Sat) + a_{HbO_{2_{940}}} \cdot HbO_2 Sat } $

Therefore the extraction of $PPG_{relaxed}$, $PPG_{extended}$ from two measurement traces at two wavelengths as well as the optical constants of oxyhemoglobin and deoxyhemoglobin at those wavelengths can be used to solve for

$HbO_2 Sat=\frac{a_{Hb_{660}} - R a_{Hb_{940}}}{R(a_{HbO_{2_{940}}}-a_{Hb_{940}}) - (a_{HbO_{2_{660}}}-a_{Hb_{660}})}$

Recording of the signal occurs with two LEDs and a photodiode while an microcontroller will need to extract maximum and minimum intensity of the signal.

For simplicity, ratios can be calculated without taking the log and $HbO_2 Sat$ can be estimated using lookup tables.

The above formulas show that pathlength for the two wavelengths is assumed to be the same, however this is not correct when other optical components become more relevant such as Melanin. Typically, fluctuations of the NIR signal are weaker than the ones in the red and the sensor should be operated so that maximum values are measured at both wavelengths by adjusting the LED current when switching between the LEDs.