6.3 Signals of the Cardiovascular System

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If one wanted to calculate the intensity for the fluoroscopy of the fingertip or the ear-

lobe, according to Equation 6.33 the layer structure of the tissue with the correspond-

ing absorption coefficients would have to be known exactly. This is practically im-

possible. Since, as will be shown below, the temporal change in intensity is used to

calculate oxygen saturation, it is sufficient to use a simplified two-layer model. The

first layer represents all absorption processes in venous vessels, fat and connective

tissue, skin, etc., which do not depend on time. The time dependence of intensity is

due to the change of thickness of arterial vessels as a result of rhythmic pulse waves.

This is represented by the second layer in terms of a time-varying thickness d2(t). With

this model, Equation 6.33 simplifies to.

I(t) = I0e^{−(α1d1+α2d2(t))} = I0e^−α1d1 e^−α2d2(t) = C(λ) e^−α2d(t) .

(6.34)

In Equation 6.34 all time-independent factors are combined to the constant C, which,

however, depends on the wavelength because of α1. α2 is the absorption coefficient of

blood. The quantity d(t) describes the time-dependent vessel thickness, but it is not

directly accessible and therefore must be eliminated. For this purpose, the auxiliary

quantity Γ is introduced as follows:

Γ =

ln ^I(λ1t1)

I(λ1t2)

ln ^I(λ2t1)

I(λ2t2)

.

(6.35)

For the auxiliary quantity Γ, therefore, a total of four measurements are required, two

measurements each at the wavelengths λ1 and λ2 at two different times t1 and t2.

Substituting Equation 6.34 into Equation 6.35 leaves,

Γ = ^α2(λ1)

α2(λ2)

(6.36)

so d(t) is dropped.

To determine oxygen saturation, a correlation must now be established with the

concentration of oxygen-saturated hemoglobin (CHbO2) and unsaturated hemoglobin

(CHb). For this purpose, the blood absorption coefficient is considered in more detail.

In general, for media consisting of several substances, the absorption coefficient is the

sum of the molar extinction coefficients ε of the individual substances multiplied by

the corresponding concentration. Transferred to blood this means

α2(λ) = εHbO2(λ)CHbO2 + εHb(λ)CHb .

(6.37)

Of course, blood does not consist only of hemoglobin, so in Equation 6.37 also the

other components would have to be included accordingly in the sum. However, these

are insignificant for further consideration and can be mentally added to the quantity

C(λ). The extinction coefficients εHbO2(λ) and εHb(λ) are well known from the literat-

ure [39, 40, 79].