1. Introduction

The pulse oximeter is one of the most widely used medical devices. The operating principle of pulse-oximetry is based on the fact that the light intensity transmitted or reflected from a finger or an earlobe varies with the pulse wave [1]. This type of measurement is called photoplethysmography (PPG). The widely accepted assumption describes the PPG signal in terms of pulsatile changes in the vascular blood volume [2]. The fact that pulsation changes in the blood volume are measurable by various methods cannot be disputed; however, this does not mean that only volumetric changes are responsible for the pulsatile appearance of optical signals.

Recently, a modified volumetric model has been proposed [3]. In this model, the influence of elastic deformations of the capillary bed on the formation of the PPG waveform was studied. On the basis of the proposed model, for example, it was successfully explained why the amplitude of the PPG increases with an increase in the externally applied pressure. In their simulations and experiments, the authors used green LED (wavelength of 510–560 nm) where strong hemoglobin absorption prevails over scattering. However, in the spectral region 660-940nm, the region of pulse-oximetry, the absorption of hemoglobin is much weaker and hence, the proposed modified volumetric has to be evaluated as well. In [4], using Monte Carlo simulation, it was calculated how the multilayered skin structure affects the pulsating volumetric signal in the region 450-1000nm. However, the authors admit that they didn’t provide direct experimental evidence of their model and they don’t exclude the possibility of complementary mechanisms of PPG formation occurring in parallel. Thus, to explain the experimental data, even within the framework of the standard volumetric model, several modifications have been proposed. It is very likely that the contribution of each of the possible mechanisms behind the modulation of the optical signal, whether they are of a volumetric nature or any other, depends on the specific conditions and place of measurement. That is why each possible mechanism must be investigated independently, so that they can be combined in the future.

In 1967, the question pertaining to the origin of PPG signals was raised for the first time when it was experimentally shown [5] that optical pulsatile components appeared in the absence of measurable volumetric changes. In 2013, a published article reported that a considerable quantity of pulsating optical signals was detected in various human bones although the number of red blood cells during the cardiac cycles remained constant [6]; nevertheless, these findings were barely discussed in the scientific literature [7]. In 2000, to explain the optical signal behavior in vivo, accounting for the RBC aggregation process or rouleau formation was proposed [8]. The same study considered the process as responsible for the increase in light transmission during aggregation and blood flow cessation in vivo. Note that the relationship between the RBC aggregation and optical signal has been well known and studied for many years in vitro [9–12]. Thus, the study in [8] assumed that the manifestation of the dependence of reflected or transmitted light on the state of aggregation of RBCs is not limited under in vitro conditions but also in vivo [13]. Under in vitro and in vivo conditions, the presence and magnitude of shear forces in blood flow must be attributed to the key elements regulating the rouleau size.

In subsequent years, a specific mechanism explicitly pertaining to the effect of RBC aggregation on the optical signal was quantitatively elaborated [14]. The qualitative understanding of the proposed model can be summarized as follows: The dynamic balance between the destruction and formation of RBC aggregates is determined by the blood velocity. The shear rate is defined as the slope of the velocity profile. This rate is high when the flow velocity is high, and the vessel diameter is small; conversely, the shear rate is low when the flow velocity is low, and the vessel diameter is large. The size of RBC aggregates is inversely proportional to the magnitude of the shear rate [15]. The increase in shear rate results in the breakup of large aggregates into smaller ones or into single RBCs. For shear rates exceeding 50 s−1, all RBC aggregates are dissociated into single erythrocytes [16]. Under normal conditions, a periodic pressure wave causes oscillations in velocity and shear forces, modulating the aggregate size. Hence, the fluctuation in the shear forces of blood flow is the main rheological factor leading to observed variations in the measured optical signals. The influence of shear forces on the size of aggregates in vitro and the semi-empirical description of this process are published elsewhere [17,18].

To describe the effect of erythrocyte aggregation on the transmission of light, two levels of modeling have to be considered. First, the change in light scattering should be expressed in terms of aggregate size. In [19], the using of Wentzel–Kramers–Brillouin (WKB) approximation, that allows the derivation of a scattering coefficient for substantially large aggregates, was proposed.

Following this approximation, radiation propagates inside the particle in the same direction as the propagation of incident radiation (low-refractive particles). The radiation wavenumber inside the particle is equal to the radiation wavenumber in the particle material. In this case, the effect of scattering is related to the radiation phase change when radiation propagates inside the particle. Based on this model, the erythrocyte aggregate can be approximated as a growing spheroid. This model is suitable for describing a long aggregate that grows due to the blood stasis process. However, in the case of blood flowing through arterioles, the assumption that the average rouleau size can exceed several erythrocytes is difficult to realize; consequently, the elongated spheroid approximation is not that applicable.

For the scattering of light by particles whose sizes are comparable to the wavelength, the Mie model affords an exact analytical solution in the case of spherical particles. Although a single erythrocyte is not spherical, it has become apparent that the Mie solution is well-suited for RBCs [20]. With the transition of one erythrocyte to an aggregate of several erythrocytes, the combined particle resembles a sphere. Therefore, the use of the Mie model in this case is even more justified than in the case of a single RBC. For this reason, approximating the growing small rouleau with an equivalent inflating sphere was proposed [14]. In this way, the change in the scattering cross-section during aggregation can be described by corresponding Mie functions. An additional level of modeling refers to the propagation of light in tissues and blood [21]. A wide variety of models ranging from the transport equations of photon flux in blood to the Monte Carlo models of light propagation in tissue and blood exists [22–25]. The selection of a suitable model depends on the practical purpose of the model. In this case, the objective is to describe the variable pulsating component of the optical signal, which is completely predetermined by changes in the optical properties of RBCs. The tissue surrounding the blood vessels is responsible for causing the multiple scattering of light; accordingly, this justifies the adoption of diffusion approximation. All of the above are the principles of the theoretical model based on which the results of the experimental part of the study were considered. As for the experimental part, the main goal was to create such conditions at which the difference between the volumetric and aggregation models will be most pronounced. The experiments are implemented to test the plausibility of the two concepts for the PPG signal: the volumetric model and aggregation model (also named scattering-driven model (SDM)) [14]. The behavior of “gamma” values for different pairs of wavelengths (for example (670 nm, 940 nm) and (590 nm, 940 nm)) as a function of the aggregate size was examined.

Experimentally, the change in the aggregate size was achieved by reducing the blood flow at the measurement site where external pressure was applied. Simultaneously, volumetric pulsations were recorded by measuring the pressure in the air cushion wrapped around the fingertip (oscillometric signal) [26].

In a series of experiments, the amplitude and phase shift between the optical and oscillometric signals were investigated under different pressure conditions. The derived experimental data were compared with the model predictions for the two discussed cases. Our stated goal was to test the plausibility of our hypothesis about the hemodynamic nature of the PPG signal by comparing the theoretical predictions of our model and the experimental results.

2. Theoretical consideration

2.1 RBC aggregation and shear rate

An RBC can be approximated as an oblate spheroid with an axial ratio of approximately 0.3 and a maximum radius of 4.1 microns. The RBCs in the plasma tend to form aggregates or rouleaux that appear similar to a stack of coins [27,28]. The so-called rouleau formation is caused by plasma macromolecules and is supposed to be a reversible aggregation process. As we will see later, the scattering of light in the blood depends on the size of the aggregates. The formation of aggregates is, in turn, a reversible process regulated by blood flow shear forces. Therefore, to evaluate the dynamics of aggregation in the blood, one should consider the behavior of the velocity profiles in the blood flow in the vessel.

The parabolic profile of blood flow in the blood vessels, as a rule, does not have a pronounced maximum. The degree of deviation from parabolicity is determined by a parameter called blunting. The blunting is dependent upon the ratio of vessel size and particle thickness [29]. Blunted profiles move closer to a parabolic profile as the diameter of the vessel decreases. As a result of a change in the blood flow velocity due to the pulsation, shear forces increase or decrease, which leads to a variation in the length of the rouleau. In order to evaluate changes in the size of aggregates, we used a simplified model that gives only a rough estimate of the effect of pulsation on the distribution of aggregates in blood vessels. In flowing blood, an inverse relationship exists between the size of red blood cell aggregates and the shear rate of blood in vivo [30].

In the simplified case, when blood flow is laminar and the vessel is cylindrical, the velocity profile, , can be approximated [31] as:

The average value of shear forces achieves maximum for r = R, namely near the walls.

For arterioles [33], the variation in velocity from systolic to diastolic phases is 1.5–2.5 mm/s. The number of red blood cells in the rouleau depends on the shear forces at any given instance. Assume that the rate of aggregate destruction in the systolic part is sufficiently fast for the disaggregation process not to lag behind the pulse velocity wave. For RBCs aggregating in the lateral direction of vessels, the average rouleau size (An) is given by [18]:

Figure 1 demonstrates the variation in the aggregate length as a function of shear rate. The figure indicates that for small values of shear forces, the extremely small variations in the shear rate considerably affect the aggregate size (NA signifies number of RBC’S in the rouleau).

 

Fig. 1. RBC aggregate size vs. shear rate.

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The PPG signal is virtually formed by averaging the signals over an ensemble of different vessels, where the shear rate distribution is defined by the vessel diameter and distribution of pulse wave velocities. In addition, note that for small arterioles, the degree of blunting approaches 4; this further increases the dependence of aggregate length on the shear forces in the central region of the vessel.

2.2 Gamma determination

The PPG signal is characterized by light intensity changes after passing through a perfused tissue. The associated pressure waves lead to periodic changes in the transmitted and reflected light. The focus of this work is on the specific mechanism that causes changes in the intensity of the measured signal. In pulse oximetry, the signal variations are converted into the “ratio of ratios” [34]. We defined ratio of ratios as “Gamma”. This value is calculated from the pulsatile (alternating current (AC)) and non-pulsatile (direct current (DC)) components of the measured signal for two wavelengths (i.e., ${\lambda _1}$ and ${\lambda _{ref}}$) [1]:

In pulse oximetry, the “gamma” value uniquely determines the percentage of oxyhemoglobin in arterial blood (SPO₂). Another parameter, called parametric slope (PS), can be regarded as virtually equivalent to “gamma” [35] and was found to be a more useful form to express “gamma” for model analysis.

Let the intensity of the measured optical signal be denoted by I (λ, t), which is a function of time and wavelength. At any given time, the derivatives of the signal can be calculated. One of the most significant characteristics of gamma is the lack of dependence on perfusion and therefore it remains unchanged in different intervals of the pulse wave.

As a rule, the ratio of signal derivatives measured at two wavelengths does not considerably fluctuate at different pulse wave intervals. The average PS value over the full pulse wave interval corresponds to the standard “gamma” (the “ratio of ratios”):

The signal intensity, I($\lambda ,t)$, passing through or reflecting from the tissue depends on the optical properties of the tissue; however, temporal pulsation dependence is determined only by the blood component. Accordingly, interest is focused on the time-dependent function of the scattering and absorption coefficients of blood. These coefficients can be expressed in terms of the scattering and absorption cross-sections [19]:

The absorption cross-section of blood depends on the hemoglobin saturation given by

The light scattering by a single erythrocyte is well described by the scattering function of Mie [20], which provides a complete solution for determining the scattering cross-section value and phase-scattering function (g). To adjust the Mie theory for aggregates, the following simplified model is used. When RBCs stick together in a chain, the new scattering particle can be approximated as a sphere (Fig. 2). As the length of the rouleau approaches 4µ, the shape of the resulting particle becomes more symmetric and more closely corresponds to the approximation of a sphere.

 

Fig. 2. Rouleaux approximated by equivalent sphere.

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The particle is considered composed of N_R erythrocytes (each with volume ${V_0}$). For a given size of small aggregates, an equivalent sphere with radius r_eff generating the same volume as the particle composed of N_R erythrocytes is produced. The effective radius, r_eff, of each aggregate is given by:

A diffusion model [23] is typically employed to describe the transmission of light through tissue. The “inverse diffusion length” or diffusion coefficient is convenient to use as a basic parameter. This coefficient [36] is defined by

According to the Mie model, the scattering cross-section and g are defined by the radius of the sphere, r_eff, and relative refractive index. It was shown [37] that an ellipsoid with a revolution of a given size, axial ratio, and orientation could be approximated by a suitable equivalent sphere. The effects of the difference between the ellipsoid and equivalent sphere can be accounted for using a corrected refractive index, ${\textrm{m}_\textrm{c}}$, for the latter [37]:

In the rouleau analysis, the foregoing factor was considered in calculating the scattering cross-section and g.

To assess the dependence of the results of the calculated gamma on the type of diffusion model, two approaches focusing on the diffusion of light in a highly scattering medium were selected. Because the measuring system is designed for the reflective signal, a model in which the intensity of the measured reflected signal is expressed in terms of the absorption, scattering and diffusion coefficient is selected [24]:

Parameter A depends on the refractive index of the medium (A =1 for n =1, and A > 1 for n > 1); r is the detector radius (r= 3 mm in this work).

Within the framework of this model, any changes in the properties of the medium can be expressed through the scattering and absorption coefficients. However, this model is not well-suited for simulating volumetric changes. Accordingly, as a second choice, a simple approximation suitable for both transmission and reflection are applied. Note that the scattering of light in tissue is considerably more isotropic than scattering using erythrocytes; consequently, the photon flux becomes virtually isotropic from the beginning. Thus, according to the simplified diffusion model, the spatial distribution of the diffusing photons is dictated by the properties of the tissue, whereas the hemoglobin in blood particles absorbs the photons. Both transmission and reflection modes can be considered as transillumination governed by diffusion, where the direction of the flux of photons does not play a significant role. The evaluation of signal change due to fluctuations in absorption and scattering caused by the blood component only is of interest. The simple exponential dependence of the diffused signal on the properties of blood is the experimental fact on which pulse oximetry is based. This is the reason for choosing a simple exponential expression of photon diffusion. Only the time-dependent behavior of the signal is relevant. Thus, the following expression governs the changes in the light intensity [25]:

Equation (13) ascribes the diffusion coefficient and volumetric changes to the time-dependent intensity modulation. x_b is volumetric changes expressed in the length units.

Let time-dependency be assigned only to ${x_b}(t )$; this specific case is called the volumetric model. According to this model, during the systolic phase, a pressure wave increases the blood volume in the tissue. For this volumetric case, Eq. (5) becomes

Using the transmission model given by Eq. (13), the gamma value defined by Eq. (14) was estimated for different thicknesses of aggregates ranging from 1 to 3 microns. If we imagine that an erythrocyte is a disk with a thickness of 1.1 microns, then the formation of an aggregate is similar to putting these disks into a pile of coins. Only the height grows and the diameter remains unchanged. Thus, 3 erythrocytes will be with the same diameter as one and with a height of 3.3 microns.

To correspond to fully oxygenated arterial blood, SPO2 is taken as 1. Figure 3(a) shows that the gamma value for the wavelength pair (670 nm, 940 nm tends to decrease with the increase in the rouleau size, where the gamma value for the pair (590 nm, 940 nm) [Fig. 3(b)] increases as the aggregate grows.

 

Fig. 3. (a) Volumetric model. Gamma)670 nm, 940 nm). (b) Gamma (590 nm, 940 nm)

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For the SDM, the gamma values for the reflection approach [Eq. (11,12)] and for the transmission model [Eq. (13)] are calculated. In reflection pulse oximetry, the same expressions for calculating gamma are applicable as in transmission geometry. This means that the exponential diffusion approximation is equally valid for both cases.

The aggregation process in the SDM is the only mechanism responsible for signal changes. This process is modeled by the inflation of sphere-like scatterers, whereas the number of scatterers drops; hence, the total hematocrit remains unchanged. The particle size and concentration (c(t)) periodically fluctuate with time (t), and ${\mu _d}(t )$ incorporates all changes that are induced by the aggregation process. Thus, unlike the volumetric model, the dependence of the optical signal on time is no longer determined not by the change in blood volume during the pulse, but rather by the fluctuation of the scattering coefficient of erythrocytes. Therefore, to calculate gamma, the ratio of time derivatives can be replaced by the ratio of partial derivatives of intensity with respect to the scattering coefficient. From Eq. (5), gamma should be written as

The figures below show the calculated gamma for the SDM from Eq. (15) for two pairs of wavelengths, (670 nm, 940 nm) and (590 nm, 940 nm), based on the transmission model [Eq. (13)].

For the reflection model [Eq. (11,12)], the “gamma” values for the SDM are shown in Fig. 5.

 

Fig. 4. SDM. (a) Gamma (670 nm, 940 nm). (b) Gamma (590 nm, 940 nm)

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Fig. 5. Reflection model. (a) Gamma (670 nm, 940 nm). (b) Gamma (590 nm, 940 nm).

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With regard to the gamma dependency on the aggregate length, the reflection and transmission models for the SDM manifest similar behaviors; however, they are diametrically opposite to the volumetric model trends.

3. Experiments

The measuring system (Fig. 6) includes a standard reflective geometry optical scheme for measuring the PPG signal. In the experiments, two light source modules were used: matrices with LED pairs of (670 nm, 940 nm) and (590 nm, 940 nm). The distance between the light source and detector was 5 mm. To acquire PPG signals, a two-channel system based on a fully integrated chip (AFE4403, Texas Instruments) was employed; the measured signals were stored in the computer memory for further processing. An additional system module includes a pressure regulator that controls the pumping of air into a special air cuff. The cuff is cylindrical with an elastic inner membrane and a rigid outer shell. The cuff volume was minimized to increase sensitivity to volume changes. Pressure pulsations were recorded by a pressure sensor and subsequently digitized and stored in the PC memory. The experimental device had a resolution of 16 bits and a sampling rate 100 Hz.

 

Fig. 6. Measuring system diagram consisting of optoelectronic and pneumatic sub-systems.

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Initially, a pair of LEDs (670 nm, 940 nm) was utilized, and the PPG signal from the fingertip was recorded for 1 min. At the same time, the air cuff was wrapped around the fingertip and a predetermined pressure was applied. The oscillometric signal and two PPG signals were simultaneously recorded. Then, the pressure in the air cuff was increased, and the experiment was repeated. The experiments continued until the applied pressure exceeded the systolic pressure of the subject by 40 Torrs. The entire series of experiments was repeated for another pair of LEDs (590 nm, 940 nm). Generally, the velocity of blood flowing into the fingertip is supposed to decrease with increasing pressure inside the air cuff [39].

The velocity decrease leads to a drop in shear forces and a corresponding increase in the average length of RBC aggregates. The recorded PPG signals were processed by a separate program, and the gamma values were averaged over a 1-min period. In addition, the PPG signal waveforms at 940 nm and the oscillometric signal were recorded. Then, the amplitudes of the AC components for each type of signal and the phase shift in-between were calculated.

4. Results and discussions

The experimental data was obtained on a large number of subjects and the results below represent the most reproducible typical pattern of measured signal and calculated “gamma” values. The comparison of the waveforms of the PPG and oscillometric signals reveals the following features. At relatively low applied pressures, the shapes of the oscillometric and PPG (940 nm) signals including the typical manifestation of the dicrotic notch are considerably similar (Fig. 7(a)). Upon reaching the over-diastolic pressure at which the non-pulsatile blood flow stops, the dicrotic form disappears in the PPG signal, whereas the oscillometric signal preserves the dicrotic feature (Fig. 7(b)). With further increases in pressure, a delay in the onset of the optical signal relative to the oscillometric signal is observed. This delay increases monotonically with increasing pressure (Fig. 7(c)).

 

Fig. 7. (a) PPG and Oscillometric signals 71 Torr. (b) PPG and Oscillometric signals at 98 Torrs. (c) Delay between Oscillometric and PPG signals.

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The AC value of the PPG and oscillometric signals was determined as the difference between the maximum and minimum values corresponding to the measured signals during the measurement interval. For example, at 60 Torrs, the AC value of the PPG signal was approximately13 units, and the DC value was approximately 220 units (Fig. 8(a)). The amplitude of oscillometric pulsations was 0.55 Torr. Upon reaching 130 Torrs, the sufficiently significant pulsation of the oscillometric channel (0.34 Torr) and virtual disappearance of the PPG signal (Fig. 8(b)) were observed.

 

Fig. 8. (a) PPG vs. Oscillometric pulse (50 Torrs). (b) PPG vs. Oscillometric pulse (130 Torrs).

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Finally, the following results show the behavior of gamma as a function of applied pressure for the two pairs of wavelengths. The experiments were repeated several times; hence, datasets from different measurements were observed. The graph shows the gamma behavior of the signal of the LED pairs, (670 nm, 940 nm) (Fig. 9(a)) and (590 nm, 940 nm) (Fig. 9(b)), as a function of applied pressure. The gamma value for the pair (670 nm, 940 nm) tends to increase, whereas for the pair (590 nm, 940 nm), the opposite tendency as a function of the applied pressure is observed.

 

Fig. 9. (a) Experimental gamma (670 nm, 940 nm). (b) Experimental gamma (590 nm, 940 nm).

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The purpose of the succeeding discussion is to compare two alternative models of the PPG signal. The volumetric model assumes that the source of the pulsation signal is the blood volume fluctuation in the vessels. According to the alternative model (i.e., SDM), the optical pulsation signal is assumed to be determined by the hemodynamic behavior of blood flow in the vessels. One of the manifestations of hemodynamics is shear force fluctuation, which leads to scattering changes in the RBCs and RBC aggregates. The first group of results is discussed to compare the simultaneously measured PPG signal (at 940 nm) and oscillometric signal associated with the volumetric behavior (Fig. 7(a)–(c)). These experiments enable the juxtaposition of the oscillometric and optical signal waveforms as a function of the applied external pressure. At low pressure levels, both waveforms exhibit similar behaviors. Upon reaching a certain threshold that approaches the diastolic pressure, the so-called dichroitic notch in the PPG signal disappears, whereas the pressure curve is preserved. According to the volumetric model, the shape of the volume change must fully correspond to the shape of the optical signal. Based on the volumetric concept, the situation in which at a certain moment the dichroitic notch disappears in the PPG but continues to be present in the oscillometric signal is difficult to explain. Another phenomenon is demonstrated in Fig. 7(c). With the increase in the applied pressure, a phase lag between the pulsatile signals of the PPG and pressure wave is observed starting from a certain threshold. This lag increases with increasing pressure and is a phenomenon that is not easy to explain in terms of the volumetric model. Moreover, Fig. 8(b) shows that although the PPG pulsation virtually vanishes, a considerably prominent manifestation of the oscillometric pulsation continues to be observed. This result is also inconsistent with the volumetric model. The following attempts to explain the same results based on the hemodynamic model. Within the framework of the aggregation model, the primary focus is on the behavior of the dynamics of blood flow. When the air pressure in the cuff reaches the diastolic point, the stationary blood flow stops and only the pulsating component continues. With the increase in external pressure beyond the diastolic pressure, blood flow only appears when the pulse wave exceeds the locally applied threshold pressure; this threshold is shown by the solid horizontal line in Fig. 7(b). It implies that if the local arterial blood flow is measured, the delay in its appearance in relation to the pressure wave can be observed. After the expected behavior of blood flow pulsation, the observation that the PPG waveform does not correspond to the volumetric pulsation starting from a certain moment is consistent. Moreover, with the increase in the external pressure, although the amplitude of the oscillometric pulsation slightly decreases, it continues to be visible after reaching the systolic threshold, whereas the measured PPG signal disappears with the blood flow. The reason for the absence of the optical signal pulsation, whereas the volumetric pulsation continues to be observed, is difficult to explain. Thus far, the presented results indicate that the main source of the PPG signal (at least for high local pressures) is hemodynamic change and not volumetric change.

In the succeeding parts of the experiments, the correspondence between the optical model simulation of gamma and experimental results was checked. According to the proposed model, the blood flow pulsation is associated with the corresponding changes in the shear forces. These changes result in variations in the size of erythrocyte aggregates. The fluctuation in the sizes of aggregates must lead to a change in the scattering of transmitted light, as observed during the measurement. According to the proposed model, the blood flow velocity increase, which can be experimentally achieved using the applied pressure, must reduce the shear forces and increase the average size of scatterers. The model’s predictions were formulated in terms of gamma values. It has been experimentally established that for the pair (670 nm, 940 nm), the gamma value whose calculation is based on the PPG signal measured at two wavelengths does not depend on the measurement geometry and amount of blood in the tissue. This experimental fact is used in pulse oximetry. The SDM model predicts an increase in gamma for the pair (670 nm, 940 nm) and a decrease in gamma for the pair (590 nm, 940 nm) with the increase in the average aggregate length (Fig. 4(a)–5(b)). Figure 9, (a) and (b), shows the experimental behavior of gamma as a function of the applied pressure measurement. For the pairs (670 nm, 940 nm) and (590 nm, 940 nm), a disagreement in the experimental result trends is observed. A strong disagreement in the gamma values for the pair (590 nm, 940 nm) is also observed.

5. Conclusion

Based on the theoretical and experimental results, the following can be concluded: