1.

Introduction

Photoplethysmography (PPG) is composed of a pulsatile component (ac) and nonpulsatile component (dc). ac is synchronized with the heart and related to arterial pulsation, while dc is related to light absorption in the tissue, vein, and diastolic arterial blood volume. dc has many physiological meanings and correlates with the ac waveform. ^{1, 2, 3, 4, 5, 6, 7, 8} According to the Lambert–Beer (or Beer–Lambert) law that explains mechanisms of PPG, PPG ac is generated by the optical path length (Δd) that changes with the increase of volume in blood vessels as they pulsate.⁹ Therefore, it can be said that PPG ac is in a consistent relationship generated by variations in dc and Δd.

In previous studies of ac, dc, and Δd, there were correlations between continuous ac and spontaneous variations of dc,¹⁰ but there has been no research on ac variation according to dc intensity. There have also been comparative studies of preop and postop blocking of sympathetic nerve systems or specific patients such as diabetics.^{3, 11, 12, 13} However, these studies only reported on experimental results and did not suggest a mathematical model that can explain the relationship between components, so they offer limited explanations of physiological results. Therefore, this study suggested an equation for the relationship between ac, dc, and components using the existing Lambert–Beer law to analyze the characteristics of PPG as a noninvasive vascular assessment tool. Results of the suggested equation were assessed through simple simulation and experiment.

2.

Material and Method

2.1.

Optical Model

Based on the Lambert–Beer law, the photodetector detects photons not absorbed into the blood. The waveform of general photoplethysmography used in many studies can be gained by inversing or subtracting the amount of light not absorbed at a constant (k).³ This is shown in Fig. 1, and the onset and peak of the photoplethysmography can be presented as Eqs. 1, 2.¹⁴

Eq. 1

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{equation} I_H = I_O {\rm e}^{ - b_{{\rm dc}} d_{{\rm dc}} } e^{ - b_{{\rm Hb} + {\rm HbO}_2 } d_{\min } } = \alpha \beta ^{d_{\min } } = k - I_{{\rm onset}}, \end{equation}\end{document} $$I_H=I_O{\mathrm{e}}^{-b_{\mathrm{dc}}{d}_{\mathrm{dc}}}{e}^{-b_{\mathrm{Hb}+{\mathrm{HbO}}_2}{d}_{\min }}=\alpha {\beta }^{d_{\min }}=k-I_{\mathrm{onset}},$$

Eq. 2

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{equation} I_L = I_O {\rm e}^{ - b_{{\rm dc}} d_{{\rm dc}} } e^{ - b_{{\rm Hb} + {\rm HbO}_2 } d_{\max } } = \alpha \beta ^{d_{\max } } = k - I_{{\rm peak}}. \end{equation}\end{document} $$I_L=I_O{\mathrm{e}}^{-b_{\mathrm{dc}}{d}_{\mathrm{dc}}}{e}^{-b_{\mathrm{Hb}+{\mathrm{HbO}}_2}{d}_{\max }}=\alpha {\beta }^{d_{\max }}=k-I_{\mathrm{peak}}.$$

Here, I _L is the small value of reflected amount of light and the peak point of PPG. I _H is the large value of the reflected amount of light and the onset point of photoplethysmography. b _dc is a function of the absorption and the attenuation constants, and equation is b _dc = ε_dc(λ)c _dc. ε_dc(λ), c _dc, d _dc each represent extinction coefficient, concentration of absorbing substance, and optical path length, respectively, that form extra-vascular photoplethysmography dc. λ stands for wavelength, and Hb and HbO₂ represent hemoglobin and oxyhemoglobin, respectively. d _min is the basal diameter of arterial vessel before pulsation and d _max is the maximum variation of arterial diameter during pulsation. The square of arterial diameter is proportionate to the amount of blood. If each ε and c are constant during a heartbeat, [TeX:] $\alpha = I_O {\rm e}^{ - b_{{\rm dc}} d_{{\rm dc}} }$ $\alpha =I_O{\mathrm{e}}^{-b_{\mathrm{dc}}{d}_{\mathrm{dc}}}$ and [TeX:] $\beta = e^{ - b_{{\rm Hb} + {\rm HbO}_2 } }$ $\beta =e^{-b_{\mathrm{Hb}+{\mathrm{HbO}}_2}}$ can be substituted into Eqs. 1, 2.

Fig. 1

Typical photoplethysmographic signal. (a) A raw signal measured from a photodetector. (b) Final signal, constant k-reflected light intensity.

PPG ac(I _ac) is I _L − I _H, the subtraction of onset from peak, and is shown in Eqs 3, 4, 5, and (6) using variations in optic path length Δd(= d _max − d _min).

Eq. 3

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{equation} I_{\rm ac} = I_L - I_H, \end{equation}\end{document} $$I_{\mathrm{ac}}=I_L-I_H,$$

Eq. 4

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{equation} = \alpha \beta ^{d_{\min } } - \alpha \beta ^{d_{\max } }, \end{equation}\end{document} $$=\alpha {\beta }^{d_{\min }}-\alpha {\beta }^{d_{\max }},$$

Eq. 5

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{equation} = \alpha \beta ^{d_{\min } } - \alpha \beta ^{\left({d_{\min } + \Delta d} \right)}, \end{equation}\end{document} $$=\alpha {\beta }^{d_{\min }}-\alpha {\beta }^{\left(d_{\min }+\Delta d\right)},$$

Eq. 6

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{eqnarray} I_{\rm ac} &=& \alpha ({\beta ^{d_{\min } } - \beta ^{d_{\min } + \Delta d} }) = \alpha \beta ^{d_{\min } } ({1 - \beta ^{\Delta d} })\nonumber\\ &=& (k - I_{{\rm onset}})\cdot({1 - \beta ^{\Delta d} }). \end{eqnarray}\end{document} $$\begin{array}{ccc}\hfill I_{\mathrm{ac}}& =& \alpha \left({\beta }^{d_{\min }}-{\beta }^{d_{\min }+\Delta d}\right)=\alpha {\beta }^{d_{\min }}\left(1-{\beta }^{\Delta d}\right)\hfill \\ & =& (k-I_{\mathrm{onset}})·\left(1-{\beta }^{\Delta d}\right).\hfill \end{array}$$

According to Eq. 6, I _ac (a component of ac) is inversely proportionate to I _onset, proportionate to arterial diameter variation Δd, and has nonlinear characteristics. I _onset is a dc component of photoplethysmography and stands for nonpulsatile components and basal blood volume before pulsation. Δd is affected by arterial stiffness and generates I _ac. When Eq. 6 is transformed like Eq. 7, Δd can be estimated.

Eq. 7

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{equation} \Delta d = {\rm abs}\left[ {\ln \left({1 - \frac{{I_{{\rm ac}} }}{{k - I_{{\rm onset}} }}} \right)} \right]. \end{equation}\end{document} $$\Delta d=\mathrm{abs}\left[\ln \left(1-\frac{I_{\mathrm{ac}}}{k-I_{\mathrm{onset}}}\right)\right].$$

Numerical simulation was conducted to analyze the characteristics of ac components according to I _onset and Δd. Constant k was set up as 10 because signals voltage range of acquisition in the most commercial product for measuring biomedical data is ±10 V.^{15, 16} I _onset increased in units of 0.001 from 0 to 10. Δd is converted approximately 0.04 mm when photoplethysmography is generated,⁹ so it was increased from 0 to 1 mm in units of 1 μm. The size of PPG ac calculated from each combination of I _onset and Δd was divided by the largest value, stabilized, and then analyzed.

2.3.

Experiment

In this experiment, respiration, ECG, and PPG were measured for 5 min in the supine position, as can be seen in Fig. 2. All signals were measured using ECG100C, PPG100C, and SKT100C (BIOPAC System, Inc., Goleta, California), with a 1 kHz sampling rate. PPG was measured on the left index finger, and using a TSD100 reflection-type sensor, it was set to 20× amplification in the 0.05 to 10 Hz bandwidth. ECG was used to identify the beat of PPG using a Lead II and amplified 2000× in the 0.5 to 35 Hz bandwidth.

Fig. 2

Experiment setting.

Respiration was used to observe the rapid change in ac and dc due to deep inspiration, and it was measured via the Nasal method using a TSD202 sensor.¹⁷ The experiment setting was maintained at room temperature.

3.

Result

Figure 3 is a numerical simulation of stabilized ac characteristics with constant parameters (dc, Δd) in Eq. 6. Figure 3a shows characteristics of ac components according to Δd when dc is constant. As the dc value increased, the ac slope (sensitivity) toward increase in Δd decreased. Figure 3b shows the response of ac component characteristics according to dc when Δd is constant. When Δd increased, the slope of ac raised in dc increased.

Fig. 3

Computed ac characteristics of proposed model equation. (a) ac-Δd characteristics with fixed dc and (b) ac-dc characteristics with fixed Δd.

Figure 4 shows the ac characteristics according to dc and Δd percentile distribution obtained from actual experiment. In Fig. 4a, when Δd was increased from the 10th percentile distribution to the 90th percentile distribution, stabilized ac increased from 1 ± 0 to 1.53 ± 0.40. The curve fitting equation was y = 0.000001368x ³ − 0.00022x ² + 0.01621x + 0.8627(adj. R ² = 0.9997, p < 0.0001). In Fig. 4b, when dc was increased from the 10th percentile distribution to the 90th percentile distribution, stabilized ac decreased from 1 ± 0 to 0.89 ± 0.21. The curve fitting equation was y = −0.3898x ³ + 0.3028x ² − 0.07596x + 1.009(adj. R ² = 0.9273, p < 0.0001).

Fig. 4

ac characteristics by increasing Δd and dc percentile distribution. (a) ac-Δd and (b) ac-dc.

4.

Discussion

This study investigated ac characteristics according to the nonpulsatile component (dc) and optical path length (Δd) to analyze characteristics of PPG components.

In the equation suggested to analyze ac characteristics, ac was dependent on dc and Δd. dc was inversely proportionate to ac, and Δd was proportionate to ac. Nonpulsatile components are related to tissue, vein, and diastolic arterial blood volume. Light absorption of other tissue, such as skin pigment, dye, and bone, were disregarded because it does not change according to blood flow, which means weak changes in continuous PPG signals.

From nonpulsatile components, venous capacity during respiratory change is connected to respiratory-induced fluctuation in the dc component, which is highly correlated to central and periphery venous pressure.^{1, 6} Venous capacity corresponds with venous return that affects blood capacity of the heart in each contraction. If venous tone in consistent in the venous return curve, venous return decreases with the increase in central venous pressure and decreases stroke volume. At this point, the autonomous nervous system increases heartbeat and arterial pressure to maintain homeostasis (maintain cardiac output).^{2, 7} PPG ac is proportionate to stroke volume, so when PPG dc is increased due to increase in central venous pressure, this increases arterial pressure, leading to a decrease in PPG ac components.^{2, 7}

The second nonpulsatile component in PPG is the arterial nonpulsatile component, which signifies the baseline blood capacity of the artery in contraction, and this baseline blood capacity is proportionate to diameter.²⁰ Arterial nonpulsatile components increase as arterial capacity or diameter increases. According to the stress-strain curve of the artery, the diameter is nonlinearly proportionate to blood pressure, and high blood pressure has high stiffness.²⁰ Therefore, an increase in arterial nonpulsatile components increases vascular stiffness, which then decreases ac. From the two cases above, ac decreases according to an increase in PPG dc shown in the proposed equation, and the experiment is in accord with existing physiological research.

ac characteristics according to dc and Δd generated transition characteristics starting from each median point. In experimental characteristics against dc, attenuation appeared differently at the 50th percentile, and as transition happened in Δd, ac increased. According to existing research,^{8, 21} on mechanical properties of normal subjects’ peripheral blood vessels, as blood pressure increases, compliance and distensibility of the artery exponentially decreased with an inflection point, and diameter exponentially increases. Therefore, it can be concluded that the transition characteristics of ac shown in attenuation and an increase of experimental results follow the mechanical properties of blood pressure-diameter.

In conclusion, an equation to explain ac characteristics was suggested, and its dependence on dc and Δd was verified through a simple simulation and experiment. Our results may be utilized as a basic analysis tool for noninvasive vascular assessment.