PPG Second Derivative (SDPPG/APG): What Each Wave Means and How to Use It

The second derivative of the photoplethysmogram — known as SDPPG or the acceleration plethysmogram (APG) — amplifies subtle inflection points in the PPG waveform that are nearly invisible in the raw signal. By differentiating the PPG signal twice, researchers and clinicians can extract quantitative features that track arterial aging, cardiovascular risk, and autonomic state with surprising precision.

What Is the SDPPG?

The PPG waveform is a smooth curve representing the volume changes in peripheral tissue with each heartbeat. The first derivative (velocity plethysmogram, VPG) shows the rate of change of that volume. The second derivative (SDPPG) shows the acceleration — how fast the rate of change is itself changing.

Mathematically:

• PPG: Volume-related optical signal over time
• VPG (first derivative): d(PPG)/dt — velocity of blood volume change
• SDPPG (second derivative): d²(PPG)/dt² — acceleration of blood volume change

The SDPPG was named the "acceleration plethysmograph" (APG) by Takazawa et al. in their landmark 1998 paper that characterized its waves and established their clinical correlates (American Journal of Hypertension, doi: 10.1016/S0895-7061(98)00025-4). The APG/SDPPG nomenclature is used interchangeably in the literature.

The Five Waves of the SDPPG

A single cardiac cycle in the SDPPG produces a characteristic pattern of five waves, labeled a through e:

Wave 'a' — Early Systolic Positive Wave

The first and largest positive wave. Corresponds to the early rapid acceleration of blood volume during ventricular ejection — the sharp rise of the PPG waveform. Wave 'a' amplitude correlates with cardiac contractility and stroke volume. It's the reference amplitude against which other waves are normalized.

Wave 'b' — Mid-Systolic Negative Wave

A negative deflection following wave 'a'. Represents deceleration of the acceleration — the waveform's rising slope is still positive, but rising less steeply. Wave 'b' corresponds to the mid-systolic deceleration as ventricular ejection velocity decreases. Its amplitude (expressed as b/a ratio) correlates inversely with arterial compliance: stiffer arteries produce a more negative b/a.

Wave 'c' — Late Systolic Re-acceleration

A second, smaller positive wave. Corresponds to the re-acceleration associated with the arrival of the reflected wave from peripheral vessels. In young, compliant arteries, this wave is clearly separated from 'a'. In older or stiffer arteries, it merges toward 'a' and may eventually disappear as a distinct feature.

Wave 'd' — Late Systolic Notch

A negative inflection just before the dicrotic notch. Corresponds to deceleration as the systolic reflected wave passes. The timing and depth of 'd' relative to 'c' help localize the dicrotic notch on the primary PPG waveform.

Wave 'e' — Diastolic Positive Wave

The final positive wave of the cycle, corresponding to the diastolic (reflected) wave on the primary PPG. Lower amplitude than 'a'; its prominence relative to 'a' changes with cardiac output and peripheral resistance.

The Aging Index (AGI): The Key SDPPG Clinical Metric

Takazawa et al. defined the aging index as:

AGI = (b - c - d - e) / a

All values are amplitudes (positive or negative) at the respective wave peaks, normalized by wave 'a' amplitude.

The AGI increases with vascular aging. Key properties:

• Young adults (20s): AGI typically negative (approximately -0.4 to -0.2)
• Middle age (40s): AGI approaches zero
• Older adults (60+): AGI positive (0.1 to 0.5+)
• AGI correlates with carotid-femoral PWV: r = 0.65-0.75 in most studies

The index effectively captures how much the systolic wave shape changes as arterial compliance decreases and the reflection wave timing shifts toward systole.

A validation study by Imanaga et al. (Hypertension Research, 1998) found that APG aging index correlated significantly with coronary artery disease severity in patients undergoing angiography, suggesting it captures clinically relevant vascular pathology beyond simple age-related changes.

Simplified Indices

Because the 5-wave pattern isn't always cleanly identifiable (waves 'c' and 'd' may merge in older subjects), simpler indices have been proposed:

b/a ratio: Simpler and more robust. The ratio of mid-systolic deceleration amplitude to early systolic acceleration. Becomes more negative with younger, more compliant arteries.

(c-b)/a ratio: Captures the systolic re-acceleration relative to early acceleration. Sensitive to peripheral reflection timing.

Second derivative area: Integrated area under positive/negative portions of SDPPG.

d/a ratio and e/a ratio: Diastolic wave indices with moderate clinical significance.

For noisy wearable signals where waves c and d are often indistinct, the simpler b/a ratio may outperform the full AGI for vascular age estimation — as demonstrated by Elgendi in Frontiers in Physiology (2012; doi: 10.3389/fphys.2012.00139).

How SDPPG Changes With Cardiovascular Risk Factors

The SDPPG/APG aging index tracks several key risk factors:

Hypertension: Elevated blood pressure directly stiffens arteries, shifting AGI positive. Effective antihypertensive treatment reduces AGI, sometimes to a greater degree than blood pressure reduction alone.

Diabetes: Glycation of arterial collagen is one of the most potent drivers of arterial stiffening. Diabetic patients show markedly elevated AGI compared to age-matched controls. Glucose control partially reverses this.

Smoking: Both acute (vasoconstriction) and chronic (endothelial damage, inflammation) effects increase AGI.

Physical fitness: Aerobically trained individuals typically have younger APG indices for their age. The difference can be equivalent to 10-15 years of vascular age.

Obesity: Central adiposity associates with insulin resistance and inflammatory cytokines that damage arterial walls, shifting AGI positive.

Computing SDPPG: Practical Considerations

The second derivative amplifies high-frequency noise. Applying it directly to a raw PPG signal produces an unusable output dominated by electrical noise. The processing pipeline requires:

1. Bandpass filtering: Filter PPG to 0.5-10 Hz or similar before differentiation. This removes baseline wander and high-frequency noise.

2. Numerical differentiation: Use a Savitzky-Golay filter for simultaneous smoothing and differentiation. A 5th-order polynomial over a 51-point window (at 250 Hz) provides good balance between noise rejection and temporal resolution. Simple central-difference methods introduce too much noise in the second derivative.

3. Cycle segmentation: Segment SDPPG into individual cardiac cycles using the foot detection of the original PPG.

4. Wave identification: Detect peaks and troughs within each cycle. Multiple algorithms exist; template-matching approaches outperform simple threshold methods in noisy signals.

5. Quality filtering: Cycles with poor SDPPG waveform quality (wave 'a' below threshold, non-physiological wave ratios) should be excluded.

The ChatPPG algorithm library includes full SDPPG extraction and aging index calculation with Savitzky-Golay differentiation and automated wave identification.

SDPPG in Research vs. Consumer Devices

SDPPG-based vascular age assessment has been commercially available in Japan for over two decades (Nihon Kohden's "vascular age" devices). Clinical evidence in Asian populations is strong; validation in Western populations is more limited but generally supportive.

Consumer wearable devices have been slower to adopt SDPPG, primarily because:

• Wrist PPG has lower amplitude and SNR than finger PPG, making 2nd-derivative waves harder to resolve
• Motion artifacts amplify significantly in the differentiation process
• Wave identification requires clean, artifact-free beats

Research-grade finger clip PPG devices remain the most reliable platform for SDPPG analysis. Emerging processing techniques (denoising with learned priors, adaptive Savitzky-Golay window selection) are improving wrist PPG SDPPG feasibility.

Frequently Asked Questions

What is SDPPG and what does it measure? SDPPG (second derivative of PPG) is the acceleration plethysmogram — the second derivative of the PPG waveform. It amplifies subtle inflection points in the PPG to reveal five waves (a-e) whose relative amplitudes encode arterial stiffness, vascular age, and cardiovascular health state.

What do the waves a, b, c, d, e in the APG represent? Wave 'a' is early systolic acceleration (cardiac output proxy), 'b' is mid-systolic deceleration (compliance indicator), 'c' is reflection wave re-acceleration, 'd' is the late systolic notch (pre-dicrotic), and 'e' is the diastolic wave. Together they map the mechanics of one complete cardiac ejection and reflection cycle.

What is the APG aging index? The aging index is (b - c - d - e) / a, where each letter represents the amplitude of that SDPPG wave. It's negative in young, compliant arteries and increases toward positive values with arterial stiffening. It correlates with carotid-femoral pulse wave velocity and predicts cardiovascular risk.

How does SDPPG differ from the raw PPG waveform? The raw PPG is a smooth volume-change signal. SDPPG is the second derivative, which amplifies rate-of-change information to reveal inflection points invisible in the raw waveform. It requires noise reduction before differentiation because high-frequency noise is amplified in the process.

Can SDPPG measure vascular age? Yes — population studies show the aging index correlates strongly with biological vascular age. Individuals with favorable lifestyle factors (fitness, non-smoking, controlled blood pressure) show younger SDPPG profiles than their chronological age. The index doesn't replace clinical arterial stiffness measurement but provides a non-invasive screening tool.

Why is wave 'c' absent in older subjects? In older adults with stiffer arteries, the reflected pressure wave returns to the measurement site during systole rather than diastole. This causes the reflection to merge with the primary systolic wave, eliminating the separate 'c' inflection. When wave 'c' cannot be identified, composite indices that require it (like the full AGI) are replaced with simpler b/a ratios.

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