From Optical Signals to Hemoglobin

In Lesson 2.2, we discussed the Modified Beer-Lambert Law and how we can associate a measurable change in optical density with an unknown change in chromophore concentration. Under the assumptions discussed, this formalism works, but there is still one problem remaining: it contains two unknowns (ΔHbO₂ and ΔHbR), and at a single wavelength it gives you only one equation. We are one equation short.

This lesson shows how adding a second wavelength provides the missing equation, how the two are solved simultaneously to produce the separate hemoglobin time series that are the primary output of fNIRS, and what those time series actually contain once you look at them carefully.

1. Why a Single Wavelength Is Not Enough?

Write out the MBLL for a single wavelength \(\lambda_1\), keeping only the two hemoglobin species as chromophores, and you have:

$$\Delta OD(\lambda_1)= \left[ \epsilon_{HbO_2}(\lambda_1)\cdot \Delta HbO_2 + \epsilon_{HbR}(\lambda_1)\cdot \Delta HbR\right] \cdot d \cdot DPF(\lambda_1)$$

We measure ΔOD(λ₁) at the detector. We know ε_HbO₂(λ₁) and ε_HbR(λ₁) from tables, and d · DPF(λ₁) from your geometry and assumed DPF. The two unknowns are ΔHbO₂ and ΔHbR. One equation, two unknowns.

We cannot solve this without additional information. Any combination of ΔHbO₂ and ΔHbR that satisfies the equation is equally valid (i.e., there are infinitely many solutions). We know that the combined absorption changed by a certain amount, but you cannot tell whether that was due to a change in HbO₂, HbR, or some mixture of the two.

The solution is to add a second measurement at a different wavelength. As long as the extinction coefficients of HbO₂ and HbR are different at the second wavelength from their ratio at the first (which will be true at any wavelength away from the isosbestic point), the second equation carries genuinely independent information, and the system can be solved.

This is why the isosbestic point (~805 nm) from Lesson 1.2 matters so much in practice. At this point, ε_HbO₂ = ε_HbR, so the two chromophores contribute identically to absorption at that wavelength. A second measurement there would add no new information: the two equations would be proportional to each other and the system would remain underdetermined. The isosbestic point is the one wavelength in the NIR window that is completely useless for separating HbO₂ from HbR.

2. Solving the problem for ΔHbO₂ and ΔHbR

With two wavelengths λ₁ and λ₂, the MBLL gives two equations:

$$\Delta OD(\lambda_1)= \left[ \epsilon_{HbO_2}(\lambda_1)\cdot \Delta HbO_2 + \epsilon_{HbR}(\lambda_1)\cdot \Delta HbR\right] \cdot d \cdot DPF(\lambda_1)$$ $$\Delta OD(\lambda_2)= \left[ \epsilon_{HbO_2}(\lambda_2)\cdot \Delta HbO_2 + \epsilon_{HbR}(\lambda_2)\cdot \Delta HbR\right] \cdot d \cdot DPF(\lambda_2)$$ Two equations, two unknowns (ΔHbO₂ and ΔHbR). Everything else is measured or known: ΔOD at the detector, ε from tables, d from the probe geometry, DPF from empirical estimates. Now this system can be solved algebraically, and the solution gives the concentration changes.

In practice, fNIRS requires us to solve this system for every time point in the recording and for every source-detector channel. You do not need to perform the algebra by hand. But understanding that this is what the software is doing (i.e.,  inverting a two-equation linear system at each time step, using tabulated extinction coefficients and an assumed DPF) is essential for understanding where errors can enter.

[If you are curious, the solution can be written compactly in matrix form (as \( \Delta c = E^{-1}(\Delta OD / (d\cdot DPF))\)), where E is a 2×2 matrix of extinction coefficients. This matrix notation is how the calculation appears in most software implementations and in the mathematical literature, though understanding the matrix algebra is not required here. What matters is that the operation is a simple linear inversion: you go from two measured optical density changes to two estimated concentration changes.]

The result at each channel is two time series: ΔHbO₂(t) and ΔHbR(t), expressed in micromolar (µM) changes from a baseline. These are the primary outputs of fNIRS.  (Note, in some cases, researchers prefer not to make choices regarding the effective path length in tissue; in this case, the time series will be expressed in µM/cm.) Everything downstream – statistical analysis, group comparisons, source reconstruction – operates on these time series.

3. Wavelength Selection Matters

We know we need at least two wavelengths, but we have not discussed which ones yet. It turns out that not all pairs of wavelengths are equally good at separating ΔHbO₂ from ΔHbR. The quality of the separation depends on how different the two equations are from each other, which in turn depends on how different the extinction coefficient ratios of HbO₂ and HbR are at the two chosen wavelengths.

An intuitive way to think about it is the following: if you choose two wavelengths where HbO₂ and HbR look almost identical in their absorption (for example, two wavelengths both close to the isosbestic point), then any small measurement noise in ΔOD(λ₁) or ΔOD(λ₂) gets amplified enormously in the estimated ΔHbO₂ and ΔHbR. The two equations are nearly dependent, and the solution is numerically unstable – a situation mathematically described as an ill-conditioned system. Conversely, wavelengths that maximize the spectral contrast between the two chromophores give a well-conditioned system where measurement noise translates into proportionally smaller concentration errors.

Practical wavelength choices

The first optimal strategy is to choose one wavelength below the isosbestic point, where HbR absorbs more than HbO₂, and one above it, where HbO₂ absorbs more than HbR. This maximizes the spectral contrast between the two equations. Common wavelength pairs used in commercial and research fNIRS systems include:

The choice of wavelengths is fixed by the hardware and cannot be changed during analysis. This means conditioning is a hardware-level decision with permanent consequences for data quality. Before acquiring data on a new system, it is worth checking that the wavelength pair provides sufficient spectral contrast to separate the chromophores you care about.

4. Signal Crosstalk

Even with well-chosen wavelengths, the separation of ΔHbO₂ and ΔHbR is not perfect in practice. A specific and practically important failure mode is crosstalk: a situation in which errors in the estimated ΔHbO₂ artificially drive spurious changes in the estimated ΔHbR (and vice versa), even when only one chromophore has actually changed.

The most common cause of crosstalk in CW fNIRS is an incorrect wavelength dependence of the assumed DPF. Recall from Lesson 2.2 that the DPF differs slightly between wavelengths because scattering is wavelength-dependent. If the DPF at λ₁ is assumed incorrectly relative to the DPF at λ₂, even by a small amount, the two equations in the MBLL system are effectively using inconsistent path length corrections. The inversion then attributes some of the apparent absorption change to the wrong chromophore.

Sassaroli and Fantini (2004) showed that a common formulation of the MBLL in the fNIRS literature contains a subtle notational error that, under certain conditions, introduces systematic crosstalk. Their correction clarified that the MBLL must use the differential pathlength at each wavelength independently, not a single averaged value, a distinction that seems minor but has measurable consequences for the accuracy of hemoglobin separation.

5. Decoding Hemoglobin Time Series

After solving the two-wavelength system, you have a time series of ΔHbO₂(t) and ΔHbR(t) at each channel. If you are doing a functional neuroimaging experiment with a clear stimulus, you might expect to see a clean response: HbO₂ rising a few micromoles above baseline about 2 seconds after stimulus onset, peaking around 6 seconds, and returning to baseline over the next 10-20 seconds; HbR moving in the opposite direction, though with smaller amplitude. This is the hemodynamic response driven by neurovascular coupling, which we will see in Unit 3.

But if you look at the raw fNIRS time series before any processing, the hemodynamic response is rarely obvious. What you actually see is a signal dominated by much larger oscillations that have nothing to do with your stimulus. Understanding where these come from and why they are so much larger than the brain signal is one of the most important conceptual steps in fNIRS.

Figure 1. Time courses of oxy- (HbO), deoxy- (HbR), and total (HbT) hemoglobin concentration changes.

The fNIRS signal is a superposition of multiple physiological processes

The hemoglobin concentration in tissue oscillates continuously, driven by several physiological processes operating at different timescales. Because fNIRS measures hemoglobin concentration changes with relatively high temporal resolution (typically, above 3 Hz), it picks up all of these processes simultaneously. Think of the raw fNIRS signal as a mixture of several waveforms, each with its own characteristic frequency:

Component	Typical frequency	Origin	Relative amplitude
Cardiac pulsation	~0.8–1.5 Hz (adults)	Each heartbeat pumps a bolus of blood through the vasculature, producing a rhythmic oscillation in local blood volume and hemoglobin concentration	Very large (often 10× the neurovascular signal)
Respiration	~0.15–0.4 Hz	Breathing modulates intrathoracic pressure, venous return, and arterial CO₂ (a potent vasodilator), all of which alter cerebral blood volume and oxygenation	Large (comparable to or larger than the neurovascular signal)
Mayer waves / slow vasomotion	~0.05–0.15 Hz	Spontaneous oscillations in arterial tone driven by the autonomic nervous system and baroreflex feedback; present in both scalp and brain vasculature	Moderate (and overlaps directly with the neurovascular band)
Neurovascular response	~0.01–0.05 Hz	The hemodynamic response to neural activity, driven by neurovascular coupling. The signal of interest in functional fNIRS.	Small (typically the weakest component in the raw signal)
Very slow drifts	<0.01 Hz	Slow changes in systemic physiology (temperature regulation, postural changes, emotional state) or instrumental drift	Variable

The practical consequence of this frequency structure is that the neurovascular signal most people are interested in studying is the weakest component in the raw fNIRS time series! It sits at the lowest frequencies, below the respiratory and cardiac oscillations that are often an order of magnitude larger. For this reason, extracting it requires signal processing (e.g., filtering, physiological-noise regression, or more sophisticated approaches), which we will touch on in Unit 5.

There is also an important consequence for what fNIRS measures beyond functional neuroimaging: the cardiac and respiratory components are not simply noise to be discarded. They carry genuine physiological information about heart rate variability, cerebrovascular reactivity, and autonomic function. In clinical monitoring applications (e.g., neonatal intensive care, surgery, rehabilitation), these components are often the signals of interest. The same measurement, the same physics, the same MBLL calculation produces a signal that is interpreted very differently depending on the research question.

In summary, the key message is that the MBLL gives you ΔHbO₂ and ΔHbR, which are honest, quantitative estimates of hemoglobin concentration changes within the limits of its assumptions. But those changes reflect everything happening in the tissue, not just the brain activity you are trying to study.

💭 Pause and Think

Mayer waves (slow vasomotion at ~0.1 Hz) overlap almost exactly with the frequency band of the neurovascular response. Why does this create a particular problem for fNIRS, compared to cardiac or respiratory contamination?

Think about what makes it possible to use a filter to remove one frequency component while keeping another. What happens when the signals you want and the noise you want to remove are at the same frequency? And how might this problem be different in a resting-state experiment compared to a blocked task design?

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📌 Key Takeaways

• A single wavelength gives one MBLL equation with two unknowns (ΔHbO₂ and ΔHbR) and cannot be uniquely solved. A second wavelength at a different extinction coefficient ratio provides a second independent equation, making the system solvable.
• The quality of the separation depends on the conditioning of the two-equation system, i.e., how different the two wavelengths’ extinction coefficient ratios are.  Wavelengths should be chosen one on each side of 805 nm to maximize spectral contrast. Poorly conditioned pairs amplify measurement noise into large concentration errors.
• Crosstalk occurs when errors in the wavelength-dependent DPF cause the concentration estimates to bleed into each other. It produces artificial correlations between ΔHbO₂ and ΔHbR and is a common source of error in CW fNIRS.
• The fNIRS time series is a superposition of multiple physiological processes: cardiac pulsation (~1 Hz), respiration (~0.3 Hz), Mayer waves (~0.1 Hz), and the neurovascular response (~0.01–0.05 Hz). The signal of interest is typically the weakest component in the raw data. Isolating it is the task of signal processing.

📚 Further Reading & Key References

1. Cope, M. & Delpy, D.T. (1988). “System for long-term measurement of cerebral blood and tissue oxygenation on newborn infants by near infrared transillumination.” Medical & Biological Engineering & Computing, 26(3), 289–294. [doi] — The first application of the two-wavelength MBLL system to extract ΔHbO₂ and ΔHbR from living tissue; the conceptual origin of the standard fNIRS calculation.
2. Sassaroli, A. & Fantini, S. (2004). “Comment on the modified Beer-Lambert law for scattering media.” Physics in Medicine and Biology, 49(14), N255–N257. [doi] — Essential reading on crosstalk: demonstrates that a common MBLL formulation introduces wavelength-dependent errors and proposes the corrected notation. Explains precisely why DPF must be specified per wavelength.
3. Strangman, G., Franceschini, M.A. & Boas, D.A. (2003). “Factors affecting the accuracy of near-infrared spectroscopy concentration calculations for focal changes in oxygenation parameters.” NeuroImage, 18(4), 865–879. [doi] — A systematic analysis of the factors that degrade accuracy in fNIRS hemoglobin estimation, including wavelength selection, DPF errors, and the layered head geometry. Highly recommended background reading.
4. Scholkmann, F. et al. (2014). “A review on continuous wave functional near-infrared spectroscopy and imaging instrumentation and methodology.” NeuroImage, 85, 6–27. [doi] — Comprehensive review covering wavelength selection, the MBLL calculation, and the physiological noise components in fNIRS signals.

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Society for Functional Near-Infrared Spectroscopy.

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