2.1.

Study Protocol

The data used in this study were acquired in a series of 11 hypercapnia and 11 hypocapnia measurements on 11 human volunteers, previously published in Ref. 27. Blood $\mathrm{C}{\mathrm{O}}_2$ content is an important regulator of cerebral blood flow (CBF),²⁸ so that an increase (decrease) in blood $\mathrm{C}{\mathrm{O}}_2$ leads to an increase (decrease) in CBF. These CBF changes should lead to parallel changes in cerebral $\left[\mathrm{Hb}{\mathrm{O}}_2\right]$ and opposite changes in [HbR] if cerebral blood volume and oxygen consumption stay unchanged.

Each measurement consisted of three $2\text{-}\min$ periods of hyper- or hypocapnia. Since the recovery of blood $\mathrm{C}{\mathrm{O}}_2$ to rest level takes some time, especially after hypocapnia, the hyper- or hypocapnia periods were separated by $4\text{-}\min$ intervals of rest. Hypercapnia was induced by having the subject breathe $\mathrm{C}{\mathrm{O}}_2$ -enriched air through a mask, hypocapnia was induced by hyperventilation, and during the rest periods the subject breathed normally. In the hypocapnia measurements, end-tidal $\mathrm{C}{\mathrm{O}}_2$ was 2.0 to 3.0% below the rest level (approximately 5%) at the end of each hyperventilation period. In the hypercapnia measurements, a similar increase was produced. The study protocol was approved by the ethical committee of Helsinki University Hospital, and informed consent was obtained from each subject before the study.

2.2.

Instrumentation

The NIRS device used in the measurements was developed in the Laboratory of Biomedical Engineering (currently the Department of Biomedical Engineering and Computational Science) at Helsinki University of Technology. ^{29, 30} The device is based on frequency domain technology, but only attenuation changes of signals were considered in this study, so the results are comparable to those obtainable with continuous wave technology.

At the beginning of each measurement, two light source fibers and ten detector fiber bundles were arranged on the forehead of the subject at $1\text{-}\mathrm{cm}$ intervals in two parallel rows (Fig. 1 ). Signal amplification was adjusted independently for each SD pair to avoid phase-amplitude cross talk at short measurement distances while maximizing signal amplification. During the measurement, light pulses were time-multiplexed at two wavelengths (760 and $830\mathrm{nm}$ ) and two source positions sequentially. Each pulse lasted for approximately $1\sec$ , resulting in a $4\text{-}\sec$ cycle $(2\text{sources}\times 2\text{wavelengths}\times 1\sec )$ . The raw data were averaged over each pulse, and data from different wavelengths were temporally aligned by interpolating the data to double sampling frequency (approximately $0.47\mathrm{Hz}$ ) and time-shifting data collected with one wavelength by one sample.

Fig. 1

The SD configuration and numbering used in the study. Half of the sources and detectors were in direct contact with the skin, and the rest were attached approximately $1.6\mathrm{mm}$ above the skin surface to allow examining the effects of the skin-fiber coupling on the NIRS signals. The skin-fiber contact did not appear to affect the results presented in this study. Source-detector pairs s1d1 and s1d3 are used for two-channel PCA and ICA (see Sec. 2.5), and s1d3 and s1d5 are used as reference channels for quantitative performance evaluation (Sec. 3.3). The corresponding fiber positions are marked with rectangles.

Light attenuation in tissue is commonly modeled in NIRS using the modified Beer-Lambert law:³

The modified Beer-Lambert law is based on the assumption of a homogeneous medium, whereas the brain is surrounded by layers of bone and scalp. The scalp and brain form two anatomically and physiologically separate compartments, resulting in a partial volume effect where a change in $\left[\mathrm{Hb}{\mathrm{O}}_2\right]$ or [HbR] in either compartment registers as a parallel but smaller change in the $\Delta \left[\mathrm{Hb}{\mathrm{O}}_2\right]$ or $\Delta \left[\mathrm{HbR}\right]$ signal. The contribution of cerebral tissue to the NIRS signal increases with SD separation, but scalp contribution is present at all SD separations. ^{21, 31}

Scalp bloodflow is affected by changes in heartbeat, respiration, and blood pressure, as well as neuronal and hormonal regulation of vasodilation and vasoconstriction. Hemodynamic changes in the scalp are typically nonlocalized, leading to a surface effect that typically affects $\Delta \left[\mathrm{Hb}{\mathrm{O}}_2\right]$ and $\Delta \left[\mathrm{HbR}\right]$ estimates on all NIRS channels. Since changes in the source-tissue coupling will also result in uniform attenuation changes in all signals recorded using that particular source, we define the term surface effect to signify all signal changes that are present in most or all of the measurement channels and do not originate from cerebral hemodynamic changes. To accurately estimate cerebral hemodynamics, the surface effect in the $\Delta \left[\mathrm{Hb}{\mathrm{O}}_2\right]$ and $\Delta \left[\mathrm{HbR}\right]$ signals has to be minimised.

The use of two sources and ten detectors produced a total of 20 $\Delta \left[\mathrm{Hb}{\mathrm{O}}_2\right]$ and 20 $\Delta \left[\mathrm{HbR}\right]$ signals at SD separations of approximately $1\text{to}5\mathrm{cm}$ . We have analyzed the signals separately with both PCA and ICA to identify and remove the surface effect. As the modified Beer-Lambert law only gives concentration changes relative to an arbitrarily chosen baseline level, the signals were zero meaned before applying PCA or ICA.

2.3.

Principal Component Analysis

PCA is a technique commonly used for reducing the dimensionality of a dataset.³² It is based on interpreting a set of $n$ measured signals ${\mathbf{x}}_k$ as a linear combination of $n$ uncorrelated components ${\mathbf{y}}_k$ :

Since $\mathbf{Y}$ can be expressed as $\mathbf{Y}={\mathbf{W}}^{-1}\mathbf{X}$ , the $k$ ’th principal component can be removed from the data by removing ${\mathbf{y}}_k$ from $\mathbf{Y}$ :

2.4.

Independent Component Analysis

Like PCA, ICA is also based on representing $n$ signals ${\mathbf{x}}_k$ as a combination of $n$ components ${\mathbf{y}}_k$ [Eq. 3]. However, in ICA the mixing matrix $\mathbf{W}$ is estimated so that the statistical independency of the components is maximized. ICA algorithms are based on various methods such as maximizing the nonGaussianity of components or maximum likelihood estimation for finding the optimal $\mathbf{W}$ . They also commonly employ a preprocessing method such as PCA or singular value decomposition (SVD) for whitening the data.³²

In this study we have chosen to use the second-order blind identification (SOBI) ICA algorithm.³³ Whereas many ICA algorithms treat the data as random variables with no time structure, the SOBI algorithm attempts to minimize the lagged covariances of the components, making it robust against noise in time-dependent signals such as NIRS data. The algorithm consists of three steps: whitening the data, e.g., by using the SVD of the data covariance matrix, computing time-delayed covariance matrices for the whitened data, and joint diagonalization of the covariance matrices. For the third step we used the joint diagonalization method presented in Ref. 34. A Matlab implementation of the method is available on the Internet.³⁵ The number of time lags used in calculating the covariance matrices was empirically set to 100.

A common feature of most ICA algorithms is that they do not provide any information on the relevance of the individual components (compare ${\lambda }_k$ in PCA).³² However, the spatial distribution of the components can still be used to identify the surface component. In Ref. 25, a coefficient of spatial uniformity (CSU) was defined for component $k$ as

2.5.

Effect of Channel Configuration on Principal and Independent Component Analysis

Previous NIRS studies using PCA and ICA have used dozens of SD pairs covering a large portion of the head surface.^{24, 25, 26} Also, they have generally attempted to detect localized changes caused by activation of a specific area of the brain (e.g., motor cortex). In this case, the surface tissue signal is expected to affect all SD pairs, while the cerebral signal is present only in a small number of pairs. In contrast, the cerebral hemodynamic changes caused by hyper- or hypocapnia would be present to some extent in a majority of the SD pairs, which might make it difficult to automatically distinguish between the surface and cerebral components.

We tested whether PCA and ICA would work if a minimal number (two) of SD pairs was used, so that one pair represented primarily surface tissue and the other contained also a cortical component. This would correspond to the monitoring of cerebral oxygenation with simple equipment in a clinical setting. We used as input for PCA and ICA $\Delta \left[\mathrm{Hb}{\mathrm{O}}_2\right]$ and $\Delta \left[\mathrm{HbR}\right]$ values from source 1 and detectors 1 and 3. Using detector 3, which probes only a relatively small volume of cortical tissue, allows benchmarking of filtering results against unfiltered signals recorded at SD separations of $3\text{to}5\mathrm{cm}$ . For reference purposes, we also carried out the analyses using all 20 SD pairs.

3.1.

Original Measurements

Figure 2 shows means and STDs of unfiltered $\Delta \left[\mathrm{Hb}{\mathrm{O}}_2\right]$ and $\Delta \left[\mathrm{HbR}\right]$ from three SD pairs over all hypercapnia measurements. These three SD pairs together provide a representative view of the waveforms of signals measured at different distances from the sources, so signals from other detectors are omitted to simplify the graphical presentation. The response to hypercapnia in the $5\text{-}\mathrm{cm}$ signals corresponds to the expected cerebral response, whereas the $1\text{-}\mathrm{cm}$ signals do not exhibit any sign of such response. The $3\text{-}\mathrm{cm}$ signals show features from both the 1- and $5\text{-}\mathrm{cm}$ signals, but distinguishing the cerebrovascular $\Delta \left[\mathrm{Hb}{\mathrm{O}}_2\right]$ reaction to hypercapnia from other fluctuations in the $3\text{-}\mathrm{cm}$ signal would be difficult without knowledge of the hypercapnia periods.

Fig. 2

Mean and STD of $\Delta \left[\mathrm{Hb}{\mathrm{O}}_2\right]$ (red) and $\Delta \left[\mathrm{HbR}\right]$ (blue) changes recorded with SD separations of (a) 1, (b) 3, and (c) $5\mathrm{cm}$ , corresponding to source 1 and detectors 1, 3, and 5 in Fig. 1. Since the choice of zero level does not affect the interpretation of the results, all $\Delta \left[\mathrm{Hb}{\mathrm{O}}_2\right]$ and $\Delta \left[\mathrm{HbR}\right]$ signals have been plotted using arbitrary offsets so that signal overlapping is minimized and vertical scales in different figures can be chosen uniformly. STDs are plotted for every fifth datapoint. Shaded areas mark periods of $\mathrm{C}{\mathrm{O}}_2$ breathing. (Color online only.)

Figure 3 shows corresponding averages over all hypocapnia measurements. The signal changes at $5\mathrm{cm}$ are consistent with the expected cerebrovascular reaction to a rapid hyperventilation-induced decrease and gradual recovery of blood $\mathrm{C}{\mathrm{O}}_2$ content. In contrast, although the $3\text{-}\mathrm{cm}$ signals also include a cerebral component, the hyperventilation periods could not be reliably identified from those signals alone.

Fig. 3

Mean and STD of $\Delta \left[\mathrm{Hb}{\mathrm{O}}_2\right]$ and $\Delta \left[\mathrm{HbR}\right]$ changes measured with source 1 and detectors (a) 1, (b) 3, and (c) 5 in the hypocapnia measurements. Shaded areas mark periods of hyperventilation.

In both hyper- and hypocapnia, the STD of the $1\text{-}\mathrm{cm}$ $\Delta \left[\mathrm{Hb}{\mathrm{O}}_2\right]$ signals is two to four times larger than the STD of the 3- and $5\text{-}\mathrm{cm}$ signals, and there are no consistent responses to hyper- or hypocapnia in the mean $1\text{-}\mathrm{cm}$ signals. Since the contribution of surface tissue to the signal decreases with increasing SD separation, these findings indicate that the surface effect is in practice independent of blood $\mathrm{C}{\mathrm{O}}_2$ content, and its time behavior varies greatly between individuals.

3.2.

Filtered Signals

We used PCA and ICA for two different channel configurations to filter (i.e., remove the surface component from) the $3\text{-}\mathrm{cm}$ signals corresponding to source 1 and detector 3. Means and STDs of the original, unfiltered signals are shown in Figs. 2b and 3b. The filtering was first carried out with two SD pairs, with means and STDs of the filtered signals shown in Figs. 4a and 4c (hypercapnia) and Figs. 5a and 5c (hypocapnia). Both PCA and ICA enhance the cerebral $\Delta \left[\mathrm{Hb}{\mathrm{O}}_2\right]$ waveform in hypercapnia, but in the case of hypocapnia, PCA enhances the cerebral waveform of both $\Delta \left[\mathrm{Hb}{\mathrm{O}}_2\right]$ and $\Delta \left[\mathrm{Hb}\mathrm{R}\right]$ , while the improvement in signal quality from ICA is negligible.

Fig. 4

Results of ICA and PCA filtering of $3\text{-}\mathrm{cm}$ hypercapnia signals using two different SD configurations.

Fig. 5

Results of ICA and PCA filtering of the $3\text{-}\mathrm{cm}$ hypocapnia signals using two different SD configurations.

When the filtering was repeated with 20 SD pairs, results from ICA were clearly inferior to PCA. We suspected that the performance of ICA might be related to the number of independent components estimated, so we reduced the number from 20 to 5 by reducing the dimensionality of the data in the whitening step of ICA. This is commonly done when the number of channels is much larger than the expected number of actual signals present in the data. The choice to use exactly five components was to some extent arbitrary, since there was in practice no difference between results obtained using three to ten components, and two components produced clearly inferior results with hypocapnia data. ICA performed slightly better after dimensionality reduction, producing in hypercapnia results equal to 20-channel PCA; but in hypocapnia, results from ICA remained inferior to PCA. The results of PCA filtering and five-component ICA filtering of the $\Delta \left[\mathrm{Hb}{\mathrm{O}}_2\right]$ and $\Delta \left[\mathrm{Hb}\mathrm{O}\mathrm{R}\right]$ signals corresponding to source 1 and detector 3 with 20 SD pairs are shown in Figs. 4c and 4d and Figs. 5c and 5d.

In most cases, signal changes and STDs are smaller in the filtered $3\text{-}\mathrm{cm}$ signals than in the unfiltered $5\text{-}\mathrm{cm}$ signals. This is an expected consequence of the partial volume effect in the modified Beer-Lambert law: since the $3\text{-}\mathrm{cm}$ signals sample a smaller volume of cerebral tissue than the $5\text{-}\mathrm{cm}$ signals, the magnitude of concentration changes in the $3\text{-}\mathrm{cm}$ signals after surface component removal should be smaller than in the original $5\text{-}\mathrm{cm}$ signals.

3.3.

Quantitative Performance Evaluation

For quantitative performance evaluation, an estimate of the cerebral response during the measurement period is required. The $5\text{-}\mathrm{cm}$ $\Delta \left[\mathrm{Hb}{\mathrm{O}}_2\right]$ and $\Delta \left[\mathrm{HbR}\right]$ signals are affected by both the surface effect and cortical hemodynamics, but since there is no consistent response in surface tissue to hyper- or hypocapnia [Figs. 2a and 3a], averaging the $5\text{-}\mathrm{cm}$ signals over all measurements effectively cancels out the surface effect and provides the best available estimate of the cerebral response. We therefore used the averages of the unfiltered $5\text{-}\mathrm{cm}$ signals [Figs. 2c and 3c] to represent the expected cerebral response $x_{\exp }$ in hyper- and hypocapnia, respectively.

To provide a quantitative measure of error, we calculated the mean squared error (MSE) between $x_{\exp }$ and the PCA or ICA filtered $3\text{-}\mathrm{cm}$ signal $x$ for the two channel configurations:

Figure 6 shows the mean and STD of the MSE criterion for the filtered $3\text{-}\mathrm{cm}$ signals and the unfiltered 3- and $5\text{-}\mathrm{cm}$ signals in the two SD configurations and breathing conditions. To find out whether the MSEs of the filtered $3\text{-}\mathrm{cm}$ signals and the unfiltered $5\text{-}\mathrm{cm}$ signals were lower than the MSE of the unfiltered $3\text{-}\mathrm{cm}$ signals in a statistically significant sense, the corresponding MSE distributions were compared using a Student’s $t$ -test for unequal sample variances. For 20-channel ICA, MSE values are shown for the five-component case, but similar results were obtained with three to ten components (compare Sec. 3.2).

Fig. 6

MSEs of the unfiltered 3- and $5\text{-}\mathrm{cm}$ and the filtered $3\text{-}\mathrm{cm}$ signals with respect to the averaged $5\text{-}\mathrm{cm}$ signal. Values for HbR have been multiplied by four to use the same vertical scale as $\mathrm{Hb}{\mathrm{O}}_2$ . Columns show average MSEs and errorbars their STDs. $\left({}^{*}\right)$ indicates that the average MSE is lower than that of the corresponding unfiltered $3\text{-}\mathrm{cm}$ signals at the $p<0.10$ level. $\left({}^{**}\right)$ indicates the same at the $p<0.05$ level.

Overall, both PCA and ICA reduced extracerebral contribution to the $3\text{-}\mathrm{cm}$ signal, but PCA always produced lower MSEs than ICA. The lowest MSEs in both breathing conditions were given by two-channel PCA. Results from two-channel ICA were almost equal to two-channel PCA in hypercapnia, but were clearly inferior in hypocapnia. In the 20-channel case, results from PCA and ICA were almost similar after the performance of ICA was improved by dimensionality reduction. Two-channel ICA performed poorer than 20-channnel ICA in hypocapnia, but better in hypercapnia.

PCA produced a statistically significant MSE reduction in six out of eight cases at the $p<0.05$ level, and in eight out of eight at the $p<0.10$ level (Fig. 6). The corresponding figures for ICA were only one out of eight cases at $p<0.05$ and two out of eight at $p<0.10$ . The MSE difference between the unfiltered 3- and $5\text{-}\mathrm{cm}$ signals was statistically significant in only one out of four cases at the $p<0.10$ level. The average MSEs of the PCA filtered $3\text{-}\mathrm{cm}$ signals were in seven out of eight cases lower than those of the unfiltered $5\text{-}\mathrm{cm}$ signals, while ICA filtered $3\text{-}\mathrm{cm}$ signals had a lower MSE than the unfiltered $5\text{-}\mathrm{cm}$ signals in four out of eight cases. However, these MSE differences were typically not statistically significant.

Qualitative analysis supports the results of the quantitative MSE analysis. The shape of the cerebral response to hypercapnia is enhanced more in the two-channel case than in the 20-channel case, especially by PCA (Fig. 4). In hypocapnia, the best result is achieved with two-channel PCA, while ICA filtering completely fails to enhance the shape of the cerebral response, regardless of channel configuration (Fig. 5).

3.4.

Comparison of the Coefficient of Spatial Uniformity and $\lambda$ Criteria

To determine how unambiguously the surface component could be identified with different methods, we examined the distributions of ${\lambda }_k$ and $\mathrm{CSU}\left(k\right)$ over the measurements. If the data originates from similar processes in all measurements, and if the mean value of ${\lambda }_2$ is much smaller than ${\lambda }_1$ , and STD of ${\lambda }_2$ is small compared to the mean, it is likely that the first PCA component originates from the same process in all measurements. Similar argumentation can be applied to the CSU criterion.

Means and STDs of ${\lambda }_2$ and CSU(2) in the two-channel configuration are shown in Table 1 . The mean values of both ${\lambda }_2$ and CSU(2) are less than half of ${\lambda }_1$ and CSU(1), respectively, and ${\lambda }_2$ is significantly smaller than CSU(2) ( $p<0.05$ using student’s $t$ -test for unequal variances).

Table 1

Means and STDs of λ2 and CSU(2) in the two-channel configuration, when λ1 and CSU(1) have been normalized to unity.

Mean and STD of the five largest $\mathrm{CSU}\left(k\right)$ and ${\lambda }_k$ in the case of 20 SD pairs are shown in Fig. 7 . The distributions of $\mathrm{CSU}\left(k\right)$ and ${\lambda }_k$ are fundamentally different, and CSU(2) and even CSU(3) are much closer to CSU(1) than ${\lambda }_2$ is to ${\lambda }_1$ . Also, the STDs are larger for $\mathrm{CSU}\left(k\right)$ than for ${\lambda }_k$ . $\mathrm{CSU}\left(k\right)$ are shown for ICA with the number of data dimensions reduced to five. Without dimensionality reduction, the mean value of CSU(2) was approximately 0.8 and that of CSU(3) approximately 0.6. This suggests that in the case of ICA, the CSU criterion for identifying the surface component is prone to error when the number of channels increases. On the other hand, in the case of PCA, the good filtering results and small mean value of ${\lambda }_2$ confirm the assumption that the first PCA component corresponds to the surface effect.

Fig. 7

Means and STDs of ${\lambda }_k$ and $\mathrm{CSU}\left(k\right)$ in the 20-channel configuration, with ${\lambda }_1$ and CSU(1) normalized to unity. For graphical simplification, data from hyper- and hypocapnia measurements were combined before calculating the distributions, and both ${\lambda }_k$ and $\mathrm{CSU}\left(k\right)$ are shown for $k⩽5$ , since only five ICA components were estimated. For each of the four groups, the leftmost dot corresponds to $k=1$ and the rightmost to $k=5$ .