There are few experimental data available in the literature with which to compare the model simulations for O2Hb and HHb, although there is more data for the ABP–CBFV phase shift. The primary source used here is the table of phase angles at 0.1 Hz for ABP–O2Hb (−24°) and ABP–HHb (−210°) given by Reinhard et al.18
Note that the ABP–CBFV phase is also given, which will be examined in detail later. The values predicted by the model here are found to be −23° and −257°, respectively. The O2Hb phase is thus in very good agreement and the HHb phase in reasonable agreement: however, note that no optimization for the hemoglobin part of the model response has been performed here. Of particular importance is the fact that the ABP–O2Hb is negative and the ABP–HHb is more negative than −180°. In theory, it would be possible to optimize over the parameter space to remove two degrees of freedom from the model, in a similar manner to that performed by Payne and Tarassenko12
: however, for reasons that will become obvious below, we simply provide a comparison between model and experiment at this point.
At very low frequencies, the HHb response to fluctuations in ABP tends toward:
and that of O2Hb toward:
The HHb response will be either positive or negative (i.e., in phase or out of phase) dependent solely upon whether αv
is less than or greater than 1. Since αv
is the ratio of two time constants, this means that if the time constant governing venous transit time (v
) is greater than the time constant governing venous outflow (Rlvv
); then HHb will be out of phase with ABP at low frequencies. For the values quoted in Payne,11
the former is some 5–6 times greater.
The parameter αv
, in fact, is also the scaling factor between fractional changes in venous flow and venous volume. It thus has significance beyond being simply the ratio between two time constants (although this additional interpretation lends further interest to its experimental value). For a simple compliant vessel, the ratio between changes in flow and volume must be at least 2 (resistance being inversely proportional to radius to the power 4 and volume being proportional to radius squared). In practice, it is bigger than 2: there has been a very large literature on this subject since the first study by Grubb et al.
who showed an exponent of 0.38 between CBV and CBF. Hence HHb is always likely to be out of phase with ABP at low frequencies.
Likewise, the O2Hb response will be out of phase with ABP at low frequencies if
Interestingly, this is strongly dependent upon the level of autoregulation present. At very high levels, the O2Hb will be out of phase with ABP, at very low levels, it will be in phase. For the values given in , Eq. (20)
is not satisfied, which thus results in O2Hb being in phase. However, it is possible in subjects with overactive autoregulation for both O2Hb and HHb to be out of phase with ABP: for the values of parameters used here, α2
would have to increase from 0.29 to 2.17. This is a very large increase, so is unlikely to be observed in practice without concurrent changes in other non-dimensional parameters. It can easily be shown that the venous O2Hb must always be in phase with ABP, but that arterial O2Hb will only be in phase if β2
. It is therefore possible for the two compartments to be out of phase with each other: however, since the venous O2Hb component dominates, due to its much larger volume, it is very unlikely that total O2Hb will be out of phase with ABP. All of these results also hold directly for the responses to changes in arterial CO2
The experimental results for phase angle quoted above were measured in a study where paced breathing was used at 6 breaths per minute (0.1 Hz). This increases the power at this frequency in the signals being recorded, enabling more robust measures of gain and phase to be measured. However, it also increases the power in the arterial CO2
concentration at this frequency, which may influence the inferred phase angle between ABP and O2Hb. It has previously been shown14
that not accounting for the effects of CO2
fluctuations can give rise to a significant error in the phase angle calculated between ABP and CBFV. Using the transfer functions derived here, it is possible to estimate the error that would be induced by CO2
fluctuations to see whether this is likely to be a significant effect.
It was shown by Peng et al.14
that this error in inferred phase angle is
denotes the amplitude of the signal, and H
and ϕ denote the gain and phase of the transfer functions, respectively, the subscripts “C” and “P” denoting the two inputs (CO2
and ABP) driving the output. The phase of the CO2
fluctuations relative to the ABP fluctuations is given by θC
. Taking the values for transfer function magnitude and phase predicted by the model and assuming that the fractional fluctuations in both ABP and CO2
are the same, as a first-order approximation, and in phase with each other, giving the smallest error, gives a phase error of 16.6°.
Since both O2Hb and HHb are directly driven by flow and volume changes, which are in turn driven by fluctuations in ABP and CO2, the compensation in phase angle due to variability in CO2 will be the same for both hemoglobin signals and for CBFV since it only directly affects the behavior of CBF: note that this makes the phase angle between the two hemoglobin signals a much more robust measure than either signal with respect to ABP in the presence of unmeasured variability.
Since there are no data in the literature to enable us to compare the effects of paced breathing on the O2Hb and HHb responses, we examine its effect on the ABP–CBFV response. As outlined above, changes in CO2 power at this frequency will have the same effect on the O2Hb and HHb responses as on the CBFV response. Any results derived for the ABP–CBFV phase angle will thus be identical for the ABP–O2Hb and the ABP–HHb responses.
There has been a series of studies by Reinhard and co-workers examining the phase angle at 0.1 Hz between ABP and CBFV in a range of subject groups and under conditions of both spontaneous and paced breathing. When examining unilateral stenosis subjects, for stenosis levels of 90–99% (43 subjects), the phase angle was found to be 20° and 39° on the ipsilateral and contralateral sides, respectively, for spontaneous breathing conditions, compared to 32° and 66° for paced breathing conditions; whereas for 100% stenosis levels (24 subjects), the corresponding phase angles were found to be 24° and 42° for spontaneous breathing and 31° and 48° for paced breathing conditions, respectively.15
They concluded that “Inter-method agreement [between the two methods] … is poor for phase,” the phase angle being consistently larger under conditions of paced breathing for both sides and for both groups, indicating both that the change in breathing conditions has a significant effect on the phase angle and that this effect is larger on the unaffected side than on the affected side.
In a similar study, performed solely under paced breathing conditions,18
a comparison between control subjects and subjects with stenosis found that the phase angle for ABP–CBFV was 64° in the controls, whereas it reduced to 34° for the ipsilateral side and remained stable at 67° on the contralateral side for the subjects with stenosis. These values are in good agreement with the results above for 90–99% stenosis levels. The corresponding results for ABP–O2Hb and ABP–HHb were −24° and −209° for the controls and −29° and −261° for the ipsilateral side and −13° and −205° for the contralateral side in the stenosis subjects. The phase angle was thus significantly altered in the ipsilateral side compared to the control group for both ABP–CBFV and ABP–HHb, but not for ABP–O2Hb. The changes in the O2Hb and HHb phase angles under conditions of stenosis are thus noticeably different, which cannot be explained purely in terms of the change in vascular reactivity. There are thus additional confounding factors present, which will need further investigation: however, given that this is the only available study in the literature, it is difficult to draw too many firm conclusions.
In another similar study, performed on stroke subjects under conditions of spontaneous breathing, it was found that there was no significant difference in ABP–CBFV phase between affected and unaffected sides, between controls and stroke subjects or between different days after treatment.16
It was thus concluded that there is no “major
disturbance of dynamic autoregulation in acute ischemic stroke.”
There is a clear difference between the results from these studies when obtained under conditions of either spontaneous breathing or paced breathing. There is also an alteration in the difference between control subjects (or unaffected hemispheres) and affected hemispheres under the different conditions: the ABP–CBFV phase is significantly larger under conditions of paced breathing in control subjects and unaffected hemispheres than under spontaneous breathing, whereas for the affected sides, there is a much smaller difference under the two breathing conditions.
The static vascular reactivity to CO2
has also been measured for the same subject groups, enabling a comparison to be made between the ipsilateral and contralateral sides in stenosis.17
In the unaffected hemispheres, the CO2
reactivity was approximately 2.1%/mmHg, but on the affected sides it was around 1.2%/mmHg (based on 58 subjects with severe unilateral stenosis). This indicates that the reactivity declines by approximately 40% under conditions of severe stenosis. This drop in reactivity under conditions of reduced flow has been previously reported by Reivich,19
and the values quoted are in very close agreement with each other. The resulting phase error would then drop to 11.5°, compared to 16.6°, as calculated by Eq. (21)
, if it is assumed that this fractional drop in reactivity is the same at 0.1 Hz. The predicted alteration in inferred phase angle under different breathing conditions and the predicted change in this alteration when considering conditions of reduced vascular reactivity are both in good qualitative agreement with the experimental data outlined above.
Two other studies have been performed to examine the effects of stroke on dynamic cerebral autoregulation.1,2
However, it is difficult to compare these results with those above, since the autoregulation response is quantified in terms of changes in AutoRegulation Index (ARI), an index developed to measure autoregulation status on a scale of 0–9.20
Deriving the phase angle for comparison with the results above and the model proposed here is thus not simple. It should also be noted that the results obtained in these two studies were calculated using only short transient events in the signals, rather than the complete frequency response. We do not thus consider these results further.
Having investigated the phase relationships between ABP and O2Hb/HHb in the context of the CO2
response, the effects of varying the model parameters are now investigated, similar to the approach set out in Payne and Tarassenko,12
where the effects of changes in autoregulation gain and time constant were quantified in terms of the IR. Here, the effects of the same parameters are characterized in terms of the phase relationships at 0.1 Hz. The variation in phase angles at 0.1 Hz with fractional nondimensional gain and time constant are shown in and , respectively.
Variation in phase angle for O2Hb and HHb with nondimensional feedback gain.
Variation in phase angle for O2Hb and HHb with feedback time constant.
Phase angle at 0.1 Hz changes by only a relatively small amount as autoregulation status changes: if feedback is abolished completely both the ABP–O2Hb and ABP–HHb phase angles decrease by only 29° and 25°, respectively. There is thus approximately a 2.5°–3° change in phase angle for every 10% drop in autoregulation feedback. The phase angle is less sensitive to changes in the feedback time constant, with a turning point: it would not be possible to infer the time constant from the phase angle. The change in phase angle at 0.1 Hz with variations in the feedback status would seem to indicate it is likely to be difficult to infer changes in autoregulation status solely by measuring this phase angle, unlike the way in which the phase angle between ABP and CBFV has been used to assess autoregulatory status. Note also that the phase angle between O2Hb and HHb, which is much less sensitive to fluctuations in CO2, is almost completely invariant with feedback gain. A further difficulty with using O2Hb and HHb as measures of autoregulation is that there are many more parameters controlling their behavior and changes in autoregulation gain are unlikely to be unaccompanied by other changes in physiological parameters.
Having investigated the effects of changes in both autoregulation status (through the model) and CO2 reactivity (through experimental data interpreted in the context of this model), we are finally able to consider the likely role of autoregulation impairment under conditions of stenosis or stroke. The effects of changes in autoregulation status on phase angle are likely to be of the order of only 10–20° for even large changes in autoregulation status, but changes in phase angle of similar or greater magnitude are observed experimentally when breathing patterns are altered from spontaneous to paced breathing. This leads us to propose the hypothesis that under conditions of stenosis or stroke, it is changes in vascular CO2 reactivity, which are well established as being caused by changes in flow, that are more significant, any changes in autoregulation status having a much smaller effect on the phase angle. It is thus likely to be difficult to infer any such changes in autoregulation status under differing physiological conditions.
However, it should be stressed that the scarcity of available data in the literature means that there are few results from which to draw such a substantive conclusion in the context of stroke, brain injury or brain trauma. Even the effects of aging on this phase angle are yet to be determined experimentally. Such studies will be extremely valuable in helping to assess the clinical potential of NIRS as a marker of changes in brain function and in helping to understand the underlying physiological processes and how they are altered under situations of physiological stress. It is likely, though, that multi-modal measurements of CBFV, O2Hb, and HHb, performed in response to a variety of different stimuli and interpreted in the context of a model such as the one proposed here, will open up the greatest possibility of interpreting cerebral dynamics fully.