The critical closing pressure contribution to dynamic cerebral autoregulation in humans: influence of arterial partial pressure of CO2

Dynamic cerebral autoregulation (CA) is often expressed by the mean arterial blood pressure (MAP)–cerebral blood flow (CBF) relationship, with little attention given to the dynamic relationship between MAP and cerebrovascular resistance (CVR). In CBF velocity (CBFV) recordings with transcranial Doppler, evidence demonstrates that CVR should be replaced by a combination of a resistance–area product (RAP) with a critical closing pressure (CrCP) parameter, the blood pressure value where CBFV reaches zero due to vessels collapsing. Transfer function analysis of the MAP–CBFV relationship can be extended to the MAP–RAP and MAP–CrCP relationships, to assess their contribution to the dynamic CA response. During normocapnia, both RAP and CrCP make a significant contribution to explaining the MAP–CBFV relationship. Hypercapnia, a surrogate state of depressed CA, leads to marked changes in dynamic CA, that are entirely explained by the CrCP response, without further contribution from RAP in comparison with normocapnia.


Introduction
In healthy individuals, rapid changes in mean arterial blood pressure (MAP) induce corresponding changes in cerebral blood flow (CBF), which subsequently return to original levels, within a few seconds, a phenomenon that has been termed dynamic cerebral autoregulation (CA) (Aaslid et al. 1989). This response of CBF to changes in MAP is predominantly controlled by myogenic mechanisms (Faraci et al. 1989), with the probable additional involvement of metabolic control (Willie et al. 2014), as expressed by the strong influence of arterial CO 2 (P aCO 2 ) on the efficiency of dynamic CA (Aaslid et al. 1989;Murkin, 2007;Hoiland et al. 2019). Both myogenic and metabolic pathways lead to alterations in vascular smooth muscle (VSM) contractility, with consequent changes in the diameter of intra-cerebral arteries, which will then control CBF due to corresponding changes in cerebrovascular resistance (CVR). By far, most physiological and clinical studies of dynamic CA have focused on the dynamic relationship between MAP and CBF (Panerai, 1998;van Beek et al. 2008;Czosnyka et al. 2009;Claassen et al. 2016), with only a few studies reporting on the temporal changes in CVR that ultimately determine the CBF response (Aaslid et al. 1989;Garnham et al. 1999;Hughson et al. 2001;Edwards et al. 2002;O'Leary et al. 2004;Panerai et al. 2006). However, a major limitation of these studies exists: adopting the definition of CVR as the ratio of MAP/CBF (or CBF velocity (CBFV), given the dominant use of transcranial Doppler ultrasound in these studies), there is the inherent assumption that the instantaneous pressure-velocity relationship passes through the origin. In other words, according to this formula, CBF becomes zero only when MAP = 0. However, the overwhelming evidence is that, similar to the lung (Burton, 1951), the cerebral circulation also shows the existence of a critical closing pressure (CrCP), whereby CBF stops at blood pressure values significantly greater than zero (Panerai, 2003). As modelled by Burton (1951), CrCP is likely to result from the balance between transmural pressure, hence reflecting the influence of intracranial pressure (ICP), and arterial wall tension, which would depend on vessel diameter, VSM tone and its activation (Burton, 1951). In fact, CrCP has been shown to be highly sensitive to changes in P aCO 2 (Panerai, 2003), and it has been suggested that it could show a greater association to metabolic mechanisms of VSM contraction, as distinct from myogenic control (Panerai et al. 2005;Panerai et al. 2012). Accepting that CrCP > 0, a more realistic model of the pressure-flow (or velocity) relationship would be CBFV = (MAP − CrCP)/RAP, where RAP is the resistance-area product. RAP represents the slope of the instantaneous pressure-velocity relationship and takes into account that mean cross-sectional blood velocity is the absolute flow divided by arterial cross-sectional area (Evans et al. 1988).
The distinct contributions of CrCP and RAP to the dynamic CBFV response to changes in MAP have been reported in previous physiological (Maggio et al. 2013) and clinical studies, including for conditions such as stroke , autonomic dysfunction (Castro et al. 2014) and pre-eclampsia (van Veen et al. 2015). For this purpose, one attractive approach is sub-component analysis, whereby any changes in CBFV can be broken down into the separate contributions of MAP, CrCP and RAP (Panerai et al. 2005). In the present study, we have merged this approach with the estimation of the CBFV response to a step change in MAP, as one of the conventional techniques that is often adopted for assessment of dynamic CA (Panerai et al. 2001). The CBFV step response approach is intrinsically linked to the autoregulation index (ARI), proposed by Tiecks et al. (1995), in the form of 10 template curves that represent values of ARI ranging from zero (absence of CA) to 9 (best observed CA). We and others have obtained estimates of the CBFV step response using frequency-or time-domain techniques to model the dynamic MAP-CBFV relationship (Panerai et al. 1998b;Edwards et al. 2004;Liu et al. 2005;Elting et al. 2014;Marmarelis et al. 2014) from which values of ARI can be obtained by comparison with the Tiecks template curve providing the best match (Panerai et al. 1998b).
Step responses have been reported for CVR ) and RAP (Panerai et al. 2006), but not for CrCP. Given the relevance of RAP and CrCP as determinants of the instantaneous blood pressure (BP)-CBFV relationship in each cardiac cycle, further investigations are needed on their contribution to the regulation of CBF, particularly their role in the dynamic response of CBF to rapid changes in MAP. In this study, we used sub-component analysis to obtain separate estimates of CBFV, RAP and CrCP step responses to changes in MAP in healthy subjects, during normocapnia and hypercapnia, to test two interrelated hypotheses. First, that during normocapnia CrCP makes a significant contribution to the dynamic CA response to sudden changes in MAP. Second, that during hypercapnia, a surrogate state for impairment of dynamic CA, the CrCP step response is the dominant sub-component to explain the deterioration in CA induced by breathing 5% CO 2 in air.

Ethical approval
All studies were approved by the UK Research Ethics Committees from Southampton and South West Hampshire (10/H0502/1), Nottingham (11/EM/0016), North East-Newcastle and North Tyneside (14/NE/1003) and the University of Leicester (jm591-c033). All procedures were conducted in accordance with the Declaration of Helsinki and all participants provided written informed consent.

Subjects and measurements
The study re-analysed recordings performed in healthy subjects, included in the Leicester database (Patel et al. 2016), who underwent measurements during normocapnia and hypercapnia, induced by breathing 5% CO 2 in air (Katsogridakis et al. 2013;Maggio et al. 2013;Llwyd et al. 2017;Minhas et al. 2018a). Subjects were 18 years of age or older without any history or symptoms of cardiovascular, neurological or respiratory disease.
Recordings were performed at the University of Leicester's Cerebral Haemodynamics in Ageing and Stroke Medicine (CHIASM) research laboratory, which was maintained at controlled temperature (20-23°C) and with minimal auditory or visual distraction. Volunteers were asked to avoid heavy exercise, caffeine, alcohol and nicotine for at least 4 h prior to measurements. CBFV was measured in both middle cerebral arteries (MCAs) using transcranial Doppler ultrasound (TCD; Vyasis Companion III, Vyasis Healthcare, Hochberg, Germany) with 2 MHz probes secured in place using a head-frame. End-tidal CO 2 (ET CO 2 ) was recorded continuously via nasal prongs (Salter Labs, Lake Forest, IL, USA) or a face mask, by a capnograph (Capnocheck Plus, Smiths Medical ASD, Inc., Norwell, MA, USA). Heart rate was derived from a three-lead electrocardiogram (ECG). BP was recorded continuously using a Finapres/Finometer device (FMS, Finapres Measurement Systems, Arnhem, Netherlands), attached to the middle finger of the left hand. Systolic and diastolic BP were measured by classical brachial sphygmomanometry before each 5 min recording. The servo-correcting mechanism of the Finapres/Finometer was switched on and then off prior to measurements.
Data were simultaneously recorded onto a data acquisition system (PHYSIDAS, Department of Medical Physics, University Hospitals of Leicester, Leicester, UK) for subsequent off-line analysis using a sampling rate of 500 samples/s.

Experimental protocol
All recordings followed a similar protocol and were performed by investigators trained to rigorous standard procedures. After resting for 20 min in the supine position with the head elevated at 30°, two 5 min recordings were performed in each subject. The first recording corresponded to baseline resting conditions with subjects breathing ambient air. In the second recording, after a 60 s period of breathing air, subjects were switched to breathing 5% CO 2 in air, through a face mask that was tightly fitted to avoid leakage, as confirmed by visual inspection of the end-tidal CO 2 waveform. After 3 min of CO 2 breathing, subjects were returned to ambient air and a further 60 s was recorded during return to normocapnia.

Data analysis
All signals were visually inspected to identify artefacts; noise and narrow spikes (<100 ms) were removed by linear interpolation. CBFV channels were subjected to a median filter and all signals were low-pass filtered with a eighth order Butterworth filter with cut-off frequency of 20 Hz. BP was calibrated at the start of each recording using systolic and diastolic values obtained with sphygmomanometry. The R-R interval was then automatically marked from the ECG and beat-to-beat J Physiol 598.24 heart rate (HR) was plotted against time. Occasional missed marks caused spikes in the HR signal; these were manually removed by remarking the R-R intervals for the time points at which they occurred. Mean, systolic and diastolic BP and CBFV values were calculated for each cardiac cycle. CrCP and RAP were calculated using the first harmonic method (Panerai et al. 2011). The end of each expiratory phase was detected in the ET CO 2 signal, linearly interpolated and resampled with each cardiac cycle. Beat-to-beat data were spline interpolated and resampled at five samples/s to produce signals with a uniform time base.
Transfer function analysis (TFA) of the MAP-CBFV relationship was performed using Welch's method (Welch, 1967) with data segmented with 102.4 s duration and 50% superposition (Claassen et al. 2016). The mean values of MAP and CBFV were removed from each segment and a cosine window was applied to minimise spectral leakage. The squared coherence function, gain and phase frequency responses were calculated from the smoothed auto-and cross-spectra using standard procedures (Panerai et al. 1998a;Claassen et al. 2016). The CBFV step response to the MAP input was estimated using the inverse fast Fourier transform of gain and phase (Bendat & Piersol, 1986). ARI, which represents dynamic CA, was extracted by using the normalised minimum square error (NMSE) fit between the CBFV step response and one of the 10 model ARI curves proposed by Tiecks et al. (1995). ARI values were only accepted if the mean squared coherence function for the 0.15-0.25 Hz frequency interval was above its 95% confidence limit, adjusted for the corresponding degrees of freedom, and the NMSE was ࣘ0.30 ). Sub-component analysis. As detailed in the Appendix, the relationship between mean CBFV, MAP, CrCP and RAP for each cardiac cycle can be expressed as a function of three distinct sub-components, representing the different contributions to a change in CBFV, that is: When this expression is normalised as a percentage, the total percent change in V MCA can then be explained by the corresponding sum of the percent changes in the sub-components, that is V MAP , V CrCP and V RAP (Appendix). For normalisation as a percentage, and also to obtain the beat-to-beat differences from the mean ( v, p, c, r; eqn (A3)), the mean values of CBFV, MAP, CrCP and RAP were obtained for the entire duration of each 5 min segment of data.
In the above expression for V MCA , if V MAP changes suddenly, as a perfect step function, the resulting changes in V MCA , V CrCP and V RAP will be the corresponding step responses for the V MAP input. Each of these three-step responses were derived by TFA, as described above for the CBFV-MAP relationship, and from the above expression it is possible to write: where SRV * MAP is an estimate of the MAP step change, based on the summation of the three other step responses (SR). Therefore, the expression above works as a 'checksum' for the correctness of the three distinct SR estimated for V MCA , V CrCP and V RAP .

Statistical analysis
As explained above, SRV MCA was only accepted based on the dual criteria of coherence above the 95% confidence limit and a NMSE ࣘ0.30 for fitting of the Tiecks model (Tiecks et al. 1995;Panerai et al. 2016). For reasons that will be discussed later, SRV CrCP and SRV RAP were only accepted if their corresponding transfer functions both had mean coherence above the 90% confidence limit (in the 0.15-0.25 Hz range), for at least one of the hemispheres, assuming that SRV MCA had also been accepted, for both normocapnia and hypercapnia. In other words, for the right or left hemisphere to be accepted, all three-step responses for that hemisphere had to be accepted for both normocapnia and hypercapnia.
With the exception of the mean coherence, all other parameters were treated as normally distributed (Patel et al. 2016). The distribution of coherence was expressed by its median (interquartile limits), all other parameters as the mean ± SD. Differences between parameters were assessed using the paired Student's t test or the Wilcoxon test. In each of the SR, values were averaged for three distinct time intervals; T1 (0.6-1.4 s), T2 (3-4 s) and T3 (7-10 s). T1 corresponds to the peak of the SRV MCA , T2 the beginning of its plateau phase, and T3 the tail of the response (Figs 1A and 2A). Differences in SR values from T1 to T2 and then at T3 were used with repeated measures ANOVA to test the effects of breathing condition (normocapnia vs. hypercapnia) and relative contribution of each SR to explain the V MCA change. SRs and ARI values were averaged for the right and left hemispheres when both were accepted. A P-value of <0.05 was adopted to indicate statistical significance.

Results
A total of 120 good quality recordings were extracted from the database for healthy subjects with measurements of CBFV, BP and ET CO 2 for both normocapnia and hypercapnia. Forty-four subjects were rejected as the strict conditions imposed for acceptance of all three SR were not met by either hemisphere, for both normocapnia and hypercapnia. The remaining 76 subjects (36 women) had mean ± SD age 43.3 ± 13.8 years, range 21-70 years.
In six of these subjects, CBFV was not recorded in the left MCA. Hypercapnia induced changes in CBFV, mean and diastolic BP, heart rate and mean CrCP (Table 1), but did not affect mean RAP in either hemisphere.
The average coherence in the frequency range 0.15-0.25 Hz for the V MAP -V MCA transfer function for both normocapnia (0.74 (0.58, 0.85)) and hypercania (0.76 (0.55, 0.86)) was higher than corresponding values for the V MAP -V RAP (0.23 (0.18, 0.33); 0.26 (0.19, 0.32)) and V MAP -V CrCP (0.29 (0.21, 0.39); 0.29 (0.22, 0.39)) transfer functions, respectively. Figure 1 presents the SRV MCA , SRV RAP and SRV CrCP for a representative subject with corresponding population averages given in Fig. 2. Noteworthily, Fig. 1D demonstrates that the 'checksum' for SRV MAP yields a temporal pattern that confirms the correctness of the three other SRs that were calculated independently. A similar result was obtained for all subjects.
Following the step change in V MAP , SRV MCA showed a sudden change, followed by a relatively rapid recovery towards baseline values, which was more pronounced during normocapnia as compared to hypercapnia (Figs 1A and 2A; P < 0.0001). The effects of hypercapnia were also reflected by the ARI, corresponding to 5.70 ± 1.58 and 4.14 ± 2.05 (P < 0.0001) for normocapnia and hypercapnia, respectively. The negative temporal pattern of the SRV RAP (Figs 1B and 2B) reflects its relevant contribution to the reduction of the SRV MCA , without a difference between normocapnia and hypercapnia. V CrCP presented a more complex step response (Figs 1C and 2C), undergoing a dip around 2-3 s after the V MAP step change, followed by a slow recovery, which was more pronounced during hypercapnia (P < 0.0001). Due to this biphasic pattern, SRV CrCP initially contributed to reduce SRV MCA , but after 5-6 s it tended to increase SRV MCA . After reaching a peak around t = 1 s, SRV MCA showed a sharp drop, up until around t = 3 s, due to an active CA (Figs 1A and 2A). On average, this drop was approximately 59% for normocapnia and 39% for hypercapnia, and it was dominated by the SRV RAP contribution (Table 2). Noteworthily, SRV CrCP had a highly significant contribution during both normocapnia and hypercapnia to explain the overall SRV MCA variance (   Values are means ± SD for n = 76 subjects, unless specified, averaged for the entire 5 min recording. P-values for paired t tests between normocapnia and hypercapnia. BP, blood pressure; CBFV, cerebral blood flow velocity; CrCP, critical closing pressure; RAP, resistance-area product; R/L, right/left MCA.

Table 2. Change in step responses (SR) of sub-components during CBFV recovery phase (1-3 s) for both normocapnia and hypercapnia
Step At a later stage of the response (t > 7 s), the two-way ANOVA indicated that the differences in SRV MCA between normocapnia and hypercapnia were solely due to SRV CrCP (P < 0.0001), without a contribution from SRV RAP (Fig. 3).
Of note, a posteriori analysis of the data, for n = 120 subjects, thus including the 44 subjects that were removed due to low TFA coherence, did not show any significant differences in comparison with the results in Figs 2 and 3 and Tables 1 and 2 (results not shown).

Discussion
In the brain, as well as in several other organs, reductions in transmural pressure can lead to the collapse of small vessels, resulting in the cessation of blood flow, thus defining the critical closing pressure of each particular circulation (Burton, 1951;Garnham et al. 1999;Panerai, 2003;Panerai et al. 2005Panerai et al. , 2011Varsos et al. 2013;Castro et al. 2014;Minhas et al. 2018aMinhas et al. , 2019. Despite evidence that CrCP and RAP, the slope of the instantaneous relationship between MAP and CBFV, might make distinct contributions to the regulation of CBF (Panerai et al. 2012), the role of CrCP in dynamic CA, as expressed by the temporal response of CBF (or CBFV) to rapid changes in MAP, has been largely ignored. Based on a relatively large group of healthy individuals, we have demonstrated that CrCP and RAP make significant, albeit distinct, contributions to the dynamic CA response, with the corollary that it is the CrCP response, and not the RAP, that explains the depression in dynamic CA efficiency observed during hypercapnia.

Main findings
Sub-component analysis (SCA) expresses changes in CBFV (V MCA ) broken down into the separate contributions of MAP (V MAP ), RAP (V RAP ) and CrCP (V CrCP ) (Panerai et al. 2005). Due to the linearity of transfer function analysis, a step change can also be parcelled out into the sum of the corresponding distinct SRs, as demonstrated by the 'checksum' in the Methods and Appendix, and in Fig. 1D. Our findings agreed with the temporal pattern of the SRV MCA , described in several previous studies (Panerai et al. 2006 and its changes due to hypercapnia Katsogridakis et al. 2013). We have also confirmed that the temporal pattern of the SRV RAP tends to follow a gradual change, similar to an exponential curve, which was reported in a few studies (Panerai et al. 2006(Panerai et al. , 2011, with a pattern similar to that described for the CVR step response as well . Of note, when the RAP or CVR step responses are calculated using standard units, the resulting curves tend to rise continually, as expected from the myogenic response, leading to a corresponding reduction in CBF. With the SCA method, though, V RAP is expressed in units of flow/velocity, with the negative sign in eqn (A9) meaning that vasoconstriction will lead to a reduction in V RAP , with consequent reductions in CBFV. For this reason, the SRV RAP showed the negative curves displayed in Figs 1B and 2B.
Our main novel finding, though, was that the SRV CrCP made a highly significant contribution to explaining the V MCA changes in response to a step change in V MAP , and that it was solely responsible for explaining the changes in SRV MCA during hypercapnia. Taken together, these two original findings lead to the acceptance of our two main hypotheses, and should have considerable impact on our understanding of the dynamic CA response in humans. As discussed below, including the role of V CrCP in future investigations could represent a paradigm shift in our ability to progress towards a more detailed assessment of dynamic CA in physiological and clinical studies.

Methodological considerations
Similarly to ARI (Tiecks et al. 1995), the majority of indices proposed for assessment of dynamic CA are encapsulated in a single number, such as the rate of regulation (RoR) (Aaslid et al. 1989), the mean flow index (Mx) (Czosnyka et al. 1996) or the pressure-reactivity index (PRx) (Czosnyka et al. 1997). Although expressing the effectiveness of CA with a single number is convenient, mainly in clinical applications, it is unlikely that the complexity of the multiple mechanisms underlying the CA response can be reduced to a unidimensional entity. In TFA of the dynamic relationship between MAP and CBFV, theoretical considerations would indicate that each harmonic of the gain and phase frequency responses behaves as a statistical independent quantity, which could offer the possibility of overcoming the dimensional limitations of other indices. However, despite an extensive literature on the application of TFA to physiological and clinical studies, segmentation of the spectral range where dynamic CA is active, from DC to approximately 0.2 Hz , into averages over the very low and low frequency ranges (Zhang et al. 1998;Claassen et al. 2016) has not led to consistent metrics that could be treated as independent indicators in different physiological conditions, or reflecting specific pathophysiological alterations in different cerebrovascular conditions (Panerai, 2008;Tzeng et al. 2012;Intharakham et al. 2019). To our knowledge, the demonstration that SRV CrCP makes a significant contribution to the dynamic CA response, and can explain the differences in SRV MCA due to hypercapnia, is the first evidence that it should be possible to quantify the phenomenon of dynamic CA in more than a single dimension. Further physiological and clinical studies are needed, though, to confirm whether the additional information that could be obtained with the joint use of SRV CrCP and SRV RAP allows greater understanding of the behaviour of CA in different phenotypes, physiological conditions or in cerebrovascular diseases of different aetiology. From a methodological perspective, it would also be useful to advance mathematical models, similar to the one proposed by Tiecks et al. for ARI (Tiecks et al. 1995), that would allow further quantification of the SRV CrCP and SRV RAP . Despite the promising possibilities offered by our preliminary findings, a word of caution is needed, given the limitations involved in the calculation of CrCP and RAP for each cardiac cycle. Non-invasive BP was measured in the finger and CBFV in the MCA. Due to the time delay in BP wave propagation, when CBFV is plotted as a function of BP for each cardiac cycle, the resulting graph shows a loop, instead of the linear relationship implied by eqn (A1). For this reason, before applying methods like linear regression analysis to estimate the CrCP and RAP parameters, the CBFV and BP waveforms need to be shifted in time, which normally yields a more linear relationship (Panerai, 2003). The first harmonic method to obtain estimates of RAP and CrCP that we adopted (Panerai et al. 2011) has the advantage of not needing compensation for the propagation time delay, but could still be affected by differences in BP waveform morphology between the finger and the MCA. These factors could be behind the much poorer signal-to-noise ratios that are often observed in time series of beat-to-beat estimates of CrCP and RAP, which were likely to have caused the low values of coherence that led to the rejection of 44 subjects in our study. With the inherent difficulty of obtaining good quality recordings in clinical settings, it is possible that an even larger fraction of patients could not provide suitable data for acceptable estimates of SRV CrCP and SRV RAP . For this reason, more work is needed into more robust methods for deriving beat-to-beat values of RAP and CrCP. One possibility, which remains to be investigated, is the alternative model proposed by Varsos et al, based on the concept of arterial impedance (Varsos et al. 2013).
The rejection of 44 subjects was also directly dependent on the confidence levels adopted for the coherence function in the frequency interval 0.15-0.25 Hz. Compared to the more classical estimates of the MAP-CBFV relationship, this relatively high number of cases with low coherence was likely to result from the difficulty of obtaining accurate estimates of CrCP and RAP for each cardiac cycle (Panerai et al. 2011). As described previously , the reasoning for adopting the 0.15-0.25 Hz frequency interval was to avoid the frequency region where dynamic CA is more likely to be non-linear (<0.15 Hz), but still maintain an acceptable signal-to-noise ratio (<0.25 Hz). Although initially derived for the MAP-CBFV relationship, from a statistical standpoint, the criterion based on confidence limits also applies to any transfer function since it is based on the use of random noise for the input and output functions (Claassen et al. 2016). A problem exists, though, when the criterion is applied to multiple transfer functions, as was our case. The 95% confidence limit for the mean coherence function in the range 0.15-0.25 Hz that we found in previous studies was approximately 0.190 when using 5 min recordings, with five segments of 102.4 s duration and 50% superposition (Claassen et al. 2016;Panerai et al. 2016Panerai et al. , 2018. However, when applying this approach to multiple transfer functions, also subjected to the condition that it must be satisfied for both normocapnia and hypercapnia, the 95% limit is likely to be much lower and further studies of this joint distribution would be needed. In the absence of this information, we adopted the 90% confidence limit of 0.150 as a very conservative threshold for acceptance of the SRV RAP and SRV CrCP . With more in-depth studies of the joint distribution of coherence for multiple SR, it is very likely that the number of subjects yielding acceptable estimates of SRV CrCP and SRV RAP will increase significantly. Nevertheless, as mentioned above, dismissing the 95% confidence criterion for coherence, and re-analysing the data with all 120 subjects included, did not show any significant differences in the main conclusions of the study.

Physiological interpretation
The temporal pattern of SRV RAP fits well with established concepts about the myogenic response of intra-cerebral vessels to a sudden change in MAP (Panerai et al. 2006(Panerai et al. , 2011. What is novel, and puzzling, is the corresponding temporal pattern of SRV CrCP . A sudden increase in MAP leads to alterations in VSM membrane permeability to potassium and calcium, which will increase VSM contraction and thus reduce vessel diameter and blood flow (Faraci et al. 1989). With reduced arterial diameters and stiffer vessel walls, CrCP will increase (Burton, 1951;Panerai, 2003), leading to the initial reduction in SRV CrCP . CrCP is also influenced by ICP (Panerai, 2003), which tends to increase with hypercapnia (Hoiland et al. 2019). What is puzzling, though, is why SRV CrCP then reaches a minimum and gradually increases continuously (Figs 1C and 2C). This behaviour of SRV CrCP , approximately 3 s after the MAP change, has two main effects. First, it counteracts the continuous reduction in SRV RAP , leading to relatively stable values of SRV MCA during this phase of the responses. Second, it contributes to the much larger values of SRV MCA observed during hypercapnia, when compared to the normocapnic response. At this stage, we can only speculate about the underlying physiological mechanisms responsible for the SRV CrCP pattern. It is generally accepted that capillaries and pre-capillary arterioles are the vessels that could collapse at small values of transmural pressure, and for this reason they represent the segment of the cerebral circulation that would contribute to changes in CrCP. Increases in flow in small arterioles elicit the release of nitric oxide by the endothelium, due to the phenomenon of shear stress, leading to vasodilatation (Hoiland et al. 2019). However, the response to changes in shear stress involves a time delay, which can be up to 30 s, but if shorter, or mediated by more rapid neurotransmitters, would fit well with the slow rise in SRV CrCP observed in Figs 1C and 2C. The finding that SRV CrCP reached more positive values during hypercapnia (Figs 1C and 2C), reinforces this interpretation, due to the more pronounced effects of shear stress that have been associated with hypercapnia (Hoiland et al. 2019). More work is needed to shed light on the determinants of the CrCP response to a step change in MAP. Changes in posture, which can alter intracranial pressure, or neural stimulation, which can also lead to vasodilatation of small arterioles, would be two interesting possibilities to be tested. Independently of the mechanistic factors determining the gradual rise in CRV CrCP in the late phase of the response, our findings confirmed the strong sensitivity of CrCP to P aCO 2 (Table 1) that has been previously reported by several studies (Panerai, 2003). This association, together with the CrCP response to neural stimulation (Panerai et al. 2005(Panerai et al. , 2012Beishon et al. 2018) has led to speculations that whilst RAP might be more sensitive to vessel diameter changes due to myogenic mechanisms van Veen et al. 2015), CrCP might be more responsive to vasomotor activity of metabolic origin.

Limitations of the study
Several important limitations of the study were addressed above, including the rejection of 44 subjects due to the low coherence in either the V MAP -V RAP or V MAP -V CrCP transfer functions.
As with any studies based on TCD, attention is needed to factors that could influence MCA diameter, with consequent disruption of a stable relationship between CBFV and absolute CBF. Hypercapnia has been shown to increase MCA diameter (Coverdale et al. 2014;Verbree et al. 2014), but only at levels of P aCO 2 much higher than those inferred from our measurements of ET CO 2 (Table 1). Nevertheless, if we accept that minor vasodilatation of the MCA might have led to CBFV underestimating CBF, any effects in our results would be minimised by the normalisation procedure inherent to SCA (Appendix).
Although the levels of ET CO 2 attained due to breathing of 5% CO 2 in air were significantly higher than those recorded during normocapnia at baseline (Table 1), when compared to the majority of studies in the literature, they should be regarded as mild hypercapnia. The relatively low values of ET CO 2 that we obtained were the result of two limitations. First, participants breathed 5% CO 2 in air at their own respiratory frequency and tidal volume. Much higher values of P aCO 2 could be achieved with the use of a computerised closed-loop end-tidal forcing system, using a higher target for ET CO 2 (Hoiland et al. 2019). Second, 5% CO 2 breathing took place for 3 min and hence its effects were diluted within recordings lasting a total of 5 min, both for its mean values (Table 1) and for the TFA and resultant estimates of SRs. The other side of this limitation, though, is that our relatively low values of ET CO 2 during hypercapnia were well below the levels where dilatation of the MCA would have raised concerns about underestimation of changes in CBF (Coverdale et al. 2014;Verbree et al. 2014). In addition, despite only mild hypercapnia being achieved, we still detected highly significant effects of P aCO 2 on SRV CrCP and it is open to question whether similar or even stronger effects would be obtained with higher levels of P aCO 2 . On the other hand, it is possible that with wider changes in P aCO 2 the RAP step responses might show significant differences, due to hypercapnia or hypocapnia, contrary to our results in Tables 1 and Figs 1-3. In this study, dynamic CA was probed by spontaneous fluctuations in MAP, but the reliability of this approach has been questioned, due to the relatively low amplitude of MAP changes observed with spontaneous variability, in comparison with alternative approaches such as the rapid release of pressurised thigh cuffs or repeated squat-stand manoeuvres (Aaslid et al. 1989;Claassen et al. 2009;Simpson & Claassen, 2018). On the other hand, spontaneous MAP variability affords minimal disruption to underlying physiological processes and it can be adopted in most clinical studies (Tzeng & Panerai, 2018). Nevertheless, further studies of the SRV CrCP and SRV RAP with different protocols to induce larger changes in MAP would be of considerable interest, to provide a better understanding of the influences of co-factors, such as sympathetic activation (Tzeng & Panerai, 2018), and to test the linearity of the responses to larger changes in MAP. J Physiol 598.24 The linear model linking CrCP and RAP to mean values of CBFV and BP for each cardiac cycle, as expressed by eqns (A2)-(A10) (Appendix) relies on a number of assumptions whose limitations need to be kept in mind. Firstly, for a single cardiac cycle, it is unlikely that the assumption of a linear instantaneous relationship between CBFV and BP would hold at very low values of diastolic BP, leading to the distinction between 'true' and 'apparent' CrCP (Panerai, 2003). However, the linear relationship has been shown to represent an acceptable model in the physiological range of diastolic BP values (Panerai, 2003;Panerai et al. 2011), but that might not be the case in the presence of severe arterial hypotension. Secondly, as also expressed by eqns (A2)-(A10), the interdependence between beat-to-beat values of CBFV, MAP, RAP and CrCP means that of these four parameters, only three are independent, and this is also the case of the relationship between the corresponding step responses. In this study, this interdependence was circumvented by obtaining separate estimates of the V MAP -V CBFV , V MAP -V RAP and V MAP -V CrCP step responses, thus leaving the fourth parameter, i.e. V MAP , as the dependent variable in eqns (A2)-(A10), only used to confirm the numerical accuracy of the three distinct transfer function calculations, as the 'checksum' illustrated in Fig. 1D. As a result of this approach, our estimates of SRV RAP and SRV CrCP can be considered as independent of each other, as demonstrated by the distinct temporal patterns in Figs 1 and 2.
Finally, we have not considered the potential effects of sex or age on estimates of SR RAP and SR CrCP . This would be worth investigating, given previous indications that RAP increases with ageing and the non-linear dependence of RAP and CrCP on P aCO 2 is influenced by sex (Minhas et al. 2018b.

Conclusions
In a relatively large number of healthy subjects, the use of TFA for assessment of dynamic CA was extended to include the separate contributions of RAP and CrCP to explain the temporal pattern of the CBFV response to a step change in MAP. Of considerable relevance, the contribution of CrCP to the CBFV step response was not only highly significant during normocapnia at rest, but during hypercapnia it became the dominant factor to explain the changes in the CBFV response, without a contribution from RAP. These findings could have considerable impact on future studies of dynamic CA in humans, but further work is needed to confirm their generalizability to different populations and the extent to which the increase in the dimensionality of CA assessment, afforded by the separate analysis of the RAP and CrCP contributions, could improve the sensitivity and specificity of clinical applications.

Sub-component analysis
For each cardiac cycle, the instantaneous relationship between arterial blood pressure (BP) and cerebral blood flow velocity (CBFV) can be approximated by (Panerai, 2003): where v i (t) and p i (t) are the instantaneous CBFV and BP for the ith cardiac cycle and CrCP i and RAP i are the corresponding critical closing pressure and resistance-area product parameters obtained by fitting a linear relationship (Panerai et al. 2011). When averaged over the entire cardiac cycle, or a longer period, eqn (A1) can be expressed as: where V i represents the mean value of v i (t) for the cardiac cycle and MAP i is the corresponding mean of p i (t).
When MAP i changes from its mean value by an amount p, it will induce changes in V i as well as changes in CrCP i and RAP i , leading to: where v, c and r represent small changes in V i , CrCP i and RAP i , respectively. Assuming that r << RAP Substituting in (A3): Neglecting second order products (r.p 0 and r.c 0) and using eqn (A1): where V MCA v. Defining: with the result and the total change in CBFV, V MCA , is now approximated as the sum of three sub-components, reflecting the separate contributions of parallel changes in MAP, CrCP and RAP. The great advantage of eq. (A10) is that each sub-component has the same units of CBFV (cm s −1 ). In other words, the transformations expressed by eqns (A1-A10) allow us to have a uniform 'currency' to compare different contributions to changes in CBFV.