Monitoring the Evolution of Asynchrony between Mean Arterial Pressure and Mean Cerebral Blood Flow via Cross-Entropy Methods

Cerebrovascular control is carried out by multiple nonlinear mechanisms imposing a certain degree of coupling between mean arterial pressure (MAP) and mean cerebral blood flow (MCBF). We explored the ability of two nonlinear tools in the information domain, namely cross-approximate entropy (CApEn) and cross-sample entropy (CSampEn), to assess the degree of asynchrony between the spontaneous fluctuations of MAP and MCBF. CApEn and CSampEn were computed as a function of the translation time. The analysis was carried out in 23 subjects undergoing recordings at rest in supine position (REST) and during active standing (STAND), before and after surgical aortic valve replacement (SAVR). We found that at REST the degree of asynchrony raised, and the rate of increase in asynchrony with the translation time decreased after SAVR. These results are likely the consequence of the limited variability of MAP observed after surgery at REST, more than the consequence of a modified cerebrovascular control, given that the observed differences disappeared during STAND. CApEn and CSampEn can be utilized fruitfully in the context of the evaluation of cerebrovascular control via the noninvasive acquisition of the spontaneous MAP and MCBF variability.


Introduction
Given the importance of the brain and its high susceptibility to hypoxic states [1], the assessment of the dynamical relationship between mean cerebral perfusion pressure, approximated by the mean arterial pressure (MAP), and mean cerebral blood flow (MCBF), usually approximated by the MCBF velocity (MCBFV) assessed via the transcranial Doppler ultrasound device [2], is of paramount relevance. This evaluation is traditionally carried out by computing the degree of linear association between MAP and MCBFV as a function of

Generalities for the Computation of CApEn and CSampEn
Given two systems X and Y, possibly interacting with each other, their mutual interactions are usually studied in real contexts by assessing the relationship between two stochastic process realizations x = {x n , 1 ≤ n ≤ N} and y = {y n , 1 ≤ n ≤ N} collected during experimental sessions. Defined as y − i = y i−1 . . . y i−m+1 the pattern formed by m − 1 past values of y and y i−1+k the k-step-ahead value of y − i where k is the translation time, with 1 ≤ k ≤ K, where K is the maximum translation time, y i−1+k = y i−1+k ⊕ y − i is the mdimensional vector obtained by concatenating y i−1+k with y − i with 1 ≤ i ≤ N − m − k + 2. Analogously, we define x − j = x j−1 . . . x j−m+1 , x j−1+k and x j−1+k = x j−1+k ⊕ x − j with 1 ≤ j ≤ N − m − k + 2. y − i and x − j can be interpreted as points of a (m − 1)-dimensional state-space built using the method of lagged coordinates, whereas y i−1+k and x j−1+k are points of a special m-dimensional space built non-uniformly [37,38] given that all the components of y − i and x − j are separated in time by k = 1, whereas the most recent sample of y i−1+k and x j−1+k are translated into the future by k when k > 1. We also define with p y i−1+k − x j−1+k ≤ r the probability that y i−1+k lies in the neighborhood of x j−1+k of size r and with p y − i − x − j ≤ r the probability that y − i lies in the neighborhood of x − j of size r, where r is the tolerance in the computation of the neighborhood and · is a metric to compute distance. In this study, the adopted metric is the maximum norm, namely the maximum of the absolute difference between corresponding scalar components [28,29].

CApEn
CApEn [28] is defined as the negative averaged logarithm of the ratio of the p( y i−1+k where · performs the average over all the reference vectors built over x and log is the natural logarithm. p y i−1+k − x j−1+k ≤ r and p y − i − x − j ≤ r are estimated as the fraction of y i−1+k in the neighborhood of the reference pattern x j−1+k and the fraction of y − i in the neighborhood of the reference pattern x − j within a tolerance r, respectively. The fractions are obtained by counting the number of y i−1+k closer than r to x j−1+k and the number of y − i closer than r to x − j and by dividing them by N − m − k + 2, respectively. The logarithm of p y i−1+k − x j−1+k ≤ r and of p y − i − x − j ≤ r was averaged over all the reference patterns built over x to obtain log p y i−1+k − x j−1+k ≤ r and log p y − i − x − j ≤ r . In agreement with [29], we adopted the "bias 0" and "bias max" strategies to deal with the possibility that of p y i−1+k − x j−1+k ≤ r and/or p y − i − x − j ≤ r could be 0 due to the lack of y i−1+k and y − i in the neighborhood of patterns x j−1+k and x − j , respectively. The "bias 0" strategy substituted the contemporaneous occurrence of p y i−1+k − x j−1+k ≤ r = 0 and p y − x − j ≤ r) = 1. The CApEn is a measure of strength of association between x and y. Values of CApEn close to 0 indicate that patterns built over x and y that were close to r in the (m − 1)-dimensional space remained similar in the m-dimensional space. Conversely, high values of CApEn indicated a certain degree of asynchrony between x and y.

CSampEn
CSampEn [29] is defined as the negative logarithm of the ratio of the averaged where · performs the average over all the reference vectors built over x. At difference with CApEn p y i−1+k − x j−1+k ≤ r and p y − i − x − j ≤ r were averaged over all the reference patterns x j−1+k and x − j , respectively, to obtain p y i−1+k − x j−1+k ≤ r and p y − i − x − j ≤ r before carrying out the logarithm. The interpretation of CSampEn follows closely that of CApEn. The advantage is the smaller bias [29] given that it is In the very unlikely case that p y i−1+k − x j−1+k ≤ r was 0, it was substituted with In the equally unlikely situation that both p

Experimental Protocol
The study was in keeping with the Declaration of Helsinki. The study was approved by the ethical review board of the San Raffaele Hospital, Milan, Italy (approval number: 68/int/2018; approval date: 5 April 2018) and authorized by the Policlinico San Donato, San Donato Milanese, Milan, Italy (authorization date: 13 April 2018). Written, signed and informed consent was obtained from all subjects.
We enrolled 30 patients (age: 66 ± 10 yrs, 23 males) undergoing SAVR at the IRCCS Policlinico San Donato, San Donato Milanese, Milan, Italy. Demographic and clinical data of the SAVR group are reported in Table 1. They did not feature either atrial fibrillation, overt autonomic nervous system pathologies or cerebrovascular diseases. We acquired electrocardiogram (ECG) from lead II (BioAmp FE132, ADInstruments, Bella Vista New South Wales, Australia), non-invasive finger arterial pressure (AP) by volume-clamp photoplethysmography (CNAP Monitor 500, CNSystems, Graz, Austria), and pulsatile cerebral blood-flow velocity (CBFV) via transcranial Doppler device (Multi-Dop X, DWL, San Juan Capistrano, CA, USA) from the left or right middle cerebral artery. Signals were sampled at 400 Hz through a commercial acquisition system (Power Lab, ADInstruments, Bella Vista New South Wales, Australia). Signals were recorded in PRE, i.e., 1 day before SAVR, and in POST, i.e., within 7 days after SAVR, at rest in supine position (REST) and during active standing (STAND). Experimental sessions lasted 10 min with REST acquired always before STAND. Seven patients were excluded due to poor quality of CBFV, as checked during the first session of the protocol (i.e., at REST in PRE). PRE analyses could be carried out in 23 subjects at REST, and in 20 individuals during STAND subjects, whereas POST data was processed for 15 and 13 patients, respectively. The decreasing number of subjects in POST compared with PRE is explained by the post-surgery physical and psychological debilitation of some patients. The difficulty in locating either the left or right middle cerebral artery in the different experimental sessions further limited the final figures. We checked that the groups examined in the PRE and POST conditions exhibited the same basic characteristics as the population reported in the Table 1 (e.g., demographic data). This consideration held for the comparison between individuals at REST and during STAND. Pharmacological treatment that might interfere with the autonomic control was preserved in POST, unless specific situations suggested the administration of additional medications (e.g., beta-blockers in case of associated coronary artery bypass graft surgery). No significant differences were found across groups in relation to pharmacological treatment that might interfere with the autonomic nervous system activity.

Extraction of Beat-to-Beat Variability Series
The ECG was exploited to facilitate the detection of diastolic fiducial points on AP. After detecting the R-wave peaks from the ECG using a threshold applied to the first derivative, the nth systole and diastole were located, respectively, as the timing of the AP maximum was observed within the nth cardiac cycle, and the timing of the AP minimum following the nth systole. The AP at the systolic and diastolic points were taken as systolic AP (SAP) and diastolic AP (DAP). The length of the nth heart period (HP) was also derived as the time interval between the nth and (n + 1)th R-wave peaks. The nth MAP was obtained as the integral of AP between the (n − 1)th and nth diastoles, and by dividing the result by Entropy 2022, 24, 80 6 of 15 the corresponding interdiastolic time interval. The nth MCBFV was computed similarly over the CBFV signal using the minima detected over the CBFV signal in the proximity of diastoles. The HP, SAP, DAP, MAP and MCBFV series were manually checked and corrected in case of missing beats or misdetections. Effects of ectopic beats or isolated arrhythmic events were mitigated via linear interpolation. Synchronous sequences lasting 256 consecutive beats were randomly selected within the whole recordings.
CApEn and CSampEn were computed over normalized series obtained by subtracting the mean to each value and dividing the result by the standard deviation σ. According to the standard setting [28,29,39] we assigned m = 3 and r = 0.2 × σ. CApEn and CSampEn were computed as a function of k with 1 ≤ k ≤ K and K = 8. CApEn and CSampEn at k = 1 were denoted as CApEn k=1 and CSampEn k=1 , respectively. Linear regression analysis of CApEn and CSampEn on k was performed on an individual basis. The slopes of the linear regression were computed, and these markers were denoted as CApEn slope and CSampEn slope .

Statistical Analysis
Two-way analysis of variance (Holm-Sidak test for multiple comparisons) was applied to CApEn and CSampEn markers to assess the effect of translation time versus k = 1 within the same period of analysis (i.e., PRE or POST) and the effect of the surgery within the same translation time (from 1 to 8). Two-way analysis of variance (Holm-Sidak test for multiple comparisons) was applied to CApEn k=1 , CSampEn k=1 , CApEn slope and CSampEn slope to detect the effect of cardiac surgery within the same experimental condition (i.e., REST or STAND) and the response to the postural challenge within the same period of analysis (i.e., PRE or POST). Statistical analysis was carried out using a commercial statistical program (Sigmaplot, v.14.0, Systat Software, Inc., Chicago, IL, USA). A p < 0.05 was always considered statistically significant. Table 2 summaries the time domain markers derived from HP, SAP, DAP, MAP and MCBFV. STAND reduced µ HP , but this effect was visible only in PRE. STAND increased µ DAP exclusively in POST. The depression of cardiovascular autonomic control was stressed by the decrease in σ 2 HP and σ 2 DAP in POST at REST. The expected increase in σ 2 SAP and of σ 2 DAP , and the expected decline in σ 2 HP with STAND was not detected in either PRE or POST. None of the time domain markers usually evaluated to typify CA, namely µ MAP , σ 2 MAP , µ MCBFV and σ 2 MCBFV varied with orthostatic challenge and/or period of analysis. In the following we reported results of CApEn derived exclusively using "bias 0" strategy. Indeed, findings relevant to the use of "bias max" strategy exhibited similar differences between periods of analysis and/or experimental conditions. The scatter plots in Figure 1 show the course of CApEn as a function of the translation time k at REST ( Figure 1a) and during STAND ( Figure 1b) in a representative subject. The trend of CApEn in PRE is given as solid circles, whereas in POST it is given as open circles. CApEn starts from lower values and the rate of the CApEn increase was faster in PRE than POST, both at REST and during STAND. In the following we reported results of CApEn derived exclusively using "bias 0" strategy. Indeed, findings relevant to the use of "bias max" strategy exhibited similar differences between periods of analysis and/or experimental conditions. The scatter plots in Figure 1 show the course of CApEn as a function of the translation time k at REST (Figure 1a) and during STAND (Figure 1b) in a representative subject. The trend of CApEn in PRE is given as solid circles, whereas in POST it is given as open circles. CApEn starts from lower values and the rate of the CApEn increase was faster in PRE than POST, both at REST and during STAND.     The vertical box-and-whisker plots of Figure 3 show CApEn as a function of the translation time k at REST (Figure 3a) and during STAND (Figure 3b). The markers of asynchrony were reported in PRE (grey boxes) and in POST (white boxes). The height of the box represents the distance between the first and third quartiles, with the median marked as a line, and the whiskers show the 5th and 95th percentiles. In PRE, CApEn increased with k: the raise compared to k = 1 was significant for k ≥ 3 both at REST and during STAND. In POST, CApEn remained stable with k and the finding held both at REST and during STAND. At REST, CApEn was significantly larger in POST than in PRE solely at k = 1 and k = 2, whereas during STAND no PRE-POST differences were visible, regardless of the values of k.    Entropy 2022, 23, x FOR PEER REVIEW 8 The vertical box-and-whisker plots of Figure 3 show CApEn as a function of translation time k at REST (Figure 3a) and during STAND (Figure 3b). The marke asynchrony were reported in PRE (grey boxes) and in POST (white boxes). The heig the box represents the distance between the first and third quartiles, with the me marked as a line, and the whiskers show the 5th and 95th percentiles. In PRE, CA increased with k: the raise compared to k = 1 was significant for k ≥ 3 both at REST during STAND. In POST, CApEn remained stable with k and the finding held bot REST and during STAND. At REST, CApEn was significantly larger in POST than in solely at k = 1 and k = 2, whereas during STAND no PRE-POST differences were vis regardless of the values of k.   The vertical box-and-whisker plots of Figure 5 show CApEnk=1 (Figure 5a) CSampEnk=1 ( Figure 5b) as a function of the experimental condition (i.e., REST STAND). The markers of asynchrony were reported in PRE (grey boxes) and in P (white boxes). The height of the box represents the distance between the first and t quartiles, with the median marked as a line, and the whiskers show the 5th and   The vertical box-and-whisker plots of Figure 3 show CApEn as a function of translation time k at REST (Figure 3a) and during STAND (Figure 3b). The marker asynchrony were reported in PRE (grey boxes) and in POST (white boxes). The heigh the box represents the distance between the first and third quartiles, with the med marked as a line, and the whiskers show the 5th and 95th percentiles. In PRE, CA increased with k: the raise compared to k = 1 was significant for k ≥ 3 both at REST during STAND. In POST, CApEn remained stable with k and the finding held bot REST and during STAND. At REST, CApEn was significantly larger in POST than in solely at k = 1 and k = 2, whereas during STAND no PRE-POST differences were vis regardless of the values of k.   The vertical box-and-whisker plots of Figure 5 show CApEnk=1 (Figure 5a) CSampEnk=1 ( Figure 5b) as a function of the experimental condition (i.e., REST STAND). The markers of asynchrony were reported in PRE (grey boxes) and in PO (white boxes). The height of the box represents the distance between the first and t quartiles, with the median marked as a line, and the whiskers show the 5th and 9 The vertical box-and-whisker plots of Figure 5 show CApEn k=1 (Figure 5a) and CSampEn k=1 (Figure 5b) as a function of the experimental condition (i.e., REST and STAND). The markers of asynchrony were reported in PRE (grey boxes) and in POST (white boxes). The height of the box represents the distance between the first and third quartiles, with the median marked as a line, and the whiskers show the 5th and 95th percentiles. CApEn k=1 increased in POST compared with PRE at REST, whereas the effect of cardiac surgery was not evident during STAND. The same conclusion held for CSampEn k=1 . In PRE, the postural challenge did not induce any modification of either CApEn k=1 or CSampEn k=1 . Conversely, in POST we observed a tendency towards a decrease in markers of asynchrony during STAND, compared with REST, and this tendency became significant in the case of CSampEn k=1 .

Results
Entropy 2022, 23, x FOR PEER REVIEW 9 percentiles. CApEnk=1 increased in POST compared with PRE at REST, whereas the e of cardiac surgery was not evident during STAND. The same conclusion held CSampEnk=1. In PRE, the postural challenge did not induce any modification of e CApEnk=1 or CSampEnk=1. Conversely, in POST we observed a tendency towards crease in markers of asynchrony during STAND, compared with REST, and this tend became significant in the case of CSampEnk=1.  Figure 6 has the same structure as Figure 5, but it shows CApEnslope (Figure 6a) CSampEnslope (Figure 6b). Similar to Figure 5, the effect of cardiac surgery was signif only at REST over both asynchrony markers. STAND did not affect either CApEnsl CSampEnslope, regardless of the period of analysis.

Discussion
The main findings of this study can be summarized as follows: (i) CApEn CSampEn allow the assessment of the asynchrony between two time series as a fun of the translation time; (ii) CApEn and CSampEn provide similar conclusions abou effect of postural challenge and cardiac surgery; (iii) none of the time domain ma characterizing MCBFV-MAP regulation detect the impact of either cardiac surgery or tural challenge; (iv) the impact of cardiac surgery on asynchrony markers is evide REST but irrelevant during STAND; (v) postural challenge has a limited impact on a chrony markers visible only over CSampEnk=1 in POST.  Figure 6 has the same structure as Figure 5, but it shows CApEn slope (Figure 6a) and CSampEn slope (Figure 6b). Similar to Figure 5, the effect of cardiac surgery was significant only at REST over both asynchrony markers. STAND did not affect either CApEn slope or CSampEn slope , regardless of the period of analysis.
Entropy 2022, 23, x FOR PEER REVIEW 9 percentiles. CApEnk=1 increased in POST compared with PRE at REST, whereas the of cardiac surgery was not evident during STAND. The same conclusion hel CSampEnk=1. In PRE, the postural challenge did not induce any modification of e CApEnk=1 or CSampEnk=1. Conversely, in POST we observed a tendency towards crease in markers of asynchrony during STAND, compared with REST, and this tend became significant in the case of CSampEnk=1.  Figure 6 has the same structure as Figure 5, but it shows CApEnslope (Figure 6a CSampEnslope (Figure 6b). Similar to Figure 5, the effect of cardiac surgery was signi only at REST over both asynchrony markers. STAND did not affect either CApEns CSampEnslope, regardless of the period of analysis.

Discussion
The main findings of this study can be summarized as follows: (i) CApEn CSampEn allow the assessment of the asynchrony between two time series as a fun of the translation time; (ii) CApEn and CSampEn provide similar conclusions abou effect of postural challenge and cardiac surgery; (iii) none of the time domain ma characterizing MCBFV-MAP regulation detect the impact of either cardiac surgery or tural challenge; (iv) the impact of cardiac surgery on asynchrony markers is evide REST but irrelevant during STAND; (v) postural challenge has a limited impact on chrony markers visible only over CSampEnk=1 in POST.

Discussion
The main findings of this study can be summarized as follows: (i) CApEn and CSam-pEn allow the assessment of the asynchrony between two time series as a function of the translation time; (ii) CApEn and CSampEn provide similar conclusions about the effect of postural challenge and cardiac surgery; (iii) none of the time domain markers characterizing MCBFV-MAP regulation detect the impact of either cardiac surgery or postural challenge; (iv) the impact of cardiac surgery on asynchrony markers is evident at REST but irrelevant during STAND; (v) postural challenge has a limited impact on asynchrony markers visible only over CSampEn k=1 in POST.

CApEn and CSampEn Allow the Assessment of the MCBFV-MAP Asynchrony as a Function of the Translation Time
In the study of cerebrovascular dynamical interactions, a reliable quantification of the degree of MCBFV-MAP association is fundamental. Indeed, the strength of MCBFV-MAP coupling is the balance between CA mechanisms that should induce a certain degree of decoupling between MAP and MCBFV variability series, given that CA aims at limiting the variability of MCBFV against MAP variations [3,4,[19][20][21], and the high level of association imposed by the pressure-to-flow [6,7] and flow-to-pressure [8][9][10][11] causal pathways. This study originally applied CApEn and CSampEn to quantify the degree of asynchrony between the MAP and MCBFV variability series, and monitor the level of MCBFV-MAP asynchrony as a function of the translation time k. CApEn and CSampEn built separate patterns of dimension m − 1 over both MAP and MCBFV variability series, and checked whether these patterns remained close after the enlargement of both patterns with an additional component k-step-ahead into the future. This feature means that CApEn and CSampEn belong to the class of state-space correspondence methods that propose to reconstruct the dynamical behaviors of two series in two separate embedding spaces, and search for the possible relationship linking the reconstructed geometrical entities [40][41][42]. The advantage of model-free approaches based on state-space correspondence lies in the possibility of the searching association under very broad hypotheses [40][41][42]. Therefore, this approach might be well suited in the context of assessment of cerebrovascular control and CA, given the nonlinear characteristics of the link between MAP and MCBFV [1,[22][23][24][25][26]. Moreover, the proposed approach has the advantage of assessing asynchrony as a function of k. Asynchrony between two stochastic processes is expected to change with the dynamical characteristic of the input-output relationship between them. Therefore, characterization of the evolution of asynchrony with k might provide new indexes describing the MCBFV-MAP relationship. In general, in stochastic systems, the difficulty in predicting the output from the input increases with prediction time, and this characteristic leads to a level of asynchrony between the input and the output increasing with k [43]. It has been demonstrated that, in the presence of a full decoupling between the input and the output, asynchrony assessed via cross-conditional entropy did not vary with k [34,35]. This finding holds even when asynchrony is evaluated via mutual predictability [40]. Moreover, it was found that the presence of a nonlinear relationship between the input and the output might influence the value of asynchrony and the rate of increase in asynchrony with k [35]. As a matter of fact, a cross-condition entropy approach capable of describing nonlinearities detected a smaller level of asynchrony and a higher rate of rise of asynchrony with k than those found over surrogate data built from the original ones by preserving only linear aspects of the dynamics and their interactions such as cross-correlation, power spectra and distributions [35].

CApEn and CSampEn Provide Similar Conclusions about the Effect of Postural Challenge and Cardiac Surgery
Both CApEn and CSampEn measure the degree of asynchrony between two time series [28,29]. Despite being defined to measure the same quality of the dynamical interactions between two time series, there are a couple of reasons that might lead CApEn and CSampEn to different conclusions. First, CApEn needs the application of correction schemes [29] to enlarge its original definition [28], that is otherwise very limited due to the highly likely occurrence of the log-of-zero situation [29] and the untrustworthiness of procedures designed to increase the number of matches [44]. Second, the CApEn is a directional marker, whereas CSampEn is not. Indeed, even though p y i−1+k − x j−1+k ≤ r and are directional markers (i.e., reversing the role of x and y modified prob- given that they are equivalent to the probability of finding pairs of vectors built over x and y, at a distance closer than r in the m-dimensional and (m − 1)-dimensional embedding spaces, respectively, namely a feature that evidently does not depend on which series is taken as x or y. Conversely, the directionality of CApEn is the consequence of the application of the logarithm to p( y i−1+k − x j−1+k ≤ r) and p( y − i − x − j ≤ r) before averaging, and to its nonlinear characteristic. Contrary to the expectations, results derived from CApEn and CSampEn were similar. Indeed, both CApEn and CSampEn markers were able to detect the impact of cardiac surgery at REST, but not during STAND, and trends with posture modification were similar. Therefore, we can conclude that in the context of evaluating cerebrovascular control in SAVR population there is no reason to privilege either approach.

Impact of CApEn and CSampEn on the Assessment of the Cerebrovascular Control in SAVR Patients
This study confirms previous results on cardiac, vascular, and cerebrovascular controls in the SAVR population [36,[45][46][47]. Time domain markers suggest that cardiac and vascular regulations are depressed after SAVR surgery, as indicated by the decline of σ 2 HP and σ 2 DAP and by the irrelevant modifications of σ 2 HP , σ 2 SAP and σ 2 DAP in response to STAND. The missing impact of STAND over σ 2 HP , σ 2 SAP and σ 2 DAP in PRE indicates that cardiovascular control is impaired already before SAVR surgery [36]. Time domain markers confirm that cerebrovascular control and CA were not affected by the postural challenge and cardiac surgery given that, in the presence of stable values of σ 2 MAP with postural and surgical stressors, σ 2 MCBFV remained unvaried [36]. Both CApEn and CSampEn were able to detect at REST the greater MCBFV-MAP asynchrony in POST, compared with PRE. The same tendency was found via squared coherence computed in the frequency bands below the respiratory one [36], but in the present study the change is significant. This finding might be taken as an indication of the postoperative improvement of CA, given that the aim of CA is to limit variability of MCBFV against MAP variations [3,4,[19][20][21]. Indeed, situations of impaired CA, such as those induced by aneurysmal subarachnoid hemorrhage, and by pharmacological autonomic blockades increased the squared coherence function between 0.04 and 0.08 Hz [3,25,26]. However, this interpretation holds in the presence of significant MAP variations. Since we observed a postsurgery sympathetic depression in our population resulting in a tendency toward lower amplitudes of MAP oscillations, this input might be insufficient to drive MCBFV changes. The unvaried modification of MCBFV-MAP asynchrony in POST compared to PRE during STAND corroborates the conclusion that cardiac surgery did not modify CA [36]. Indeed, during STAND the amplitude of the MAP oscillations tended to increase, compared with REST in our population [36], and this increase might be able to drive MCBFV oscillations, thus revealing that the degree of association between MAP and MCBFV was not changed after cardiac surgery.
Both CApEn and CSampEn detected a smaller rate of increase in MCBFV-MAP asynchrony with translation time k at REST in POST compared with PRE. The reduced increase in asynchrony with k between MAP and MCBFV variability might indicate a more deterministic relation between MCBFV and MAP, or a more limited ability to reach the condition of MCBFV-MAP uncoupling in POST than in PRE. Given that the MCBFV-MAP link is the result of the integrate action of multiple mechanisms comprising chemoreflex, neuronal metabolism, neurovascular coupling, CA and autonomic control [5], the tendency toward a more deterministic MCBFV-MAP relation might indicate a postoperative loss of cerebrovascular complexity and, consequently, an impairment of cerebrovascular regulation. Given that the goal of CA is to limit the variability of MCBFV despite changes in MAP [3,4,[19][20][21], the tendency towards a reduced rate of increase in the MCBFV-MAP decoupling with k might suggest an impairment of cerebrovascular control. However, again the insufficient perturbing action of MAP might be responsible for this finding. This conclusion is corroborated by the missed postoperative variations of CApEn slope and CSampEn slope during STAND, when MAP changes tended to be more important.
The effect of STAND on asynchrony markers is limited, and this conclusion is in agreement with several studies stressing the negligible impact of the postural challenge over cerebrovascular regulation [48][49][50]. Indeed, solely CSampEn k=1 indicated an effect of STAND, and this influence was detected solely in POST. However, since a similar tendency was suggested by CApEn k=1 as well, and given that it was evident only in POST, future studies carried out should investigate more deeply the effect of posture change in this population by considering a larger number of subjects.

Limitations of the Study and Future Developments
A possible limitation of the study is the missed random allocation of the subject in each condition. Indeed, the allocation of a subject in each group is exclusively based on the quality of the recording of AP and CBFV. This criterion might have biased the study toward a special subset of our original group, despite the preservation of the general characteristic of the overall population. Another possible limitation of the study is linked to the dependence of the autonomic response on the number of postoperative days [51]. In the present study the timing of POST varies from 4 to 7 days. This peculiar setting is likely to have increased the variance of the markers, but it is unlikely to have biased the study towards a specific conclusion given that the autonomic control is expected to remain influenced by surgery for some days [51]. One possible confounding factor is the significant fraction of subjects under beta-adrenergic blockade in our population. However, the impact of beta-blockers on CA is more controversial [52,53] than that of alpha-blockers [25]. The impact of beta-blocker therapy on the PRE-POST and/or REST-STAND comparisons is expected to be limited, given that the fraction of subjects under beta-blocker therapy is similar in all conditions. Future studies should test this approach in subjects who developed stroke during SAVR surgery, to test whether the method could indicate a CA impairment.

Conclusions
The link between MAP and MCBFV spontaneous fluctuations resulting from the action of cerebrovascular control mechanisms was explored via cross-entropy approaches. Crossentropy metrics have the possibility to interpret possible nonlinear interactions between the two series and monitor the evolution of the degree of the MCBFV-MAP asynchrony with the translation time. The approach was found to be useful to typify cerebrovascular regulation. More specifically, since the effect of cardiac surgery on cross-entropy markers observed at REST disappeared during STAND, we conclude that cardiac surgery did not alter the state of the cerebrovascular regulation in SAVR population.  Informed Consent Statement: Written informed consent was obtained from all subjects involved in the study.

Data Availability Statement:
The data presented in this study are available on request from the corresponding author (i.e., A.P.) upon permission of the IRCCS Policlinico San Donato. The data are not publicly available because they contain information that could compromise the privacy of research participants.