Alteration of cerebrovascular haemodynamic patterns due to atrial fibrillation: an in silico investigation
Stefania Scarsoglio, Andrea Saglietto, Matteo Anselmino, Fiorenzo Gaita, Luca Ridolfi
AAlteration of cerebrovascular haemodynamicpatterns due to atrial fibrillation: an in silico investigation
S. Scarsoglio ∗ , A. Saglietto , M. Anselmino , F. Gaita and L. Ridolfi Department of Mechanical and Aerospace Engineering, Politecnico di Torino,Torino, Italy Division of Cardiology, Department of Medical Sciences, Citt`a della Salutee della Scienza Hospital, University of Turin, Torino, Italy Department of Environmental, Land and Infrastructure Engineering, Po-litecnico di Torino, Torino, Italy
Abstract
There has recently been growing evidence that atrial fibrillation(AF), the most common cardiac arrhythmia, is independently associ-ated with the risk of dementia. This represents a very recent frontierwith high social impact for the number of individuals involved and forthe expected increase in AF incidence in the next 40 years. Although anumber of potential haemodynamic processes, such as microembolisms,altered cerebral blood flow, hypoperfusion and microbleeds, arise asconnecting links between the two pathologies, the causal mechanismsare far from clear. An in silico approach is proposed that combines insequence two lumped-parameter schemes, for the cardiovascular systemand the cerebral circulation. The systemic arterial pressure is obtainedfrom the cardiovascular system and used as the input for the cerebralcirculation, with the aim of studying the role of AF on the cerebralhaemodynamics with respect to normal sinus rhythm (NSR), over a5000 beat recording. In particular, the alteration of the haemody-namic (pressure and flowrate) patterns in the microcirculation duringAF is analysed by means of different statistical tools, from correlationcoefficients to autocorrelation functions, crossing times, extreme val-ues analysis and multivariate linear regression models. A remarkablesignal alteration, such as a reduction in signal correlation (NSR, about3 s; AF, less than 1 s) and increased probability (up to three to fourtimes higher in AF than in NSR) of extreme value events, emergesfor the peripheral brain circulation. The described scenario offers a ∗ Corresponding author: [email protected] a r X i v : . [ phy s i c s . f l u - dyn ] M a y umber of plausible cause-effect mechanisms that might explain theoccurrence of critical events and the haemodynamic links relating toAF and dementia. Atrial fibrillation (AF), leading to an irregular and faster heart rate, is themost common tachyarrhythmia with an estimated number of 33.5 millionindividuals affected worldwide in 2010 [1], and its incidence is expected todouble within the next 40 years [2]. Besides thromboembolic transient is-chaemic attack (TIA) and strokewhose risk is increased fivefold in patientswith AF [3] and is associated with both cerebral impairment and demen-tia [4] - it has been recently observed that AF is independently associatedwith cognitive decline through a range of different potential haemodynamicmechanisms, such as silent cerebral infarctions (SCIs) as a result of microem-bolization [5, 6], altered cerebral blood flow [7], hypoperfusion [8] and mi-crobleeds, whose repetition increases the risk of intracerebral haemorrhagicevents and dementia by five times [9].Although representing a currently debated topic [10], there is growingevidence that AF - independently of clinically relevant events - enhancesthe risk of dementia and cognitive deficit [8, 9, 11]. Several different kindsof observational works - such as meta-analyses [12]-[14], reviews [15, 16],cross-sectional [17]-[19], cohort and longitudinal [20]-[25] studies - confirman independent association between AF and cognitive decline at differinggrades of severity. Only a few with critical limiting aspects and potentialsources of bias, such as small population [26], very high rate of loss duringfollow-up [27] and very elderly subjects (aged 85 and older) [28], found nosignificant relation between AF and cognitive impairment.However, most of the above observational studies can only show an asso-ciation between AF and cognitive impairment and not a causal relation basedon haemodynamics for any of the known potential mechanisms. Recently,the role of SCIs in cognitive function during AF has been assessed throughmagnetic resonance imaging [6]. Although some of the haemodynamic con-sequences of AF, such as lower diastolic cerebral perfusion and decreasedblood flow in the intracranial arteries, have been reported ( [8, 9, 15]and ref-erences therein), the linking mechanisms with cognitive impairment remaintheoretical or mainly undetermined.To the best of our knowledge, the specific impact of the altered AF heartrate on the cerebral haemodynamics is still in great part unexplored. Infact, currently adopted clinical techniques in the field of cerebral haemody-namics - such as transcranial doppler ultrasonography - lack the resolvingpower to give any insights on the micro-vasculature, in terms of flow andpressure signals. In particular, little is known about the consequences of AF2reatment on the evolution of cognitive decline. So far, only a few studieshave examined the potential benefits from AF treatment in reducing cogni-tive impairment. Increased cognitive dysfunction was found to relate to lesseffective oral anticoagulation treatment [29], while AF patients who under-went catheter ablation had a lower risk of dementia than those who did not[30]. These studies, though not prospective and with biased information,give insights that specific treatments for AF could modify the risk of de-mentia. The intriguing recent hints offered by literature encourage a deepercomprehension of the AF effects on hypoperfusion and irregular cerebralblood flow, which is still lacking [31].The efficiency of a mathematical modelling approach to describe thecerebral circulation has been widely recognized and in silico haemodynam-ics is currently a rising field of research, e.g. [32, 33]. In a previous work,we obtained the first exploratory results by adopting two lumped-parametermodels for the cardiovascular and cerebral circuits, which highlighted theonset of critical events - such as hypoperfusions and hypertensive events -at the arteriolar and capillary levels during AF [34]. Aim of the presentwork is to understand and analyze - through a systematic and extensivesignal analysis - possible haemodynamic-based causal relations underlyingthe occurrence of such critical events, for which AF may imply cognitivedysfunction. The statistical tools exploited here are borrowed from classi-cal time-series analysis and include cross-correlation functions between theinput pressure/flow rate signals and the corresponding downstream signals,auto-correlation functions in different cerebral regions, distribution of con-secutive time lapses spent above/below the mean value of the pressure andflow rate temporal signals, detection of minimum and maximum haemody-namic values per beat, quadrant analysis, and multivariate linear regressionmodels for the haemodynamic variables (averaged by beat). A model-basedestimation of these critical events can offer useful hints for the assessment ofsome of the AF treatments, in particular rate and rhythm control strategies,as it can suggest priority treatment to minimize neurodegenerative changes.To isolate single cause-effect relations and ascertain which AF-driven vari-ation mostly affects the cerebral circulation and should therefore be takenunder strict control, a comparative signal analysis (in terms of pressure andflow rate time-series) is proposed between normal sinus rhythm (NSR) andAF signals over a 5000 beat recording. The paper is organized as follows.In the Materials and Methods Section the stochastic modelling, consistingof the cardiovascular and cerebral systems, is introduced. The followingsection (Pressure and Flow Rate Signal Analysis) proposes a collection ofdifferent statistical tools to carry out a systematic study of the signal varia-tion. In the Discussion, a summarizing framework explaining the reasons forAF-induced changes in the micro-circulation is given. Conclusions remarkthat the emergence of critical events in AF turns out to be caused by thesignal alteration - especially in terms of correlation, memory and complexity3 induced by AF in the micro-vascular haemodynamics.
The stochastic modelling of the AF-induced cerebral haemodynamics hasbeen recently proposed [34] and consists of three sequential steps. Figure 1describes the modelling process adopted (panels 1, 2, 3) and shows a repre-sentative pressure time series obtained as outputs (panel 4). The proposedstochastic algorithm combines two different lumped models in sequence: thecardiovascular model is exploited to obtain the systemic arterial pressure, P a , which is then used as the forcing input for the next cerebral model. • Building the RR intervals.
We recall that RR [s] is the temporal in-terval between two consecutive heart beats, while the heart rate, HR ,is the number of heart beats per minute. Normal sinus and fibril-lated beating are modelled via artificially built RR intervals based onNSR and AF beating features (see the details in [35]). Normal RR heart beats are extracted from a correlated pink Gaussian distribu-tion (mean µ = 0 . σ = 0 .
06 s), which is thetypical distribution observed during sinus rhythm for RR [35]. AFbeatings are instead extracted from an exponentially Gaussian modi-fied (EGM) distribution (mean µ = 0 . σ = 0 . γ = 7 .
24 Hz), which is unimodal and represents themost common AF distribution (60-65% of the cases) [36, 37]. Theexponential contribution is responsible for the uncorrelated nature ofthe AF beating. Comparison between NSR and AF is proposed atthe same mean heart rate (75 beats per minute (b.p.m.)) to facilitateanalysis of the results. 5000 beats are extracted and then simulatedfor both configurations in order to achieve the statistical stationarityfor the main statistics of the outcomes (the 5000 RR beats extractedin NSR and AF conditions are reported in the first panel of Fig. 1). • Cardiovascular model.
Once the RR extraction is complete, a lumpedcardiovascular model is used to obtain the systemic arterial pressure( P a ). The model was first proposed [38] to describe, through a Wind-kessel approach, the complete cardiovascular system. It includes thesystemic and venous circuits together with the four cardiac chamberswhich are actively modelled. By means of a network of compliances,resistances, and inductances, the cardiovascular dynamics is expressedin terms of pressures, flow rates, volumes, and valve opening angles.After being validated during resting conditions over more than 30 clini-cal datasets [35, 39], the model has been exploited to study left valvular4igure 1: Scheme of the in silico approach (the figure should be read clock-wise starting from the left bottom panel 1). (1) Cardiovascular model: 5000extracted RR records in NSR (blue) and AF (red), and examples of P a timeseries obtained through the cardiovascular model. (2) Cerebral circulation:sketch of the cerebral vasculature forced by the P a input, which is obtainedfrom the cardiovascular model described in panel 1. (3) Cerebral mathemat-ical model: R , resistance; C , compliance; Q , flow rate; P , pressure. The leftICA-MCA pathway is highlighted in red and is composed of P a , Q ICA,left , P MCA,left , Q MCA,left , P dm,left , Q dm,left , P c and Q pv . (4) Examples of pres-sure time series. Representative resulting time series for the pressure alongthe ICA-MCA pathway, in NSR (blue) and AF (red) conditions, obtainedfrom the cerebral model described in panel 3. In panels 2, 3, 4, the colouredboxes refer to different cerebral regions: large arteries (light blue), distalarteries (green), capillary/venous circulation (yellow).5iseases [40] during AF and the effect of increased heart rate in restingconditions [41] and under exercise [42]. To account for AF conditions,both atria are considered as passive (while in NSR they actively con-tract). We point out that the cardiovascular and cerebral models arecombined in sequence: once the systemic arterial pressures, P a , areobtained from the cardiovascular scheme in NSR and AF conditions,they are then used as forcing inputs for the cerebral model. Examplesof P a time series are reported in the first panel of Fig. 1. • Cerebral model.
Zero-dimensional modelling for the cerebrovasculardynamics has been proposed [43] to study the whole (arterial and ve-nous) cerebral circulation (Panels 2 and 3, Fig. 1). Similarly to thecardiovascular model, a framework of resistances ( R , [mmHg s/ml])and compliances ( C , [ml/mmHg]) accounts for the dissipation effectsand the elastic properties of vessels, respectively. The cerebral circula-tion is expressed in terms of pressure ( P , [mmHg]), volume ( V , [ml]),flow rate ( Q , [ml/s]), and can be divided into three principal regions:large arteries, distal arterial circulation, capillary-venous circulation.The first section is composed of the afferent arteries and the circleof Willis, while the six main cerebral arteries link this region to thedownstream distal circuit. The distal arterial circulation includes thepial circulation and the intracerebral arteries-arterioles, and is splitinto six regional districts, independently controlled by autoregulationand CO reactivity. The cerebrovascular control mechanisms are in-dividually described by means of first-order low-pass dynamics, actingto directly maintain the physiological flow rate level. The consequentautoregulation mechanisms of vasodilatation and vasoconstriction areruled by a temporal variation of the distal compliances, C , and re-sistances, R . A unique pressure downstream from the distal regionrepresents the capillary pressure. The cerebral venous circulation isdefined by two-element Windkessel modelling, while the cerebrospinalfluid circulation is formed at the level of cerebral capillaries. In thefollowing, a single pathway (internal carotid artery (ICA) – middlecerebral artery (MCA)) highlighted in Fig. 1 (Panel 3) is studied asrepresentative of the blood flow and pressure distributions from largearteries to the capillary-venous circulation: left internal carotid artery( P a and Q ICA,left ), middle cerebral artery ( P MCA,left and Q MCA,left ),middle distal region ( P dm,left and Q dm,left ), capillary-venous circula-tion ( P c and Q pv ). Examples of the pressure time series of the ICA-MCA pathway are shown in Fig. 1, Panel 4. More details on thecerebral model are offered elsewhere [34].6 Pressure and Flow Rate Signal Analysis
The analysis, involving a record of 5000 beats (for both NSR and AF), fo-cuses on the pressure and flow rate time series along the ICA-MCA pathwayand can be divided in two main parts: (i) analyses of the continuous timeseries and (ii) beat-by-beat analyses. In the first set, the signal is continuousand defined by all the temporal instants of the time series. In the secondset, the signal is discretized and one per beat data are obtained. Therefore,discrtized time series are composed of 5000 elements, corresponding to the5000 beats simulated. The i-th element may contain the average values ( Q and P ), as well as the maximum ( Q max and P max ) and minimum ( Q min and P min ) values of the related haemodynamic variables computed over the i-th beat. NSR AF( P a , P MCA,left ) 1.00 1.00( P a , P dm,left ) 0.83 0.76( P a , P c ) 0.83 0.65( Q ICA,left , Q
MCA,left ) 0.99 0.98( Q ICA,left , Q dm,left ) 0.87 0.72( Q ICA,left , Q pv ) 0.88 0.80Table 1: Linear correlation coefficient, ρ , for NSR and AF conditions. Sig-nals are normalized with respect to their mean and standard deviation val-ues, as follows: x n = ( x − µ x ) /σ x . For mean and standard deviation values,please refer to [34].The linear correlation coefficient is calculated between the signal enter-ing into the brain ( P a ) and the signals downstream up to the capillary re-gion ( P MCA,left , P dm,left , P c ). Analogous computation is performed for flowrates, that is, Q ICA,left with respect to Q MCA,left , Q dm,left and Q pv . In Ta-ble 1 the linear correlation coefficient, ρ , between couples of haemodynamicsignals is reported in NSR (II column) and AF (III column) conditions. Inboth NSR and AF conditions the correlation - which remains very high inthe middle cerebral artery section - is damped towards the distal circulation.However, the damping is by far more relevant in the fibrillated condition. Atthe capillary-venous level, the correlation in AF is decreased by up to 21%with respect to NSR for the pressure, while in the distal region is decreasedby up to 17% for the flow rate. The key aspect emerging here is that AFhaemodynamic signals in the deep cerebral circulation are more prone than7igure 2: Autocorrelation functions (thin curves), R ( τ ), and correspondingenvelopes (thick curves), R env ( τ ), as functions of the delay time, τ . Pressure(left, panels a and c) and flow rate (right, panels b and d) signals from largearteries (top, panels a and b) to the capillary/venous region (bottom, panelsc and d). NSR: blue, AF: red.the corresponding NSR signals to lose their temporal interdependence withrespect to the large artery circulation. Peripheral signals differ much morefrom the corresponding input signals during AF than during NSR.Fig. 2 shows the autocorrelation functions, R ( τ ), together with the corre-sponding envelopes, R env ( τ ), of pressures and flow rates at the large arterieslevel (Fig. 2a,b) and in the capillary-venous region (Fig. 2c,d) for both NSR(blue) and AF (red) (for details, see the Appendix A). NSR autocorrelationsin all cerebral regions display quasi-repetitive patterns (with period of about0.8 s), and a decay in amplitude over the delay axis. The coherence timesreported in Appendix A provide evidence that NSR signals maintain long-term memory (around about 4 beats) through the whole ICA-MCA pathway(temporal coherence even increases a bit towards the distal/capillary circu-lation). The picture is substantially different in AF. For the input signals, R ( τ ) still shows a remaining quasi-periodicity although the decay rate is veryhigh. In the capillary region, the decrease in R ( τ ) resembles the behaviour8igure 3: Examples of T cr evaluation for an illustrative portion of the P dml,left time series (a: NSR, b: AF). T cr intervals are indicated in green.of random signals. The coherence times, τ c , in the AF regime confirm aloss of memory, with values varying between 1.11 and 0.87 s. Towards thedeep cerebral circulation the signal memory deteriorates even more so thatcapillary/venous haemodynamic signals during AF reveal short-time mem-ory features ( τ c < Quantification of the consecutive time lapses spent by each variable aboveor below a certain threshold is introduced here through the crossing time, T cr : it represents the temporal interval spent by the haemodynamic variableabove or below the threshold indicated by the mean value in NSR. Fig. 3(panels a and b) displays representative examples of T cr intervals for P dm,left ,during NSR and AF. The T cr intervals are found throughout the wholetemporal series to evaluate how AF influences the duration of excursionsfrom the reference mean value in NSR.As the crossing times, T cr , are computed over the whole temporal series,we can then evaluate their probability density functions (PDFs) in NSR andAF (Fig. 4). During NSR in the large artery region (blue curves, panelsfrom a to d), T cr values are narrowly centered around the mean value, whichis half of the average beat, i.e. 0.4 s, thereby showing a stable oscillationof the signals around their mean values. Going towards the distal/capillaryregion (blue curves, panels from e to h), the mean values do not substantiallyvary, but the variability around them increases, revealing wider probabilitydensity functions. In AF conditions, in the large artery region (red curves,panels from a to d), the mean values are comparable to those observed duringAF, while the standard deviation values are increased by 3 to 4 times. In the9igure 4: PDFs of the crossing times, T cr : NSR (blue), AF (red). Insertsshow the percentile analysis of the T cr values, demonstrating to which AFpercentile each NSR percentile corresponds.10eep circulation (red curves, panels from e to h), the T cr mean values increasewith respect to NSR and the standard deviation values increase by up to 3-4times with respect to the AF large artery region. The PDFs display muchmore pronounced right tails and lose the symmetry shown during NSR. Asdisplayed in the example of Fig. 3, the increased importance of the right-tails implies that the AF signals lose their periodicity around the mean valueand spend long times (up to 2-3 s) consecutively well above or below thephysiological threshold, without crossing it.Complementary information is related to the percentile analysis. To high-light the AF-induced changes at the cerebral level, during NSR we computedthe percentiles, from the 5th to the 95th (separated by 5ths), of the differ-ent quantities analyzed, conferring on these percentiles the role of referenceNSR thresholds. In AF, we then evaluated to which percentile each of thenineteen NSR thresholds corresponds, thus quantifying how AF modifies theprobability of reaching extreme values. An example is reported in Fig. 4,panel a: the T cr value indicated by the 5th percentile in NSR correspondsto the 37th percentile in AF. This means that a value which is extremelylow and rarely reached in NSR becomes common and frequently attained inAF.The inserts in Fig. 4 report the percentile analysis performed on the T cr val-ues and represent to which AF percentile (red) each NSR percentile (blue)corresponds. It can be noted that in the large artery region low NSR per-centiles (5-20 %) correspond to quite high AF percentiles (35-50 %), espe-cially for the pressure. This means that shorter T cr values are more likely tooccur in AF than in NSR. This picture does not apply to the distal/capillarycirculation, where, in addition, high NSR percentiles (95-80 %) relate tomuch lower AF percentiles (70-50 %). Thus, in the deep circulation, T cr values are statistically longer in AF than in NSR. This scenario is similarlyobserved for both pressure and flow rate and demonstrates in AF an in-creased probability of having long temporal ranges where excursions fromthe baseline haemodynamic value can develop.The combined analysis of PDFs and percentile variation of the crossing time T cr reveals in the distal cerebral region a higher probability of extreme valueevents, such as hypoperfusions or hypertensive peaks, since the pressure andflow rate signals remain above or below their reference values for much longer(and consecutively). With the following beat-by-beat analysis we will be ableto specify which kind of critical events may emerge, whether below (hypo)or above (hyper) the NSR haemodynamic thresholds.11 .2 Beat-by-beat analysis Minimum ( Q min and P min ) and maximum ( Q max and P max ) values overa cardiac beat are considered here, recalling that these are instantaneoushaemodynamic values. In Appendix B, the mean and standard deviationvalues of the 5000 minimum and maximum values are reported for thehaemodynamic variables. Fig. 5 presents the PDFs of the minimum andmaximum values for pressures and flow rates at the large arteries level (Fig.5a,b) and in the capillary-venous region (Fig. 5c,d) in NSR (blue) and AF(red). In NSR maximum and minimum PDFs are narrowly centered aroundthe relative mean values and the coefficients of variation ( c v = σ/µ ) arewell below 0.1, with values which do not significantly vary along the ICA-MCA pathway. In AF, the mean values do not essentially vary with respectto NSR (apart from P a ), while standard deviation values are significantlylarger, leading to c v values often above 0.1.The inserts in Fig. 5 exhibit percentile variations in AF with respect toNSR, by focusing on the maximum values for pressures (Fig. 5a,c) and min-imum values for flow rates (Fig. 5b,d). Although the input pressure is morelikely to present hypotensive events, partially due to an averagely lower P a in AF [35], on the contrary the probability of hypertensive events increasesalong the ICA-MCA pathway, with a maximum at the capillary level (95% in NSR corresponds to less than 70 % in AF). For flow rates we concen-trate on the percentile variations of the minima, as possible quantification ofhypoperfusive events. No significant differences emerge when moving fromlarge arteries towards the deep cerebral circulation (Fig. 5, panels b, d, f, h).Contrary to hypertensive events, which are mainly linked to the instanta-neous maximum pressure values reached, hypoperfusions are more related tothe temporal persistence of the flow rate below the physiological thresholds[34]. Therefore, pressure maxima are indicators of increased hypertensiveevents, while flow rate minima - being markers of low instantaneous flowrate - are not analogously symptomatic of hypoperfusions. An improvedinterpretation can be gained through the analysis of mean values per beat,which is offered in the following sections. The mean values per beat are computed for pressure ( P i ) and flow rate( Q i ) over the 5000 cardiac periods ( i = 1 , ..., P and Q , of the complete temporal signals: P ∗ i = P i − P and Q ∗ i = Q i − Q . In Fig. 6, flow rate-pressure scatter plotsare reported in NSR and AF conditions for the internal carotid artery (panela), middle cerebral artery (panel b), middle distal region (panel c), and thecapillary-venous region (panel d), together with a linear regression data fit-12igure 5: Probability density functions of the maximum ( P max and Q max )and minimum ( P min and Q min ) values, NSR: blue, AF: red. Pressure (left,panels a and c) and flow rate (right, panels b and d) from large arteries(top, panels a and b) to the capillary/venous region (bottom, panels c andd). Inserts represent the percentile analysis, showing how percentiles in NSRare modified in AF. Percentiles of maxima are reported for pressures, whilepercentiles of minima are shown for flow rates.ting for each condition with the corresponding coefficient of determination, R . Data dispersion is high at the large arteries level ( R < . R values around 0.97,with a strict direct proportionality between Q ∗ pv and P ∗ c . This implies that,at the capillary-venous level, hypertensive events are strictly concomitantwith hyperperfusions, while hypotensive episodes occur during hypoperfu-sions. The present behaviour is observed in both NSR and AF; however, therange of AF data is much wider, reaching extreme values.The inserts in Fig. 6a-d represent the PDFs of the mean flow rates, Q ∗ i ,confirming the enhanced variability going to the peripheral regions, whichis 3 times higher than in the large arteries. Moreover, standard deviationvalues, σ , are about 4 times higher in AF than in NSR at each region. Thiscombined increase in variability leads to extremely high data dispersion inthe micro-circulation during AF and underlies the mechanisms promoting13igure 6: Analysis of the mean values per beat (blue: NSR, red: AF).Scatter plots: (a) large arteries P ∗ a and Q ∗ ICA,left ; (b) middle cerebral artery P ∗ MCA,left and Q ∗ MCA,left ; (c) middle distal region P ∗ dm,left and Q ∗ dm,left ; (d)capillary-venous region P ∗ c and Q ∗ pv . Inserts represent the PDFs of mean flowrate values, Q ∗ i . Coefficients of determination, R , and the linear fittings arecomputed for each configuration.the presence of hypoperfusions. Multivariate linear regression models are shown here for the mean values P i and Q i in the four regions, having as regressors the preceding beats. Thecurrent mean value is indicated as P and Q , while with RR − i we refer tothe i-th preceding beat. The models are formalized as follows: P = a + N (cid:88) i =1 α i RR − i , (1) Q = b + N (cid:88) i =1 β i RR − i , (2)14igure 7: Multivariate linear regression models in NSR (blue) and AF (red)conditions, for pressure ((a,c), P a and P c ) and flow rate ((b,d), Q ICA,left and Q pv ). Coefficients of determination, R , as a function of the number ofregressors, N .where a and b are the intercept values, α i and β i represent the coefficientsof the linear multivariate model, while N is the number of regressors. Bychoosing N , a multivariate model is built with N regressors. The numberof regressors was tested up to N = 6, leading to 6 models for pressure and 6for flow rate, at each cerebral region and rhythm condition (96 models arecomputed in total). With N = 1, the multivariate linear regression modelturns into a univariate linear regression model.In Fig. 7, the coefficients of determination, R , are shown for each modelas a function of the number of regressors, N , in NSR and AF conditionsfor pressures and flow rates at the large arteries level ( P a and Q ICA,left )and in the micro-circulation ( P c and Q pv ). In general, AF presents lower R values than NSR for the multivariate models, while for the univariatemodels ( N = 1) AF shows higher R values than NSR for all flow rates andcapillary pressures. Moreover, in both NSR and AF, R values are higherin the peripheral regions than the large arteries, since the signal averageamplitude decreases going downstream (e.g., in AF P a,max − P a,min = 42mmHg, P c,max − P c,min = 6 mmHg).15igure 8: Coefficients α i and β i of the linear multivariate models, i = 1 , ...N ,with N = 4. (a) Pressures, P dm,left and P c ; (b) flow rates, Q dm,left and Q pv .NSR: blue, AF: red.For pressures, in the large arteries region the univariate models capturemost of the correlation, having R > . RR − and RR − are sufficient to accurately predict the current pressure level, P , at the cerebral entrance. At the capillary level, instead, R reaches aplateau for N = 4, meaning that 4 preceding beats are necessary to retainthe present haemodynamic content. For flow rates, in the large arteriesregion univariate models are not very informative ( R < . RR − and saturated with RR − . Inthe peripheral region, univariate models gain relevance exceeding R = 0 . R .For both pressures and flow rates, the number of preceding beats necessaryto fully describe the current state increases towards the micro-circulation.It should, however, be noted that, in all the regions and rhythm conditions,the multivariate model with 4 regressors ( N = 4) represents the maximumcorrelation level obtained. In fact, beyond this threshold, by adding fur-ther regressors the prediction of the current state is not improved. We cantherefore conclude that RR beats are significant regressors and the presenthaemodynamic state has memory of the past to the extent of about 4 beats.In other words, 4 consecutive beats are sufficient to predict the next pressureand flow rate levels.To better explore the role and weight of the preceding beats, the mul-tivariate models retaining the maximum correlation level (4 regressors) arenow analyzed in detail, with the related coefficients α and β shown in NSRand AF for the distal and capillary-venous sections (see Fig. 8). Coeffi-cients α and β are always negative with large absolute values, meaningthat a substantial contribution to potential hypoperfusions and hypotensiveepisodes is linked to the length of RR − . With a long RR − beating, the16erms β RR − and α RR − become predominant, leading to a low flow rateand low pressure levels. Coefficients related to RR − are more variable,since α and β are positive in the capillary-venous region, while in the dis-tal region α are negative and β are close to zero. Coefficients α and β present moderate positive values, and this scenario is found again with noconsiderable variation for the α and β coefficients.Based on the regression coefficients, the most dangerous RR combinationscan be finally studied, i.e. those configurations which are able to minimizeflow rate (hypoperfusion) and maximize pressure (hypertensive episode).We only consider the AF condition, where the RR beating is uncorrelated.In NSR, instead, the beating is correlated and closely varies around 0.8 s,therefore all coefficients have the same weight as they refer to beats whichall remain strictly around 75 b.p.m. • Hypoperfusions can be obtained with the following quadruplets: – Q dm,left : RR − long beat, RR − any beat, RR − short beat, RR − normal/short beat; – Q pv : RR − long beat, RR − normal/short beat, RR − shortbeat, RR − normal/short beat. • Hypertensive episodes may occur with the following quadruplets: – P dm,left : RR − : short beat, RR − short/normal beat, RR − nor-mal/long beat, RR − long beat; – P c : RR − short beat, RR − any beat, RR − long beat, RR − normal/long beat.The least probable configuration is the distal hypertension, since a se-quence of consecutive beats with decreasing duration (from long to short)has to occur, while the most probable combinations are represented by distalhypoperfusion and capillary-venous hypertension. In fact, to obtain one ofthese two conditions at the current state, it is sufficient to have a long (orshort) last beat and a short (or long) third to last beat, which is quite aplausible circumstance in AF. By means of different statistical tools, the signal analysis so far describednot only underlines the increasing impact of AF on the cerebral micro-circulation, but also suggests a coherent framework explaining why AF in-duces such evident haemodynamic changes. The key point is how differentlythe cerebral regions respond to the alteration of the cardiac rhythm.To better understand this, we forced the cerebral modelling with an ideal-ized input. A sinusoidal input signal (period T = 0 . P a ,17igure 9: (a) Sinusoidal P a signal (mean 100 mmHg, amplitude 20 mmHg)abruptly interrupted at its maximum (top panel) and pressure response inthe downstream cerebral districts ( P MCA,left , P dm,left , P c ). (b) Temporaldelays, T d , normalized with respect to the corresponding value in the firstregion ( P MCA,left ) along the ICA-MCA pathway.with mean value 100 mmHg and four different amplitudes (100 ± ±
10 mmHg, 100 ±
15 mmHg, 100 ±
20 mmHg). When at the maximum orminimum amplitude the signal is abruptly interrupted and instantaneouslyjumps to the mean value, then maintaining this steady state: the first caseis defined as an up-mean jump (an example with an amplitude of 20 mmHgis reported in Fig. 9a, top panel), while the second represents a down-meanjump. This approach is borrowed from the theory of dynamical systems,where the system is excited by an external impulsive forcing to understandits response time. In this case, the system involved is the cerebral circulationand reacts in the different downstream districts as reported in Fig. 9a (fromtop, P a , to bottom, P c ). Two basic remarks arise: (i) due to the inertiaof the system, the signal in the downstream sections does not immediatelyreach the steady level, but it goes on oscillating with a damped amplitudebefore recovering the equilibrium state, and (ii) the transient damping be-haviour considerably varies along the ICA-MCA pathway. To associate aquantitative measure with the transient dynamics, in every district down-stream from the carotid entrance we evaluate the time lapse, T d , necessaryto reach the steady constant levels of pressure. T d represents the tempo-ral delay or latency to recover the equilibrium constant state in responseto a sudden and abrupt variation, and it is identified by | d P/ d t < (cid:15) | (here (cid:15) = 10 − ) at each section. The latency T d at the entrance ( P a level) is0, since the jump is instantaneous, while it has a finite value immediatelydownstream. In Fig. 9b, we report the time delay for the pressure over theICA-MCA pathway normalized with respect to the corresponding value atthe first region, P MCA,left . For each of the four jump amplitudes (5 mmHg,180 mmHg, 15 mmHg, 20 mmHg), the two up-mean and down-mean jumpsgive similar results, thus the average value between them is taken. Sinceabsolute values of T d depend on the threshold (cid:15) and the jump amplitude, weconsider the time delay normalized with respect to the upstream district.In fact, the focus is not on the specific value assumed by the latency but onits variation towards the micro-circulation.It can be noted that the normalized behaviours of T d /T d ( P MCA,left ) donot practically depend on the jump amplitude. In the micro-circulationthe latency in recovering the equilibrium state is about five times greaterthan that at the beginning of the middle cerebral artery. The longer delay isdue to the interplay between the different mechanical features of the cerebralsystem, which here is modelled as an electric circuit, composed by a networkof resistances and compliances (Panel 3 of Fig. 1). These mechanical andstructural properties make the inertia of the system increase when enteringthe cerebral circulation towards the micro-vasculature. As a consequence,when a disturbance at the carotid level propagates into the cerebral vesselnetwork, the distal and capillary regions remain altered for longer. Thebehaviour is analogous to that of a system of springs in series and parallel,which is externally excited at one end: each spring stiffness combines withthe others, and, at a point far from the perturbed end, the damping of theoscillation is lengthened, even if the external perturbation is ceased.The synthetic alteration of the carotid signal here described is a limitcase, but it relates well to the fundamental mechanism underlying the resultsdescribed in the above signal analysis. In fact, AF leads to an irregularRR series, which in turn promotes - through the systemic circulation - acollection of in-series pressure disturbances at the carotid level. Each of theseperturbations singularly produces an alteration of the cerebral circulation.The higher mechanical inertia in the peripheral districts explains why hereduring AF right tails of the crossing time, T cr , become important (Section3.1.2). In fact, every P a modification leads to a downstream signal excursionfrom the physiological threshold (i.e., mean value in NSR). The consecutivetime lapse spent above or below this threshold allows the signal to reachmaximum, mean or minimum values which definitely exceed the NSR range(Sections 3.2.1 and 3.2.2). When the signal is uninterruptedly above thethreshold, hyperperfusions and hypertensive events are promoted. Whenthe contrary holds, hypotensive episodes and hypoperfusions occur.The continuous sequence of transient perturbations at the carotid en-trance represented by the AF beating does not allow the system to recoverthe physiological state before another disturbance arrives. The uncorrelatednature of AF beating enhances the complexity of the deep cerebral signaland reduces its predictability, since a disturbance can lead the system to thesame or opposite direction with respect to the previous perturbation. Asa consequence, the signal periodicity breaks towards the micro-circulation,provoking a decrease in the correlation and a reduction in the coherence19ime (Section 3.1.1). Although the predictive grade remains satisfactory,the increased signal complexity and uncertainty make the regression modelsperform less well in AF than in NSR (Section 3.2.3). Up to three or fourpreceding beats are necessary to averagely describe the current haemody-namic state and this temporal range is governed by the combined interplaybetween the superposition of different transient disturbances introduced intothe system by the AF and the intrinsic latency of the system (Section 3.2.3).However, characterization of the present haemodynamic state in the firstcerebral regions requires in general fewer preceding beats than in the capil-lary regions, that is, signal predictability deteriorates towards the peripheralregions. This aspect furthermore strengthens the basic mechanism describedthroughout the Discussion. Due to the mechanical features of the differentcerebral districts and their reciprocal interconnection, the micro-circulationsuffers much more and for longer from the AF-induced haemodynamic al-terations. The limiting aspects of the present work are related to the computational hy-potheses. The modelling of the cardiovascular system providing the pressureinput, P a , for the cerebral dynamics does not account for the baroreceptormechanisms in the short-term. Moreover, AF is simulated assuming an un-correlated beating and no atrial contraction, but with no increase in theconstant baseline value of elastance with respect to NSR. Additionally, nolong-term remodelling effects are captured and no reduced ventricular con-tractility is assumed. In the cerebral modelling, NSR and AF configurationssolely differ by the entrance inputs, P a , while the remaining haemodynamicframework is set as in healthy conditions. Several haemodynamic mechanisms have been recently proposed for the as-sociation between AF and cognitive dysfunction independent of clinicallyrelevant events. However, definitive clinical evidence is still missing, and,at the present stage, an in silico approach can be valuable in providingand addressing new haemodynamic-based suggestions for primary medicaltreatments.Through an accurate and diversified signal analysis the present work showsa range of possible symptoms for the alteration of the haemodynamic pat-terns during AF in the cerebral micro-circulation. AF signals in the distal-capillary circulation lose their temporal interdependence and predictability,becoming more complex and revealing short-memory features. The cross-ing time analysis displays an increased probability of extreme value events20hich, through the beat-by-beat analysis, results in hypertensive and hy-poperfusive episodes. The RR beats turn out to be good haemodynamicregressors. In particular, the role of the preceding beat in the current vascu-lar state is crucial, while up to four consecutive RR beats are necessary tofully describe the averaged haemodynamic level of the next beat. Exploitingthis outcome, the worst haemodynamic configuration occurs with a long (orshort) last beat and a short (or long) third to last beat, which is rathercommon during AF [36, 37, 45]. The intrinsic structural latency revealedby the cerebral circulation is plausibly disturbed by AF and exacerbates theobserved scenario.The framework here described can offer physically-based hints explainingwhy critical events, such as hypertensive or hypoperfusive episodes, are morelikely to occur in the cerebral peripheral regions during AF, thereby furtherstrengthening the haemodynamic link between AF and cognitive decline.
Authors’ contributions
SS and LR designed the study, developed the mathematical and computa-tional models, performed the statistical analyses, and drafted the paper. Allauthors conceived the study, contributed to the interpretation of the dataand manuscript writing, and gave final approval for publication.
Appendix A. Autocorrelation function, R ( τ ) , andcoherence time, τ c The autocorrelation function, R ( τ ), representing the correlation of the signalwith itself at different temporal lags τ , detects repeating temporal patternsand periodicity. Through its envelope, R env ( τ ), the autocorrelation functionis used as a standard measure of the coherence time, τ c [44]: τ c = (cid:90) + ∞−∞ | R env ( τ ) | dτ (A1)The coherence time, τ c , quantifies the degree of temporal correlation ofthe signal: long-term coherent signals have autocorrelation functions witha slow rate of decay, while short-term memory signals (such as random sig-nals) show very rapidly decaying autocorrelation functions. Within bioelec-tromagnetic signals, long-term refers to coherence times equal to or greaterthan 1-2 s [44]. Coherence time values, τ c , are reported in Table 2, for bothNSR and AF conditions along the ICA-MCA pathway.21SR AF P a P c Q ICA,left Q pv τ c , for the haemodynamic signals in the largearteries ( P a and Q ICA,left ) and in the capillary/venous region ( P c and Q pv ),for NSR (II column) and AF (III column). Appendix B. Minimum and maximum values anal-ysis
Table 3 presents the mean and standard deviation values of the 5000 mini-mum and maximum values for the haemodynamic variables.NSR AFVariable Minimum Maximum Minimum Maximum P a [mmHg] 77.91 ± ± ± ± P MCA,left [mmHg] 76.59 ± ± ± ± P dm,left [mmHg] 53.38 ± ± ± ± P c [mmHg] 21.69 ± ± ± ± Q ICA,left [ml/s] 2.25 ± ± ± ± Q MCA,left [ml/s] 1.90 ± ± ± ± Q dm,left [ml/s] 2.99 ± ± ± ± Q pv [ml/s] 9.70 ± ± ± ± P max and P min ) and flow rates ( Q max and Q min ). References [1] Piccini JP, Daubert JP 2014 Atrial fibrillation and sudden cardiac death:is heart failure the middleman?
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