From time-series to complex networks: Application to the cerebrovascular flow patterns in atrial fibrillation
aa r X i v : . [ phy s i c s . d a t a - a n ] S e p From time-series to complex networks: application to the cerebrovascularflow patterns in atrial fibrillation
Stefania Scarsoglio, a) Fabio Cazzato, and Luca Ridolfi Department of Mechanical and Aerospace Engineering, Politecnico di Torino, Torino,Italy Medacta International SA, Castel San Pietro, Switzerland Department of Environmental, Land and Infrastructure Engineering, Politecnico di Torino, Torino,Italy
A network-based approach is presented to investigate the cerebrovascular flow patterns during atrial fibrilla-tion (AF) with respect to normal sinus rhythm (NSR). AF, the most common cardiac arrhythmia with fasterand irregular beating, has been recently and independently associated with the increased risk of dementia.However, the underlying hemodynamic mechanisms relating the two pathologies remain mainly undeterminedso far; thus the contribution of modeling and refined statistical tools is valuable. Pressure and flow rate tem-poral series in NSR and AF are here evaluated along representative cerebral sites (from carotid arteries tocapillary brain circulation), exploiting reliable artificially built signals recently obtained from an in silico ap-proach. The complex network analysis evidences, in a synthetic and original way, a dramatic signal variationtowards the distal/capillary cerebral regions during AF, which has no counterpart in NSR conditions. Atthe large artery level, networks obtained from both AF and NSR hemodynamic signals exhibit elongated andchained features, which are typical of pseudo-periodic series. These aspects are almost completely lost towardsthe microcirculation during AF, where the networks are topologically more circular and present random-likecharacteristics. As a consequence, all the physiological phenomena at microcerebral level ruled by periodic-ity - such as regular perfusion, mean pressure per beat, and average nutrient supply at cellular level - canbe strongly compromised, since the AF hemodynamic signals assume irregular behaviour and random-likefeatures. Through a powerful approach which is complementary to the classical statistical tools, the presentfindings further strengthen the potential link between AF hemodynamic and cognitive decline.
The paper presents a network-based perspectiveto investigate the cerebrovascular flow patternsduring atrial fibrillation (AF) with respect to nor-mal sinus rhythm (NSR). There has been recentlygrowing evidence that AF, the most common car-diac arrhythmia with faster and irregular beat-ing, is independently associated to the increasedrisk of dementia. The topic has a high social im-pact given the number of individuals involved andthe expected increasing AF incidence in the nextforty years. Although several mechanisms try toexplain the relation between the two pathologies,causality implications are far from being clear.In particular, little is known about the distal andcapillary cerebral circulation. Thus, the contri-bution of modeling and statistical tools is valu-able. Here, we exploit the powerful techniquesof complex network theory to better character-ize the cerebrovascular patterns (in terms of insilico pressure and flow rate time-series) duringAF with respect to NSR. Each cerebral region(from large carotid arteries to capillary districts)is represented through a network, whose topo-logical features evidence the differences betweenarrhythmic and normal beating, thus highlightingin the micro-circulation plausible mechanisms forcognitive decline during AF. a) Electronic mail: [email protected]
I. INTRODUCTION
The growing size of numerical data coming from multi-scale simulations, highly-resolved imaging and computa-tional fluid dynamics approaches requires refined quan-titative tools to appropriately analyze biomedical signals.Complex network theory, by combining elements fromthe graph theory and statistical physics, offers an in-novative and synthetic framework to handle and inter-pret complex systems with a huge number of interact-ing elements . In the last decades, beside the well-established applications to Internet, World Wide Web,economy and social dynamics , complex networks havebeen employed in a variety of physical and engineer-ing systems, e.g. from hydrology and climate dynam-ics (e.g., ), turbulent and fluid flows (e.g., ), tobiomedical applications (e.g., ). The network-basedapproach has been extensively proposed for the charac-terization of time series which - by means of algorithmsbased on recurrence plot , visibility graph , correlationmatrix and pseudo-periodicity - are converted intocomplex networks.We here present a network-based approach, which re-lies on the correlation matrix and exploits the pseudope-riodic framework proposed for the electrophysiologicalsignals , to study the cerebrovascular flow patternsduring atrial fibrillation (AF). This cardiac pathology,characterized by irregular and accelerated heart-beating,is the most common arrhythmia with an estimated num-ber of 33.5 million individuals affected worldwide in2010 . Beside the well-known disabling symptoms (suchas palpitations, chest discomfort, anxiety, decrease ofblood pressure, limited exercise tolerance, pulmonarycongestion) which deteriorate the quality of life, AF isrelated to thromboembolic transient ischemic attacks and increases the risk of suffering an ischemic stroke byfive times . Through a collection of hemodynamic mech-anisms, such as silent cerebral infarctions , alteredcerebral blood flow , hypoperfusion and microbleeds,an independent association between AF and cognitive de-cline has been recently observed.Although an increasing number of very different obser-vational studies suggests this possible association (e.g.,among the most recent ), to the best of our knowl-edge none of them definitely states a causal relationbetween AF hemodynamics and cognitive impairment.Recent works highlight AF consequences on the cere-bral circulation (e.g. lower diastolic cerebral perfu-sion and decreased blood flow in the intracranial ar-teries), but the causal connections still remain mostlyundetermined . Moreover, current techniques -such as transcranial doppler ultrasonography and in-tracranial pressure measures - are difficult to be ob-tained and usually fail in capturing the cerebral micro-vasculature fluid dynamics. For all these reasons, lit-tle is known about AF effects on altered pressure levelsand irregular cerebral blood flow. However, an accuratecerebral hemodynamic mapping would be helpful in re-vealing AF feedbacks on brain circulation. For exam-ple, anomalous pressure levels could be symptomatic topredict haemorrhagic events such as microbleeds, whileimpair blood flow repartition can locally alter brain oxy-genation.Awaiting direct clinical evidences which are nowadayslacking, the efficiency of the computational hemodynam-ics is a promising branch , as it can be extremelyhelpful in isolating single cause-effect relations and un-derstanding which AF features lead to specific hemody-namic changes. In recent works, AF and normal sinusrhythm (NSR) cerebral hemodynamic signals were in sil-ico simulated and compared , revealing a dramati-cally altered scenario during AF, where critical events- such as hypoperfusions and hypertensions - are morelikely to occur in the peripheral brain circulation.The aim of the present work is to exploit a validatedmodeling algorithm to analyze, through some suitableadvanced metrics adopted in the complex network the-ory, how the peripheral cerebral circulation is altered byAF, in particular, in those regions where in vivo mea-sures are still missing. Hemodynamic signals of pressureand flow rate over 1000 heartbeats in NSR and AF areevaluated along four representative cerebral sites (inter-nal carotid artery, middle cerebral artery, distal region,capillary-venous district), exploiting the in silico data re-cently obtained . A total number of sixteen time series is thus analyzed. Each time series is then transformedinto a network, associating with each temporal segment anode and identifying possible links among nodes throughthe linear correlation coefficient. If the statistical interde-pendence between two nodes is above a sufficiently highthreshold, a link between the two nodes exists. The re-sulting sixteen networks are then studied with the net-work metrics, revealing in the peripheral circulation dif-ferent topological features between NSR and AF. II. METHODS
A stochastic modeling approach is adopted to repro-duce pressure and flow rate signals in the cerebral circu-lation during NSR and AF. Two lumped-parameter mod-els describing the cardiovascular and cerebral circulationsare exploited in series and stochastically forced throughartificially-built RR series, where RR (s) is the temporalrange between two consecutive heart beats. The cardio-vascular and cerebral models have been previously cal-ibrated and extensively validated as long as literaturedata are available to mimic realistic NSR and AFconditions for the cerebrovascular circulation. Thus, thevalidated in silico outputs are the object of the presentwork and used to understand how the cerebral fluid dy-namics and the temporal structure of the hemodynamicsignals modify during AF, especially in the peripheral re-gions where clinical measures are still lacking. This goalis achieved through a refined and innovative method forsignal analysis, based on the complex network theory,by broadening and strengthening what recently observedwith more classical statistical tools . A. Cerebral time series: pressures and flow rates in NSRand AF
The NSR and AF signals have been acquired from arecent computational hemodynamic study , which con-sists of an algorithm made of three subsequential steps.In Figure 1, the modeling approach adopted (panels 1,2, 3) is described and representative pressure time seriesobtained as outputs (panel 4) are displayed. The presentcomputational framework combines two different lumpedmodels in sequence: the cardiovascular model is run toobtain the systemic arterial pressure, P a , which is nextexploited as forcing input for the cerebral model.To obtain the cerebral time series, the first step is to ar-tificially generate the RR intervals, in NSR and AF con-ditions. We used artificially-built RR-intervals to avoidthe patient-specific details (e.g., sex, age, weight, car-diovascular diseases, ...) inherited by real RR beating.To purely catch the overall impact of a fibrillated cardiacstatus, we consider the same healthy young adult configu-ration, first forced through NSR and then by AF rhythm.Being normal beating a remarkable example of pinknoise , RR beats during NSR have been extracted from FIG. 1. Schematic algorithm of the modeling approach. (1) Cardiovascular model. 1000 extracted RR intervals in NSR (blue)and AF (red), and examples of P a time series obtained through the cardiovascular model. (2) Cerebral circulation. Scheme ofthe cerebral vasculature forced by the P a signal. (3) Cerebral model. R : resistance, C : compliance, Q : flow rate, P : pressure.The left ICA-MCA pathway is evidenced in red and is composed by P a , Q ICA,left , P MCA,left , Q MCA,left , P dm,left , Q dm,left , P c and Q pv . (4) Example of time series. Representative pressure time series along the ICA-MCA pathway, in NSR (blue) andAF (red) conditions. a pink-correlated Gaussian distribution (mean µ = 0 . σ = 0 .
06 s), which is the typical dis-tribution observed during sinus rhythm . Differentlyto the white noise which is uncorrelated, the adoptionof a pink noise introduces a temporal correlation, whichis a common feature of the normal beating . TheAF distribution is commonly unimodal (60-65% of thecases) and is fully described by the superposition oftwo statistically independent times , RR = ϕ + η . ϕ isextracted from a correlated pink Gaussian distribution, η is instead drawn from an uncorrelated exponential dis-tribution (rate parameter γ ). The resulting AF beats arethus obtained from an uncorrelated Exponentially Gaus-sian Modified distribution (mean µ = 0 . σ = 0 .
19 s, rate parameter γ = 7 .
24 Hz). TheAF beating results less correlated and with a higher vari-ability than NSR, as clinically observed . The RRparameters of both configurations are suggested by theavailable data , and by considering that the coeffi-cient of variation, cv , is around 0.24 during AF . Sincethese RR time sequences have been validated and testedover clinically measured beating , we adopt them asthe most suitable and reliable RR time-series to modelNSR and AF conditions. Moreover, both RR series have been chosen with the same mean heart rate (75 bpm) tofacilitate comparison between the two conditions. Moredetails on RR extraction are reported elsewhere .1000 cardiac cycles, shown in Fig. 1 (first panel), are con-sidered for each configuration. This allows us to achievestatistically stable results. Namely, we tested that otherextractions of the same number of beats give results anal-ogous to those described in the following sections III andIV.As second step, a lumped cardiovascular model wasrun using the RR signals as input, to obtain the signalsof systemic arterial pressure, P a , in NSR and AF condi-tions (Fig. 1, first panel), which will be the forcing in-puts of the cerebral model. The cardiovascular modeling,first proposed by Korakianitis and Shi and then widelyadopted in the computational hemodynamics , consistsof a network of electrical components - such as compli-ances, resistances and inductances - and describes thewhole systemic and pulmonary circulation, with an activerepresentation of the four heart chambers. Parameterswere fitted to reproduce the physiological hemodynamicsof a young healthy man, by providing results which arein agreement with the expected real behavior . More-over, by suitably changing the parameters, the compu-tational approach is able to capture the main featuresof different cardiovascular pathologies, such as hyperten-sion and valvular diseases . During AF, the modelwas first tuned and extensively validated in resting con-ditions over more than thirty clinical studies . Then,it has been successfully adopted to evaluate different AFaspects, such as the concomitant presence of left valvulardiseases and the role of increased heart rate at rest and under effort . To mimic the absence of the atrialkick, which is observed in AF, left and right atria areimposed as passive.In the end, the P a signals are introduced in a lumpedparameter model which simulates the entire cerebralcirculation (Fig.1, second panel). The cerebral modelis based on electrical counterparts and accurately de-scribes the cerebral circulation up to the peripheral andcapillary regions. It is able to reproduce several dif-ferent pathological conditions characterized by hetero-geneity in cerebrovascular hemodynamics and can be di-vided into three main sections: large arteries (Fig. 1,light blue box), distal arterial circulation (Fig. 1, greenbox), and capillary/venous circulation (Fig. 1, yellowbox). The computational approach has been validatedmainly over mean flow rates up to the middle cerebralcirculation , since current clinical techniques for pres-sure and flow rate measures lack the resolving power togive any insights on the micro-vasculature. For this rea-son, the cerebral modeling can be a useful predictive toolto understand how the hemodynamic signals change to-wards the micro-circulation in presence of AF . Theleft vascular pathway ICA-MCA (i.e., internal carotidartery - middle cerebral artery) evidenced in Fig. 1(Panel 3, red path) is here analyzed as representativeof the blood flow and pressure distributions from largearteries to the capillary-venous circulation: left inter-nal carotid artery ( P a and Q ICA,left ), middle cerebralartery ( P MCA,left and Q MCA,left ), middle distal district( P dm,left and Q dm,left ), capillary-venous circulation ( P c and Q pv ). Pressure time series of the left ICA-MCApathway are reported as exemplificative of NSR and AFbehaviors, in the fourth panel of Fig. 1. More detailson the cerebral model are offered elsewhere . All thecerebral pressure and flow rate time series selected tobuild the corresponding networks have a frequency of 250Hz and are composed, as previously mentioned, by 1000heartbeats. B. Complex network metrics
Some fundamental concepts of the complex networktheory are here summarized , by recalling only the mea-sures and definitions which are relevant to the presentanalysis.A network is defined by a set V = 1 , ..., N of nodes and aset E of links { i, j } . We assume that a single link existsbetween a pair of nodes and no self-loops can occur. The adjacency matrix , A : A ij = ( , if { i, j } / ∈ E , if { i, j } ∈ E, (1)accounts whether a link is active or not between nodes i and j . The network is considered as undirected, thus A is symmetric, moreover A ii = 0 since self-loops are notallowed.The normalized degree centrality of the i − th node isdefined as k i = N P j =1 A ij N − , (2)and represents the number of neighbors of node i , normal-ized over the total number of possible neighbors ( N − eigenvector centrality , measuring the influence of thenode i in the network, is given by x i = 1 λ X k A ki x k , (3)where A ki is the adjacency matrix and λ is its largesteigenvalue (in modulus) . In matrix notation, we have: λx = xA, (4)where the centrality vector x is the left-hand eigenvectorof the adjacency matrix A related to the eigenvalue λ .The assortativity coefficient , r , is defined as r = 1 σ q X jk jk ( e jk − q j q k ) , (5)where q j is the distribution of the remaining degree, thatis the number of edges leaving the node other than theone we arrived along. e jk is the joint probability distri-bution of the remaining degrees of the two nodes, j and k (for undirected networks, e jk = e kj ). This quantityfollows the rules: P jk e jk = 1 and P j e jk = q k . Theassortativity coefficient is the Pearson correlation coeffi-cient of the degree between pairs of linked nodes, thus r ∈ [ − , r = 1, the network has perfect assor-tative mixing patterns, meaning that high-degree nodestend to connect each other (e.g., rich-club effect). When r = 0 the network is non-assortative or uncorrelated,which is typical of random graphs, while at r = − local clustering coefficient of node i is lc i = e (Γ i ) k i ( k i − , (6)where Γ i is the set of first neighbors of i , e (Γ i ) is thenumber of edges connecting the vertices within the neigh-borhood Γ i , and k i ( k i − / i , 0 ≤ lc i ≤
1. The local clustering coefficientgives the probability that two randomly chosen neighborsof i are also neighbors. The global clustering coefficient is the mean value of lc i , lc = P Ni =1 lc i /N .The betweenness centrality of node k is bc k = X i,j = k σ ij ( k ) σ ij , (7)where σ ij are the number of shortest paths connectingnodes i and j , while σ ij ( k ) represents the number ofshortest paths from i to j , across node k . If node k iscrossed by a large number of all existing shortest paths(i.e., high bc k values), then it can be considered an im-portant mediator for the information transport in thenetwork.The closeness centrality of node i is cc i = N − N P j =1 d ij (8)where the shortest path length, d ij , is the minimum num-ber of edges that have to be crossed from node i to node j ,with i, j ∈ V ( d ii = 0). If i and j are not connected, themaximum topological distance in the graph d ij = N − ≤ cc i ≤
1. According to this definition,node i has a high closeness centrality value when it istopologically close to the rest of the network. C. Building the networks: from time-series to complexnetworks
To transform the time series into complex networks,the approach proposed for pseudo-periodic series hasbeen adopted. The complete temporal signal is dividedinto sequential cycles according to the RR intervals,which represent the portions of series corresponding toeach beat. Fig. 2a shows an example of pressure tem-poral series divided into 5 time segments, according tothe RR beating. Every temporal segment is associatedto a node of the network, with the convention that node i corresponds to the i − th beat ( i ∈ [1 , R is built, where the element R ij represents the max-imum value of the linear correlation coefficient betweenthe i − th and j − th temporal segments. The maximumcorrelation value is taken when the two segments have t [s] P a [ mm H g ] FIG. 2. (a) Example of pressure time series at the internalcarotid entrance, P a . Each RR interval is transformed into anode. (b) Example of network built from the pressure timeseries reported in panel (a), by taking R ij ≥ τ , where τ is theninth decile of the matrix R . different temporal lengths, by shifting the shortest signal-ing segment all along the length of the longest segment.Each matrix R is symmetric ( R ii = 1 by definition) andhas dimension n x n , where here n = 1000 is the numberof the heartbeats (i.e., nodes) analyzed. Exploiting thesymmetry of the R matrix, the correlation coefficientsto be evaluated are the matrix elements above the di-agonal and correspond to the number of possible links, n ( n − /
2. In the example of Fig. 2b, the correlationmatrix is (5x5), thus we have 10 correlation coefficients.To compare pairs of cycles, the maximum value of thecorrelation coefficients has been selected among other dis-tance criteria. Two other distance measures have beenchecked. The first analyzed is the phase space distance ,defined as M ij = min l l i , l j ) min( l i ,l j ) X k =1 || X k − Y k +1 || , (9)where l i and l j are the lengths of the cycle i and j , while X k and Y k are the k − th elements (e.g., hemodynamicsignals) of the cycles i and j , respectively. M representsthe minimum value of the sum of the modules of thedifferences between the samples of all the possible pairsof cycles. The second measure is based on the meanvalue distance. For each cycle, i , a mean value of thehemodynamic signal is computed, x i . Then, the distancematrix, D , is built considering the distance, d i , betweenthe mean value of the cycle i and the average value of thecomplete signal. The element D ij is defined as | d i − d j | .The three distance measures here introduced lead to sim-ilar results, however the most significative to evidence thedifference between NSR and AF conditions turned out tobe the correlation matrix, R . Therefore, results in Sec-tions III and IV are presented only by using the linearcorrelation.To define the adjacency matrix, A , and the correspond-ing network, a link between nodes i and j exists whether R ij ≥ τ , where τ is the ninth decile of the matrix R (com-puted excluding the diagonal). In so doing, each resultingnetwork is undirected ( A ij = A ji ) and unweighted, since A ij = 1 (unitary weight) as long as R ij ≥ τ . In the ex-ample of Fig. 2b, the ninth decile of the corresponding R matrix is 0 . τ , has been long discussed incorrelation-based networks and is a non-trivial aspectof building the network. It should represent a good com-promise between a very high degree of correlation and asuitable network dimension. Our goal is to compare thenetworks here built, which represent different cerebral re-gions (from the internal carotid to the capillary regions)in physiological (NSR) and pathological (AF) conditions.Therefore, we preferred using a threshold which, case bycase, through the ninth decile accounts for the maximumcorrelation of the local dynamics considered, rather thana unique threshold (e.g., τ = 0 .
8) fixed for all the con-figurations, which could be not equally meaningful in allthe districts and conditions. Since τ is the only arbitraryparameter involved in the network building, a sensitivityanalysis with different τ percentile values (namely, 85 th and 95 th percentiles) is reported in the Appendix A.The described mapping of time series into networks hasbeen achieved for pressure and flow rate series duringNSR and AF in the 4 aforementioned cerebral regions.Thus, 16 networks are built and analyzed starting fromthe corresponding 16 hemodynamic series. III. RESULTS
Outcomes for the metrics introduced in Section IIBare here presented for the 16 networks. Apart from theassortativity which is a global network parameter, eachnode of a network has a value for the analyzed metrics(degree centrality, eigenvector centrality, local clusteringcoefficient, betweenness centrality and closeness central-ity). To synthesize this large amount of information, webuild the probability density function (PDF) and the cu-mulative function (CDF) distributions for all the met-rics of each network. Then, at the same district and for ( )
P = (a)P a P MCA,left P dm,left P c P ii Eigenvector CentralityDegree CentralityLocal ClusteringBetweennessCentralityClosenessCentrality (b)
FIG. 3. (a) Example of PDF difference matrix, P , computedover the betweenness centrality, bc , for the pressure signal. Irow and column: large arteries; II row and column: middlecerebral artery; III row and column: distal middle artery; IVcolumn and row: capillary/venous circulation. (b) Diagonalvalues of P matrixes for all the metrics considered ( k , x , bc , cc , lc ) along the ICA-MCA pathway (pressure signal). the same hemodynamic signal (pressure and flow rate)we compare NSR and AF conditions. We recall that,given a beating condition (NSR or AF), results shownin the following are insensitive to the specific RR val-ues composing the sequence of 1000 cardiac cycles, sincethis number of beats allows the statistical stationary tobe reached. In fact, we checked that other extractionswith the same number of beats give analogous trends,with negligible differences with respect to the differencesobserved between NSR and AF.A first concise outcome is revealed by the matrix, P ,defined as P ij = Z D | p i ( x ) − p j ( x ) | dx, i, j = 1 , ..., , (10)where p i and p j are two PDFs to be compared, while D = D i ∪ D j is the union of the domains D i and D j ,where the PDFs are defined. The values of the subscripts i and j vary from 1 to 4 according to the cerebral districtsconsidered (1: large arteries, 2: middle cerebral region,3: distal region, 4: capillary/venous circulation). P , foreach metric and hemodynamic signal, accounts for thearea of the difference (in module) between pairs of PDFdistributions along the different districts in NSR and AFconditions. P has the same dimension [4x4] as the cere-bral regions studied. In particular, each element of thediagonal of P represents the area of the difference be- cc P CC ( cc ) P a P MCA,left P dm,left P c (a) NSR 0 1000 2000 3000 4000 5000 bc P B C ( b c ) P a P MCA,left P dm,left P c (b) NSR0 0.2 0.4 0.6 cc P CC ( cc ) P a P MCA,left P dm,left P c (c) AF 0 1000 2000 3000 4000 5000 bc P B C ( b c ) P a P MCA,left P dm,left P c (d) AF FIG. 4. Pressure signals. Cumulative density function (CDF) distributions of the closeness centrality (left panels) and betwen-ness centrality (right panels), during NSR (top panels) and AF (bottom panels) conditions. The four cerebral districts ( P a , P MCA,left , P dm,left , P c ) are highlighted with different colors. tween two NSR and AF PDFs in the same region (from P for the large arteries to P for the capillary/venouscirculation). The upper triangular part of P takes intoaccount the difference between two PDFs at different dis-tricts in NSR (e.g., P represents the area of the differ-ence between the PDF at large arteries and the PDF atthe distal region during NSR). The lower triangular partof P expresses the difference between two PDFs at dif-ferent districts in AF (e.g., P represents the area of thedifference between the PDF at the middle cerebral re-gion and the PDF at the capillary/venous region duringAF). Since the PDF has unitary area, the area of thedifference between the two compared PDFs can vary be-tween 0 (when the distributions coincide) and 2 (whenthe distributions have completely different domains). Anexample of P matrix is computed over the betweennesscentrality in Fig. 3a for the pressure signal.By considering the diagonal values (from top to bottom)of all the P matrices, one can infer how much each met-ric is significant to capture AF-induced variations (withrespect to NSR) along the ICA-MCA pathway, as re-ported in Fig. 3b. All the metrics studied are displayedfor the pressure signal (similar results are found for theflow rate), where on the y-axis lie the P ii values, whileon the x-axis the cerebral stations are localized. Fig. 3bshows that the differences between healthy and fibrillatedconditions are minimal in large arteries and increase inthe distal and capillary circulation. Both the degree and the eigenvector centrality indicators, however, are notoptimal metrics in this context, since they do not sig-nificantly inherit structural signaling variations inducedby AF. The local clustering coefficient, lc , together withthe closeness, cc , and the betweenness, bc , centralitiesare the best metrics to highlight the differences betweenNSR and AF.The analysis of P matrices is exploratory since allowsus to discern the most useful metrics from those whichhere are not so meaningful. To this end, in the followingwe only focus on lc , bc and cc distributions, as well as onthe assortative mixing. Moreover, the preliminary exam-ination of P matrices reveals the presence of importantvariations, in absolute terms, between NSR and AF whengoing towards the microcirculation. However, evaluating P matrices is not sufficient, as we are not able to observewhether the metrics increase or decrease during AF withrespect to NSR along the ICA-MCA pathway. The bc and cc distributions are thus analyzed in a more exten-sive way through their CDFs. CDFs are shown insteadof PDFs because they are less sensitive to oscillations ofhigh-tail values and can be more easily interpreted.Figure 4 shows the CDFs of the closeness centrality(left) and betweenness centrality (right), starting fromthe pressures signals. Top and bottom panels refer toNSR and AF conditions, respectively. In each panel thedistributions at the four cerebral districts are reported.Figure 5 is organized similarly to Fig. 4, but the results cc P CC ( cc ) Q ICA,left Q MCA,left Q dm,left Q pv (a) NSR 0 1000 2000 3000 4000 5000 bc P B C ( b c ) Q ICA,left Q MCA,left Q dm,left Q pv (b) NSR0 0.2 0.4 0.6 cc P CC ( cc ) Q ICA,left Q MCA,left Q dm,left Q pv (c) AF 0 1000 2000 3000 4000 5000 bc P B C ( b c ) Q ICA,left Q MCA,left Q dm,left Q pv (d) AF FIG. 5. Flow rate signals. Cumulative density function (CDF) distributions of the closeness centrality (left panels) andbetwenness centrality (right panels), during NSR (top panels) and AF (bottom panels) conditions. The four cerebral districts( Q ICA,left , Q MCA,left , Q dm,left , Q pv ) are highlighted with different colors. are obtained with the flow rate signals.For both pressure and flow rate, during NSR the distri-butions of cc and bc (top panels of Fig. 4 and 5) assumesimilar values along the ICA-MCA pathway. During AF,closeness centrality dramatically increases towards thedistal and capillary/venous regions (Fig. 4c and 5c),with significantly higher values reached especially for thepressure (Fig. 4c). On the contrary, betweenness central-ity in AF tends to meaningfully decrease when enteringthe microcirculation (Fig. 4d and 5d), for both pressureand flow rate. The two hemodynamic signals (pressureand flow rate) confirm that in normal conditions no rele-vant variation occurs along the ICA-MCA pathway, whileduring AF closeness centrality increases and betweennesscentrality decreases.Apart from local clustering and assortativity coeffi-cients, the other metrics ( k , x , bc , cc ) here discussed aremeasures of a node prominence in a network. However,as observed so far, these measures behave differently andtheir trends along the cerebral pathway reveal non-trivialbehaviors . Degree and eigengvector centrality param-eters remain almost constant when entering the cerebralregions. As the two metrics represent a similar degree ofnode centrality, it is reasonable they are quite correlated.On the contrary, closeness and betweenness measures ofcentrality present opposite trends along the ICA-MCApathway.At the large arteries level in AF conditions, the network has lower cc values and higher bc values with respectto the microcirculation region. Low cc and high bc val-ues mean that the network is topologically elongated andchained. Each node (i.e., beat) is generally linked to theprevious and the next nodes (beats), as well as to otherfar nodes. On average, every node has a low closenesscentrality since it has direct links (i.e., low shortest path)only with its neighborhood, while the shortest paths withthe rest of the network are in general high. Moreover,each node is equally important with respect to the oth-ers for the information transmission. In fact, given twonon-directly connected nodes of the network, informationhas to necessarily pass through the intermediate nodes(beats) between them. This last aspect implies a high bc value for almost all the nodes.In the microcirculation region during AF, the networkhas higher cc values and lower bc values with respect tothe cerebral input. This configuration means that a nodeis more incline to connect with nodes which are not notits precedent and subsequent nodes. On average, eachnode needs a limited shortest path to reach all the othernodes of the network. As a consequence, cc value is high.On the contrary, since information has not to cross all theintermediate nodes between pairs of nodes, the between-ness centrality is averagely decreased. Being the shortestpaths in general shorter than at large arteries, the streamof information now excludes several nodes which were in-stead crossed at the carotid entrance. The network is P a P MCA,left P dm,left P c r NSRAF Q ICA,left Q MCA,left Q dm,left Q pv r NSRAF (a)(b)
FIG. 6. Assortativity coefficient, r , along the ICA-MCA path-way for pressure (panel a) and flow rate (panel b) signals.Blue: NSR, red: AF. P a P MCA,left P dm,left P c l c AFNSR Q ICA,left Q MCA,left Q dm,left Q pv l c AFNSR (a)(b)
FIG. 7. Local clustering coefficient, lc , along the ICA-MCApathway for pressure (panel a) and flow rate (panel b) signals.Mean values lc , are reported with open circles, while trianglesindicate lc ± σ lc , where σ lc is the standard deviation value ofthe lc distribution. Blue: NSR, red: AF. topologically more circular, with non-consecutive distantbeats (nodes) which often share a link.Since during AF degree and eigenvector centrality dis-tributions remain basically constant along the ICA-MCApathway, this entails that on average each node main-tains the same number of links with the other nodes.What makes the difference is the link topology. At thecarotid entrance, the links of a node are created withthe surrounding nodes as well as with long-range links(beats). Going towards the capillary/venous region, thelinks connecting consecutive nodes are almost all brokenand substituted with long-range links (i.e., links betweentemporally distant beats).Evidence of this different link distribution during AFalong the cerebral circulation is given by the assorta-tivity coefficient, r , along the ICA-MCA pathway (Fig.6). We recall that r measures the tendency of a net-work to present link between similar ( r = 1) or dissimi-lar ( r = −
1) nodes. An assortativity value close to zeromeans that none of the above trends is evidenced, links emerge with no preference between similar or dissimilarnodes, thus the network is uncorrelated. During NSR,both pressure and flow rate reveal a quite high assorta-tivity (0.65-0.7) which is maintained constant along theICA-MCA pathway (blue curves, panels a and b). InAF condition, the assortativity has analogous values asin NSR at the carotid entrance, with a significant droptowards the distal and capillary/venous circulation (redcurves, panels a and b). In the peripheral regions, linksare no more between nodes sharing the same properties(i.e., degree centrality), but spurious long-range links arepredominant. The network here resembles the featuresof a random uncorrelated graph.We conclude the Results Section with a comment onthe local clustering coefficient. In Fig. 3b, diagonal el-ements of the P matrix showed a slight increase of thismetric towards the capillary region. In Fig. 7, the meanvalues, lc , are reported for both pressure and flow ratesignals during NSR and AF, together with the dispersiondue to the standard deviation values, lc ± σ lc , where σ lc isthe standard deviation value of the lc distribution. It canbe noted that, the global local clustering values are quiteconstant along the ICA-MCA pathway in NSR as well asin AF. However, in the distal and capillary regions, thedata dispersion increases (for both NSR and AF), beinghigher during AF than NSR. Thus, going towards the mi-crocirculation during AF, the neighborhood of each nodecan be either almost fully connected or poorly connected.This is a further symptom of the increased variability andunpredictability induced by AF on the peripheral hemo-dynamic signals. IV. DISCUSSION
As evidenced in Section III, NSR and AF networks inthe cerebral microcirculation present significantly differ-ent structures. To better highlight the topological fea-tures, we graphically represent the pressure (Fig. 8) andflow rate (Fig. 9) networks at the beginning and to-wards the end of the ICA-MCA pathway in both NSRand AF conditions. Networks are visualized through theopen-source software package
Gephi , with respect tothe closeness centrality values. Red color correspondsto low cc values, while blue-colored nodes correspond tohigh cc values. In addition, the size of a node is propor-tional to its cc value.Let us first consider the networks obtained by the pres-sure signals, in Fig. 8. At the large arteries level (Fig.8, bottom panels), the NSR and AF networks present anelongated and almost planar shape, which is the typicalfeature of networks based on pseudo-periodic series, withsimilar mean closeness centrality values, cc ( cc : 0.257(NSR), 0.247 (AF)). In the capillary region, the NSRnetwork is slightly less elongated (Fig. 8, top left panel),but the mean closeness centrality value ( cc = 0 . cc = 0 . FIG. 8. Pressure signals. Networks at the internal carotid level (bottom panels, P a ) and in the capillary district (top panels, P c ), in NSR (left panels) and AF (right panels) conditions. Closeness centrality values are reported through the color (fromred, cc = 0, to blue, cc = 1) and are expressed as proportional to the size of the nodes. pletely altered (Fig. 8, top right panel). The network as-sumes a markedly circular and three-dimensional shape,which is usually encountered in random networks, withan average closeness centrality value ( cc = 0 . cc = 0 . cc = 0 . cc = 0 . cc = 0 . Q ICA,left net-work ( cc = 0 . cc = 0 . bc , cc , lc , r ) agreein locating this alteration. During AF, the input sig-nals ( P a and Q ICA,left ) exhibit pseudo-periodic features,which are almost completely lost towards the microcir-culation, where instead P c and Q dm,left signals revealrandom-like characteristics.The hemodynamic consequences of this substantial alter-ation can be of high biomedical impact. All the physi-ological phenomena at microcerebral level ruled by pe-riodicity - such as regular perfusion, mean pressure perbeat, average nutrient supply at cellular level - are notguaranteed and can be strongly compromised, since theAF hemodynamic signals assume irregular behaviour andrandom-like features. The cardiovascular implications ofthe highlighted alteration surely deserve to be furtherquantified through clinical evidences, although invasiveand accurate measurements are still difficult to be accom-1 FIG. 9. Flow rate signals. Networks at the internal carotid level (bottom panels, Q ICA,left ) and in the distal district (toppanels, Q dm,left ), in NSR (left panels) and AF (right panels) conditions. The size of the nodes is proportional to the closenesscentrality value, while color spans from red ( cc = 0) to blue ( cc = 1). plished, also because of the signal complexity induced bythe heart rate variability. Awaiting necessary in vivo val-idation, the network-based hints here emerged can plau-sibly explain the hemodynamic mechanisms leading tocognitive impairment in presence of persistent AF. V. CONCLUSIONS
The network analysis performed over cerebral hemo-dynamic signals highlighted that the degree centrality,which is usually the most intuitive and firstly analyzedmetric, is here not much informative. Other local (i.e.,eigenvector centrality) and mesoscopic (i.e., local clus-tering coefficient) measures are not crucial in discerningNSR and AF hemodynamic features. A deeper examina-tion of the adjacency matrix was necessary, especiallyin terms of global metrics. In particular, the mark-ers of betweenness and closeness centrality as well asthe assortativity coefficient turned out to be meaning-ful. From the combined analysis of the network met-rics through their probability distributions, during AF itemerges that towards the peripheral cerebral circulationthe closeness centrality increases, while the betweeness centrality is reduced. The AF pressure and flow ratenetworks change from an elongated shape (which is char-acteristic of pseudo-periodic series) in the large arteryregion to a circular-like shape (which is a feature of ran-dom series) in the capillary-venous districts.The complex network analysis evidences in a syntheticand innovative way how hemodynamic signals in the cere-bral microcirculation are deeply altered by AF. This re-sult, on one hand, confirms that the complex networktheory can be successfully extended to explore otherpathological biomedical signals in complex geometries,such as stenotic flows across aortic valve and flow dynam-ics in brain aneurysms. On the other hand, the presentfindings further strengthen the always more evident linkbetween AF hemodynamic and cognitive decline, througha powerful approach which is complementary to the clas-sical statistical tools.
Appendix A: Sensitivity Analysis
A sensitivity analysis on the threshold value τ is hereperformed, recalling that τ is the only arbitrary param-eter involved for building the network. Beside the 90 th P a P MCA,left P dm,left P c P ii kbccc(a) 85 th th th th th th Q ICA,left Q MCA,left Q dm,left Q pv P ii kbccc(b) 85 th th th th th th FIG. A1. Diagonal values of P matrixes for the degree, close-ness and betweenness centrality measures ( k , cc , bc ) alongthe ICA-MCA pathway considering three percentile values:85 th (dashed-dotted line), 90 th (thick line with symbols), 95 th (dashed line). (a) pressure, (b) flow rate. percentile of the R matrix, other two values have beenchosen to set the threshold τ , namely the 85 th and 95 th percentiles. In Fig. A1, the diagonal of P matrix is re-ported for the degree centrality, closeness centrality andbetweenness centrality for the three percentile values, forboth pressure (top panel) and flow rate (bottom panel)signals. It can be noted that, apart from the specificvalues, the trend of the P ii values remain unaltered bychanging the threshold τ . In fact, for all the percentilevalues considered, the degree centrality, k , remains al-most constant towards the peripheral circulation, whilebetweenness and closeness centrality values experiencesignificative variations with respect to large arteries. Theproposed percentile (90 th ) is therefore a good choice tocatch the network behavior which, despite the specificvalues assumed at each cerebral district, turns out to besubstantially insensitive to the threshold adopted. J. A. Etchings and K. Buetow,
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