Complex analysis of intracranial hypertension using approximateentropy*
Roberto Hornero, PhD; Mateo Aboy, PhD; Daniel Abasolo, MS; James McNames, PhD;Wayne Wakeland, PhD; Brahm Goldstein, MD, FCCM
Elevated intracranial pressure(ICP) following acute braininjury may be due to cerebraledema, cerebral hyperperfu-
sion, and/or intra- or extracranial hemor-rhage. Intracranial hypertension (ICH)has been defined as elevated ICP �25 mmHg for �5 mins in the absence of externalnoxious stimuli (1). ICH may be associ-ated with secondary brain injury due todecreased cerebral perfusion pressureand cerebral ischemia.
Current intensive care unit (ICU)monitoring devices only allow for display
of the ICP waveform with a digital vari-able display of the averaged 3- to 10-secmean ICP value (2). ICP alarms are typi-cally set for sustained periods of ICP inaccordance with the preceding definitionof ICH. Thus, by the time a bedside ICPalarm goes off, the ICP may have beendangerously elevated for seconds to min-utes, depending on the alarm settingsand type of ICP elevation. Conversely,very brief elevations in ICP lasting �3–10secs may be missed altogether.
We have previously demonstrated thatelevated ICP due to severe traumaticbrain injury (TBI) is associated withchanges in physiologic signal metrics de-rived from the electrocardiogram, arte-rial pressure waveform, and ICP wave-form that suggests uncoupling ofautonomic cardiovascular regulatorymechanisms (3–5). In addition, we re-ported preliminary data showing a break-down in long-range correlation behaviorof heart rate fluctuations, as measured bythe nonlinear scaling exponent, �, a non-linear complexity statistic calculated bydetrended fluctuation analysis (6, 7). Thisline of research suggests that metrics to
analyze the ICP signal other than thetime-averaged mean may have physio-logic and clinical significance and thatthere are measurable differences in signalmetrics between patients with ICH.
In this study, the objective was to de-termine whether there were immediatechanges in the ICP signal before, during,and after an acute elevation in ICP (aso-called “ICP spike”) using metrics otherthan the time-averaged mean value. Wehypothesized that measures of ICP com-plexity would decrease during periods ofICH compared with baseline values whenthe ICP was within physiologic normalranges, providing evidence for decom-plexification during an acute and finiteperiod of severe physiologic stress (8). Totest this hypothesis, we measuredchanges in the complexity of the ICP sig-nal, estimated by the approximate en-tropy (ApEn), as patients progressed froma stable state of normal ICP (�25 mmHg) to ICH (ICP �25 mm Hg for �5mins in the absence of external noxiousstimuli) (9), and then back toward pre-ICH levels.
*See also p. 245.From ETSI-Telecomunicación de Valladolid, Uni-
versity of Valladolid, Spain (RH, DA); Biomedical SignalProcessing Laboratory, Portland State University, Port-land, OR (MA, JM); Oregon Institute of Technology,Portland, OR (MA); Systems Science PhD Program,Portland State University, Portland, OR (WW); and Di-vision of Pediatric Critical Care, Pediatrics, OregonHealth & Science University, Portland, OR (BG).
No authors have any financial interests or conflictsrelated to the work herein.
Objective: To determine whether decomplexification of intra-cranial pressure dynamics occurs during periods of severe intra-cranial hypertension (intracranial pressure >25 mm Hg for >5mins in the absence of external noxious stimuli) in pediatricpatients with intracranial hypertension.
Design: Retrospective analysis of clinical case series over a30-month period from April 2000 through January 2003.
Setting: Multidisciplinary 16-bed pediatric intensive care unit.Patients: Eleven episodes of intracranial hypertension from
Interventions: None.Measurements and Main Results: We measured changes in the
intracranial pressure complexity, estimated by the approximateentropy (ApEn), as patients progressed from a state of normalintracranial pressure (<25 mm Hg) to intracranial hypertension.We found the ApEn mean to be lower during the intracranial
hypertension period than during the stable and recovering periodsin all the 11 episodes (0.5158 � 0.0089, 0.3887 � 0.077, and0.5096 � 0.0158, respectively, p < .01). Both the mean reductionin ApEn from the state of normal intracranial pressure (stableregion) to intracranial hypertension (�0.1271) and the increase inApEn from the ICH region to the recovering region (0.1209) weredetermined to be statistically significant (p < .01).
Conclusions: Our results indicate that decreased complexity ofintracranial pressure coincides with periods of intracranial hy-pertension in brain injury. This suggests that the complex regu-latory mechanisms that govern intracranial pressure may bedisrupted during acute periods of intracranial hypertension. Thisphenomenon of decomplexification of physiologic dynamics mayhave important clinical implications for intracranial pressuremanagement. (Crit Care Med 2006; 34:87–95)
Approximate entropy is a metric toestimate system complexity. A low valueof ApEn indicates predictability, regular-ity, or a quantitatively less complex state,whereas high ApEn indicates unpredict-ability, irregularity, and greater complex-ity (10). ApEn is a computable measurethat can be used to determine changingsystem complexity from time series and ispotentially capable of classifying complexsystems from a relatively small amount ofdata. It has been used mainly in the anal-ysis of heart rate variability (11), endo-crine hormone release pulsatility (12),and the impact of pulsatility on the en-semble orderliness of neurohormone se-cretion (13). Furthermore, ApEn hasbeen applied to studies discriminatingatypical electroencephalograph (14) andrespiratory patterns (15) from normativecounterparts, to estimate depth of anes-thesia (16), and to examine the time andfrequency structure of Parkinson’s dis-ease tremor (17, 18). Preliminary evi-dence suggests that when applied to anal-ysis of electroencephalographs, the ApEnis predictive of epileptic seizure (19).ApEn has also been used to study theconnection between panic disorder andrespiration dynamics (20), to investigatechanges during stages of consciousness,and to associate such alterations withbrain function (21). This measure ofcomplexity, however, has not been ap-plied directly to the ICP signal to evaluatechanges in system complexity during dif-ferent pathologic states of brain injury. In
this article, the term complexity refers tothe estimated complexity as measured bythe ApEn metric (see Appendix 1 for def-initions).
MATERIALS AND METHODS
Patients and Patient Management andData Acquisition. The study protocol was re-viewed and approved by the Institutional Re-view Board at Oregon Health and Science Uni-versity. The requirement for informed consentwas waived.
This study included 11 ICP spikes fromseven patients with brain injury admitted tothe pediatric ICU at Doernbecher Children’sHospital. The patients’ age, gender, mecha-nism and description of injury, admissionGlasgow Coma Scale score (22), survival, andGlasgow Outcome Score (23) are listed in Ta-ble 1. Management of the six patients withsevere traumatic brain injury (TBI) followedthe recently published “Guidelines for theAcute Medical Management of Severe Trau-matic Brain Injury in Infants, Children, andAdolescents” (1). One patient with a history ofcraniosynostosis and severe headaches was ad-mitted for ICP monitoring.
ICP Database and ICP Spike Detection.Data for this study were obtained from thephysiologic signal library in the Complex Sys-tems Laboratory (2). The database consisted of76 GB of ICP data collected from 93 patientsfrom 1998 to 2003. ICP was monitored con-tinuously using a ventricular catheter or pa-renchymal fiberoptic pressure transducer (In-tegra NeuroCare, Integra LifeSciences,Plainsboro, NJ). The ICP monitor was con-nected to a Philips Merlin patient monitor
(Philips, Best, Netherlands) that sampled theICP and arterial blood pressure signals at 125Hz. An HPUX workstation automatically ac-quired these signals through a serial data net-work, and they were stored in files on CD-ROM. Detailed description of the dataacquisition system used in the Complex Sys-tems Laboratory was previously reported (24).
The following criteria were used to detectICP spikes. The criteria are specified in termsof three nonoverlapping segments of the meanICP signal: a 300-sec stable region, a 10- to300-sec transition zone, and a 20-sec region ofICH. Similar criteria were reported in an ear-lier study (25).
1. The difference between the minimumvalue in the critical region and the max-imum value in the stable region was �10mm Hg. This ensured that the detectoronly detected significant elevations of�10 mm Hg that occur over a period of�5 mins.
2. The minimum ICP value in the criticalregion was �20 mm Hg. This ensuredthat each ICP spike was large enough tobe clinically significant (26).
3. The mean ICP was in the range of 0–100mm Hg in the stable region and �150mm Hg in the ICH region. These criteriawere used to limit artifact from beingdetected as ICP spikes.
4. Each ICP spike was separated from pre-ceding ICP spikes by �5 mins. This en-sured that a single long elevation in ICPwas not detected as two separate ICPspikes.
Table 1. Subject characteristics
Subject Age GenderMechanism and Description of
Brain InjuryAdmission
GCS Survival GOS
1 9.5 F Fall off horseDepressed skull fracture, IPH,
cerebral edema
7 Y 4
2 4.5 M MVASkull fracture, SDH, cerebral edema
3 Y 3
3 8 F Status post-craniosysostosis repairat age 2 yrs, ICP monitoring forheadaches
15 Y NA
4 4.75 F MVASkull fracture, IPH, cerebral edema
5 Y 2
5 11.5 M MVASkull fracture, SDH
8 Y 3
6 12.5 M Gunshot woundIPH, SDH, cerebral edema
4 Y 4
7 15.8 M Fall off skateboardDepressed skull fracture, IPH,
cerebral edema
3 Y 3
Mean � SD 9.5 � 4.1 6 � 4 3 � 1
GCS, Glasgow Coma Scale score; GOS, Glasgow Outcome Score; IPH, intraparenchymal hematoma; MVA, motor vehicle accident; ICP, intracranialpressure; NA, not applicable; SDH, subdural hematoma.
88 Crit Care Med 2006 Vol. 34, No. 1
To estimate the mean ICP, a low-pass filterwas applied with a cutoff frequency sufficientlylow to eliminate the pulsatile components ofthe ICP signal due to respiration and heartbeats. The noncausal low-pass filter had a cut-off frequency of 0.22 Hz and zero phase delay.To decrease the computational load, the signalwas decimated by a factor of 225 using a non-causal eight-order low-pass Chebyshev type Ifilter. This changed the effective sampling ratefrom 125 Hz to 0.556 Hz. The criteria de-scribed in the previous section were then ap-plied to every sample of the decimated signalto determine whether a spike occurred.
ICP spikes that met the specified criteriawere visually screened for artifact. This screenwas based on a plot of the ICP signal spanning20 mins before and 30 mins after the leadingedge of the spike and a spectrogram of thesame segment. The visual screen eliminatedcandidate spikes if a) they contained artifact;b) there was an abrupt drop in the ICP signalconsistent with cerebrospinal fluid with-drawal; c) the signal was clipped; or d) the ICPspike was part of the preceding episode ofintracranial hypertension. If there were noproblems detected during the 50-min record,the ICP spike was included in the study.
The automatic spike detection algorithmfound 166 ICP segments that met our criteriafor an acute spike. During the visual screen wefound that 31 of the segments contained arti-fact, 28 of were actually periods of cerebralspinal fluid drainage when the ventriculos-tomy catheter was turned “off” to the pressuremonitor and created a “false” spike. An addi-tional 95 segments were “clipped” at the max-imum range of the patient monitor (i.e., the
top or bottom of the ICP waveform was cut offcreating signal artifact not suitable for analy-sis—this technical problem has since beenresolved). One segment was identified as asecond detection of a single ICH episode. Theend result was 11 clean records of ICH de-tected from seven different patients that wereused for analysis. An example of a detectedICH region is shown in Figure 1. Figure 2shows an example of the ICP segments usedfor analysis including the stable, ICH, andrecovering regions.
Approximate Entropy. Approximate en-tropy (ApEn) is a family of variables and sta-tistics introduced as a quantification of regu-larity in time-series, initially motivated byapplications to short and noisy data sets. It wasfirst proposed by Pincus (10) in 1991, and itsbiological and physiologic applications arespreading rapidly (27, 28). Several propertiesof ApEn facilitate its utility for empirical timeseries analysis (29):
1. ApEn is nearly unaffected by noise belowa de facto specified filter level (r).
2. ApEn can be applied to time series of�50 points with good reproducibility.
3. ApEn is finite for stochastic, noisy deter-ministic, and composite processes.
4. Increasing values of ApEn correspond tomore irregularity in the time series or tointuitively increasing process complex-ity.
The potential uses of ApEn to provide newinsights in epidemiologic settings are consid-erable from a complementary perspective to
that given by more classic statistical methods.It appears that ApEn has potential widespreadutility to practical data analysis and clinicalapplication due to the salient features it bears.Moreover, when applied to the analysis of bio-medical time series, ApEn does not show theimportant drawbacks (e.g., very long data se-quences needed to estimate them accurately,data must be stationary) that many widelyapplied nonlinear methods (correlation di-mension, first positive Lyapunov exponent)have. Webber (30) recently pointed out someof the limitations associated with ApEn. Evenconsidering these limitations, for analysis ofrelatively short time series and intrasubjectcomparisons, the advantages of using ApEncompared with other nonlinear metrics sug-gest that it is a valid choice.
ApEn is scale invariant, is model indepen-dent, evaluates both dominant and subordi-nate patterns in data, and discriminates seriesfor which clear feature recognition is difficult;notably it detects changes in underlying epi-sodic behavior not reflected in peak occur-rences or amplitudes (31). ApEn assigns anonnegative number to a time series, withlarger values corresponding to more complex-ity or irregularity in the data. It has two user-specified variables: a run length m and a tol-erance window r. Briefly, ApEn measures thelogarithmic likelihood that runs of patternsthat are close (within r) for m contiguousobservations remain close (within the sametolerance width r) on subsequent incrementalcomparisons. ApEn has two user-specifiedvariables: a run length m and a tolerance win-dow r. It is important to consider ApEn(m,r)—or ApEn(m, r, N), where N is the numberof points of the time series—as a family ofvariables: Comparisons between time seriessegments can only be made with the samevalues of m and r (29). Appendix 2 includes adescription of the algorithm used to calculatethe ApEn metric.
Before ApEn estimation, the ICP signalswere filtered to eliminate the low-frequencycomponents (baseline trend) and remove themean pressure (DC component). We used ahigh-pass equi-ripple FIR filter with a cutofffrequency of 0.5 Hz. This guaranteed that ourApEn estimates obtained from the ICP signalwere based exclusively on the ICP pulse mor-phology, since both the mean ICP and thebaseline trend were eliminated with by thehigh-pass filter. Figures 2 and 3 illustrate thismethodology. Each filtered ICP signal waswindowed into segments of 20 secs in dura-tion. ApEn was estimated for each segment.We used normalized variables of m � 1 and r� 20% of the segments’ time series SD (13).Normalizing r in this manner gives ApEn atranslation and scale invariance, in that it re-mains unchanged under uniform processmagnification, reduction, or constant shift to
Figure 1. Illustration of the criteria for an acute elevation in intracranial pressure (ICP). The signaldivided into three segments: a 5-min stable region (SR), a 10- to 300-sec transition zone (TZ), and a20-sec critical region (CR). The stable and critical regions had a required separation of �10 mm Hg.
89Crit Care Med 2006 Vol. 34, No. 1
higher or lower values (29). Several studies(10, 31, 32) have demonstrated that these in-put variables produce good statistical repro-ducibility for ApEn for time series of length n� 60, as considered herein.
Statistical Analysis. Statistical analysis wasaimed at determining the statistical signifi-
cance of the mean reduction in ApEn duringthe ICH period for each of the 11 episodes. Weconsidered the episodes to be independent.Assuming independence is reasonable becauseacute spikes occur very sporadically. Havinginformation from one spike does not provideany a priori information about the next spike.
Mean ApEn values were obtained for 2-minwindows immediately before, during, and afterthe ICH period. We used bootstrap to estimatethe standard error of the means for each ofthese states. We performed a nonparametrichypothesis test to determine statistically sig-nificant mean ApEn reductions based on boot-strap (33). The main advantage of using thenonparametric bootstrap technique is that itcan be used to assess the statistical signifi-cance of these reductions without making anyassumptions about the distribution of themean ApEn reductions. The nonparametricbootstrap hypothesis testing involves comput-ing a bootstrap confidence interval for thedifference of mean ApEn. Equality of the meanApEn is obtained if zero is a possible value inthe confidence interval. Results were consid-ered to be statistically significant if p � .01.
RESULTS
Table 2 shows the estimated meanApEn during the stable, ICH, and recov-ering regions for each of the 11 episodes.Table 3 shows the estimated standard er-rors corresponding to the ApEn meansshown in Table 2. The mean ApEn waslower during the ICH period than duringthe stable and recovering period in all the11 episodes (p � .01, Table 2). The meansacross all the 11 episodes during the sta-ble, ICH, and recovering regions wereestimated to be 0.5158 � 0.0089, 0.3887� 0.077, and 0.5096 � 0.0158, respec-tively. Both the mean reduction in ApEnfrom the state of normal ICP (stable re-gion) to the ICH region (�0.1271) andthe increase in ApEn from the ICH regionto the recovering region (0.1209) weredetermined to be statistically significant(p � .01). Figure 4 shows histograms ofthe mean ApEn for the stable, ICH, andrecovering regions. Figure 5 shows a plot
Figure 2. Illustration of the type of intracranial pressure (ICP) spike analyzed in this study. The lightgray is the ICP signal waveform sampled at 125 Hz, and the dark line is the moving average ICPpressure. All ICP segments studied were 50 mins long, with the onset of the ICH period synchronizedat minute 20.
Figure 3. Before approximate entropy (ApEn) estimation, the intracranial pressure (ICP) signals werefiltered to eliminate the low-frequency components (baseline trend) and remove the mean pressure(DC component). This figure illustrates the results of this operation. Processing the high-passed ICPsignal ensures that the ApEn metric obtained is based entirely on the ICP beat morphology, since theICP mean pressure information is eliminated in this operation.
Table 2. Mean approximate entropy of the intra-cranial pressure signal for the stable, intracranialhypertension, and recovering regions
Standard errors for each of the columns areshown in Table 3.
90 Crit Care Med 2006 Vol. 34, No. 1
of the normalized ApEn for each of the 11episodes and the median ApEn across allthe episodes. The estimated ApEn de-creases as patients progressed from a sta-ble state of normal ICP to ICH. The levelof complexity begins to return withinminutes as ICP drops below 20–25 mmHg. Figure 6 shows the ApEn vs. ICP fortwo representative patients during thestudy period.
DISCUSSION
Our main finding was that ApEn of theICP signal decreased as patients pro-gressed from a state of normal ICP to astate of ICH. As larger ApEn values cor-respond to increased complexity or irreg-
ularity in the data, this indicates thatdecreased complexity of ICP coincideswith episodes of ICH. We also noted thatthe level of complexity begins to increaseand return toward baseline levels withinminutes as ICP drops below 20–25 mmHg. These findings suggest that the com-plex regulatory mechanisms that governICP are disrupted during acute rises inICP and return toward baseline with res-olution of the acute elevation in pressure.Although we only analyzed one patientwithout severe TBI as a cause of ICH, theresults were similar in this patient, sug-gesting that the regulatory mechanismsthat govern ICP are independent of thetype of injury.
Physiologic Implications. Godin andBuchman (34) suggested that the patho-genesis of multiple organ dysfunctionsyndrome from the systemic inflamma-tory response syndrome may be due toerosion of interconnections among or-gan systems. Loss of variability in theheartbeat and systemic blood pressurehas been demonstrated during the sep-sis syndrome in experimental modelsand septic patients (35–37). Return ofvariability was reported by Ellenby et al.(38) to be associated with recouplingbetween the autonomic nervous andcardiovascular systems during septicshock. Our results suggest that analo-gous physiologic mechanisms occur
during ICH, at least during short peri-ods of observation.
Zwiener et al. (39) found evidence ofimpaired short-term (25- to 60-sec peri-ods) dynamics between respiratory move-ments and fluctuations in heart rate andblood pressure in brain-injured patientscompared with controls. Furthermore,the impairments in short-term dynamicswere inversely proportional to the degreeof neurologic injury and were not affectedby concomitant analgesic or sedativemedications. These authors concludedthat the diminished coherence was indic-ative of organ system uncoupling or de-complexification consistent with previousreports (8, 34).
Zweiner et al. (39) further suggestedthat the short-term dynamics of coher-ence and coordination of multiple organsystem activity depend on brainstem af-ferent activities. They pointed to researchby Langhorst et al. (40) and Schulz et al.(41), who found that the degree and pat-tern of rhythmic fluctuations in brain-stem neurons were related to respiratoryor cardiovascular functions and that thejoint influence on the efferent neuroau-tonomic activity to lung, heart, and bloodvessels depended on brainstem afferentactivity. As the main stimuli to the brain-stem in these experiments were from pe-ripheral somatic and visceral origins, theconclusion is that these experimentswere an example of autonomic adapta-tions to changed external conditions, apostulated rationale for the advantages ofcomplexity in healthy organisms (34, 39).Thus, uncoupling between lungs, heart,and the vascular system decreases theirfunctional performances and results inworse outcome.
Zweiner et al. (39) also pointed out,and we agree, that it is not clear whetherour findings of decomplexification repre-sent a primary physiologic process oronly an epiphenomenon. However, theresults of the current study, using moresensitive nonlinear methodologies, sug-gest the former. A recent study by Rassiaset al. (28) (and accompanying editorial)of decreased physiologic variability fol-lowing experimental human endotoxemiaseem to lend support to this interpreta-tion and also suggest that nonlinear dy-namic models of human pathophysiologymay have diagnostic and therapeutic im-plications (28, 30).
The six patients with severe TBI hadan evolving clinical course. It is clinicallyaccepted that maximal ICP generally oc-curs 24–48 hrs following severe TBI. If
Table 3. Standard error of the mean approximateentropy for the stable, intracranial hypertension,and recovering regions
Figure 4. Histogram approximating the sampling distribution of the mean approximate entropy(ApEn) across all the patients for the stable, critical (intracranial hypertension, ICH), and recoveringregions obtained using bootstrap. Note the statistically significance reduction in mean ApEn duringthe ICH period.
91Crit Care Med 2006 Vol. 34, No. 1
the systemic inflammatory response syn-drome to multiple organ dysfunction syn-drome analogy applies in TBI, then futurestudies need to examine if repeated epi-sodes of decreased complexity in ICP, in-dicative of loss of regulatory controlmechanisms, result in worsening braininjury, cerebral edema, neuronal celldeath, and, in extreme cases, eventuallybrain death. In addition, there has beenmuch discussion about what is the besttreatment threshold value of ICP to usein severe TBI (1, 26, 42). Loss of physio-logic complexity of the ICP signal may bea sensitive and specific method for deter-mining exactly at what value elevated ICPbecomes physiologically dangerous.
Clinical Implications—Measures ofCerebral Autoregulation (CAR). Investi-gators have suggested that diminishedcomplexity results in a decreased abilityto respond to external perturbations (34,43). Under normal conditions, cerebralautoregulatory mechanisms maintain ce-rebral blood flow to the brain constantover a wide range of systemic arterialpressures, with typical values between 45and 65 mL/100 mg of brain tissue persecond despite variations in blood pres-sure as large as 100 mm Hg (44). Localcerebral autoregulation delivers a rela-tively constant cerebral perfusion pres-sure in response to fluctuations in ICPsuch as with postural change and cough.
It is clear that acute ICH is a result of
failure of normal cerebral autoregulatorymechanisms to compensate for over-whelming changes in cerebral volume(hemorrhage or edema), external pres-sure (depressed skull fractures), cerebralblood volume (cerebral hyperperfusion),or cerebral spinal fluid (obstructive hy-drocephalus). Although speculative, ourfindings suggest that, similar to sepsis,the ICP waveform exhibits diminishedcomplexity and increased regularity dur-ing periods of ICH when CAR has failed.Conversely, when CAR is intact, the ICPbecomes more complex and irregular.Thus, our findings suggest that ApEn ofthe ICP waveform may provide an indi-rect measure of CAR.
Other investigators have proposed anumber of indirect measures to deter-mine whether CAR is intact. Physiologicstressors or challenges such as changesin PaO2, PaCO2, arterial blood pressure,and intracranial volume (via ballooncatheter) have all been reported to corre-late to various degrees with CAR. Thesetechniques require manipulation of oneor more physiologic variables with someinherent risk to the patient. More re-cently, investigators have examined thetransfer function and phase shift betweencerebral blood flow and ICP as a dynamicmeasure of CAR (16, 45, 46). These mea-sures typically use Doppler blood flowvelocity in the middle cerebral artery thatmay be difficult to maintain for continu-
ous measurements in the ICU environ-ment. Direct analysis of the ICP wave-form using ApEn or some othernonlinear metric may provide an easierand less risky alternative for determina-tion of CAR. Currently there are a fewindexes of potential clinical interest ob-tained by direct analysis of the ICP signal(e.g., RAP, pressure-reactivity). The RAPindex (index of compensatory reserve) isdefined as the correlation coefficient (R)between the ICP pulse amplitude (AMP)and mean pressure (P). This index is ob-tained by calculating the linear correla-tion between consecutive, time averagedata points of AMP and ICP (usuallyabout 40 of such samples are used) andindicates the degree of correlation be-tween AMP and mean ICP over short pe-riods of time (�4 mins). This can be usedto estimate the state of the pressure-volume compensatory reserve. AnotherICP-derived index is the pressure-reactiv-ity index. This index has been shown tocorrelate well with indexes of autoregu-lation based on transcranial Doppler ul-trasonography. In conjunction, these in-dexes can be use to indirectly estimatethe CAR or deranged cerebrospinal com-pensatory reserve (47).
Interpretation of Approximate En-tropy. Since it is not possible to directlymeasure the complexity of individual or-gan systems, approximate measures mustbe used. There are several methods toestimate the complexity of systems fromanalysis of time series. In this work, themeaning of the term complexity is re-stricted to the estimated complexity bythe ApEn metric (10). We chose to useApEn because it was specifically designedas a technique to determine changingsystem complexity from short time se-ries. Furthermore, contrary to the fre-quently used correlation dimension mea-sure of complexity, ApEn does notassume an underlying deterministicmodel or chaos and is capable of classify-ing complex systems that include cha-otic, stochastic, and composite processes;these last processes are likely models forcomplicated biological data sets. In thegeneral stochastic, noisy deterministic,or composite setting, the statistical accu-racy of the correlation dimension mea-sure of complexity is typically very poor(10, 48). Because dynamic mechanics ofmost biological signals remain unde-fined, a suitable statistic of complexity forthese signals must be more cautious toaccommodate general classes of pro-cesses and their much more diffuse re-
Figure 5. Approximate entropy (ApEn) (normalized) for each of intracranial hypertension (ICH)episodes (light gray) and mean and median across all ICH episodes (dark). ApEn decreases as patientsprogressed from a stable state of normal ICP to a state of ICH. This indicates that decreased complexityof ICP coincides with episodes of ICH in TBI.
92 Crit Care Med 2006 Vol. 34, No. 1
constructed dynamics (26). Moreover, arobust estimate of ApEn can be obtainedby using a number of points several or-ders of magnitude lower than that neededto estimate accurately the correlation di-mension or the first positive Lyapunovexponent (27).
To aid in interpretation of the ApEnvalues obtained in this study, we per-formed a series of tests with syntheticsignals with known characteristics. Fromthese test we concluded that a) the ApEnincreases as the frequency and the num-
ber of harmonics of a sinusoidal signalincrease; b) ApEn is correlated with noisebandwidth, increasing as the noise band-width increases (ApEn is lower in the caseof colored noise than for white noise); c)typical values of ApEn for sinusoidal sig-nals range from 0.001 to 0.07 (increasingas the numbers of harmonics increases)and 1.4 to 2 for white noise; and d) theApEn of ICP ranges from 0.05 to 1.5,correlating with heart rate variability,noise power, and pulse morphologychanges.
The fact that ApEn correlates with thenumber of harmonics in periodic andquasi-periodic signals has important im-plications in the context of ICP analysis,since the number of harmonics of thesignal directly relates to the morphologyof the ICP beats. Investigators have doc-umented specific variations in the ICPbeat morphology, which correspond tospecific alterations in the cerebral vascu-lar system, CSF circulation, and respira-tion. These morphology variations maybe used to measure the progression ofdisease. For instance, Pornoy and Choop(48) stated that in patients who do nothave a cerebral edema or expandingmass, the ICP beat shows an initial sharprise and subsequent downward slope sim-ilar to the arterial pulse, but as the ex-panding mass or edema develops, the ICPpulse becomes more rounded. As the ICPpulse becomes more rounded, the ampli-tude of the higher frequency sinusoidalcomponents also decreases.
In the light of our results, ApEn can beinterpreted as a “summarizing metric”that combines information such as heartrate variability, number of harmonics,and time morphology variability. Our re-sults support the hypothesis that ApEn isinversely related and negatively corre-lated to acute elevations in ICP. Further-more, our results also support the hy-pothesis that there is information in theICP pulse that correlates with severity ofdisease, since ApEn was calculated fromhigh-pass filtered ICP signals where themean ICP trend was removed prior toApEn calculation.
Limitations. The data presented in thisstudy came from only seven patients. More-over, in our statistical analysis we assumedthe 11 ICH episodes were independent. Fur-ther study is warranted in a larger and morediverse population of patients with ICH froma variety of diseases and injuries. Additionally,
O ur results indi-
cate that de-
creased com-
plexity of intracranial
pressure coincides with peri-
ods of intracranial hyperten-
sion in brain injury.
Figure 6. Top, approximate entropy (ApEn) vs. intracranial pressure (ICP) for patient 1, episode 1. Notedecreased ApEn associated with both ICP spikes in pressure �20 mm Hg. Bottom, ApEn vs. ICP forpatient 3, episode 1. Note decrease in ApEn during ICP spikes with return to baseline levels duringnormal ICP.
93Crit Care Med 2006 Vol. 34, No. 1
there was no direct or indirect measure ofcerebral autoregulation during periods ofICH. Finally, the dataset was blinded to treat-ments used in response to ICH, so it is notpossible to differentiate what effects, if any,therapies such as osmotic diuresis, cerebralspinal fluid drainage, mild hyperventilation,or elevation of the head-of-the bed may havehad on ICP analysis.
ACKNOWLEDGMENTSWe thank Matthew Goldstein for help
in the preparation of this manuscript.
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APPENDIX 1: GLOSSARY OFTERMS
Approximate entropy: A metric thatquantifies the regularity or unpre-dictability of a time series.
Aliasing: The apparent conversion ofhigh-frequency signals to low-fre-quency signals due to an insufficientsample rate.
Bandpass filter: A filter that elimi-nates low- and high-frequency com-ponents of a signal but retains anintermediate range.
Bandwidth: The range of frequenciesspanned by a signal. When applied tobandpass filters, this describes therange of frequencies that are allowedto pass through the filter.
Bootstrap: A computer-basedmethod introduced in 1979 as acomputational technique for esti-mating the standard error of an es-timator.
Complexity: In this context of thisarticle, complexity is defined as theapproximate entropy (ApEn) of a sig-nal segment.
Deterministic: Signals or systemsthat can be described by an explicitmathematical relationship.
Harmonics: Frequencies that are in-teger multiple of the fundamentalfrequency.
High-frequency noise: Many types ofartifact in physiologic signals con-tain significant power at high fre-quencies. This noise is often emittedby medical equipment near the pa-tient.
High-pass filter: A filter that elimi-nates low-frequency components of asignal but retains high-frequencycomponents.
Linear interpolation: The process ofestimating a value of a signal orfunction between two intermediatevalues using a line between the twopoints.
Low-pass filter: A filter that elimi-nates high-frequency components ofa signal but retains low-frequencycomponents.
Lyapunov exponent: Metric used toquantify the divergence of a dynam-ical system from perturbed initialconditions.
Noncausal filter: Filter that uses fu-ture values of the input.
Nonlinear: Any system or devicewhose behavior is governed by a setof nonlinear equations. These sys-tems do not produce an output thatis proportional to the input, in gen-eral.
Stochastic: Signals that cannot be de-scribed to any reasonable accuracy byexplicit mathematical relationshipsand must be studied statistically.
APPENDIX 2: ALGORITHM
Given N data points from a time series�x(n)� � x (1), x (2), . . . , x(N), tocompute ApEn, the following steps aretaken:
1. Form m-vectors X (1)�X(N�m�1)defined by: X(i) � [x(i), x(i�1), . . . ,x(i�m�1)], i � 1�N�m�1. Thesevectors represent m consecutive xvalues, commencing with the ithpoint.
2. Define the distance between X(i)and X(j), d[X(i),X(j)], as the maxi-mum absolute difference betweentheir respective scalar compo-nents:
d�Xi,X j� �max
k � 1,2, . . . m�xi � k
� 1 � x j � k � 1� [1]
3. For a given X(i), count the numberof j (j � 1�N�m�1, j�i) such thatd[X(i), X(j)]�r, denoted as Nm(i).Then, for i � 1�N�m�1,
Crmi � Nmi/N � m � 1 [2]
The Crm(i) values measure within a tol-
erance r the regularity, or frequency, ofpatterns similar to a given one of win-dow length m.
4. Compute the natural logarithmof each Cr
m(i) and average it
mr �1
N � m � 1 �i�1
N�m�1
lnCrmi
[3]
over i �m (r) represents the averagefrequency that all the m point patternsin the sequence remain close to eachother.
5. Increase the dimension to m�1.Repeat steps 1 to 4 and findCr
