Patent Application: US-24773888-A

Abstract:
the statistical properties of a biological signal intermittently obscured by a relatively independent biological process are estimated by noting the time intervals during which the obscuring process is occurring . from the time intervals , a windowing function is constructed which makes a transition from one towards zero as the process obscures the signal and a transition towards one as the process terminates . thereafter , a statistical characterization is performed in which the windowing function is used to weight the relative contributions of corresponding segments of the biological signal . in preferred embodiments , the statistical properties include the autocorrelation function and power spectrum of biological signals such as heart rate and blood pressure . the invention is applicable also to estimating the crosscorrelation and transfer function of a first signal intermittently obscured by a first process and a second signal intermittently obscured by a second process in which each of the signals are relatively independent of each of the obscuring processes . when the biological signal is heart rate , the obscuring process may be the occurrence of ectopic beats . intervals which contain ectopic beats are treated as though they are missing . in this way , data which might otherwise be unusable by virtue of ectopic beats can become useful for assessing the status of the biological system .

Description:
first of all , the theory on which the present invention is based will now be presented . assume that one would like to know the power spectra and transfer function of two biological signals x [ n ] and y [ n ] ( e . g ., hr and bp ), but that we do not have full information on x [ n ] and y [ n ]. the versions of x [ n ] and y [ n ] available to us are intermittently obscured by some interfering process ( e . g ., ectopic beats ). the series v [ n ] indicates the parts of x [ n ] that were obscured , and w [ n ] indicates the parts of y [ n ] that were obscured . all four time series are assumed to be time aligned and that each one consists of n samples taken at intervals of t s seconds . in specific : x [ n ] is the time series that we would have observed if parts of the first signal had not been obscured . y [ n ] is the time series that we would have observed if parts of the second signal had not been obscured . v [ n ] is a time series which is defined to be 0 during times in which x [ n ] was obscured and 1 otherwise . it is assumed that v [ n ] is independent of x [ n ] and y [ n ]. w [ n ] is a time series which is defined to be 0 during times in which y [ n ] was obscured and 1 otherwise . it is assumed that w [ n ] is independent of x [ n ] and y [ n ]. the conventional estimate of the autocorrelation of x [ n ] in the absence of an obscuring signal is ## equ1 ## with the addition of an obscuring signal we introduce the modified autocorrelation estimate ## equ2 ## where n k is defined as the number of x [ n ] x [ n + k ] terms for which v [ n ]= v [ n + k ]= 1 , and where the restricted sum is taken only over those n k terms . when no data is missing equations ( 1 ) and ( 2 ) are equivalent . equation ( 2 ) is a special case of a more general formula involving the autocorrelations r x [ k ] and r v [ k ]. it can be shown that if x [ n ]= v [ n ] x [ n ], where x [ n ] and v [ n ] are independent , and if r v [ k ]≠ 0 , then ## equ3 ## is an asymptotically unbiased estimate of r x [ k ] [ 10 ]. by direct analogy , the autocorrelation estimate for y is ## equ4 ## in addition , we introduce the relationship for the crosscorrelation estimate given by ## equ5 ## where ## equ6 ## and is analogous to equation ( 2 ), and ## equ7 ## it should be noted that , although in the definitions given above , v [ n ] and w [ n ] are defined to be 0 or 1 , this condition is not a necessary restriction . equations ( 3a - c ) are valid even if v [ n ] and w [ n ] take on intermediate values between 0 and 1 . if the definition of the problem at hand is such that use of intermediate values makes sense , then they can be used . that is , the windowing functions v [ n ] and w [ n ] make a transition from one toward zero as the process obscures the signal and a transition towards one as the process terminates . next , computation of the correlation estimates will be discussed . we start with the values x [ n ], y [ n ], v [ n ], and w [ n ] for n = 1 , . . . n . the first step is to de - trend x [ n ] and y [ n ]. although this biases the estimated correlations , the bias is minor and affects mostly the very low frequency region of the spectrum . failure to remove large low frequency trends can subsequently cause a substantial increase in the variance of the entire spectrum or transfer function . when possible , it is best to compute the trend for x [ n ] using only values of x [ n ] for which v [ n ]= 1 . once the trend is computed x [ n ] is adjusted by subtracting from it the product of v [ n ] and the trend at sample n . a similar detrending is carried out with y [ n ] and w [ n ]. we then use fft - based convolution to compute ( n - k ) r x [ k ], ( n - k ) r y [ k ], ( n - k ) r xy [ k ], ( n - k ) r v [ k ], ( n - k ) r w [ k ], and ( n - k ) r vw [ k ]. from these we compute r x [ k ], r y [ k ], and r xy [ k ] according to equations ( 3a - c ). values of k for which r [ k ]≅ 0 , where r [ k ] can be either r v [ k ], r w [ k ], or r vw [ k ], we set r [ k ]= r [ k - l ]. this kind of singular condition rarely occurs except for large values of k , and is usually of little consequence in the estimation . with the autocorrelation and crosscorrelation estimates in hand , one can then estimate the spectra s x ( f ), s y ( f ), and transfer function h xy ( f ). these estimates can be made using parametric or fft - based techniques [ 11 , 12 ]. using fft - based techniques , the relevant power spectra and transfer function estimates are : ## equ8 ## where dtft stands for the discrete time fourier transform and q [ k ] is a windowing function chosen to achieve a desired level of spectral smoothing . the coherence k ( f ) and impulse response h ( t ) are then estimated as ## equ9 ## where q ( f ) is a low - pass filter chosen to eliminate the high frequency portion of h xy ( f ) beyond which k ( f ) is substantially decreased . parametric techniques like autoregressive ( ar ) or autoregressive moving average ( arma ) models can compute the spectra , transfer function , and impulse response directly from the auto - and crosscorrelation without the use of the fft . parametric techniques have the advantage of weighting the major spectral features more heavily and hence usually require fewer degrees of freedom to specify the result . in addition , parametric transfer functions estimates can be forced to yield a causal relationship between the input and output . fft - based methods , on the other hand , require more degrees of freedom to represent the result , but weight all parts of the spectrum equally . we present the estimation of the hr power spectrum as an example of the method described above . we assume that we have already processed the ecg data , and that we have at our disposal an annotation stream consisting of beat types ( normal or ectopic ) and times - of - occurrence . such a stream of annotations is typical of what would be produced by many present - day clinical ecg analysis instruments . in the case of hr , the obscuring events are the ectopic beats . below , we first describe the method by which we would generate the time series hr [ n ] and v [ n ]. afterwards we give an example of how a spectral estimate derived using the new method compares to the estimate derived from a conventional strategy which uses splining to fill in the ectopic intervals . we have previously reported [ 13 ] a computationally efficient algorithm which takes as its input { t i }, the times - of - occurrence of the beats , and produces hr [ n ], a discrete instantaneous hr estimate at a sampling frequency f s = 1 / t s . the algorithm is based on the integral pulse frequency model ( ipfm ) of san modulation . in the model , the influence of the ans is represented by a modulating function m ( t ) which is the rate at which the san approaches the next beat firing . the ipfm model is appropriate since integration of the modulation function m ( t ) is analogous to the charging of the phase 4 transmembrane potential of the san cells . for the simple model with no lockout after firing , m ( t ) is proportional to the instantaneous heart rate . it has been shown that for the ipfm model , hr [ n ] yields a better spectral estimate for m ( t ) ( i . e ., lower harmonic and intermodulation distortion ) than estimates based on other time series [ 13 ]. we have extended our algorithm to handle ectopic beats ( see fig1 ). in fig1 a the function m ( t ) represents the level of the combined influences modulating the san . the san integrates m ( t ). when the integral reaches a threshold , a beat is produced and the san begins integrating again . the goal of hr spectral estimation is to recover the spectrum of m ( t ). fig1 b is the ecg corresponding to m ( t ) from fig1 a . each triangle in fig1 b represents one beat . the beat labelled e is an artificially introduced ectopic beat which did not originate in the san ; e can be considered an atrial or ventricular ectopic beat . in fig1 c , the function hr ( t ) is the instantaneous hr tachometer estimated from the rr intervals of the ecg . note that the value of hr ( t ) for two intervals bounded by the ectopic beat is unknown . fig1 d is the estimate of hr [ n ] given by hr [ n ]/ v [ n ] and fig1 e is the windowing function v [ n ]. both hr [ n ] and v [ n ] were calculated using t s = 0 . 5 , which corresponds to a sampling frequency of f s = 2hz . note that fig1 c at 5 . 5 seconds shows a rectangular window used in deriving the corresponding point in fig1 d . at zero and 10 . 5 seconds such an estimate was not possible since v [ n ]= 0 . as before , we first compute rr i = t i + 1 - t i for the ith interval , and estimate the instantaneous hr tachometer as ## equ10 ## we then compute discrete samples from the low - pass ( anti - alias ) filtered tachometer but , instead of computing ## equ11 ## as we do in the case of no ectopic intervals , we compute ## equ12 ## where v ( t ) is a windowing function defined by ## equ13 ## equations ( 6 ) and ( 7 ) represent samples of waveforms which have been convolved with a rectangular filter of width 2t s , and sampled at times nt s . when there are no ectopic intervals , hr [ n ]= hr [ n ]. observing that hr ( t ) is piece - wise constant , we can write equation ( 7 ) as ## equ14 ## where δt i is the part of rr i contained in the interval [( n - 1 ) t s , ( n + 1 ) t s ]. for example , if 20 % of rr i is in the interval , then δt i = 0 . 2rr i . if rr i is an ectopic interval , we set δt i = 0 . defining v [ n ] as ## equ15 ## we can approximate hr [ n ] as ## equ16 ## for output series with t s shorter than the mean rr interval , we often extend the duration of the ectopic interval to the next multiple of t s . this modification is accomplished by setting v [ n ]= hr [ n ]= 0 for v [ n ]& lt ; 1 , and has the advantage of turning equation ( 12 ) into an equality . the disadvantage is that a smaller fraction of the available hr ( t ) is used for estimation ; however , when t s is shorter than the mean rr interval , the additional data declared &# 34 ; missing &# 34 ; is usually small . it should be noted that the rectangular filter implicit in equation ( 7 ) multiplies the spectrum of hr ( t ) by ## equ17 ## hence any spectrum computed using hr [ n ] and v [ n ] must be compensated by dividing with f ( f ). furthermore , since the f ( f ) approaches zero at f s / 2 , the hr spectrum cannot be considered valid much beyond f s / 4 . fig2 compares a spectrum computed according to the method of the invention ( labelled &# 34 ; new &# 34 ; in the figure ) to a spectrum computed from a hr [ n ] series filled in by splining . fig2 a is a five minute simulated hr [ n ] signal sampled at t s = 0 . 5 seconds . the signal has the spectral content typical of the high frequency , respiration - induced component of hr . immediately under hr [ n ] is a bar code which represents the ectopic intervals . the dark part of the bar indicates the &# 34 ; missing data &# 34 ;, and accounts for 45 % of the 5 minutes . the ectopic interval pattern is not simulated , but is taken from an actual clinical ecg containing substantial ectopy . we first computed the true spectrum of the 5 minute hr [ n ] using all 600 points of hr [ n ]. we then used the &# 34 ; missing data &# 34 ; pattern for v [ n ] and computed the spectral estimate s hr ( f ) described here . the true and estimated ( by the method of the invention ) spectra are shown in fig2 b . this estimated spectrum compares favorably to the true spectrum . we then computed a spectral estimation derived from a splined version of hr [ n ]. splining is a commonly used ad hoc &# 34 ; fix &# 34 ; for obscured data . in the splined version , obscured values of hr [ n ] were filled in by linear splines joining bordering regions where hr [ n ] was known . the spectral estimate from splined hr [ n ] and the true spectrum are shown in fig2 c . note that the splined spectrum not only underestimates the 0 . 2 - 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