Patent Application: US-201313801426-A

Abstract:
the present invention relates generally to a method for computer - aided quantitative diagnosis of cerebrovascular and neurodegenerative diseases via a vasomotor reactivity index which is computed on the basis of a computational model of the dynamic nonlinear inter - relationships between beat - to - beat time - series measurements of cerebral blood flow velocity , arterial blood pressure and end - tidal co2 . this model is obtained by means of a method pioneered by the inventors and may incorporate additional physiological measurements from human subjects . its purpose is to provide useful information to physicians involved in the diagnosis and treatment of cerebrovascular and neurodegenerative diseases with a significant neurovascular component by offering quantitative means of assessment of the effects of the disease or medication on cerebral vasomotor reactivity . initial results from clinical data have corroborated the diagnostic potential of this approach .

Description:
the modeling methodology required by the invention utilizes the key concept of principal dynamic modes ( pdm ) which has been pioneered by the inventors and has been elaborated in a recent monograph [ 46 ]. while one embodiment is illustrated in this application , many variations exist that will not limit the general applicability of the method or compromise the integrity of the requisite data . the method comprises the following computational steps : the pdms of each subject are estimated from the collected beat - to - beat time - series data of mean cerebral blood flow velocity , mean arterial blood pressure and end - tidal co2 ( obtained by a variety of non - invasive methods from human subjects in a clinical setting ) over several minutes . the beat - to - beat measurements are pre - processed to remove artifacts and they are re - sampled evenly over time , typically at 2 samples per second . very low frequency trends or cycles are removed prior to processing of the time - series data to obtain a dual - input dynamic nonlinear volterra model , following the methodology pioneered by the inventors [ cited publications 45 - 46 ]. typically , two inputs are used : one representing blood co2 tension through its surrogate end - tidal co2 measurements ( etco2 ) and the other being the measurements of beat - to - beat mean arterial blood pressure ( mabp ). the output variable is the beat - to - beat measurements of mean cerebral blood flow velocity in the middle cerebral artery . an illustrative example of such time - series data is shown in fig1 , where the variations of mean cerebral blood flow velocity are represented by transcranial doppler measurements in the middle cerebral artery . in the presented embodiment , the modeling task commences with the estimation of a 2 nd order volterra model of this dual - input system using laguerre expansions of the kernels with 5 basis functions for the mabp ( input # 1 ) and 3 basis functions for the etco2 ( input # 2 ). this results in 45 free parameters ( including a constant baseline term ) for the dual - input 2 nd order volterra model , which can be adequately supported ( in terms of estimation accuracy ) by a minimum of 4 min of time - series data . typically , 6 min of time - series data are collected and analyzed . the proposed procedure estimates the kernels of the dual - input 2nd order volterra model that has the form [ 46 ]: where p ( t ) denotes the mabp input , x ( t ) denotes the etco2 input and y ( t ) denotes the mean cerebral blood flow velocity ( mcbfv ) output . the modeling task involves the estimation of the unknown volterra kernels of the model { k p , k x , k pp , k xx , k px } from given input - output data p ( t ), x ( t ) and y ( t ). this task is facilitated immensely by laguerre expansions of the kernels : where { b j ( τ )} denote the orthogonal laguerre function basis . other bases can be used as well . such kernel expansion yields the following nonlinear input - output relation which involves linearly the laguerre expansion coefficients { a r } and { c r }: where the signals v j ( t ) and z j ( t ) are the convolutions of the laguerre basis function b j with the respective input , and ε ( t ) denotes possible measurement or modeling errors . the fact that the laguerre expansion coefficients enter linearly in the nonlinear volterra model of equation ( 3 ) allows their estimation via least - squares regression ( a simple , robust and stable numerical procedure ). having estimated the laguerre expansion coefficients , we can construct the volterra kernel estimates using equation ( 2 ) and compute the model prediction for any given input using equation ( 1 ) or ( 3 ). this procedure applies to higher order volterra models as well . the introduction of the concept of principal dynamic modes ( pdms ) has allowed the practical estimation of nonlinear models of higher order as in the subject application . briefly stated , the use of pdms is an efficient basis and allows us to write the output equation ( 3 ) as : where { u n ( t )} and { u m ( t )} are the pdm outputs ( i . e . convolutions of the input with the respective pdm ) for the mabp and etco2 inputs , respectively , and { f n [ u n ]}, { f m [ u m ]} are the static nonlinearities associated with each pdm , termed associated nonlinear functions ( anfs ). the anfs are typically given polynomial form ( cubic in this application ). the “ cross terms ” in equation ( 4 ) are pair products of { u h } and { u m } that have significant correlation with the output . the coefficients of the selected cross - terms are estimated , along with c 0 and the coefficients of the ( cubic ) anfs via least - squares regression of equation ( 4 ). the computation of the pdms from the kernel estimates employs singular value decomposition ( svd ) of a rectangular matrix composed of the 1st order kernel estimates ( as column vectors ) and the 2nd order self - kernel estimates as block sub - matrices , weighted by the root - mean - square value of the respective input . a block - diagram of the pdm - based model of the dual input system of cerebral hemodynamics is shown in fig2 . following the estimation of the pdms for each subject in the reference group of control subjects , we compute the “ global pdms ” for each input as the most significant “ singular vectors ” ( corresponding to the largest “ singular values ”) resulting from svd analysis of the rectangular matrix containing the pdms for all subjects in the reference group weighted by the respective singular values . it is important to note that the waveforms of the global pdms were not affected significantly when different sets of control subjects were randomly selected for the reference group . this fact corroborates the premise of the existence of global pdms for this system , which corroborates the proposition that the pdm - based model is generalizable ( i . e . applicable to all subjects ) and , therefore , potentially useful for diagnostic purposes . an illustrative example of obtained global pdms following the outlined procedure for a set of 16 control subjects is given in fig3 . to complete the development of the pdm - based nonlinear model of a subject , we must further estimate the associated nonlinear function ( anf ) of this subject for each global pdm , which is a static nonlinearity applied to the convolution of the input signal with the respective global pdm . the anfs are subject - specific and contain the differentiating information among subjects that is valuable for clinical diagnosis . completion of the pdm - based model also requires the estimation of the coefficients of the cross - terms in the model of equation ( 4 ) that are composed of pair products of pdm outputs . the cross - terms account for the inter - modulation effects between the two inputs as they affect the output . the model output prediction is composed of the sum of all anf outputs and cross - terms , along with a constant baseline value , as indicated in the block - diagram of fig2 . the coefficients of the six cubic anfs ( one for each of the six pdms ) and the significant cross - terms are estimated via least - squares fitting of the input - output data according to equation ( 4 ). the estimated coefficients of the anfs and the cross - terms are distinct for each subject and can be used to quantify uniquely its cerebral hemodynamics , offering a potential diagnostic tool for ad ( or other diseases with a cerebrovascular component ). it was found that cubic anfs are adequate for this system . an illustrative example of the anfs obtained for the two sets of three global pdms for the mabp and etco2 inputs in a control subject are shown in fig4 . considerable inter - subject variability was observed in the form of the anfs corresponding to the two sets of global pdms for the two inputs of this model . however , the critical finding , relevant to the utility of this invention , is that this variability remained within well - defined bounds for each of the two groups of control subjects ( cs ) and ad patients ( ap ), and furthermore the functional characteristics of these two groups ( defined largely by the respective anfs ) were rather distinct and allowed clear delineation of the two groups . to quantify this important fact of significant vasomotor reactivity differences between the two groups in a practical manner that can have clinical utility , we propose the use of a scalar vasomotor reactivity index ( vmri ) that is computed as the difference of the time - averages of the model - predicted mean cerebral blood flow velocity in response to pulse changes in the end - tidal co2 input over 30 sec ( positive pulse response minus negative pulse response ) normalized by the respective input pulse amplitude , while the arterial pressure input is kept at baseline . the vmri is expressed in units of cm / sec / mmhg . an illustrative example of the model - predicted mean cerebral flow responses to pulse changes in co2 for a control subject and an alzheimer &# 39 ; s patient is given in fig5 . a model of cerebral hemodynamics that accounts for the mutual interdependence of blood pressure and flow , as dictated by the navier - stokes equation governing fluid dynamics , can be constructed in a closed - loop configuration that includes two input - output model components , a and b , as depicted in fig6 . model component a has inputs of pressure and co2 , and output of flow velocity ( as in the aforementioned model ), while model component b has inputs of flow velocity and co2 , and output of pressure . this closed - loop model is driven by two external “ disturbance ” signals of flow velocity and pressure , which are the computed residuals of the model prediction by the input - output models a and b , respectively . this closed - loop model configuration can be used to compute the vmri on the basis of the predicted intra - loop flow velocity for a pulse change of the co2 variable . from fig6 , we can derive the closed - loop equations : which are nonlinear stochastic integral equations . for each a and b model , we have : f ( t )= f 0 + σf i pf └ u i pf ( t )┘+ σ f j cf └ u j cf ( t )┘+ σ c k , l pc u k pf ( t ) u l cf ( t )+ f d ( t ) ( 7 ) p ( t )= p 0 + σf i fp └ u i fp ( t )┘+ σ f j cp └ u j cp ( t )┘+ σ c k , l fc u k fp ( t ) u l cp ( t )+ p d ( t ) ( 8 ) where the signals u ( t ) are convolutions of the input signals with the pdms , the functions ƒ [·] are polynomials , and f 0 , p 0 are the baseline values of pressure and flow velocity respectively ( i . e . their values when there are no systemic disturbances ). simulations of this closed - loop model for pulse changes of co2 allow computation of the vmri as described earlier . the elements of this invention are the sequential methodological and computational steps that capture the dynamic nonlinear relationship governing cerfebral hemodynamics in healthy subjects and ad patients or patients with other cerebrovascular and neuro - degenerative diseases with a significant neurovascular component . the invention can be implemented in various ways , different from the presented preferred embodiment . for instance , different kernel expansion bases ( other than laguerre ) can be used for the estimation of the initial volterra model , which may be of various orders ( other than second ) and with multiple inputs ( not the two used in the preferred embodiment ). likewise , the estimation of the subject pdms can be accomplished with alternate methods ( e . g . using iterative procedures like the laguerre - volterra network [ 46 ]) and the computation of the global pdms can be performed with methods other than svd . the utilized model may not be based on pdms . most importantly , the model - based definition and computation of the appropriate vmri used for diagnostic purposes may take various forms of quantification of vasomotor reactivity , other than the one described in the preferred embodiment . the presented procedural steps of the invented method must be performed in sequence and in adherence to the underlying technical requirements . the resulting vmri is used for quantitative clinical diagnosis and treatment monitoring in ad or other cerebrovascular and neurodegenerative diseases with a significant neurovascular component . what has been described and illustrated herein is a preferred embodiment of the invention along with some of its variations . the terms , descriptions and figures used herein are set forth by way of illustration only and are not meant as limitations . those skilled in the art will recognize that many variations are possible within the spirit and scope of the invention in which all terms are meant in their broadest , reasonable sense unless otherwise indicated . any headings utilized within the description are for convenience only and have no legal or limiting effect . 1 . sellke f w , s seshadri , h c chui , r t higashida , r lindquist , p m nilsson , g c roman , r c petersen , j a schneider , c tzourio , d k arnett , d bennett , c ladecola , u launer , s laurent , o l lopez , d nyenhuis , p b gorelick , a scuteri , s e black , c decarli , s m greenberg ( 2011 ). vascular contributions to cognitive impairment and dementia : a statement for healthcare professionals from the aha / asa . stroke 42 ( 9 ): 2672 - 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