Patent Abstract:
a method for providing a representation of the distribution , fluctuation and / or movement of electrical activity through heart tissue , said method comprising : obtaining an ecg of the heart comprising said tissue ; obtaining a model of the heart geometry ; obtaining a model of the torso geometry ; relating the measurements per electrode of the ecg to the heart and torso geometry and estimating the distribution , fluctuation and / or movement of electrical activity through heart tissue based upon a fastest route algorithm , shortest path algorithm and / or fast marching algorithm .

Detailed Description:
since the inception of electrocardiography [ 33 , 34 ] several methods have been developed aimed at providing more information about cardiac electric activity on the basis of potentials observed on the body surface . the differences between these methods relate to the implied physical description of the equivalent generator representing the observed potential field . the earliest of these are the electric current dipole , a key element in vectorcardiography [ 35 , 36 ] and the multipole expansion [ 37 ]. neither of these source models offer a direct view on the timing of myocardial activation and recovery , or other electro - physiologically - tinted features . from the 1970s onwards , the potential of two other types of source descriptions have been explored [ 38 , 39 ]. this development stemmed from increased insight in cardiac electrophysiology and advances in numerical methods and their implementation in ever more powerful computer systems . the results of both methods are scalar functions on a surface . the solving of the implied inverse problem may , accordingly , be viewed as a type of functional imaging , which has led to their characterisation as , e . g ., “ noninvasive electrocardiographic imaging ( ecgi )” [ 40 ] or “ myocardial activation imaging ” [ 41 ]. a brief characterisation of both methods is as follows . the first surface source model is that of the potential distribution on a closed surface closely surrounding the heart , somewhat like the pericardium , referred to here as the pericardial potential source ( pps ) model . the model is based on the fact that , barring all modelling and instrumentation errors , a unique relation exists between the potentials on either of two nested surfaces , one being the body surface , the other the pericardium , provided that there are no active electric sources in the region in between . it was first proposed at duke by martin and pilkington [ 42 ]; its potential has subsequently been developed by several other groups , e . g ., [ 43 , 44 , 45 , 46 , 47 ]. the second type of surface source model evolved from the classic model of the double layer as an equivalent source of the currents generated at the cellular membrane during depolarisation , described by wilson et al . [ 48 ]. initially , this current dipole layer model was used to describe the activity at the front of a depolarisation wave propagating through the myocardium [ 49 , 50 ]. later , salu [ 51 ] expressed the equivalence between the double layer at the wave front and a uniform double layer at the depolarised part of the surface bounding the myocardium , based on solid angle theory [ 52 ]. this source description has been explored by others , e . g ., [ 31 , 41 , 53 , 54 , 55 ]. this application describes recent progress made in inverse procedures based on the second type of surface source model : the equivalent double layer on the heart &# 39 ; s surface as a model of the electric sources throughout the myocardium ; we refer to it as the edl model . in contrast with the pps model , the edl source model relates to the entire surface bounding the atrial or ventricular myocardium : epicardium , endocardium and their connection at the base . as mentioned in the previous paragraph , the edl was initially used for modelling the currents at the depolarizing wave front only . based on the theory proposed by geselowitz [ 56 , 57 ], it was found to be also highly effective for describing the cardiac generator during recovery ( the repolarisation phase of the myocytes ). the transmembrane potentials ( tmps ) of myocytes close to the heart &# 39 ; s surface act as the local source strength of the double layer . several examples of its effectiveness in forward simulations have appeared in the literature [ 58 , 59 , 60 , 61 ]. a description of the model in inverse procedures is seen in the paper by modre et al . [ 62 ], dedicated to the atrial activation sequence . in our application the tmps &# 39 ; wave forms were specified by an analytical function involving just two parameters , markers for the timing of local depolarisation and repolarisation . these parameters were found by using a standard parameter estimation method , minimizing the difference between observed body surface potentials ( bsps ) and those based on the source description . since the body surface potentials depend non - linearly on these parameters , a non - linear parameter estimation technique is required , which demands the specification of initial estimates . it is here that some novel elements are reported on , and a major part of this application is devoted to their handling . the initial parameters for the timing of local depolarisation were typically based on the fastest route algorithm , taking into account global properties of anisotropic propagation inside the myocardium . those pertaining to recovery were based on electrotonic interaction as being the driving force for the spatial differences in the local activation - recovery interval . in the methods section , the entire inverse procedure is described and a justification given of those model parameters that are treated as constants during the optimisation procedure . in the results section , examples of inverse estimated ventricular activation and recovery sequences are presented : those of a healthy subject , a wpw patient and a brugada patient during an ajmaline provocation test . the value , application and limitations of this approach are considered in the discussion . for the situation in which the electrical activity of the heart is defined , the potentials on the body surface can be simulated . this procedure is known as the forward problem in electrocardiography [ 63 ]. to obtain a solution to the inverse problem the electrical activity of the heart needs to be related to measured ecg signals on the body surface [ 39 ]. clearly the non - invasive method of cardiac activation and recovery times is of physiological and clinical relevance . any solution method to such inverse problem requires a solution to the associated forward problem . in order to solve the forward problem , one needs to compute the potential differences at the body surface that result from the electric currents generated by the heart . for this purpose a realistic description of the volume conductor is needed , incorporating the shape and conductivities of the relevant tissues . the standard procedure to obtain such a volume conductor model is to detect the contours of the relevant tissues in a set of mr images of the subject involved and reconstruct triangulated computer models of these tissues [ 64 , 65 , 66 , 67 ]. based on earlier research regarding the simulation of the qrs complex [ 66 , 68 ], we have chosen to include in our volume conductor model the blood filled cavities within the heart , the lungs , and the rest of the torso with conductivity values 0 . 6 s / m , 0 . 04 s / m and 0 . 2 s / m respectively . we have used the boundary element method to compute the transfer function based on the specified volume conductor model for relating the source activity to the potentials at the body surface . due to the large number of cardiac cells in the human heart ( approximately 10 10 ) a computer model is typically not able to simulate the activity of all these cells nor can the full complexity of all interactions between ionic currents be incorporated . any non - invasive imaging method therefore needs to postulate a source model , representing the electrical activity of the heart . the earliest of these is the electric current dipole , a key element in vectorcardiography [ 35 , 36 ], and the multipole expansion [ 37 ]. neither of these source models offer a direct view on the timing of myocardial activation and recovery , or other electro - physiologically related features . from the 1970s onwards , the potential of two other types of source descriptions have been explored . this development followed from an increased insight into cardiac electrophysiology . the results of both methods use source descriptions at the surface of the heart . solving the implied inverse problem may accordingly be viewed as a type of functional imaging , which has led to their characterisation as , e . g ., “ non - invasive electrocardiographic imaging ( ecgi )” [ 40 ] or “ myocardial activation imaging ” [ 41 ]. there is a serious complication involved in the use of both models : the associated inverse problem is ill - posed , i . e . small deviations in the measured ecg data may result in completely different outcomes of activation and recovery times [ 39 ]. a solution to the inverse problem can therefore only be obtained through regularisation of the source parameters such that these parameters express desired properties . a brief characterisation of both methods is as follows . the first distributed surface source model is that of the potential distribution on a surface closely surrounding the heart , somewhat like the pericardium , referred to here as the pericardial potential source ( pps ) model . the model is based on the fact that , barring all modelling and instrumentation errors , a unique relation exists between the potentials on either of two nested surfaces , one being the body surface , the other the pericardium , provided that there are no active electric sources in the region in - between . it was first proposed at duke [ 42 ]; its potential has subsequently been developed by several other groups , e . g ., [ 43 , 44 , 46 , 47 , 69 ]. the second model , used in our research , is based on the macroscopic equivalent double layer ( edl ) model [ 70 ], applicable to the entire electrical activity of the atria and ventricles , at any time instant [ 71 ]. this source model stems from the classic model of the double layer as an equivalent source of the currents generated at the cellular membrane during depolarisation , described by wilson et al . [ 48 ]. initially , this current dipole layer model was used to describe the activity at the front of a depolarisation wave propagating through the myocardium [ 49 , 50 ]. later , salu [ 51 ] expressed the equivalence between the double layer at the wave front and a uniform double layer at the depolarised part of the surface bounding the myocardium , based on solid angle theory [ 52 ] ( see fig4 ). more recently , geselowitz [ 56 , 57 ] has shown , using a bidomain model , that the actual current source distribution within the heart is equivalent to a double layer at the surface of the myocardium with a strength proportional to the local transmembrane potential ( tmp ) [ 72 ]. the waveform of the transmembrane potential at each location on the myocardial surface is described by two parameters : the local activation and recovery time . consequently , the source parameters consist of the activation and recovery times . the relation between the source parameters and the source strength is a non - linear relation . in a previous study by huiskamp et al . [ 73 ] the initial activation times were estimated from the time integral of the measured qrs complex . another initial estimate , introduced by huiskamp and greensite , is based on the critical point theorem [ 29 , 41 , 74 ]. both initialisation methods , however , lack a direct link to the electrophysiology of the heart . an initial estimate for the atrial or ventricular activation can be obtained based on the propagation of the electrical activation inside the myocardium . in our application , for multiple activation sequences produced based on a propagation model the corresponding ecgs are computed . the activation sequence whose computed ecg matches the actual ecg the best is used as the initial estimate . as mentioned before , the activation of myocardial tissue in the healthy human heart is initiated by the sinus node in the atria and by the his - purkinje system in the ventricles . in pathological cases , as may be encountered in a clinical setting , the initiation may occur anywhere within the atria or ventricles . these electrophysiological and clinical facts require the activation modelling to be able to combine activation sequences originating from multiple locations ( foci ), e . g . as is the case for the his - purkinje initiated activation . furthermore , the method has to be able to determine the location of a focus anywhere in the atria or ventricles . several sophisticated models are available to simulate cardiac activation [ 59 , 75 , 76 , 77 , 78 ]. however , most of these models require many hours to compute a single activation sequence . these models cannot be used in the inverse procedure , because of the vast number of activation sequences that needs to be tested . an approach that is able to simulate an activation sequence within a second is a cellular automaton model [ 79 , 80 ]. these cellular automaton models involve that a volume description of the myocardium is required instead of the surface description used in our approach . the latter used the fastest - route algorithm , based on the shortest path algorithm [ 81 , 82 ], while using a surface description of the heart only . the shortest path algorithm , was designed by dijkstra to compute the path with minimal length between any two nodes in a graph [ 83 ]. a well - known application is the route planning algorithm in any car navigation system , with the roads representing the graph . the applications and implementation issues of the fastest - route algorithm in modelling cardiac activation for the application of finding an initial estimate in the inverse procedure are the main topics of interest of this application . the simulation of ecg signals generated by atrial activity , and consequently the non - invasive estimation of atrial activation , requires a realistic volume conductor model . in a preliminary study , the influence of certain inhomogeneities within the thorax ( lungs and the intra - cardiac blood volume ) in the forward simulation of the body surface potentials generated by atrial electric activity is studied [ 84 ]. in good approximation , atrial activation can be compared to a huygens wave spreading with uniform velocity in all directions . the first application of the shortest path algorithm ( spa ), assuming uniform velocity , was therefore applied to generate atrial activation sequences [ 85 ]. to keep the setup as simple as possible atrial wall thickness was discarded , revealing the concept of prominent routes . such routes show the intensity in which of the atrial nodes are utilised in the various atrial paths generated by the shortest path algorithm . within the ventricles anisotropic propagation is known to play a prominent role . consequently it has to be incorporated in the application of spa to generate ventricular activation sequences . in the fastest route algorithm ( fra ), which is based on the spa , is introduced . in the fra , inhomogeneous propagation velocities within the ventricles can be incorporated . the fra is used to simulate the effect a local reduction of the propagation velocity on the overall activation sequence and the simulated ecg . the results shown in this application are based on data recorded in three subjects . the nature of this data is summarised below . more details can be found in previous publications [ 29 , 64 , 85 , 86 ], in which essentially the same material was used . in each patient , a 64 - lead ecg was recorded . of course more leads ( up to a point ) may provide higher resolution . typically beyond 200 leads the increase in resolution does not justify the addition of leads . also less leads may be used , particularly when more database information can be incorporated . less than the standard 12 leads may lead to results with limited resolution . for each subject , mri - based geometry data was available from which individualised numerical volume conductor models were constructed , incorporating the major inhomogeneities in the conductive properties of the thorax , i . e . the lungs , the blood - filled cavities and the myocardium . the first subject ( nh ) is a healthy subject [ 85 ]. this subject was included to illustrate intermediate and final results of the described inverse procedure . the other two subjects are added to illustrate clinical applications . the second subject ( wpw ) is a wpw patient for whom previously estimated activation times have been published [ 29 , 64 ]. the recorded ecgs included episodes in which the qrs displayed the typical wpw pattern , i . e ., a fusion beat in which the activation is initiated at both the av node and the kent bundle . the location of the latter was determined invasively . the ecgs were also recorded after an av - nodal block had been induced by a bolus administration of adenosine , resulting in an activation sequence solely originating from the kent bundle . the third case was a brugada patient in whom ecg data were recorded during infusion of a sodium channel blocker ( ajmaline ) [ 86 ], 10 bolus infusions of 10 mg , administered at one - minute intervals [ 87 ], which changes the activation and / or recovery sequence . the beats selected for analysis were : the baseline beat 5 minutes prior to infusion and the beat after the last bolus had been administered . for each subject , the number of nodes representing the numerical representation of the closed surface ( endo - and epicardium ) of the ventricles and the observed qrs durations are listed in table 1 . the time course of the tmp acting as the local edl source strength was specified by the product of three logistic functions , the functions of the type l ⁡ ( t ; τ , β ) = 1 1 + ⅇ - β ⁡ ( t - τ ) . ( 1 ) describing a sigmoidal curve with maximum slope β / 4 at t = τ . the first of these functions described phase ( 0 ) of the tmp time course [ 20 ], the depolarisation phase , as with δ the timing of local depolarisation and the factor β setting the steepness of the upstroke . local repolarisation refers to the period during which the tmp moves toward its resting state ( phases 3 and 4 ), a process that may take up to some hundred ms . the tmp wave form during this period was described as r ( t ; ρ )= l ( t ; ρ , β 1 ) l ( t ; ρ , β 2 ), ( 3 ) in which ρ sets the position of the inflection point of the tmp &# 39 ; s down slope and β 1 and β 2 are constants setting its leading and trailing slope . note that the action potential duration , α , defined as represents the time interval between the marker used for the timing of local repolarisation ρ and the timing of local depolarisation δ ; it may be interpreted as the activation recovery interval [ 88 ]. in summary , the waveform specifying the strength s ( t ; δ , ρ ) of the local edl was note that this function depends on two parameters only . the constants β 1 and β 2 were found by fitting s ( t ; δ , ρ ) to the stt segment of the rms ( t ) curve of the 64 ecgs of the individual subjects , as is described in [ 61 ] and motivated in [ 89 ]. examples of the tmp wave forms obtained from ( 5 ) are shown in fig5 . these may be shifted in time as appropriate in individual cases . moreover they may be scaled in amplitude to an arbitrary , uniform level in the application to non - ischemic tissue for which , based on solid angle theory applied to a closed double layer , a uniform strength produces no external field . based on the edl source description , with its local strength at position { right arrow over ( x )} on the surface of the ventricular myocardium ( s v ) taken to be the local transmembrane potential v m ( t ,{ right arrow over ( x )}), the potential φ ( t ,{ right arrow over ( y )}) generated at any location { right arrow over ( y )} on the body surface is φ ( t ,{ right arrow over ( y )} )=∫ s v b ( { right arrow over ( y )},{ right arrow over ( x )} ) v m ( t ,{ right arrow over ( x )} ) d ω ( { right arrow over ( y )},{ right arrow over ( x )} ), ( 6 ) in which dω ({ right arrow over ( y )},{ right arrow over ( x )}) is the solid angle subtended at { right arrow over ( y )} by the surface element ds ({ right arrow over ( x )}) of s v and b ({ right arrow over ( y )}, { right arrow over ( x )}) is the transfer function expressing the full complexity of the volume conductor ( geometry and tissue conductivity ). previous studies [ 31 , 70 ] indicated that an appropriate volume conductor model requires the incorporation of the heart , blood cavities , lungs and thorax . in this study , the conductivity values σ assigned to the individual compartments were : thorax and ventricular muscle : 0 . 2 s / m , lungs : 0 . 04 s / m and blood cavities : 0 . 6 s / m . the complex shape of the individual compartments within the volume conductor model does not permit one to determine b ({ right arrow over ( y )},{ right arrow over ( x )}) by means of an analytical method . instead , numerical methods have to be used . in this study we used the boundary element method ( bem ) [ 38 , 90 ]. another suitable method may be the finite element method ( fem ). by means of the bem , while using formula ( 5 ) for describing the tmp , the potential at any node l of the discretised ( triangulated ) body surface was computed as ϕ ⁡ ( t , ℓ ) = ∑ n ⁢ b ⁡ ( ℓ , n ) ⁢ s ⁡ ( t ; δ n , ρ n ) , ( 7 ) with n the number of nodes of the triangulated version of s v . for each moment in time this amounts to the pre - multiplication of the instantaneous column vector ( source ) s by a ( transfer ) matrix b , which incorporates the solid angles subtended by source elements as viewed from the nodes of the triangulated surface , scaled by the relative jump in ( σ i + − σ i − )/( σ i + + σ i − ) of the local conductivity values σ i + and σ i − at either sides of the interfaces i considered [ 38 , 90 ]. the timing of local depolarisation and repolarisation was treated as a parameter estimation problem , carried out by minimizing in the least squares sense with respect to the parameters δ and ρ , the difference between the potentials computed on the basis of ( 7 ) and the corresponding body surface potentials v ( t , l ) observed in the subjects studied . since the source strength depends non - linearly on the parameters , the minimisation procedure needs to be carried out iteratively , for which we used a dedicated version of the levenberg - marquardt algorithm [ 91 ] the subsequent steps of this procedure alternated between updating the δ and ρ estimates . updating δ was carried out on the basis of solving arg min δ (∥ v − φ ( δ ; ρ )∥ f 2 + μ 2 ∥ lδ ∥ f 2 ). ( 8 ) matrix l represents a numerical form of the surface laplacian operator [ 92 ], which serves to regularise the solution by guarding its ( spatial ) smoothness , μ 2 its weight [ 31 ] in the optimisation process and ∥ ∥ f 2 the square of the frobenius norm . matrix l is the laplacian operator normalized by the surface [ 31 ]. consequently lδ is proportional to the reciprocal of the propagation velocity . as the propagation velocity of activation and recovery are in the same order of magnitude the same value for μ can be used for activation and recovery . updating ρ was based on the same expression after interchanging δ and ρ in the regularisation operator ( latter part of ( 8 )). since the problem in hand is non - linear , initial estimates are required for both . in previous studies , related to the activation sequence ( δ ) only , the initial estimates involved were derived exclusively from the observed bsps [ 31 , 53 , 54 , 55 ]. the method employed here is based on the general properties of a propagating wave front . from this initial estimate for depolarisation , an estimate of the initial values of the repolarisation marker , ρ , is worked out by including the effect of electrotonic interaction on the repolarisation process . during activation of the myocardium , current flows from the intracellular space of the depolarised myocytes to the intracellular space of any of its neighbours that are still at rest ( polarised at their resting potential ). the activation of the latter takes about 1 ms and is confined to about 2 mm . the boundary of this region ( the activation wave front ) propagates toward the tissue at rest until all of the myocardium has been activated . the propagation can be likened to the huygens process . the local wave front propagates in directions dominated by the orientation of local fibres , at velocities ranging from 0 . 3 m / s across fibres to 1 m / s along fibres . under normal circumstances , ventricular depolarisation originates from the bundle of his , progresses through the purkinje system , from which the myocardium is activated [ 93 ]. in humans this purkinje system is mainly located on the lower ⅔ of the endocardial wall [ 14 , 15 ]. in other cases , ventricular activation originates from an ectopic focus or from a combination of the activity of the his - purkinje system and an ectopic focus . the initial estimate of the inverse procedure was based on the identification of one or more sites of initial activation , from which activation propagates . this includes normal activation of the myocardium , which can be interpreted as originating from several foci representing the endpoints of the purkinje system . the activation sequence resulting from a single focus was derived by using the fastest route algorithm . the fastest route algorithm ( fra ) determines the fastest route between any pair of nodes of a fully connected graph [ 94 ]. the term ‘ fully connected ’ signifies that all nodes of the graph may be reached from any of the other ones by travelling along line segments , called edges , that directly connect pairs of nodes . in the current application the term edges not only refers to the edges of the triangles constituting the numerical representation of s v , but also to the paths connecting epicardial and endocardial nodes [ 95 ], provided that the straight line connecting them lies entirely within the interior of s v . the structure of the graph is represented by the so - called adjacency matrix , a , which has elements a ij = 1 if nodes i and j are connected by an edge , otherwise a ij = 0 . by specifying velocity v ij for every edge ( i , j ), i . e ., for every non - zero element of a , the travel time t ij along the edge is t i , j = d i , j v i , j , ( 9 ) in the mri based geometry of the ventricles the node distances d i , j are known . in the application of ( 9 ), the velocities along the edges need to be specified . for want of proper estimates on the myocardial penetration sites of the purkinje system of the individual subjects , as will be case in the ultimate , clinical application of the proposed method , relatively crude edge velocity estimates were used . values reported for the anisotropy ratio v l / v t of longitudinal and transverse - fiber velocities show a wide range : from 2 to 6 [ 3 , 96 , 97 , 98 ]. in this study two related velocities were used : the velocity in directions along the ventricular surface and the one in directions normal to the local surface , v l and v t , respectively . their ratio was taken to be 2 , the lower end of the range , selected in order to account for transmural rotation of the fibers [ 18 ]. for transmural edges that were not normal to the local surface , the travel time t i , j was taken to be t i , j = d 2 v ℓ 2 + h 2 v t 2 = 1 v ℓ ⁢ d 2 + 4 ⁢ h 2 , ( 10 ) with d and h the lengths of the projections of the edge along s v and normal to it , respectively . this procedure approximates locally elliptical wave fronts . the factor 4 appearing on the right in ( 10 ) results from the assumed anisotropy ratio 2 . infinite travel times are assigned to pairs ( i , j ) that are not connected by an edge . the entire set of t i , j constitutes a square , symmetric matrix a t , on the basis of which the fra computes the shortest travel time between arbitrary node pairs . the results form a square , symmetric travel time matrix t . the element j of any row i of matrix t was interpreted as the activation time at node j resulting from focal activity at node i only . a search algorithm was designed , aimed at identifying one focus , or a number of foci , which identified the activation sequence yielding simulated body surface potentials that most closely resemble the recorded ones . if a focus at node i is taken to be activated at time t i , the other nodes will be depolarised at δ j = t i + t i , j . if multiple foci are considered , the activation sequence is computed by the “ first come , first served ” principle : if k foci are involved , the depolarisation time δ j is taken to be δ j = min k ( t k + t k , j ), k = 1 . . . k . ( 11 ) as is shown on the righthand side of ( 10 ), v t scales the elements of t , and consequently of δ . at each step the intermediate activation sequence , δ , v t was approximated by max ( δ )/ t qrs , with t qrs the qrs duration ( see table 1 ). for nodes known to represent myocardial tissue without purkinje fibers the maximum velocity was set at 0 . 8 m / s . no maximum velocity was defined for nodes in a region potentially containing purkinje fibers , the nodes in the lower ⅔ of the endocardial surfaces of the left and right ventricles . for any activation sequences δ tested , ecgs were computed from ( 7 ) at each of the 65 electrode positions ( 64 lead signals + reference ). note that , with pre - computed matrices b and t , this requires merely the multiplication of the source vector by matrix b . the linear correlation coefficient r between all elements of the simulated data matrix φ and those of the matrix of the measured ecgs , v , was taken as a measure for the suitability of δ for serving as an initial estimate . the lead signals were restricted to those pertaining to the qrs interval ( about 100 samples spaced at 1 ms ). when taking a single node i of the n nodes on s v as a focus , this results in n basic activation sequences δ i , i = 1 . . . n , and corresponding values r i . the node exhibiting the maximum r value was selected as a focus . the entire procedure was carried out iteratively . during the first iteration , the value t i = 0 was used , corresponding to the timing of onset qrs . in any subsequent iteration k , the accepted values for t i were set at 90 % of their activation times found from the previous iteration . for the nodes corresponding to the above described ‘ purkinje system ’ the values for t i were set at 40 % of the previous activation times . the purkinje systems is largely insulated from the myocardial tissue . the propagation velocity in this system ranges between 2 and 4 m / s . the 40 % value of t i represents a 2 . 5 times higher velocity than the one used in the previous activation sequence , which is usually around 1 m / s . within the myocardium the differences in velocity are much smaller , limited to approximately 0 . 7 - 1 m / s . note that a focus can be selected more than once , in which case its activation time decreases . the iteration process , identifying a focus at each step , was continued until the observed maximum value of r decreased . in contrast to the situation during the activation of the myocardium , local recovery may take up to some hundred ms , while the repolarisation process starts almost directly after the local depolarisation . similar to the situation during activation , throughout the recovery period current flows from the myocytes to their neighbours . the spatial distribution of these currents is not confined to some “ repolarisation ” boundary , but instead is present throughout all regions that are “ recovering ”. even so , some measure of the timing of local recovery at node n can be introduced , such as the marker ρ n introduced previously , and its distribution over s v can be used to quantify the timing of the overall recovery process . the intracellular current flowing toward a myocyte is positive when originating from neighbours that are at a less advance state of recovery , thus retarding the local repolarisation stemming from ion kinetics . conversely , the current flows away when originating from neighbours at a more advanced stage of recovery , thus advancing local repolarisation . the size of the two domains determines the balance of these currents , and thus the magnitude of the electrotonic interaction . at a site where depolarisation is initiated , the balance is positive , resulting locally in longer action potential durations than those at locations where depolarisation ends . the extent of the two domains is determined by the location of the initial sites of depolarisation and overall tissue geometry [ 99 ]. as a consequence , local action potential duration , a , is a function of the timing of local depolarisation and , expressed in the notation of ( 4 ), so is ρ ( δ )= δ + α ( δ ). literature reports on the nature of the function α ( δ ) as observed through invasive measurements are scarce . in some reports [ 22 , 24 ] a linear function was suggested , for which a slope of − 1 . 32 was reported . the function used in our study involved the subtraction of two exponential functions . for small distances between a local depolarisation ( its source ) and local ending of activation ( its sink ) this function becomes linear in approximation [ 99 ]. the initial estimate for ρ was found from ρ ( δ )= δ + α ( δ ), with δ the initial estimate of the timing of depolarisation . the value of α n at any node n was computed as α n = α + κ n ( e − x n / ξ − e ( 1 - x n )/ ξ ), ( 12 ) with α expressing the mean activation recovery interval , κ n ={ circumflex over ( δ )} n −{ hacek over ( δ )} n the difference between the depolarisation times of the closest sink and source , x n =( δ n −{ hacek over ( δ )} n ) κ n and ξ a dimensionless shape constant [ 99 ]. sources and sinks of activation were identified as nodes of s v for which all neighbors in a surrounding region of 2 cm were activated earlier or later , respectively . if more than one source and sink was found within a distance of 4 cm , the average value of the parameters x n and κ n was assigned to the node n involved . the value of α was found through lining up ρ = δ + α with the apex of the rms ( t ) signal of the observed ecg signals [ 100 ]. all lower case bold variables denote vectors , all upper case bold variables refer to matrixes . the differences between simulated and recorded potential data are quantified by using the rd measure : the root mean square value of all matrix elements involved relative to those of the recorded data . in addition , the linear correlation coefficient r between all elements of simulated and reference data are used . the differences between simulated and recorded potential data are quantified by using the rd measure : the root mean square value of all matrix elements involved relative to those of the recorded data . in addition , the linear correlation coefficient r between all elements of simulated and reference data is used . for all three cases considered , the weight parameter of the regularisation operator , μ ( see ( 8 )), was empirically determined and set to 1 . 5 10 − 4 , both while optimizing activation and recovery . the upstroke slope , β ( see ( 1 )), was set to 2 , resulting in an upstroke slope of 50 mv / ms . the parameter ξ ( 12 ) was tuned such that , for the healthy subject ( nh ), the linear slope of α ( δ ) was − 1 . 32 ( see franz et al . [ 24 ]). this resulted in a value of 7 . 9 · 10 − 3 for ξ . the initial activation estimated by means of the focal search algorithm is shown in fig6 a . in total 7 foci were found in four regions , the first one in the mid - left septal wall , the next on the lower right septal wall and some additional ones on the left and right lateral wall . the result obtained from the non - linear optimisation procedure based on this initial estimate is shown in fig6 b . the initial estimated activation recovery intervals ( ari ), derived from the initial activation sequence ( fig6 a ) and the use of equation ( 12 ) are shown in fig7 a . note that areas activated early indeed have a longer ari than the areas activated late . after optimisation ( fig7 b ) the global pattern is similar to the initial one , with a minimally reduced range ( from 182 - 320 ms to 176 - 300 , see also table 3 ). this can also be observed in fig8 , in which the local initial and final ari values are plotted as a function of activation time . consequently , the accompanying reduction in the linear regression slope between initial and estimated aris and activation times is also smaller ( table 3 ). the average of the estimated aris is 7 ms shorter than the initial aris . in general the estimated ari values in the right ventricle shorten more compared to the initial estimate whereas the ari values in the left ventricle and septum prolong slightly ( fig8 ). the resulting recovery times show very little dispersion ( 49 ms , table 3 ). the right ventricle starts to repolarise first , whereas the ( left ) septum repolarises last ( fig9 ). both the left and right ventricle show a prevailing epi - to endocardial direction of recovery . the measured ecgs and the simulated ecgs based on the estimated activation and recovery times , match very well during both the qrs and the stt segment ( see fig1 ), as indicated by the small rd value ( 0 . 12 , table 1 ). for the fusion beat the first focus identified by the focal search algorithm was the kent bundle ( fig1 a ; see fisher et al [ 29 ]) subsequently 3 focal areas were determined : one on the lower left septal area and two on the right ventricular wall . the estimated repolarisation times have a small dispersion ( 32 ms , see fig1 b ), which is in agreement with the fact that the heart is activated from both the kent bundle and the his - purkinje system . for the beat in which the av node was blocked a single focus was found at approximately the same location where the kent bundle was found on the basis of the fusion beat . the resulting ecgs of both beats is shown in fig1 . examples of the inversely computed timing of depolarisation and repolarisation of the brugada patient are shown in fig1 . these related to two time instants during the procedure : at baseline and just after the last infusion of ajmaline . the effect of the ajmaline on the ecg can be observed in the rms signal of the recorded bsps ( insets fig1 ): the qrs broadened and the st segment became slightly elevated following the administration of ajmaline . the first focal area was found on the left side of the septum for the baseline beat and at peak ajmaline . for the baseline beat two additional foci were found , one on the left and one on right lateral wall ( see fig1 a ). the estimated activation patterns of both analysed beats are similar , although the activation times of the ajmaline beat were later near the left and right base of the heart ( fig1 a / b ). the estimated repolarisation times of the baseline beat show a dominant epi - to endocardial recovery sequence . after the last bolus of ajmaline , the transmural repolarisation difference in the left ventricle remained almost unchanged , though slowly shifted with time . large differences , however , are found in the right ventricle ( fig1 d ). the accompanying aris initially show a dispersion of 116 ms increasing up to 193 ms . this expanded range is mainly caused by the very early recovery in the outflow tract area ( see fig1 d ). the corresponding ecgs of both beats are shown in fig1 . for each step in the inverse procedure the correlation and rd values are calculated between the measured ecg and the simulated ecg ( see table 2 ). for all subjects the resulting inverse procedure rd values were small . the initial estimates , however , showed high rd (& gt ; 0 . 7 ) values despite the fact that the corresponding correlation was well above 80 %. the linear slope of α ( δ ) in the initial solution was close to − 1 . 32 for most subjects . after optimisation the slope values decreased for all cases , except in the brugada patient at peak ajmaline ( table 3 ). the computation time used by the inverse procedure ranged between ½ a minute ( bg ) up to 23 minutes ( nh ), depending on the number of nodes ( table 1 ) used in the mesh in the heart &# 39 ; s geometry ( table 4 ). table 4 computation times of the focal search algorithm and the optimisation procedure . focal search optimisation computation # computation subject # scans time s iterations time s nh 9 233 10 1143 wpw ( fusion ) 4 61 17 208 ( av block ) 2 23 85 993 bg ( at 1 min ) 6 6 23 31 ( at 10 min ) 1 1 36 46 some foci in the multi - focal search were optimised more than once , resulting in more scans than foci . within the optimisation procedure one iteration includes the optimisation of depolarisation times ( δ ) and repolarisation times ( ρ ). 1 . 3 . 5 amplitude estimation of the local transmembrane potential ( tmp ) amplitude based on the st segment of the ecg for one brugada patient ecg data were recorded during infusion of a sodium channel blocker ( ajmaline ) [ 101 ], 10 bolus infusions of 10 mg , administered at one - minute intervals [ 102 ], which changes the activation and / or recovery sequence . the beats selected for analysis were : the baseline beat 5 minutes prior to infusion and the beat after the last bolus had been administered . the effect of the ajmaline is expected , for some cases , to be rather local on the epicardium of the right outflow tract [ 103 ]. the mechanism is thought to be a structural change in this area , causing a current to load mismatch between the cardiac cells , i . e . a single cell has to activate too many neighboring cells . as a consequence the tmp amplitude drops and the propagation velocities decreases . a drop in amplitude results in an change of the st segment level . the method proposed uses therefore only the first 40 ms ( this might also longer or shorter ) after the qrs complex to estimate the local tmp amplitude on the heart . although this is a linear problem , an can therefore be solved directly through an pseudo inverse procedure , we used the levenberg - marquardt algorithm in an iterative procedure to determine the tmp amplitudes . this procedure is the same as used in optimizing the activation and recovery times . the results in fig1 show that this procedure identifies a small area on the epicardial outflow tract . initially , at base line no deviation in tmp amplitude is found . after the first infusion of ajmaline a small drop in tmp is found decreasing up to 35 % at peak ajmaline . for all of the three cases presented , the inversely estimated timing of activation and recovery agreed well with available physiological knowledge . the resulting ecgs closely matched the measured ecgs ( rd ≦ 0 . 19 , correlation ≧ 0 . 98 , table 2 ). the quality of the results and the required computation time hold promise for the application of this inverse procedure in a clinical setting . in previous studies the required initial estimates were derived exclusively from the observed bsps [ 31 , 53 , 54 , 55 ]. the robustness of these initial estimates was limited in the sense that small variations in the parameters of the first estimate lead to quite different outcomes of the inverse procedure . in the study presented here , the initial estimates are based on knowledge about the electrophysiology of the heart . the results show that this improves the quality of the inverse procedure significantly . the major elements of the inverse procedure are discussed below . a first improvement in the initial estimation procedure was the incorporation of global anisotropic propagation in the simulation of ventricular activation . when using an uniform velocity the estimated activation wave revealed earlier epicardial activation for subject nh in the anterior part of the left ventricle [ 104 ]. although no data on individual fiber orientation was available , the global handling of trans - mural anisotropy , estimated using common accepted insights [ 18 , 105 , 106 ], improved the overall performance . a second improvement concerns the selection of foci in the multi - foci search algorithm . within this algorithm , the correlation r between measured and simulated ecgs was used , instead of the rd values used previously . the idea to investigate the appropriateness of the correlation arose from the observation that the overall morphology of the simulated wave forms closely corresponded to the measured data , in spite of relatively high rd values . this was first observed in an application to the relatively simple atrial activation sequence , frequently involving just the “ focus ” in the sinus node region [ 85 ]. subsequently , it also proved to be effective in applications to the ventricles . each simulated activation sequence of the initial multi - foci search was scaled by an estimation of the global propagation velocity ( v l ) derived from the qrs duration , taking into account the differences in propagation velocity in myocardial and purkinje tissue . the qrs duration was derived from the rms ( t ) curve , computed from all leads referred to a zero - mean signal reference [ 25 ]. this produces the optimal estimate of the global onset and completion of the activation process . this initial estimation procedure proved to be very insensitive to slight variations in parameters settings . this can be observed from the first subject presented , nh , yielding an initial estimate that agrees well with literature data [ 16 , 28 , 107 ]. the final activation times , resulting from the subsequently applied inverse procedure , globally resemble those of the initial activation sequence ( fig7 a / b ). further visual inspection of the final activation sequence also showed no unphysiological phenomena . the inverse solutions for the wpw patient have been published by fisher et al . [ 29 ]. in their report , the solution presented solution was based an initial estimate derived from using the critical point theorem [ 41 ]. the solution for the fusion beat clearly identified the actual , invasively determined location of the accessory pathway location , but the accompanying initiation of activation at the septum and the right ventricle were not found . in our current inverse procedure , the initial estimated activation sequence of the same beat ( see fig1 a ) the accessory pathway location was identified in the first run of the multi - focal search , followed by locations from where the ventricle normally is activated [ 16 ]. the estimated activation sequence thus not only shows the correct position of the kent bundle , but also a true fusion type of activation resulting from early activation in the right ventricle and left septum , as is to be expected in this situation . for the situation in which the av node was blocked a single focus was found at the approximate location of the kent bundle ( see fig1 b ). a reduction in propagation velocity of 20 - 40 % can be found [ 108 ] after the administration of ajmaline , which is reflected in the estimated activation sequences before and after ajmaline administration ( compare fig1 a & amp ; b ). note that the earliest site of activation in both sequences are approximately the same . these activation patterns are similar , suggesting that the sodium channel blocker has a global influence on the propagation velocity within the heart ( fig1 a / b ) and a more pronounced effect in the right basal area , in accordance with linnenbank et al . [ 86 ] in previous studies only the cardiac activation times were estimated from body surface potentials [ 21 , 57 ]. in the current study the recovery sequence , as quantified by the timing of the steepest down slope of the local transmembrane at the heart &# 39 ; s surface , is included as well . for the initial estimate of the ventricular recovery sequence , ρ , a quantification of the effect of electrotonic interaction on the recovery process is used , expressed by its effect on the local the activation recovery interval ( ari ). few invasive data are available on the ventricular activation recovery intervals [ 109 ]. generally these are derived from potentials measured on the endocardial and epicardial aspects of the myocardium [ 40 , 107 , 110 ]. the linear regression slope of the α ( δ ) curves for all 5 cases was within the range as found by franz et al . [ 24 ] (− 1 . 3 ± 0 . 45 ). the estimated slopes revealed slightly smaller values in the right ventricle than those in the left ventricle and septum ( fig8 ). these results suggest that electrotonic interaction is a major determinant of the action potential duration . consequently the ari depends on the activation time , resulting in similar patterns for ari and activation times . these findings are in contrast with the ari values based on the pps source model found by ramathan et al . [ 40 ], in which local ari is almost completely uncoupled from the local activation time . the differences in ari values estimated by both methods can be attributed to the fact that the local tmp waveforms cannot be extracted uniquely from the , more global , electrograms used in the pps based inverse procedure to extract local recovery times . the dispersion in the recovery times found was smaller than those of the activation times , in agreement with the negative slopes observed for the α ( δ ) function . the ranges of the activation times found were about twice as large as those of the repolarisation times for the normal cases ( nh and bg ( 1 min ), table 3 ). the apex - to - base differences in the recovery times were small ( 20 - 30 ms ), which is in accordance to literature data [ 40 ]. at several sites the local transmural recovery differences were more substantial ( fig1 - 13 ). such large transmural recovery differences ( frequently referred to by the misnomer recovery gradients ), were found throughout the ventricles in all ‘ normal ’ subjects ( bg baseline and nh ), but not in the right ventricle of the brugada patient ( bg ) after the administration of ajmaline . it is unknown whether the recovery times of the brugada patient match reality , but the locations having the largest deviations in recovery time do match common knowledge [ 87 ]. an explanation for the short ari value ( and the advanced activation ) might lie in the fact that this area is not activated at all due to structural changes [ 111 ], an option not permitted by the presented inverse procedure . the description of the transmembrane potential waveform used for driving the edl source model ( 5 ), fig5 , proved to be adequate . when testing more refined variants only minor differences in the resulting isochrone patterns were observed , which is an indication of the robustness of the inverse procedure . to further illustrate the invention a summary of a typical procedure involving the methods according to the invention is reproduced below . the inverse procedure requires the measurement of ecg signals at several ( e . g . 64 ) locations on the body surface , a body surface map ( bsm ). the locations of these electrodes have to be recorded accurately to minimize modeling errors . the reference used in this for the bsm can be any reference electrode , such as the wilson central terminal ( average of the 3 extremity leads ). by default the average of all signals is used as a reference . by taking the root mean square ( rms ) of all signals ( see fig1 b ) the fiducial points of the p wave , qrs and t wave can be found automatically [ 112 ]. other methods to determine the fiducial points automatically may be applied . this rms signal of the qrst sequence is used to estimate the downslopes of the repolarization phase [ 70 ]. the t wave part of the t wave is integrated and fitted by two logistic functions that are multiplied ( see fig1 ). the volume conductor model contains at least the geometries of the thorax , lungs , blood cavities , major blood vessels and the heart . these geometries are constructed from images , for instance mri ( see fig1 ), ct or echo . the edges of the relevant parts are automatically detected ( see fig1 ), from which the geometry of each of the parts is reconstructed . from these contours the geometry is obtained . notice that the placement of the electrodes should be accurately recorded in this volume conductor model ( see fig2 ). the used electrical source model is the equivalent double layer [ 109 ]. this source model is located on the surface of the heart . currently 7 parameters are used to describe the local transmembrane potential ( tmp ), see fig2 . the number of parameters can be extended , such that a spike and notch are incorporated in the shape of the tmp ( see section 1 . 3 ). the three ingredients described in the previous chapters are integrated in the inverse procedure . from the discretized volume conductor model a transfer matrix ( a ) is created , relating the currents generated in the heart (“ the source ”) to potentials on the body surface . from the source description ( s ) the contribution of a part of the heart is obtained at any time instant . multiplying both matrices results in simulated body surface potentials at any position on the thorax . where δ , ρ , . . . are the source parameters . the simulated ecg signals at the electrode positions of the bsm lead system are compared to the measured ecgs signals . due to the non - linear nature of the used source model ( equivalent double layer ) the associated inverse problem is non - linear and requires an initial estimate for each of the used source parameters . the initial estimate for the repolarization and a plateau slope are derived from the ecg ( see section record ecg ). the initial estimate from the depolarization moments is obtained from the fastest route algorithm , which subsequently is used to derive an initial estimate for the repolarization times . this is described in full detail in section 1 . 2 . 4 and 1 . 2 . 5 . the amplitude or resting potential is estimated from the st segment or tp segment of the measured ecg . due to the fact that the ecg signals are baseline corrected a drop in tmp amplitude has the same effect as a rise in resting potential . the drop in tmp amplitude might also be temporal due to a prolonged notch in the tmp as in brugada patients ( see fig2 ). all effects are best visible when the electrical activity of the heart is minimal , i . e . the st segment for the ventricles , and the end p wave till the beginning of the qrs complex for the atria . the associated estimation problem is nearly linear in nature , i . e . the tmp amplitude / resting potential has to be estimated from the a part of the st segment . the initial estimate for this problem is that the amplitude is homogeneous for the whole heart . the final result is obtained through the optimization procedure as described section 1 . 2 . 3 . for the activation and recovery sequence of the ventricles see section 1 . 3 . the activation sequence of the atria can be obtained using the same method ( see fig2 ) the found amplitudes for an brugada patient are also described in section 1 . 3 . fig1 reconstruction of the heart of a healthy 22 year old male . the activation starts in the sinus node region , is delayed in the av node ( not shown ). after a fast propagation of the activation along special fibers , all myocytes of the ventricle are activated . the color scale indicates the elapsed time in ms . fig2 transmembrane potentials ( tmp ) of pacemaker cells in the sinus node ( panel a ), or ventricular cells ( panel b ). the pacemaker cells slowly but continuously depolarise until a threshold is reached ( black dotted line ), initiating the process of fast depolarisation . this depolarisation is immediately followed by the repolarisation , after which the whole cascade starts again . in panel b two different ventricular tmps are shown . these tmps are divided in 5 phases : 0 ) depolarisation , 1 ) early repolarisation , 2 ) plateau phase , 3 ) repolarisation , and 4 ) rest state . the red ( r ) waveform is commonly found in the endocardial cells whereas the blue ( b ) line is found in epicardial cells . the main differences are found in the early repolarisation phase . fig3 an ecg signal with the p wave , ( atrial activation ), qrs complex ( activation of the ventricles ), and the t wave ( recovery of the ventricles ). fig4 three equivalent double layers with the same solid angle , and consequently the same external potentials , e . g . on the body surface . panel a ) actual double layer at some moment during isotropic ventricular activation . panel b ) equivalent double layer with the same solid angle but now with at an anisotropic propagating activation wave . panel c ) equivalent double layer at the ventricular surface . fig5 examples of tmp wave forms based on formula ( 5 ), the constants δ and ρ specify two different timings of activation and recovery ; the corresponding activation recovery intervals are α = ρ − δ . fig6 : estimated activation sequences of the ventricles of a healthy subject . panel a : the initial estimate resulting from the multi - focal search algorithm ; the white dots indicate some of the foci that were identified . panel b : the result of the subsequently applied non - linear optimisation procedure . isochrones are drawn at 10 ms intervals . the ventricles are shown in a frontal view ( left ) and basal view ( right ). fig7 estimated activation recovery intervals at the ventricular surface of a healthy subject . initial and final aris as generated in the inverse procedure . panel a : the initial ari estimate . panel b : the ari distribution after optimisation . remaining legend as in fig6 . fig8 initial and final aris as generated in the inverse procedure . initial estimation of the aris ( panel a ) and final estimated aris ( panel b ) as a function of the respective depolarisation sequence . the solid black line indicates the linear regression line . three areas within the heart have been identified by different markers , right ventricle ( green crosses ), left ventricle ( red circles ) and ( left and right ) septum ( blue dots ). fig9 recovery sequence as obtained from the inverse procedure . remaining legend as in fig6 . fig1 standard 12 - lead ecg ; in solid blue : the measured data ; in dashed black : in black the simulated ecg based on the estimated activation and recovery times . fig1 the results of the inverse procedure for a fusion beat , i . e . activation initiated by a kent bundle and the his - purkinje system ( panel a ) and for a kent - bundle - only beat ( panel b ). the estimated activation sequences are shown on the left , the recovery sequences on the right . the white dot indicates the position of the kent bundle as observed invasively . remaining legend as in fig6 . fig1 the simulated ( dashed black lines ) and measured ecgs for a fusion beat ( blue ( b ) lines ), i . e . activation initiated by a kent bundle and the his - purkinje system and a beat for which the av node was blocked by adenosine , leaving only the kent bundle intact ( red ( r ) lines ). fig1 activation ( panel a and b ) and recovery ( panel b and d ) of two beats in a brugada patient during an ajmaline provocation test . panel a ) and c ) show the activation respectively the recovery sequence estimated from the baseline ecg . panel b ) and d ) show the activation respectively the recovery sequence just after the last infusion of the ajmaline . color scale is the same for panel a & amp ; b and panel c & amp ; d . remaining legend see fig6 . fig1 the simulated ( dashed black lines ) and measured ( blue ( b ) lines ) ecgs for the baseline beat of the brugada patient , and the simulated ( dashed black lines ) and measured ( red ( r ) lines ) ecgs of the beat at peak ajmaline . fig1 the action potential amplitude , estimated from the first 40 ms of the st segment ( starting after the j point ). first 1 baseline beats top left , subsequently after the first infusion of ajmaline every next minute . notice the local reduction of the ap amplitude ( up to 65 %), in the outflow tract area ( red color ). the color scales on the are fixed between 0 . 7 and 1 . fig1 left panel : the body surface potentials recorded at 64 electrode positions and the root mean square ( rms ) signal ( grey ) on top . the fiducial points of start qrs and end t wave are indicated by a *. right panel : example of bsm lead system . fig1 left panel : black crosses (+) indicate the fiducial points ; onset qrs , end qrs , and end t wave , the dot indicates the peak of the t wave . the green line ( g ) is the fitted t wave by derivative of the two logistic curves . right panel : the integral of the t wave ( black ) and the fitted integral curve by two logistic curve ( grey line ). fig1 example a number of mri images taken from a heart . fig1 top panel : contours drawn around the left and right ventricle , indicated by blue lines ( b ). lungs are indicated by green dots ( g ), atria and venea cava by red dots ( r ). bottom panel : all contours of the required tissues . fig2 reconstructed thorax and heart model with the position of the electrodes fig2 : tmp with the handlers in light grey . top panel : tmp with the handlers at the depolarization moment , repolarization moment , amplitude and resting potential . bottom panel : handlers for depolarization slope , plateau an repolarization slope . fig2 three different tmp waveforms : black normal tmp waveform without notch ( endocardial node ), dashed black : tmp waveform with deep and prolonged notch ( associated with e . g . brugada syndrome ), grey : tmp with a rise in tmp resting potential ( associated with ischemia ). fig2 estimate of the activation sequence of the atria . atria are in ap view . 1 kléber , a . g . intercellular communication and impulse propgation . cardiac electrophysiology from cell to bedside . d . p . zipes and j . jalife . philadelphia , saunders . 2004 , 213 - 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