research article simulation of ectopic pacemakers...

Research ArticleSimulation of Ectopic Pacemakers in the Heart Multiple EctopicBeats Generated by Reentry inside Fibrotic Regions

Bruno Gouvecirca de Barros1 Rodrigo Weber dos Santos1

Marcelo Lobosco1 and Sergio Alonso2

1Graduate Program in Computational Modeling Federal University of Juiz de Fora 36036-330 Juiz de Fora MG Brazil2Department of Applied Physics Universitat Politecnica de Catalunya-BarcelonaTech Barcelona Spain

Correspondence should be addressed to Rodrigo Weber dos Santos rodrigoweberufjfedubr

Received 8 February 2015 Revised 8 May 2015 Accepted 17 May 2015

Academic Editor Dobromir Dobrev

Copyright copy 2015 Bruno Gouvea de Barros et alThis is an open access article distributed under theCreativeCommonsAttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited

The inclusion of nonconducting media mimicking cardiac fibrosis in two models of cardiac tissue produces the formation ofectopic beats The fraction of nonconducting media in comparison with the fraction of healthy myocytes and the topologicaldistribution of cells determines the probability of ectopic beat generation First a detailed subcellular microscopic model thataccounts for the microstructure of the cardiac tissue is constructed and employed for the numerical simulation of action potentialpropagation Next an equivalent discrete model is implemented which permits a faster integration of the equations This discretemodel is a simplified version of the microscopic model that maintains the distribution of connections between cells Both modelsproduce similar results when describing action potential propagation in homogeneous tissue however they slightly differ in thegeneration of ectopic beats in heterogeneous tissue Nevertheless both models present the generation of reentry inside fibrotictissues This kind of reentry restricted to microfibrosis regions can result in the formation of ectopic pacemakers that is regionsthat will generate a series of ectopic stimulus at a fast pacing rate In turn such activity has been related to trigger fibrillation in theatria and in the ventricles in clinical and animal studies

1 Introduction

Cardiac arrhythmias are malfunctions of the coordinateddynamics of heart electric activity In the extreme case whenmany cardiac myocytes are strongly desynchronized con-traction is impairedThis underlying complex spatiotemporaldynamics is called fibrillation and can take place in theatria or ventricles Reentry that is the reappearance of theaction potential wave after its first passage or spiral wave isbelieved to be a necessary but not sufficient condition forfibrillation Once one or more spiral waves exist they caninteract or a single one can become unstable and in eithersituation this may lead to fibrillation Spiral generation spiralbreakup instabilities and fibrillation have been extensivelystudied by the use of computational models [1ndash3] clinicaldata or animal models [4ndash7] One of the most commonprotocols to generate a spiral wave is via an ectopic beatan additional stimulus generates a second wave that will

interact with the sinus wave and lead to the genesis of a spiralwave or reentry This process is well understood and manyknown situations lead to the appearance of an ectopic beat inpathological or physiological conditions Just to name two ofthem in a healthy heart an external impactmay stretch someregion of the heart and via stretch-activated ion channelscardiacmyocytesmay coordinately generate action potentialsthat may initiate a second wave [8] In an infarct region gapjunctional remodeling fibroblast to myocyte coupling andfibrosis can significantly slow down wave propagation whenit enters the infarct region so that the wave leaves the regiondetached from the sinus wave that is when a second wave orectopic beat appears [9]

There are many different protocols for the generationof fibrillation Fast pacing is a common protocol used inboth computational and animal models and experiments[5 6 10] We can refer to a recent work [11] where atrialfibrillation was studied in an accurate model of human atria

Hindawi Publishing CorporationBioMed Research InternationalVolume 2015 Article ID 713058 18 pageshttpdxdoiorg1011552015713058

2 BioMed Research International

In the mentioned work it was only possible to induce atrialfibrillation by using a burst of at least six consecutive ectopicbeats separated by 130ms or via continuous fast periodicpacing (also with an ectopic beat period of 130ms) Unfor-tunately differently from the case of a single ectopic beatthere are notmanywell-understood examples or explanationsfor ectopic pacemakers that is a region that would generatemultiple ectopic beats at a frequency much higher (near anorder of magnitude) than the sinus rhythm In particularnew evidences suggest that two mechanisms (combined ornot) may play a very important role in the generation ofectopic pacemakers abnormal intracellular calcium (Ca2+)signalling and microreentry (reentry inside a fibrotic region)[12]

In this work we were able to generate an ectopic pace-maker in computer simulations of cardiac electrophysiologyby assuming a single hypothesis for substrate the existence ofa region with a mixture of excitable (healthy myocytes) andnonexcitable (fibrosis for instance) areas where each areahas a minimum size of a single myocyte This fibrotic regionhas been named before as microfibrosis [13ndash15] not mainlybecause of its whole size but rather due to the microscopicseparation of these two phases myocytes and fibrosis whichas mentioned before is at the scale of a single myocyte

The mechanism of this ectopic pacemaker is simple Thetwo different phases (excitable and nonexcitable) form amazeor labyrinth for wave propagation Propagation becomesfractionated by the zig-zag like or tortuous pathways it hasto follow Macroscopic propagation is considerably sloweddown Microscopically the topology of the maze allows theelectric wave to reexcite the fibrotic tissue before the waveleaves itTherefore reentry and spiral waves are formed insidethe fibrotic regionThis spiral-like wave will continuously tryto generate ectopic beats whenever it touches the border ofthe fibrotic region

Evidences that support this thesis can be found in previ-ous computational animal and clinical studies Many animalstudies have associated regions near fibrosis as triggers forventricular tachycardia fibrillation [9] or atrial fibrillation[16] Macroscopically it has been shown that propagationvelocity in the border zone of infarct may drop from 051msto 009ms [9]Microscopically propagation near fibrosis hasbeen associated with fractionated electrograms or complexfractionated atrial electrograms (CFAE) in both animal [17]and clinical studies [18] These findings have remarkablychanged the management and prognostics of patients withatrial fibrillation via a new procedure of catheter ablation ofatrial fibrillation that is guided byCFAEmapping [19] Recentclinical studies of guided CFAE ablation revisited the benefitsand risks of this procedure and clearly suggest that furtherinvestigation on this topic is required [20ndash22]

Computer simulations gave valuable insights into how themicrostructure of cardiac tissue specifically microfibrosisand CFAEs are related [13 15] These previous studies haveused realist electrophysiology single myocyte models anddetailed microscopic description to quantitatively representthe fine structure of cardiac tissue that is the models werebased on subcellular discretizations of cardiac tissue

Figure 1 In the center of this figure we present the fibrotic regionthat is simulated in this work The presence of reentry inside thefibrotic region may constantly reexcite the surrounding tissue andtherefore act as an ectopic pacemaker

However up to now a clear link between the microstruc-ture remodeling of cardiac tissue (microfibrosis) and a the-oretical substrate for cardiac arrhythmia (fibrillation) is stillmissing Evidences of the importance of tissue heterogeneityspecifically of nonconductive cells on spiral genesis andbreakup were already presented in a theoretical study usingsimple models based on cellular automata [23] RecentlyAlonso and Bar [24] have presented for the first timesimulations based on simple discrete and isotropic modelsof cardiac tissue that could generate ectopic pacemakersThe mechanism was the one explained before reentry wasgenerated inside a heterogeneous tissue (composed of activeor nonactive discrete cells) In addition Alonso and Bar [24]have described how the probability of reentry depends on aspecific feature of the topology of the discrete network thatwas used to model cardiac tissue the percentage of nonactivecells phi

The focus of this work is on bridging the gap betweenthe quantitative and more complex models used to simu-late microfibrosis and CFAEs [13] and the qualitative andsimplified models used to relate microfibrosis and ectopicpacemaker [24] For this purpose two different models wereused and both are as complex and realistic as those usedbefore in [13] that is both use realistic description of singlemyocyte electrophysiology and account for the microstruc-ture of cardiac tissue such as gap junction distribution andtissue anisotropy (1) a microscopic model with subcellulardiscretization of 8 120583m as described before in [25] and (2) adiscrete equivalent model that to the best of our knowledgehas never been presented before It is important to highlightthat in this paper our models focus on the electric dynamicsinside the ectopic pacemaker or microfibrosis region (seeFigure 1) In the near future we will study how this activitypropagates to the outside of this fibrotic region and ultimatelyinfluence the surrounding (nonfibrotic) regions of the heart

Our simulation results support the thesis that themicrostructure of a fibrotic tissue may indeed generate per se

BioMed Research International 3

(a) (b)

Plicate gap junctionIntriplicate gap junctionCombined plicate gap junction

(c)

Figure 2 (a) Six basic units being combined to form a larger tissue (b) Basic unit of cardiac myocyte distribution based on a total of 32cells (c) This figure presents an example of how different gap junctions are distributed in three neighboring myocytes that belong to thebasic unitThere are only 5 possible types of connections between neighboring volumes that are membrane which indicated no-flux betweenneighboring volumes cytoplasm which indicates the neighboring volumes are within the same cell three possible types of gap junctionsplicate intriplicate and combined plicate

an ectopic pacemaker via reentry inside the microfibrosisAlso as in [24] the probability of a fibrotic region to becomean ectopic pacemaker depends on the percentage of fibrosisin the tissue Using our realistic models we could quantifythat ectopic pacemakers were easier to be generated with 120601between 42 and 50 Our results also gave new insightsand suggest that in addition to the percentage of fibrosis120601 the probability of a fibrotic region to become an ectopicpacemaker also depends on the direction of the approachingplane wave This is in agreement with the recent resultspresented in [15] where it was shown that the waveformsof CFAEs depend on both the structure or topology of thefibrosis and the direction of the approaching propagatingwave

2 Methods

21 A Detailed Microscopic Model of Mouse Ventricular Tissue

211 Modeling Cardiac Microstructure We developed a two-dimensional model that is based on the previous workof Spach and collaborators [26 27] that accounts for themicrostructure of cardiac tissue gap junction heterogeneousdistribution and discretizations of 8 120583m times 8 120583m A basictemplate for myocyte connections was developed and ispresented in Figure 2(b) This basic unit accounts for theconnection of a total of 32 cardiac myocytes with differentshapes and numbers of neighboring cells

In the previous model of Spach and collaborators [2627] the basic template consisted of 33 myocytes with

a discretization of 10 120583m times 10 120583m whereas in our templatewe have 32 myocytes with a discretization of 8 120583m times 8 120583mThese small modifications to the original model proposedin [26 27] were adopted to better describe the values forcell length and width reported in the recent literature Themean and SD (standard deviation) values for cell length andwidth are 1209 plusmn 278 120583m and 183 plusmn 35 120583m respectivelyThese values are close to those reported in the literature[28] (length = 140 120583m and width = 19120583m) [29] (length =134 120583m and width = 18 120583m) and [30] (length = 100 120583m andwidth = 1732 120583m) On average each cell connects to theother 6 neighboring myocytes Our two-dimensional modelconsiders a homogeneous depth 119889 = 10 120583m [26 27]

This basic unit was created in such a way that it allows thegeneration of larger tissue preparations via the connectionsof multiples instances of it Figure 2(a) presents how this canbe achieved

Figure 2(c) presents an example of how the connectionsbetween different myocytes can be arranged The code wasdeveloped in a flexible way so that it allows the user to setup for each discretized volume Vol

119894119895(with area = ℎ times ℎ)

conductivity or conductance values for the north (120590119909119894119895+12

)south (120590

119909119894119895minus12) west (120590

119909119894minus12119895) and east (120590

119909119894+12119895) volume

faces These can be any nonnegative values In this workwe set the discretization ℎ to 8120583m In addition based onthe work of Spach and collaborators [26 27] we choseonly 5 possible types of connections between neighboringvolumes that are membrane (120590

119898= 00) cytoplasm (120590

119888=

04 120583S120583m) gap junction plicate (119866119901= 05 120583S) intriplicate


(119866119894= 033 120583S) and combined plicate (119866

119888= 0062 120583S) where

we use 120590 for conductivity and 119866 for conductance For thesimulations presented in this work the distribution of thedifferent gap junctions within the 32 myocytes was notrandomly generated Instead the gap junction distributionof the basic template unit was manually chosen to reproducethe distribution presented before in [26 27] With thissetup and conductivity values we found that conductionvelocity along the fibers was around 410 120583mms (LP) and was130 120583mms transversal to fiber direction (TP) This results ina ratio LPTP of 032 which is close to the conduction ratioreported in [26] It is worth mentioning that the values oflongitudinal and transversal conduction velocities observedin experiments vary significantly (longitudinal conductionvelocity is between 30 and 70 cms under normal or controlconditions) Indeed this large variation of measurements aredue tomany factors such as the experimental technique usedtemperature pacing protocol species and heart region Ourvalue for LP conduction velocity (41 cms) is within the rangeof those reported in the literature rat left atria [31] (42 plusmn17 cms) dog left ventricle [32] (63 plusmn 10 cms) mouse left andright ventricle [33] (35 plusmn 3 cms) and guinea pig left and rightventricle [34] (52 plusmn 2 cms)

212 The Heterogeneous Monodomain Model Action poten-tials propagate through the cardiac tissue because the intra-cellular space of cardiac cells is electrically coupled by gapjunctions In this work we do not consider the effects ofthe extracellular matrix Therefore the phenomenon can bedescribedmathematically by a reaction-diffusion type partialdifferential equation (PDE) calledmonodomainmodel givenby

120573119862119898

120597119881 (119909 119910 119905)

120597119905+ 120573119868ion (119881 (119909 119910 119905) 120578 (119909 119910 119905))

= nabla sdot (120590 (119909 119910) nabla119881 (119909 119910 119905)) + 119868stim (119909 119910 119905)

120597120578 (119909 119910 119905)

120597119905= f (119881 (119909 119910 119905) 120578 (119909 119910 119905))

(1)

where 119881 is the variable of interest and represents the trans-membrane voltage that is the difference between intracellu-lar to extracellular potential 120578 is a vector of state variableswhich also influence the generation and propagation of theelectric wave and usually includes the intracellular concen-tration of different ions (K+ Na+ Ca2+) and the permeabilityof different membrane ion channels 120573 is the surface-volumeratio of heart cells 119862

119898is the membrane capacitance 119868ion the

total ionic current which is a function of 119881 and a vectorof state variables 120578 119868stim is the current due to an externalstimulus 120590 is the monodomain conductivity tensor Weassume that the boundary of the tissue is isolated that is no-flux boundary conditions were imposed (n sdot120590nabla119881 = 0 on 120597Ω)

In this work the Bondarenko et al model [35] thatdescribes the electrical activity of left ventricular cells ofmice was considered to simulate the kinetics of 119868ion in (1)The Bondarenko et al model (BDK) was the first modelpresented for mouse ventricular myocytes [35] The ioniccurrent term 119868ion in this model consists of the sum of 15

transmembrane currents In short Bondarenkorsquos model isbased on an ordinary differential equation (ODE) with 41differential variables that control ionic currents and cellularhomeostasis In this model most of the ion channels arerepresented by Markov chains (MCs)

213 Numerical Discretization in Space and Time The finitevolume method (FVM) is a mathematical method used toobtain a discrete version of partial differential equationsThismethod is suitable for numerical simulations of various typesof conservation laws (elliptical parabolic or hyperbolic) [36]Like the finite element method (FEM) the FVM can be usedin several types of geometry using structured or unstructuredmeshes and generates robust numerical schemes

The development of the method is intrinsically linked tothe concept of flow between regions or adjacent volumesthat is it is based on the numerical calculation of net fluxesinto or out of a control volume For some isotropic problemsdiscretized with regular spatial meshes the discretizationobtained with the FVM is very similar to the one obtainedwith the standard finite difference method (FDM) Detailedinformation about the FVM applied to the solution ofmonodomain can be found in [37 38]

The reaction and diffusion parts of the monodomainequations were split by employing the Godunov operatorsplitting [39] Therefore each time step involves the solutionof two different problems a nonlinear system of ODEs

120597119881

120597119905=

1119862119898

[minus119868ion (119881 120578) + 119868stim]

120597120578

120597119905= f (119881 120578)

(2)

and a parabolic PDE

120573(119862119898

120597119881

120597119905) = nabla sdot (120590nabla119881) (3)

For the discretization of the nonlinear system of ODEswe note that its stiffness demands very small time steps Forsimple models based on Hodgkin-Huxley formulation thisproblem is normally overcomeby using theRush-Larsen (RL)method [40] However for the most modern and complexmodels that are highly based on MCs the RL methodseems to be ineffective in terms of allowing larger timesteps during the numerical integration For the case of theBondarenko et al model we tested both methods Euler andRL and both demanded the same time step Δ119905

119900= 00001ms

due to stability issues Since the RLmethod is more expensiveper time step than the Euler method in this work we usedthe simple explicit Euler method for the discretization of thenonlinear ODEs

Since the spatial discretization of our model ℎ isextremely small the CFL [41] condition that assures numeri-cal stability is very restrictiveTherefore for the PDE we usedthe unconditionally stable implicit Euler scheme In additionthis allowed us to use a longer time step for the numericalsolution of the PDE (Δ119905

119901gt Δ119905

119900) The time derivative


0

50

100

150

y(120583

m)

0 100 200 300 400 500 600

x (120583m)

(a)

0

50

100

150

y(120583

m)

0 100 200 300 400 500 600

x (120583m)

(b)

Figure 3 Elementary tissue unit in the microscopic (a) and the discrete (b) model The same color code of each cell is employed in bothmeshes Lines in the discrete model represent the connection between the linked cells

presented in (3) which operates on 119881 is approximated by afirst order implicit Euler scheme

The diffusion term in (3) must be discretized in spaceFor the space discretizationwe considered a two-dimensionaluniform mesh consisting of regular quadrilaterals (calledldquovolumesrdquo) Located in the center of each volume is a nodeThe quantity of interest 119881

119894119895is associated with each node of

the mesh (119894ℎ 119895ℎ) where ℎ is the spatial discretizationAfter defining the mesh geometry and dividing the

domain in control volumes the specific equations of the FVMcan be presented Equation (3) can be integrated spatiallyover an individual volume Vol

119894119895of size ℎ2119889 Substituting

the discretizations of the equations and decomposing theoperators as described by (2) and (3) yield

119862119898

119881lowast

119894119895minus 119881119899

119894119895

Δ119905119900

= minus 119868ion (119881119899

119894119895 120578119899)

120578119899+1

minus 120578119899

Δ119905119900

= f (120578119899 119881119899 119905)

(4)

(120590119894+12119895 +120590119894minus12119895 +120590119894119895+12 +120590119894119895minus12 +120572)119881

119899+1119894119895

minus120590119894119895minus12119881

119899+1119894119895minus1 minus120590119894+12119895119881

119899+1119894+1119895 minus120590119894119895+12119881

119899+1119894119895+1

minus120590119894minus12119895119881

119899+1119894minus1119895 = 120572119881

lowast

119894119895

(5)

where 120572 = (120573119862119898ℎ2)Δ119905119901 119899 is the current step lowast is an

intermediate step and 119899+1 is the next time step In addition 120590can stand for any of the gap junction conductance (119866

119901119866119894119866119888)

divided by the depth 119889 or for any conductivity value (120590119888 120590119898)

defined for each volume face as described in Section 211Thisdefines the equations for each finite volume Vol

119894119895 First we

solve the nonlinear system of ODEs associated with (4) 119873119900

times until we have 119873119900Δ119905119900= Δ119905119901 then we solve the linear

system associated with (5) to advance time by Δ119905119901

22 An Equivalent Discrete Model of Mouse Ventricular Tissue

221 A Discrete Model Recent studies on discrete or dis-continuous nature of propagation of the action potentialhave avoided the computational challenges that arise frommicroscopic models through the development and use ofdiscrete models where each cardiac myocyte is representedby a single point (cell) contact with neighboring myocytes

via discrete conductivities [24 42] This description hasenabled the study of the effects that random distributionsof conductivities have on conduction velocity and on theformation of patterns of reentry in cardiac tissue

Discrete models were introduced by Keener in [43] todescribe the electrical propagation in a 1D cable with nc cellsconnected to the case of low coupling of gap junctions In thismodel cells are considered isopotentials Recently in a workof Costa and Dos Santos [44] a comparison between discreteand microscopic models of a 1D cable of cells was presented

In this work an equivalent discrete model to the 2Dmicroscopic model presented in Section 21 was developedFor this using the cardiac microstructure presented inSection 21 and identifying the neighbors (links) of each cellthe conductance between each link was calculated and a newequivalent discrete model was generated

In the microscopic model we had a uniform two-dimensional mesh consisting of regular quadrilaterals (calledvolumes) For the discrete model each myocyte is repre-sented by a single node or cell of a network and each cell hasdifferent geometry (Vol

119894= 119860119894lowast 119889) and a different number of

neighboring cells (nn119894) Figure 3 shows the resulting discrete

topology for a basic unit of our microscopic meshThe equations that govern the propagation of action

potentials in the cardiac tissue are similar to those used inthe microscopic model presented in Section 21

Vol119894120573(119862119898

120597119881119894

120597119905+ 119868ion (119881119894 120578119894))

= (

nn119894sum

119895=1119866119894119895lowast (119881119895minus119881119894))+ 119868

stim119894

120597120578119894

120597119905= f (119881

119894 120578119894)

(6)

for each node 119894 of the discrete topology where 119866119894119895

is theequivalent conductance between cell 119894 and cell 119895

The time discretization is done in the same way as inthe microscopic model (Section 21) As before the reactionand diffusion parts of the monodomain equations were splitby employing the Godunov operator splitting [39] There-fore each time step involves the solution of two different


problems a nonlinear system of ODEs and a coupled linearsystem for the discrete diffusion equation

119862119898

119881lowast

119894minus 119881119899

119894

Δ119905119900

= minus 119868ion (119881119899

119894 120578119899)

120578119899+1

minus 120578119899

Δ119905119900

= f (120578119899 119881119899 119905) (7)

(

nn119894sum

119895=1119866119894119895+120572)119881

119899+1119894

minus

nn119894sum

119895=1(119866119894119895119881119899+1119895

) = 120572119881lowast

119894 (8)

where 120572 = (120573119862119898119860119894119889)Δ119905

119901 119899 is the current step lowast is an

intermediate step and 119899+ 1 is the next time stepThis definesthe equations for each cell 119894

222 Calculation of the Conductance between Neighbor-ing Cells To close the description of the discrete modelpresented in the previous section we need to define theconductance values of the connections that is we need thevalues for 119866

119894119895in (8)

This is done by applying the concept of combinationof resistors in parallel to the microstructure defined inSection 211 for each pair of neighboring cells From theassociation of resistors we have

1119877119894119895

=

ng

sum

119896=1(

1119877119894119895119896

) (9)

where 119877119894119895

is the equivalent resistance and ng is the amountof gap junctions that connects cell 119894 to cell 119895 The term 119877

119894119895119896is

calculated as a combination of resistors in series

119877119894119895119896

= dist119894119895119896

lowast1

(120590ℎ)+

1119866119896

(10)

where dist119894119895119896

is the amount of volumes connected by cyto-plasm that the current need to cross to go from cell 119894 to cell119895 passing through gap junction 119896 which is calculated by theequation

dist119894119895119896

= (10038161003816100381610038161003816pr119894119909minus gp119894119896119909

10038161003816100381610038161003816+10038161003816100381610038161003816pr119894119910minus gp119894119896119910

10038161003816100381610038161003816)

+ (10038161003816100381610038161003816pr119895119909minus gp119895119896119909

10038161003816100381610038161003816+10038161003816100381610038161003816pr119895119910minus gp119895119896119910

10038161003816100381610038161003816)

(11)

where pr119894is a two-dimensional coordinate that represents

the volume of the reference point of the cell 119894 and gp119894119896

is atwo-dimensional coordinate that represents the volume thatcontains the gap junction connection 119896 in cell 119894 It shouldbe noted that these coordinates are index integers of thevolumes in the mesh so they do not have unit length Thuseach volume connected by the cytoplasm or gap junctionrepresents a resistor

In a simpleway the termdistlowast(1(120590ℎ)) represents the totalcytoplasmic resistance that the current faces to go from cell 119894to cell 119895 or vice versaThe term 1119866

119896represents the resistance

of the gap junctionThe conductivity values cytoplasm 120590 andconductance 119866 for each type of gap junction are the same asdefined in Section 211

120601 = 015

Figure 4 Process for the generation of one hundred of differenttissue models with the same fraction 120601 of fibrosis

Thuswe can now calculate the conductance119866119894119895that is the

inverse of the resistance

119866119894119895=

1119877119894119895

(12)

23 Modeling Thousands of Fibrotic Regions The sophisti-cated method presented in [15] to model fibrosis involvesa huge pipeline based on histology acquisition image pro-cessing segmentation and mesh generation The purposeof our work is to assess the probability of a certain fibrotictissue to become an ectopic pacemaker and to study how thisdepends on the fraction of fibrosis 120601 To achieve this goal itis necessary to perform thousands of simulations of differentmicrofibrosismodelsTherefore themethodpresented in [15]does not fit to our purpose We could however proceed withour study by using a similar method as presented in [13]

For a given value of 120601 we randomly generate a microfi-brosis tissue by transforming some of the myocytes of ourmodel in fibrosis that is nonconducting material This wasperformed by removing all the gap junctions of the chosenmyocytes so that any propagation into or out of this myocyteis blocked One hundred of realizations of different microfi-brosis but with the same percentage of nonactive cells 120601were simulated This process is represented by Figure 4 Thevalue of 120601 was varied from 40 to 55 in steps of 1 Aftereach microfibrosis model is created using the microscopicmodel the equivalent discrete model is generated followingthe procedure explained in Section 22 For all the resultspresented in the next section the modeled cardiac tissues haddimensions of 1 cm times 1 cm

3 Results

We employ numerical simulations to compare the two mod-els described in Section 2 under two different situationscompletely homogeneous distribution of healthy cells and


Table 1 Average of execution times (in seconds) for simulations ofa 10 cm times 10 cm tissue for 10ms Each case (line of table) was runthree times

Implementation model Computingcores

Totalexecutiontime (s)

Sequential-microscopic 1 546507Multi-GPU-microscopic 64 + 16GPUs 1302Multi-GPU-discrete 64 + 16GPUs 65

the randomly inclusion of nonconducting cells First wecompare the dynamics of wave propagation and functionalreentry in both models under homogeneous conditionsThen we employ both models to study the problem ofanatomical reentry in heterogeneous cardiac tissue

The main advantage of the discrete model is the fastexecution times when compared to those of the microscopicmodel As we will present in the next sections this will beof extreme importance since we will present the simulationsof over three thousands of different microfibrosis modelsTable 1 compares the execution times of the parallel imple-mentations of the microscopic versus the discrete modelsThe simulations were executed on a parallel environmentcomposed of 64 CPU cores and 16 GPUs as previouslydescribed in [25] The parallel implementation and envi-ronment were able to reduce the execution time of themicroscopic model from more than 6 days to 22 minutes Inaddition the parallel implementation of the discrete modeltakes only near one minute for the execution of the samesimulation

31 Wave Propagation in HomogeneousModels of Cardiac Tis-sue In order to compare bothmodels we performnumericalsimulations under homogeneous conditions and compare thespatiotemporal dynamics of the action potential waves

311 TravelingWaves inHomogeneousModels The twomod-els described in the previous section gives rise to equivalentwaves of action potential under the same conditions Fora quantitative comparison see Figure 5 An initial centeredperturbation produces a concentric wave which propagatesto the border of the medium Its elliptic shape is due to theanisotropy of the tissue which allows the wave to propagatefaster in the longitudinal direction (to the left or to theright see Figure 5) than in the vertical direction (bottom-up) The good agreement in the qualitative comparisonbetween the snapshots at the same time shown at the topof Figure 5 is confirmed by the quantitative measurement ofthe action potential in a single cell of the tissue see bottomof Figure 5 Under homogeneous conditions both modelsproduce quantitatively the same types of waves because theconnectivities among cells are equivalent (see Section 222)and therefore velocity of propagation is the same

312 Functional Reentry in Homogeneous Models The activ-ity of the membrane action potential wave can remain inthe tissue if there is a reentry This can be produced by

the S1-S2 protocol which consists in the reexcitation of thetissue (ectopic beat) after the pass of a normal traveling wave[25] In Figure 6 an example of rotating spiral wave of actionpotential in the microscopic model of the tissue is shownThere is a repeated reexcitation of the tissue due to the reentryof the spiral wave and its rotation around a free end

The action potential measured in a single cell (markedwith a black square in the snapshots of the simulations)is shown in the bottom part of Figure 6 and comparedwith the equivalent simulation with the discrete model Theagreement is remarkable taking into account the differentlevels of approximation used by the two models Thuscomplex dynamics is well captured by the discrete model incomparison with the microscopic model

32 Wave Propagation in Heterogeneous Models of CardiacTissue Further comparisons between both models for het-erogeneous conditions at the tissue level are shown in thenext sections Following we remove a fraction of cells thatis we randomly choose some cells which are then completelyisolated from the rest of the tissue and cannot be excitedanymore Such cells represent the effect of the different typesof heterogeneities in the tissue damaged cells damagedconnections fibrosis extracellular matrix or dead cells Wedefine as control parameter the fraction of isolated cells120601 that is 120601 = 0 corresponds to the homogeneous limitdiscussed in the previous section and 120601 = 1 to a completelyinert piece of tissue

321 Anatomical Reentry in the Discrete Model In the dis-crete model for small fraction 120601 propagation is possible anda wave can travel with a reduced velocity from one side of thetissue to the other For large value of 120601 there are no pathwaysfor the wave to propagate and the excitation disappearswithout the activation of the tissue For intermediate valuesof 120601 the propagation looks fractionated but it is usuallystill coherent and travels to the other side of the tissueHowever depending on the random realization of the processof isolation of cells waves can be broken and it producesthe appearance of reentry into the already excited tissuesee Figure 7 A wave of action potential propagates fromthe left to the right in a tissue with a large fraction ofremoved cells following the diverse open pathways in thetissue The propagation is highly fractionated but the wavearrives to the end of the tissue However after the left-to-right propagation a piece of the wave reenters again into thetissue already excited and produces the formation of repeatedreentry equivalent to a spiral wave The dynamics is stableand the reentry persists for large times although it couldeventually disappear depending on the random realization[24] Similar patterns of microscopic reentry can be foundalso by initiating the propagation from the bottom of thetissue see Figure 8

We have done a systematic study for the evaluation ofthe optimal fraction 120601 where the probability of reentry ismaximal We run 100 numerical simulations with differentrandom realization for each value of the fraction 120601 The valueof 120601was varied from 40 to 55 in steps of 1This allowed


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Figure 7 Reentry of action potential generated by the heterogeneous structure of the tissue in the discrete model (a) Generation of thetraveling wave in the left part of the tissue (119905 = 2ms) (b)Wave rapidly propagates to the right following the direction of the fibers (119905 = 44ms)(c) A piece of thewave reenters into the tissue (119905 = 82ms) (d)Newwaves are generated and eventually a repeated reentry is formedmimickinga spiral wave (119905 = 122ms)

us to estimate for each value of 120601 the probability of reentrythat is the ratio of simulations with reentry to the totalnumber of simulations

The results of these simulations are shown in Figure 9A reentry was identified whenever there was still activity(depolarization) at 120ms after the initial stimulus for thecase of longitudinal propagation or at 300ms for the caseof transversal propagation Therefore here we did not makeany difference between sustained or nonsustained reentriesThe values of 120 and 300ms were chosen by checking thata plane wave reaches the end of the tissue for the case of120601 = 045 after 60ms (240ms) for longitudinal (transverse)propagation Therefore in both cases we checked whetherthere was still activity 60ms after the wave was alreadysupposed to be extinguished

Figure 9 shows a systematic difference of the resultsdepending on the directions of propagation of the wavesIn the LP direction the probability of reentry concentratesaround 120601 = 048 however when waves propagate trans-versely to the fiber direction (TP) the domain of reentryis broader and the distribution is flatter and above 10 for

120601 between 042 and 048 Nevertheless in both cases theprobability of reentry goes to zero shortly after 120601 sim 05 seeFigure 9

322 Anatomical Reentry in the Microscopic Model Thenumerical simulations using the microscopic model takemuch longer execution times than those based on the discretemodel The realization of good statistics with such modelbecomes a very hard task For this reason we have chosenthe topologies from some selected simulations in the discretemodel and implemented them in the microscopic model tocompare the results of both models

We first chose 30 topologies that did not generate reentryin the discrete model but those were under values of thefraction 120601 of high probability of reentry We observed thatnone of these simulations using the fine microscopic modelgave rise to the appearance of any reentry Therefore in thiscase there was a 100 of match between the two models

We have also performed 30 (15 TP plus 15 LP simulations)realizations selecting topologies which produced reentry inthe discrete model For the LP case 15 simulations were


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Figure 8 Reentry of action potential generated by the heterogeneous structure of the tissue in the discrete model (a) Generation of thetraveling wave in the bottom part of the tissue (119905 = 10ms) (b) Wave rapidly propagates to the upper part of the tissue transversely to thefibers direction (119905 = 80ms) (c) A piece of the wave reenters into the tissue (119905 = 150ms) (d) New waves are generated and eventually arepeated reentry is formed mimicking a spiral wave (119905 = 220ms)

executed five for each value of 120601 = 045 048 051 For theTP case 15 simulations were also executed five for each valueof 120601 = 042 045 048 Therefore for both cases we tookthe value of 120601 that maximizes the probability of reentry andtwo values of 120601 around this value but with low probability ofreentry see Figure 9

While 19 realizations out of 30 gave rise to reentry 11realizations did not show any reentry It produces a 63of agreement between the two models in the prediction ofreentries once the reentry appears in the discrete modelThese results are shown in Figure 10The five cases of reentryin the discrete model for each value of 120601 reduce to 2ndash4reentries in the microscopic model

An example for a case of agreement between the twomodels is shown in Figure 11 The comparison betweenthe microscopic ((a)-(c)) and the discrete ((b)-(d)) modelproduces a qualitative agreement in the propagation of thewave and in the localization of the reentry although thereentrant waves are different in shape

On the other hand some realizations using the sametopologies for bothmodels gave rise to different final dynam-ics Figure 12 shows the comparison between two simulations

with the two models While we observe a reentry in thediscrete model no reentry is observed in the simulation withthe microscopic model The wave initially propagates in thetwo models following a very similar dynamics howeverlater on the simulation propagation stops in the microscopicmodel whereas in the discrete model it persists as a reentrywave

To further investigate these differences between micro-scopic and discrete results we have performed another set ofsimulationsWe have simulated two different fibrotic regionsboth with 120601 = 048 with longitudinal stimulation (left-right)for the duration of 400ms and another two different fibroticregions both with 120601 = 045 with transversal stimulation(bottom-up) for the duration of 800ms These four newscenarios were simulated with both discrete and microscopicmodels that is a total of 8 simulations Tables 2 and 3 presentthe results of this comparison For each cell (discrete model)or discretized volume (microscopic model) we classifiedthe electrical activity observed during the simulations asNA (no activity) A (a single AP no reentry) S (sustainedreentry multiple APs until the end of the simulation) or NS(nonsustained reentry multiple APs but activity stops dies


Table 2 Left-right 120601 = 048 Each cell (discrete model) or discretized volume (microscopic model) was classified as NA (no activity) A (asingle AP no reentry) S (sustained reentry multiple APs until the end of the simulation) or NS (nonsustained reentry multiple APs butactivity stops dies out before the end of the simulation)The table presents the percentage of area of the tissue for each category In additionfor the category NS we present the time the reentry dies out or stops

Type Run 1 Run 2Microscopic Discrete Microscopic Discrete

NA 541 517 55 535OA 183 162 156 144S 192 321 173 209

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at 223ms)112 (stopsat 217ms)

Table 3 Top-down 120601 = 045 Each cell (discrete model) or discretized volume (microscopic model) was classified as NA (no activity) A(a single AP no reentry) S (sustained reentry multiple APs until the end of the simulation) or NS (nonsustained reentry multiple APs butactivity stops dies out before the end of the simulation)The table presents the percentage of area of the tissue for each category In additionfor the category NS we present the time the reentry dies out or stops


NA 609 598 627 611OA 143 138 212 206S 176 199 71 99NS 72 (stops at 391ms) 65 (stops at 389ms) 9 (stops at 368ms) 84 (stops at 365ms)

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Figure 9 Dependence of the probability of reentry 119875(120601) of actionpotential generated by the heterogeneous structure of the tissuein the discrete model on the fraction of nonconnected cells 120601Comparison between propagation along the fibers (left-right) andperpendicularly to the fibers (top-down)

out before the end of the simulation) In addition for thecase ofNS activity we have alsomeasured the time the reentrydied out Table 2 (left-right simulations) andTable 3 (bottom-up simulations) present the percentage of tissue classified asNA A S and NS We observe a very good match betweenthe results of the discrete and the microscopic models forthree out of the four cases studied (both in terms of theclassification and the duration of the reentry for the NS case)

However for one particular case (Run 1 Table 2) the discretemodel overestimates the area of the tissue that sustainsreentry (near one third of the tissue) in comparison to thearea where reentry is sustained using the microscopic model(only one fifth of the tissue)

4 Discussion

41 Bridging the Gap between Ectopic Pacemakers Formationand CFAEs in Microfibrosis Our simulation results presentthe generation of reentry inside fibrotic tissues This kind ofreentry restricted to microfibrosis regions can result in theformation of ectopic pacemakers that is regions that willgenerate a series of ectopic stimulus at a fast pacing rate Inturn such activity has been related to trigger fibrillation bothin the atria and in the ventricles in clinical and animal studies[5 6 10]

In addition our results quantitatively described how thepercentage of fibrosis (120601) influences the probability of aregion to become an ectopic pacemaker Reentry activityinside a fibrotic region was identified in cardiac tissues witha percentage of fibrosis (or nonconductingmaterial) between41 and 52 see Figure 9 It is interesting to mention that[45] reports that in postmortem atrial tissues from patientswho died of cardiovascular causes the extent of fibrosis wasstrongly correlated with the history of atrial fibrilation (AF)The mentioned work highlights this correlation presentingthree histologies (1) with a fibrosis extent of 51 of thetissue from a patient with a history of permanent AF (2)with a fibrosis extent of 14 from a patient with a historyof paroxysmal AF (3) with a fibrosis extent of 5 froma patient with no history of AF These numbers are in


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Figure 11 Reentry of action potential generated by the heterogeneous structure of the tissue in the microscopic ((a)-(c)) and the discrete((b)-(d)) model ((a)-(b)) Wave rapidly propagates to the right following the direction of the fibers in the microscopic (a) and in the discretemodel (b) for 119905 = 50ms ((c)-(d)) A piece of the wave reenters into the tissue in the microscopic (c) and in the discrete model (d) for119905 = 160ms

agreement with our findings Our results suggests that withless than 40 of fibrosis the conduction velocity of the prop-agating wave can be substantially reduced but without thegeneration of reentry inside the fibrotic region This patterncould potentially generate a single ectopic beat enough forparoxysmal AF For permanent or sustained AF the work of

[11] suggest the need for an ectopic pacemaker And indeedthe fibrosis extent of 51 reported in [45] and associated withpermanent AF is within our predicted values of 120601 (between41 and 52) that has led to the generation of reentry insidethe fibrotic region that is to the formation of an ectopicpacemaker


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Figure 12 Reentry ((b)-(d)) and nonreentry ((a)-(c)) of action potential generated by the heterogeneous structure of the tissue in themicroscopic ((a)-(c)) and the discrete ((b)-(d)) model ((a)-(b)) Wave rapidly propagates to the right following the direction of the fibersin the microscopic (a) and in the discrete model (b) for 119905 = 40ms The wave does not reenter into the tissue in the microscopic model (c)for 119905 = 100ms A piece of the wave reenters into the tissue in the discrete model (d) for 119905 = 100ms

Finally the computational models used here are verysimilar to those used in [13] that also simulated microfibrosiswith the focus on the study of CFAEs [13] In [13] it wasshown that decoupling of myocytes or increase of fibrosisextent was associated with complex pattern of propagationdiscrete propagation and CFAEs Since we have studiedthe generation of ectopic pacemakers with a very similarformalism as used in [13] we have a clear theoretical evidencethat brings together ectopic pacemakers and CFAEs for thecase of propagation in fibrotic tissues The results presentedhere and the new models developed in this work may help inthe translational and extremely complex task of relating basicelectrophysiological and structural features of cardiac tissueto clinical and relevant procedures such as catheter ablationof atrial fibrillation guided by CFAE mapping [19]

42 Extending the Percolation Threshold for Anisotropic Car-diac Tissue The maximum in the probability of reentry hasbeen previously associated with the percolation thresholdof the network or topology of the tissue [24] Here asalready mentioned the average number of first neighbors of

the network (see Figure 3(b)) is around 6 and the topologicalstructure of the cell network is similar to an hexagonalnetwork whose percolation threshold is 120601 = 05 Howeverthe network used here is not exactly hexagonal and representsan anisotropic tissue The anisotropic conduction does notonly elongates the symmetric view in one direction but alsomodifies the distribution of the first neighbors As a resultthe percolation threshold changes In addition as suggestedby the results of Figure 9 it is likely that the percolationthreshold not only depends on the topology of the networkor tissue but also on the direction of the propagating waveTo verify this hypothesis we have numerically calculated theprobability of connection between two opposing sides of thetissue 119862(120601) using the same method as proposed in [24] forthe two different propagating directions TP and LP Figure 13presents the results for the probability of connection 119862(120601)for the two cases TP and LP From these two distributions wecan obtain that in the LP direction the percolation thresholdis 120601 = 0527 whereas for the TP direction it is 120601 = 047

Therefore we conclude that the percolation threshold foranisotropic cardiac tissue depends not only on the topology


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of the connection between the myocytes but also on thedirection of propagation with respect to the cardiac fiberorientation Nevertheless as presented in Figure 9 the rangeof values for percolation threshold a pure topological oranatomical metric is still very important Even in the case oftissue anisotropy this range of values for percolation thresh-old (047ndash052) marked the percentage of fibrotic tissue 120601needed for a tissue to have a high probability to generatereentry and become an ectopic pacemaker

In addition similar to Figure 9 Figures 14 and 15 presenthow the conduction velocity (CV) depends on 120601 for the caseof LP and TP propagation directions respectivelyThe figurespresent the results of average and STD of the calculatedvalues of CV We can observe the existence of extremely slowpropagating waves for 120601 near the percolation threshold

43 A New Discrete Model for Cardiac Tissue We developedand tested a new discrete model which is a reduced andsimplified version of a complexmicroscopicmodel To obtainthe same physiological conditions we carefully tuned theparameters of the connectivities As a result of the fitting

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traveling and spiral waves propagate with very similar patternand velocities in both models discrete and microscopicunder homogeneous conditions that is we have observeda quantitative match of the two models Therefore we canconclude that the new developed discrete model is a goodalternative for the microscopic one under homogeneousconditions since it offers simulations that are much faster interms of execution times than those using the microscopicmodel

We have also compared the microscopic and the discretemodels under heterogeneous conditions Both models haveshown the same qualitative behavior when the probability ofreentry was studied on fibrotic tissues However quantitativedifferences were also clearly present Out of the 60 simula-tions that we have chosen to compare the two models in 49the two models matched (in terms of reentry generation)that is 82 of agreement However within the cases wherereentries were observed using the discrete model 30 casesonly 19 also generated reentry when using the microscopicmodel which drops the matches to 63

Therefore our preliminary results suggest that the dis-cretemodel overestimates the probability of reentry if we takethe microscopic model as our gold standard We have takena close look at the simulations where the two models did notmatch Figure 16 presents an example of these comparisonsIn panels (a) (b) and (c) we observe a successful reentrypropagation (from right to left in the tissue) using thediscrete model In panels (d) (e) and (f) we observe thecorresponding situation using the microscopic model wherereentry dies out Comparing panels (a) and (d) we observethat current spreads outmore easily in themicroscopicmodel(panel (d)) than in the discrete model (panel (a)) This isdue to the continuous nature of the microscopic model Asa result current sink is larger in the microscopic model thanin the discrete model This classical source-sink mismatch[46] blocks the propagation of the reentry in the microscopicmodel On the other hand the lumped nature of the discretemodel seems to help propagation by reducing the sink area

Nevertheless we reinforce that even in these cases ofmicrofibrosis simulation we can claim that the results of thediscretemodelmatched qualitatively those of themicroscopicmodel with accuracy between 60 and 80 Since this studyinvolved the simulations of more than three thousands of


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Figure 16 Reentry in the microscopic model in comparison with the discrete model Panels (a) (119905 = 75ms) (b) (119905 = 80ms) and (c)(119905 = 90ms) successful reentry propagation (from right to left of the tissue) using the discrete model Panels (d) (119905 = 75ms) (e) (119905 = 80ms)and (f) (119905 = 90ms) reentry dies out using the microscopic model due to source-sink mismatch

different fibrotic tissues we note that these statistics wouldnot be possible to generate in a timely fashion without the aidof the new discrete model

44 Limitations and Future Work All the simulations shownin this study were carried out in a completely damaged areaFibrosis is typically localized in the heart and the damagedarea coexists with healthy surrounding In [24] two differentsimulations were performed one considered a completelydamaged area that is a fibrotic region as performed herein this work and another larger setup considered a fibroticregion surrounded by healthy tissue In [24] it was shownthat most of the reentries generated inside the fibrotic regionbut not all of them could propagate to the healthy tissueIn future studies we will use the same methods presentedhere to simulate the setup of a fibrotic region surrounded byhealthy tissue and quantify as performed here how featuresof fibrosis are related to the generation of ectopic pacemakers

We have fixed the size of the simulation area to 1 times1 cm2 However the probability to find an ectopic beat isproportional to the probability to have locally the adequateconditions for a source-link mismatch [46] Such probabilitydepends on the size of the system the larger the system thehigher the probability to obtain an ectopic beat In futurestudies on a systematic variation of the domain of integrationwill help to identify a minimum size for the generation ofreentry

Two different dynamics have been previously obtainedin [24] sustained and nonsustained reentries The last caseproduce a single ectopic beat without any repetition and

with a mild effect in the cardiac muscle The first one isassociates with ectopic pacemakers In this work we didnot quantitatively differentiate between sustained and non-sustained reentries In future works longer simulations willbe performed to better classify how many of the simulatedreentries are sustained and how many are not

We have employed the Bondarenkomodel [35] formouseventricle A natural next step in terms of translational scienceis to replace this model with a human atrial or ventricularmodel In addition other important electrophysiologicalchanges should be taken into account by the models inorder to simulate fibrosis In future works a more completedescription of fibrosis may include for example gap junctionremodeling as well as myocyte-fibroblast coupling [47]

Finally we want to note that we considered two dimen-sional slabs of tissue It is usually a strong simplification ofthe geometry of the tissue In the near future we will considerthree-dimensional simulations In this case the introductionof an additional dimension will probably change the topolog-ical conditions of the grid (percolation threshold will changefrom 05 to larger values) but not the phenomenologyaround the percolation threshold as it was anticipated in [24]

5 Conclusions

We developed and tested a new discrete model which isa reduced and simplified version of a detailed subcellularmicroscopic model that accounts for the microstructure ofthe cardiac tissue including distribution of cells and gap


junctions Our preliminary results suggest a very good agree-ment between the simulations of the new discrete model andthose obtained by the detailed and subcellular microscopicmodel The new discrete model was used for the simulationof fibrotic regions Our results quantitatively described howthe percentage of fibrosis (120601) influences the probability ofa region to become an ectopic pacemaker Reentry activityinside a fibrotic region was identified in cardiac tissueswith a percentage of fibrosis (or nonconducting material)between 41 and 52 In addition our results suggest thatthe probability of reentry inside a fibrotic region not onlydepends on 120601 but also on the direction of propagationwith respect to the cardiac fiber orientation that is tissueanisotropy also plays an important role

In summary our simulation results present the genera-tion of reentry inside fibrotic tissues Therefore we were ableto generate an ectopic pacemaker in computer simulations ofcardiac electrophysiology by assuming a single microstruc-tural or anatomical hypothesis for substrate the existenceof a mixture of excitable and nonexcitable regions The twodifferent phases (excitable and nonexcitable) form a mazeor labyrinth for wave propagation Propagation becomesfractioned and the topology of the maze allows the electricwave to reexcite the fibrotic tissue before the wave leaves itOur results can be combined with previous works that used avery similar computationalmodel to study CFAEsThereforewe present a clear theoretical evidence that brings togetherectopic pacemakers and complex fractionated electrogramsfor the case of propagation in fibrotic tissues thatmay becometriggers of fibrillation

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors acknowledge fruitful discussions with Pro-fessor Markus Bar The authors also acknowledge finan-cial support of the Federal University of Juiz de Fora forexchange visits CAPES CNPq FINEP FAPEMIG and theGrant 888810650022014-01 of the Brazilian program Sciencewithout borders Sergio Alonso acknowledges support byMINECO of Spain under the Ramon y Cajal program withthe Grant no RYC-2012-11265

References

[1] A Panfilov and P Hogeweg ldquoSpiral breakup in a modifiedFitzHugh-Nagumo modelrdquo Physics Letters A vol 176 no 5 pp295ndash299 1993

[2] A Karma ldquoSpiral breakup in model equations of action poten-tial propagation in cardiac tissuerdquo Physical Review Letters vol71 no 7 pp 1103ndash1106 1993

[3] M Bar and M Eiswirth ldquoTurbulence due to spiral breakup in acontinuous excitable mediumrdquo Physical Review E vol 48 no 3pp R1635ndashR1637 1993

[4] D E Krummen and S M Narayan ldquoMechanisms for theinitiation of human atrial fibrillationrdquoHeart Rhythm vol 6 no8 pp S12ndashS16 2009

[5] J E Finet D S Rosenbaum and J K Donahue ldquoInformationlearned from animal models of atrial fibrillationrdquo CardiologyClinics vol 27 no 1 pp 45ndash54 2009

[6] G Laurent G Moe X Hu et al ldquoExperimental studies of atrialfibrillation a comparison of two pacing modelsrdquoThe AmericanJournal of PhysiologymdashHeart and Circulatory Physiology vol294 pp H1206ndashH1215 2008

[7] L K Bhatt K Nandakumar and S L Bodhankar ldquoExperi-mental animal models to induce cardiac arrhythmiasrdquo IndianJournal of Pharmacology vol 37 no 6 pp 348ndash357 2005

[8] M R Franz ldquoStretch pulses and cardiac arrhythmias in rathearts is it a good modelrdquo Experimental Physiology vol 94 no10 pp 1052ndash1053 2009

[9] W R Mills N Mal F Forudi Z B Popovic M S Penn andK R Laurita ldquoOptical mapping of late myocardial infarctionin ratsrdquo American Journal of PhysiologymdashHeart and CirculatoryPhysiology vol 290 no 3 pp H1298ndashH1306 2006

[10] J N Weiss A Karma Y Shiferaw P-S Chen A Garfinkel andZ Qu ldquoFrom pulsus to pulseless the saga of cardiac alternansrdquoCirculation Research vol 98 no 10 pp 1244ndash1253 2006

[11] C Tobon C A Ruiz-Villa E Heidenreich L Romero FHornero and J Saiz ldquoA three-dimensional human atrial modelwith fiber orientation Electrograms and arrhythmic activationpatterns relationshiprdquo PLoS ONE vol 8 no 2 Article IDe50883 2013

[12] N Voigt and D Dobrev ldquoCellular and molecular correlates ofectopic activity in patients with atrial fibrillationrdquo Europace vol14 no 5 pp v97ndashv105 2012

[13] V Jacquemet and C S Henriquez ldquoGenesis of complex frac-tionated atrial electrograms in zones of slow conduction acomputer model of microfibrosisrdquo Heart Rhythm vol 6 no 6pp 803ndash810 2009

[14] A M Kim and J E Olgin ldquoMicrofibrosis and complex frac-tionated atrial electrogramsrdquo Heart Rhythm vol 6 no 6 pp811ndash812 2009

[15] F O Campos T Wiener A J Prassl et al ldquoElectroanatom-ical characterization of atrial microfibrosis in a histologicallydetailed computer modelrdquo IEEE Transactions on BiomedicalEngineering vol 60 no 8 pp 2339ndash2349 2013

[16] N Ono H Hayashi A Kawase et al ldquoSpontaneous atrial fibril-lation initiated by triggered activity near the pulmonary veinsin aged rats subjected to glycolytic inhibitionrdquo The AmericanJournal of PhysiologymdashHeart and Circulatory Physiology vol292 no 1 pp H639ndashH648 2007

[17] J Kalifa K Tanaka A V Zaitsev et al ldquoMechanisms of wavefractionation at boundaries of high-frequency excitation in theposterior left atrium of the isolated sheep heart during atrialfibrillationrdquo Circulation vol 113 no 5 pp 626ndash633 2006

[18] K Nademanee J McKenzie E Kosar et al ldquoA new approachfor catheter ablation of atrial fibrillation mapping of theelectrophysiologic substraterdquo Journal of the American College ofCardiology vol 43 no 11 pp 2044ndash2053 2004

[19] K Nademanee E Lockwood N Oketani and B GidneyldquoCatheter ablation of atrial fibrillation guided by complexfractionated atrial electrogram mapping of atrial fibrillationsubstraterdquo Journal of Cardiology vol 55 no 1 pp 1ndash12 2010

[20] K Kumagai T Sakamoto K Nakamura et al ldquoCombined dom-inant frequency and complex fractionated atrial electrogram


ablation after circumferential pulmonary vein isolation of atrialfibrillationrdquo Journal of Cardiovascular Electrophysiology vol 24no 9 pp 975ndash983 2013

[21] J Chen Y Lin L Chen et al ldquoA decade of complex fractionatedelectrograms catheter-based ablation for atrial fibrillation liter-ature analysis meta-analysis and systematic reviewrdquo IJC Heartamp Vessels vol 4 pp 63ndash72 2014

[22] S Magnani D Muser W Chik and P Santangeli ldquoAdjunctablation strategies for persistent atrial fibrillationmdashbeyondpulmonary vein isolationrdquo Journal ofThoracic Disease vol 7 no2 pp 178ndash184 2015

[23] G Bub A Shrier and L Glass ldquoSpiral wave generation inheterogeneous excitable mediardquo Physical Review Letters vol 88no 5 pp 581011ndash581014 2002

[24] S Alonso and M Bar ldquoReentry near the percolation thresholdin a heterogeneous discrete model for cardiac tissuerdquo PhysicalReview Letters vol 110 no 15 Article ID 158101 2013

[25] B Gouvea de Barros R Sachetto Oliveira W Meira MLobosco and R Weber dos Santos ldquoSimulations of complexand microscopic models of cardiac electrophysiology poweredby Multi-GPU platformsrdquo Computational and MathematicalMethods inMedicine vol 2012 Article ID 824569 13 pages 2012

[26] M S Spach and J F Heidlage ldquoThe stochastic nature of cardiacpropagation at a microscopic level electrical description ofmyocardial architecture and its application to conductionrdquoCirculation Research vol 76 no 3 pp 366ndash380 1995

[27] M S Spach and R C Barr ldquoEffects of cardiac microstructureon propagating electrical waveformsrdquo Circulation Research vol86 no 2 pp e23ndashe28 2000

[28] A M Gerdes S E Kellerman J A Moore et al ldquoStructuralremodeling of cardiac myocytes in patients with ischemiccardiomyopathyrdquo Circulation vol 86 no 2 pp 426ndash430 1992

[29] R E Tracy and G E Sander ldquoHistologically measured car-diomyocyte hypertrophy correlates with body height as stronglyas with bodymass indexrdquoCardiology Research and Practice vol2011 Article ID 658958 9 pages 2011

[30] S F Roberts J G Stinstra and C S Henriquez ldquoEffect ofnonuniform interstitial space properties on impulse propaga-tion a discretemultidomainmodelrdquoBiophysical Journal vol 95no 8 pp 3724ndash3737 2008

[31] T-A Matsuyama H Tanaka T Adachi Y Jiang H Ishibashi-Ueda and T Takamatsu ldquoIntrinsic left atrial histoanatomyas the basis for reentrant excitation causing atrial fibrilla-tionflutter in ratsrdquo Heart Rhythm vol 10 no 9 pp 1342ndash13482013

[32] B Taccardi B B Punske E Macchi R S MacLeod and P RErshler ldquoEpicardial and intramural excitation during ventricu-lar pacing Effect ofmyocardial structurerdquoTheAmerican Journalof Physiology Heart and Circulatory Physiology vol 294 no 4pp H1753ndashH1766 2008

[33] M Stein T A B van Veen R N W Hauer J M T de Bakkerand H V M van Rijen ldquoA 50 reduction of excitability but notof intercellular coupling affects conduction velocity restitutionand activation delay in the mouse heartrdquo PLoS ONE vol 6 no6 Article ID e20310 2011

[34] R Veeraraghavan M E Salama and S Poelzing ldquoIntersti-tial volume modulates the conduction velocity-gap junctionrelationshiprdquo The American Journal of PhysiologymdashHeart andCirculatory Physiology vol 302 no 1 pp H278ndashH286 2012

[35] V E Bondarenko G P Szigeti G C L Bett S-J Kim and RL Rasmusson ldquoComputer model of action potential of mouse

ventricular myocytesrdquo American Journal of PhysiologymdashHeartand Circulatory Physiology vol 287 no 3 pp H1378ndashH14032004

[36] R Eymard T Gallouet and R Herbin ldquoFinite volume meth-odsrdquo in Handbook of Numerical Analysis vol 7 pp 713ndash1018North-Holland Amsterdam The Netherlands 2000

[37] D M Harrild and C S Henriquez ldquoA finite volume model ofcardiac propagationrdquo Annals of Biomedical Engineering vol 25no 2 pp 315ndash334 1997

[38] Y Coudiere C Pierre and R Turpault ldquoA 2d3d finite volumemethod used to solve the bidomain equations of electrocardi-ologyrdquo in Proceedings of the Conference on Scientific Computing(ALGORITMY rsquo09) pp 1ndash10 Podbanske Slovakia March 2009

[39] J Sundnes Computing the Electrical Activity in the HeartSpringer 2006

[40] S Rush and H Larsen ldquoA practical algorithm for solvingdynamic membrane equationsrdquo IEEE Transactions on Biomedi-cal Engineering vol 25 no 4 pp 389ndash392 1978

[41] J C Strikwerda Finite Difference Schemes and Partial Differen-tial Equations Society for Industrial Mathematics 2004

[42] S AlonsoM Bar andAV Panfilov ldquoEffects of reduced discretecoupling on filament tension in excitable mediardquo Chaos vol 21no 1 Article ID 013118 2011

[43] J P Keener ldquoThe effects of gap junctions on propagation inmyocardium a modified cable theoryrdquo Annals of the New YorkAcademy of Sciences vol 591 pp 257ndash277 1990

[44] C M Costa and R W Dos Santos ldquoLimitations of thehomogenized cardiac Monodomain model for the case of lowgap junctional couplingrdquo in Proceedings of the 32nd AnnualInternational Conference of the IEEE Engineering in Medicineand Biology Society (EMBC rsquo10) pp 228ndash231 Buenos AiresArgentina September 2010

[45] P G Platonov L B Mitrofanova V Orshanskaya and S YHo ldquoStructural abnormalities in atrial walls are associated withpresence and persistency of atrial fibrillation but not with agerdquoJournal of the American College of Cardiology vol 58 no 21 pp2225ndash2232 2011

[46] A G Kleber andY Rudy ldquoBasicmechanisms of cardiac impulsepropagation and associated arrhythmiasrdquoPhysiological Reviewsvol 84 no 2 pp 431ndash488 2004

[47] Y Xie A Garfinkel P Camelliti P Kohl J NWeiss and Z QuldquoEffects of fibroblast-myocyte coupling on cardiac conductionand vulnerability to reentry a computational studyrdquo HeartRhythm vol 6 no 11 pp 1641ndash1649 2009

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


MEDIATORSINFLAMMATION

of


Behavioural Neurology

EndocrinologyInternational Journal of



Disease Markers


BioMed Research International

OncologyJournal of



Oxidative Medicine and Cellular Longevity


PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of



Computational and Mathematical Methods in Medicine

OphthalmologyJournal of


Diabetes ResearchJournal of



Research and TreatmentAIDS


Gastroenterology Research and Practice


Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom


In the mentioned work it was only possible to induce atrialfibrillation by using a burst of at least six consecutive ectopicbeats separated by 130ms or via continuous fast periodicpacing (also with an ectopic beat period of 130ms) Unfor-tunately differently from the case of a single ectopic beatthere are notmanywell-understood examples or explanationsfor ectopic pacemakers that is a region that would generatemultiple ectopic beats at a frequency much higher (near anorder of magnitude) than the sinus rhythm In particularnew evidences suggest that two mechanisms (combined ornot) may play a very important role in the generation ofectopic pacemakers abnormal intracellular calcium (Ca2+)signalling and microreentry (reentry inside a fibrotic region)[12]

In this work we were able to generate an ectopic pace-maker in computer simulations of cardiac electrophysiologyby assuming a single hypothesis for substrate the existence ofa region with a mixture of excitable (healthy myocytes) andnonexcitable (fibrosis for instance) areas where each areahas a minimum size of a single myocyte This fibrotic regionhas been named before as microfibrosis [13ndash15] not mainlybecause of its whole size but rather due to the microscopicseparation of these two phases myocytes and fibrosis whichas mentioned before is at the scale of a single myocyte

The mechanism of this ectopic pacemaker is simple Thetwo different phases (excitable and nonexcitable) form amazeor labyrinth for wave propagation Propagation becomesfractionated by the zig-zag like or tortuous pathways it hasto follow Macroscopic propagation is considerably sloweddown Microscopically the topology of the maze allows theelectric wave to reexcite the fibrotic tissue before the waveleaves itTherefore reentry and spiral waves are formed insidethe fibrotic regionThis spiral-like wave will continuously tryto generate ectopic beats whenever it touches the border ofthe fibrotic region

Evidences that support this thesis can be found in previ-ous computational animal and clinical studies Many animalstudies have associated regions near fibrosis as triggers forventricular tachycardia fibrillation [9] or atrial fibrillation[16] Macroscopically it has been shown that propagationvelocity in the border zone of infarct may drop from 051msto 009ms [9]Microscopically propagation near fibrosis hasbeen associated with fractionated electrograms or complexfractionated atrial electrograms (CFAE) in both animal [17]and clinical studies [18] These findings have remarkablychanged the management and prognostics of patients withatrial fibrillation via a new procedure of catheter ablation ofatrial fibrillation that is guided byCFAEmapping [19] Recentclinical studies of guided CFAE ablation revisited the benefitsand risks of this procedure and clearly suggest that furtherinvestigation on this topic is required [20ndash22]

Computer simulations gave valuable insights into how themicrostructure of cardiac tissue specifically microfibrosisand CFAEs are related [13 15] These previous studies haveused realist electrophysiology single myocyte models anddetailed microscopic description to quantitatively representthe fine structure of cardiac tissue that is the models werebased on subcellular discretizations of cardiac tissue

Figure 1 In the center of this figure we present the fibrotic regionthat is simulated in this work The presence of reentry inside thefibrotic region may constantly reexcite the surrounding tissue andtherefore act as an ectopic pacemaker

However up to now a clear link between the microstruc-ture remodeling of cardiac tissue (microfibrosis) and a the-oretical substrate for cardiac arrhythmia (fibrillation) is stillmissing Evidences of the importance of tissue heterogeneityspecifically of nonconductive cells on spiral genesis andbreakup were already presented in a theoretical study usingsimple models based on cellular automata [23] RecentlyAlonso and Bar [24] have presented for the first timesimulations based on simple discrete and isotropic modelsof cardiac tissue that could generate ectopic pacemakersThe mechanism was the one explained before reentry wasgenerated inside a heterogeneous tissue (composed of activeor nonactive discrete cells) In addition Alonso and Bar [24]have described how the probability of reentry depends on aspecific feature of the topology of the discrete network thatwas used to model cardiac tissue the percentage of nonactivecells phi

The focus of this work is on bridging the gap betweenthe quantitative and more complex models used to simu-late microfibrosis and CFAEs [13] and the qualitative andsimplified models used to relate microfibrosis and ectopicpacemaker [24] For this purpose two different models wereused and both are as complex and realistic as those usedbefore in [13] that is both use realistic description of singlemyocyte electrophysiology and account for the microstruc-ture of cardiac tissue such as gap junction distribution andtissue anisotropy (1) a microscopic model with subcellulardiscretization of 8 120583m as described before in [25] and (2) adiscrete equivalent model that to the best of our knowledgehas never been presented before It is important to highlightthat in this paper our models focus on the electric dynamicsinside the ectopic pacemaker or microfibrosis region (seeFigure 1) In the near future we will study how this activitypropagates to the outside of this fibrotic region and ultimatelyinfluence the surrounding (nonfibrotic) regions of the heart

Our simulation results support the thesis that themicrostructure of a fibrotic tissue may indeed generate per se


(a) (b)


(c)



2 Methods









)south (120590

119909119894119895minus12) west (120590

119909119894minus12119895) and east (120590

119909119894+12119895) volume


119898= 00) cytoplasm (120590

119888=




119888= 0062 120583S) where



120573119862119898

120597119881 (119909 119910 119905)

120597119905+ 120573119868ion (119881 (119909 119910 119905) 120578 (119909 119910 119905))


120597120578 (119909 119910 119905)

120597119905= f (119881 (119909 119910 119905) 120578 (119909 119910 119905))

(1)









120597119881

120597119905=

1119862119898

[minus119868ion (119881 120578) + 119868stim]

120597120578

120597119905= f (119881 120578)

(2)

and a parabolic PDE

120573(119862119898

120597119881

120597119905) = nabla sdot (120590nabla119881) (3)


119900= 00001ms



119901gt Δ119905



0

50

100

150

y(120583

m)

0 100 200 300 400 500 600

x (120583m)

(a)

0

50

100

150

y(120583

m)

0 100 200 300 400 500 600

x (120583m)

(b)









119862119898

119881lowast

119894119895minus 119881119899

119894119895

Δ119905119900

= minus 119868ion (119881119899

119894119895 120578119899)

120578119899+1

minus 120578119899

Δ119905119900

= f (120578119899 119881119899 119905)

(4)

(120590119894+12119895 +120590119894minus12119895 +120590119894119895+12 +120590119894119895minus12 +120572)119881

119899+1119894119895

minus120590119894119895minus12119881

119899+1119894119895minus1 minus120590119894+12119895119881

119899+1119894+1119895 minus120590119894119895+12119881

119899+1119894119895+1

minus120590119894minus12119895119881

119899+1119894minus1119895 = 120572119881

lowast

119894119895

(5)



119901119866119894119866119888)



119894119895 First we














Vol119894120573(119862119898

120597119881119894

120597119905+ 119868ion (119881119894 120578119894))

= (

nn119894sum

119895=1119866119894119895lowast (119881119895minus119881119894))+ 119868

stim119894

120597120578119894

120597119905= f (119881

119894 120578119894)

(6)






119862119898

119881lowast

119894minus 119881119899

119894

Δ119905119900

= minus 119868ion (119881119899

119894 120578119899)

120578119899+1

minus 120578119899

Δ119905119900

= f (120578119899 119881119899 119905) (7)

(

nn119894sum

119895=1119866119894119895+120572)119881

119899+1119894

minus

nn119894sum

119895=1(119866119894119895119881119899+1119895

) = 120572119881lowast

119894 (8)

where 120572 = (120573119862119898119860119894119889)Δ119905




119894119895in (8)


1119877119894119895

=

ng

sum

119896=1(

1119877119894119895119896

) (9)

where 119877119894119895


119894119895119896is


119877119894119895119896

= dist119894119895119896

lowast1

(120590ℎ)+

1119866119896

(10)

where dist119894119895119896


dist119894119895119896

= (10038161003816100381610038161003816pr119894119909minus gp119894119896119909

10038161003816100381610038161003816+10038161003816100381610038161003816pr119894119910minus gp119894119896119910

10038161003816100381610038161003816)

+ (10038161003816100381610038161003816pr119895119909minus gp119895119896119909

10038161003816100381610038161003816+10038161003816100381610038161003816pr119895119910minus gp119895119896119910

10038161003816100381610038161003816)

(11)







120601 = 015




119866119894119895=

1119877119894119895

(12)



3 Results


















0

minus80

(mV

)

0

minus80

(mV

)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

(a)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0

minus80

(mV

)

0

minus80

(mV

)

(b)

4000000

2000000

000000

minus2000000

minus4000000

minus6000000

minus8000000

minus10000000

Pote

ntia

l diff

eren

ce

0 30 60 90 120 150 180 210 240 270 300

Time (ms)

MicroscopicDiscrete

(c)



0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

(A) (B)

(C) (D)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)

(a)

MicroscopicDiscrete

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

minus90

Pote

ntia

l diff

eren

ce

0 15 30

45

60

75

90

105

120

135

150

165

180

195

210

225

240

Time (ms)

(b)



0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

(a) (b)

(c) (d)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)










0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

(a) (b)

(c) (d)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)











NA 541 517 55 535OA 183 162 156 144S 192 321 173 209






04

041

042

043

044

045

046

047

048

049

050

051

052

053

120601

Left-rightTop-down

000

500

1000

1500

2000

2500

3000

3500

P(120601)

()





4 Discussion




042 045 048012345

reen

trie

sN

umbe

r of


051 054

120601

(a)

042 045 048012345

reen

trie

sN

umbe

r of


051 054

120601

(b)


0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

(a) (b)

(c) (d)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)





0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

(a) (b)

(c) (d)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)







0

20

40

60

80

100


04 042 044 046 048 05 052 054 056 058120601

C(120601)


119888= 0466 and 120601

119888= 0527 for TP and LP


050100150200250300350400450

Spee

d (120583

mm

s)

000 01 020 030 040 060050

120601

Left-right





0

20

40

60

80

100

120

140

Spee

d (120583

mm

s)

000 010 020 030 040 050 060

120601

Top-bottom







(a) (b)

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

(mV

)

(c)

(d) (e)

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

(mV

)

(f)









5 Conclusions







Acknowledgments


References
























































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















(a) (b)


(c)



2 Methods









)south (120590

119909119894119895minus12) west (120590

119909119894minus12119895) and east (120590

119909119894+12119895) volume


119898= 00) cytoplasm (120590

119888=




119888= 0062 120583S) where



120573119862119898

120597119881 (119909 119910 119905)

120597119905+ 120573119868ion (119881 (119909 119910 119905) 120578 (119909 119910 119905))


120597120578 (119909 119910 119905)

120597119905= f (119881 (119909 119910 119905) 120578 (119909 119910 119905))

(1)









120597119881

120597119905=

1119862119898

[minus119868ion (119881 120578) + 119868stim]

120597120578

120597119905= f (119881 120578)

(2)

and a parabolic PDE

120573(119862119898

120597119881

120597119905) = nabla sdot (120590nabla119881) (3)


119900= 00001ms



119901gt Δ119905



0

50

100

150

y(120583

m)

0 100 200 300 400 500 600

x (120583m)

(a)

0

50

100

150

y(120583

m)

0 100 200 300 400 500 600

x (120583m)

(b)









119862119898

119881lowast

119894119895minus 119881119899

119894119895

Δ119905119900

= minus 119868ion (119881119899

119894119895 120578119899)

120578119899+1

minus 120578119899

Δ119905119900

= f (120578119899 119881119899 119905)

(4)

(120590119894+12119895 +120590119894minus12119895 +120590119894119895+12 +120590119894119895minus12 +120572)119881

119899+1119894119895

minus120590119894119895minus12119881

119899+1119894119895minus1 minus120590119894+12119895119881

119899+1119894+1119895 minus120590119894119895+12119881

119899+1119894119895+1

minus120590119894minus12119895119881

119899+1119894minus1119895 = 120572119881

lowast

119894119895

(5)



119901119866119894119866119888)



119894119895 First we














Vol119894120573(119862119898

120597119881119894

120597119905+ 119868ion (119881119894 120578119894))

= (

nn119894sum

119895=1119866119894119895lowast (119881119895minus119881119894))+ 119868

stim119894

120597120578119894

120597119905= f (119881

119894 120578119894)

(6)






119862119898

119881lowast

119894minus 119881119899

119894

Δ119905119900

= minus 119868ion (119881119899

119894 120578119899)

120578119899+1

minus 120578119899

Δ119905119900

= f (120578119899 119881119899 119905) (7)

(

nn119894sum

119895=1119866119894119895+120572)119881

119899+1119894

minus

nn119894sum

119895=1(119866119894119895119881119899+1119895

) = 120572119881lowast

119894 (8)

where 120572 = (120573119862119898119860119894119889)Δ119905




119894119895in (8)


1119877119894119895

=

ng

sum

119896=1(

1119877119894119895119896

) (9)

where 119877119894119895


119894119895119896is


119877119894119895119896

= dist119894119895119896

lowast1

(120590ℎ)+

1119866119896

(10)

where dist119894119895119896


dist119894119895119896

= (10038161003816100381610038161003816pr119894119909minus gp119894119896119909

10038161003816100381610038161003816+10038161003816100381610038161003816pr119894119910minus gp119894119896119910

10038161003816100381610038161003816)

+ (10038161003816100381610038161003816pr119895119909minus gp119895119896119909

10038161003816100381610038161003816+10038161003816100381610038161003816pr119895119910minus gp119895119896119910

10038161003816100381610038161003816)

(11)







120601 = 015




119866119894119895=

1119877119894119895

(12)



3 Results


















0

minus80

(mV

)

0

minus80

(mV

)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

(a)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0

minus80

(mV

)

0

minus80

(mV

)

(b)

4000000

2000000

000000

minus2000000

minus4000000

minus6000000

minus8000000

minus10000000

Pote

ntia

l diff

eren

ce

0 30 60 90 120 150 180 210 240 270 300

Time (ms)

MicroscopicDiscrete

(c)



0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

(A) (B)

(C) (D)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)

(a)

MicroscopicDiscrete

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

minus90

Pote

ntia

l diff

eren

ce

0 15 30

45

60

75

90

105

120

135

150

165

180

195

210

225

240

Time (ms)

(b)



0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

(a) (b)

(c) (d)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)










0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

(a) (b)

(c) (d)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)











NA 541 517 55 535OA 183 162 156 144S 192 321 173 209






04

041

042

043

044

045

046

047

048

049

050

051

052

053

120601

Left-rightTop-down

000

500

1000

1500

2000

2500

3000

3500

P(120601)

()





4 Discussion




042 045 048012345

reen

trie

sN

umbe

r of


051 054

120601

(a)

042 045 048012345

reen

trie

sN

umbe

r of


051 054

120601

(b)


0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

(a) (b)

(c) (d)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)





0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

(a) (b)

(c) (d)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)







0

20

40

60

80

100


04 042 044 046 048 05 052 054 056 058120601

C(120601)


119888= 0466 and 120601

119888= 0527 for TP and LP


050100150200250300350400450

Spee

d (120583

mm

s)

000 01 020 030 040 060050

120601

Left-right





0

20

40

60

80

100

120

140

Spee

d (120583

mm

s)

000 010 020 030 040 050 060

120601

Top-bottom







(a) (b)

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

(mV

)

(c)

(d) (e)

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

(mV

)

(f)









5 Conclusions







Acknowledgments


References
























































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of


















119888= 0062 120583S) where



120573119862119898

120597119881 (119909 119910 119905)

120597119905+ 120573119868ion (119881 (119909 119910 119905) 120578 (119909 119910 119905))


120597120578 (119909 119910 119905)

120597119905= f (119881 (119909 119910 119905) 120578 (119909 119910 119905))

(1)









120597119881

120597119905=

1119862119898

[minus119868ion (119881 120578) + 119868stim]

120597120578

120597119905= f (119881 120578)

(2)

and a parabolic PDE

120573(119862119898

120597119881

120597119905) = nabla sdot (120590nabla119881) (3)


119900= 00001ms



119901gt Δ119905



0

50

100

150

y(120583

m)

0 100 200 300 400 500 600

x (120583m)

(a)

0

50

100

150

y(120583

m)

0 100 200 300 400 500 600

x (120583m)

(b)









119862119898

119881lowast

119894119895minus 119881119899

119894119895

Δ119905119900

= minus 119868ion (119881119899

119894119895 120578119899)

120578119899+1

minus 120578119899

Δ119905119900

= f (120578119899 119881119899 119905)

(4)

(120590119894+12119895 +120590119894minus12119895 +120590119894119895+12 +120590119894119895minus12 +120572)119881

119899+1119894119895

minus120590119894119895minus12119881

119899+1119894119895minus1 minus120590119894+12119895119881

119899+1119894+1119895 minus120590119894119895+12119881

119899+1119894119895+1

minus120590119894minus12119895119881

119899+1119894minus1119895 = 120572119881

lowast

119894119895

(5)



119901119866119894119866119888)



119894119895 First we














Vol119894120573(119862119898

120597119881119894

120597119905+ 119868ion (119881119894 120578119894))

= (

nn119894sum

119895=1119866119894119895lowast (119881119895minus119881119894))+ 119868

stim119894

120597120578119894

120597119905= f (119881

119894 120578119894)

(6)






119862119898

119881lowast

119894minus 119881119899

119894

Δ119905119900

= minus 119868ion (119881119899

119894 120578119899)

120578119899+1

minus 120578119899

Δ119905119900

= f (120578119899 119881119899 119905) (7)

(

nn119894sum

119895=1119866119894119895+120572)119881

119899+1119894

minus

nn119894sum

119895=1(119866119894119895119881119899+1119895

) = 120572119881lowast

119894 (8)

where 120572 = (120573119862119898119860119894119889)Δ119905




119894119895in (8)


1119877119894119895

=

ng

sum

119896=1(

1119877119894119895119896

) (9)

where 119877119894119895


119894119895119896is


119877119894119895119896

= dist119894119895119896

lowast1

(120590ℎ)+

1119866119896

(10)

where dist119894119895119896


dist119894119895119896

= (10038161003816100381610038161003816pr119894119909minus gp119894119896119909

10038161003816100381610038161003816+10038161003816100381610038161003816pr119894119910minus gp119894119896119910

10038161003816100381610038161003816)

+ (10038161003816100381610038161003816pr119895119909minus gp119895119896119909

10038161003816100381610038161003816+10038161003816100381610038161003816pr119895119910minus gp119895119896119910

10038161003816100381610038161003816)

(11)







120601 = 015




119866119894119895=

1119877119894119895

(12)



3 Results


















0

minus80

(mV

)

0

minus80

(mV

)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

(a)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0

minus80

(mV

)

0

minus80

(mV

)

(b)

4000000

2000000

000000

minus2000000

minus4000000

minus6000000

minus8000000

minus10000000

Pote

ntia

l diff

eren

ce

0 30 60 90 120 150 180 210 240 270 300

Time (ms)

MicroscopicDiscrete

(c)



0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

(A) (B)

(C) (D)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)

(a)

MicroscopicDiscrete

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

minus90

Pote

ntia

l diff

eren

ce

0 15 30

45

60

75

90

105

120

135

150

165

180

195

210

225

240

Time (ms)

(b)



0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

(a) (b)

(c) (d)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)










0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

(a) (b)

(c) (d)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)











NA 541 517 55 535OA 183 162 156 144S 192 321 173 209






04

041

042

043

044

045

046

047

048

049

050

051

052

053

120601

Left-rightTop-down

000

500

1000

1500

2000

2500

3000

3500

P(120601)

()





4 Discussion




042 045 048012345

reen

trie

sN

umbe

r of


051 054

120601

(a)

042 045 048012345

reen

trie

sN

umbe

r of


051 054

120601

(b)


0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

(a) (b)

(c) (d)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)





0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

(a) (b)

(c) (d)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)







0

20

40

60

80

100


04 042 044 046 048 05 052 054 056 058120601

C(120601)


119888= 0466 and 120601

119888= 0527 for TP and LP


050100150200250300350400450

Spee

d (120583

mm

s)

000 01 020 030 040 060050

120601

Left-right





0

20

40

60

80

100

120

140

Spee

d (120583

mm

s)

000 010 020 030 040 050 060

120601

Top-bottom







(a) (b)

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

(mV

)

(c)

(d) (e)

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

(mV

)

(f)









5 Conclusions







Acknowledgments


References
























































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















0

50

100

150

y(120583

m)

0 100 200 300 400 500 600

x (120583m)

(a)

0

50

100

150

y(120583

m)

0 100 200 300 400 500 600

x (120583m)

(b)









119862119898

119881lowast

119894119895minus 119881119899

119894119895

Δ119905119900

= minus 119868ion (119881119899

119894119895 120578119899)

120578119899+1

minus 120578119899

Δ119905119900

= f (120578119899 119881119899 119905)

(4)

(120590119894+12119895 +120590119894minus12119895 +120590119894119895+12 +120590119894119895minus12 +120572)119881

119899+1119894119895

minus120590119894119895minus12119881

119899+1119894119895minus1 minus120590119894+12119895119881

119899+1119894+1119895 minus120590119894119895+12119881

119899+1119894119895+1

minus120590119894minus12119895119881

119899+1119894minus1119895 = 120572119881

lowast

119894119895

(5)



119901119866119894119866119888)



119894119895 First we














Vol119894120573(119862119898

120597119881119894

120597119905+ 119868ion (119881119894 120578119894))

= (

nn119894sum

119895=1119866119894119895lowast (119881119895minus119881119894))+ 119868

stim119894

120597120578119894

120597119905= f (119881

119894 120578119894)

(6)






119862119898

119881lowast

119894minus 119881119899

119894

Δ119905119900

= minus 119868ion (119881119899

119894 120578119899)

120578119899+1

minus 120578119899

Δ119905119900

= f (120578119899 119881119899 119905) (7)

(

nn119894sum

119895=1119866119894119895+120572)119881

119899+1119894

minus

nn119894sum

119895=1(119866119894119895119881119899+1119895

) = 120572119881lowast

119894 (8)

where 120572 = (120573119862119898119860119894119889)Δ119905




119894119895in (8)


1119877119894119895

=

ng

sum

119896=1(

1119877119894119895119896

) (9)

where 119877119894119895


119894119895119896is


119877119894119895119896

= dist119894119895119896

lowast1

(120590ℎ)+

1119866119896

(10)

where dist119894119895119896


dist119894119895119896

= (10038161003816100381610038161003816pr119894119909minus gp119894119896119909

10038161003816100381610038161003816+10038161003816100381610038161003816pr119894119910minus gp119894119896119910

10038161003816100381610038161003816)

+ (10038161003816100381610038161003816pr119895119909minus gp119895119896119909

10038161003816100381610038161003816+10038161003816100381610038161003816pr119895119910minus gp119895119896119910

10038161003816100381610038161003816)

(11)







120601 = 015




119866119894119895=

1119877119894119895

(12)



3 Results


















0

minus80

(mV

)

0

minus80

(mV

)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

(a)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0

minus80

(mV

)

0

minus80

(mV

)

(b)

4000000

2000000

000000

minus2000000

minus4000000

minus6000000

minus8000000

minus10000000

Pote

ntia

l diff

eren

ce

0 30 60 90 120 150 180 210 240 270 300

Time (ms)

MicroscopicDiscrete

(c)



0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

(A) (B)

(C) (D)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)

(a)

MicroscopicDiscrete

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

minus90

Pote

ntia

l diff

eren

ce

0 15 30

45

60

75

90

105

120

135

150

165

180

195

210

225

240

Time (ms)

(b)



0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

(a) (b)

(c) (d)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)










0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

(a) (b)

(c) (d)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)











NA 541 517 55 535OA 183 162 156 144S 192 321 173 209






04

041

042

043

044

045

046

047

048

049

050

051

052

053

120601

Left-rightTop-down

000

500

1000

1500

2000

2500

3000

3500

P(120601)

()





4 Discussion




042 045 048012345

reen

trie

sN

umbe

r of


051 054

120601

(a)

042 045 048012345

reen

trie

sN

umbe

r of


051 054

120601

(b)


0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

(a) (b)

(c) (d)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)





0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

(a) (b)

(c) (d)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)







0

20

40

60

80

100


04 042 044 046 048 05 052 054 056 058120601

C(120601)


119888= 0466 and 120601

119888= 0527 for TP and LP


050100150200250300350400450

Spee

d (120583

mm

s)

000 01 020 030 040 060050

120601

Left-right





0

20

40

60

80

100

120

140

Spee

d (120583

mm

s)

000 010 020 030 040 050 060

120601

Top-bottom







(a) (b)

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

(mV

)

(c)

(d) (e)

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

(mV

)

(f)









5 Conclusions







Acknowledgments


References
























































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of


















119862119898

119881lowast

119894minus 119881119899

119894

Δ119905119900

= minus 119868ion (119881119899

119894 120578119899)

120578119899+1

minus 120578119899

Δ119905119900

= f (120578119899 119881119899 119905) (7)

(

nn119894sum

119895=1119866119894119895+120572)119881

119899+1119894

minus

nn119894sum

119895=1(119866119894119895119881119899+1119895

) = 120572119881lowast

119894 (8)

where 120572 = (120573119862119898119860119894119889)Δ119905




119894119895in (8)


1119877119894119895

=

ng

sum

119896=1(

1119877119894119895119896

) (9)

where 119877119894119895


119894119895119896is


119877119894119895119896

= dist119894119895119896

lowast1

(120590ℎ)+

1119866119896

(10)

where dist119894119895119896


dist119894119895119896

= (10038161003816100381610038161003816pr119894119909minus gp119894119896119909

10038161003816100381610038161003816+10038161003816100381610038161003816pr119894119910minus gp119894119896119910

10038161003816100381610038161003816)

+ (10038161003816100381610038161003816pr119895119909minus gp119895119896119909

10038161003816100381610038161003816+10038161003816100381610038161003816pr119895119910minus gp119895119896119910

10038161003816100381610038161003816)

(11)







120601 = 015




119866119894119895=

1119877119894119895

(12)



3 Results


















0

minus80

(mV

)

0

minus80

(mV

)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

(a)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0

minus80

(mV

)

0

minus80

(mV

)

(b)

4000000

2000000

000000

minus2000000

minus4000000

minus6000000

minus8000000

minus10000000

Pote

ntia

l diff

eren

ce

0 30 60 90 120 150 180 210 240 270 300

Time (ms)

MicroscopicDiscrete

(c)



0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

(A) (B)

(C) (D)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)

(a)

MicroscopicDiscrete

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

minus90

Pote

ntia

l diff

eren

ce

0 15 30

45

60

75

90

105

120

135

150

165

180

195

210

225

240

Time (ms)

(b)



0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

(a) (b)

(c) (d)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)










0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

(a) (b)

(c) (d)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)











NA 541 517 55 535OA 183 162 156 144S 192 321 173 209






04

041

042

043

044

045

046

047

048

049

050

051

052

053

120601

Left-rightTop-down

000

500

1000

1500

2000

2500

3000

3500

P(120601)

()





4 Discussion




042 045 048012345

reen

trie

sN

umbe

r of


051 054

120601

(a)

042 045 048012345

reen

trie

sN

umbe

r of


051 054

120601

(b)


0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

(a) (b)

(c) (d)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)





0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

(a) (b)

(c) (d)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)







0

20

40

60

80

100


04 042 044 046 048 05 052 054 056 058120601

C(120601)


119888= 0466 and 120601

119888= 0527 for TP and LP


050100150200250300350400450

Spee

d (120583

mm

s)

000 01 020 030 040 060050

120601

Left-right





0

20

40

60

80

100

120

140

Spee

d (120583

mm

s)

000 010 020 030 040 050 060

120601

Top-bottom







(a) (b)

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

(mV

)

(c)

(d) (e)

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

(mV

)

(f)









5 Conclusions







Acknowledgments


References
























































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of
































0

minus80

(mV

)

0

minus80

(mV

)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

(a)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0

minus80

(mV

)

0

minus80

(mV

)

(b)

4000000

2000000

000000

minus2000000

minus4000000

minus6000000

minus8000000

minus10000000

Pote

ntia

l diff

eren

ce

0 30 60 90 120 150 180 210 240 270 300

Time (ms)

MicroscopicDiscrete

(c)



0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

(A) (B)

(C) (D)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)

(a)

MicroscopicDiscrete

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

minus90

Pote

ntia

l diff

eren

ce

0 15 30

45

60

75

90

105

120

135

150

165

180

195

210

225

240

Time (ms)

(b)



0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

(a) (b)

(c) (d)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)










0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

(a) (b)

(c) (d)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)











NA 541 517 55 535OA 183 162 156 144S 192 321 173 209






04

041

042

043

044

045

046

047

048

049

050

051

052

053

120601

Left-rightTop-down

000

500

1000

1500

2000

2500

3000

3500

P(120601)

()





4 Discussion




042 045 048012345

reen

trie

sN

umbe

r of


051 054

120601

(a)

042 045 048012345

reen

trie

sN

umbe

r of


051 054

120601

(b)


0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

(a) (b)

(c) (d)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)





0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

(a) (b)

(c) (d)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)







0

20

40

60

80

100


04 042 044 046 048 05 052 054 056 058120601

C(120601)


119888= 0466 and 120601

119888= 0527 for TP and LP


050100150200250300350400450

Spee

d (120583

mm

s)

000 01 020 030 040 060050

120601

Left-right





0

20

40

60

80

100

120

140

Spee

d (120583

mm

s)

000 010 020 030 040 050 060

120601

Top-bottom







(a) (b)

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

(mV

)

(c)

(d) (e)

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

(mV

)

(f)









5 Conclusions







Acknowledgments


References
























































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

(A) (B)

(C) (D)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)

(a)

MicroscopicDiscrete

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

minus90

Pote

ntia

l diff

eren

ce

0 15 30

45

60

75

90

105

120

135

150

165

180

195

210

225

240

Time (ms)

(b)



0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

(a) (b)

(c) (d)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)










0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

(a) (b)

(c) (d)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)











NA 541 517 55 535OA 183 162 156 144S 192 321 173 209






04

041

042

043

044

045

046

047

048

049

050

051

052

053

120601

Left-rightTop-down

000

500

1000

1500

2000

2500

3000

3500

P(120601)

()





4 Discussion




042 045 048012345

reen

trie

sN

umbe

r of


051 054

120601

(a)

042 045 048012345

reen

trie

sN

umbe

r of


051 054

120601

(b)


0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

(a) (b)

(c) (d)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)





0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

(a) (b)

(c) (d)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)







0

20

40

60

80

100


04 042 044 046 048 05 052 054 056 058120601

C(120601)


119888= 0466 and 120601

119888= 0527 for TP and LP


050100150200250300350400450

Spee

d (120583

mm

s)

000 01 020 030 040 060050

120601

Left-right





0

20

40

60

80

100

120

140

Spee

d (120583

mm

s)

000 010 020 030 040 050 060

120601

Top-bottom







(a) (b)

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

(mV

)

(c)

(d) (e)

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

(mV

)

(f)









5 Conclusions







Acknowledgments


References
























































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

(a) (b)

(c) (d)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)










0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

(a) (b)

(c) (d)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)











NA 541 517 55 535OA 183 162 156 144S 192 321 173 209






04

041

042

043

044

045

046

047

048

049

050

051

052

053

120601

Left-rightTop-down

000

500

1000

1500

2000

2500

3000

3500

P(120601)

()





4 Discussion




042 045 048012345

reen

trie

sN

umbe

r of


051 054

120601

(a)

042 045 048012345

reen

trie

sN

umbe

r of


051 054

120601

(b)


0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

(a) (b)

(c) (d)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)





0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

(a) (b)

(c) (d)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)







0

20

40

60

80

100


04 042 044 046 048 05 052 054 056 058120601

C(120601)


119888= 0466 and 120601

119888= 0527 for TP and LP


050100150200250300350400450

Spee

d (120583

mm

s)

000 01 020 030 040 060050

120601

Left-right





0

20

40

60

80

100

120

140

Spee

d (120583

mm

s)

000 010 020 030 040 050 060

120601

Top-bottom







(a) (b)

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

(mV

)

(c)

(d) (e)

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

(mV

)

(f)









5 Conclusions







Acknowledgments


References
























































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

(a) (b)

(c) (d)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)











NA 541 517 55 535OA 183 162 156 144S 192 321 173 209






04

041

042

043

044

045

046

047

048

049

050

051

052

053

120601

Left-rightTop-down

000

500

1000

1500

2000

2500

3000

3500

P(120601)

()





4 Discussion




042 045 048012345

reen

trie

sN

umbe

r of


051 054

120601

(a)

042 045 048012345

reen

trie

sN

umbe

r of


051 054

120601

(b)


0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

(a) (b)

(c) (d)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)





0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

(a) (b)

(c) (d)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)







0

20

40

60

80

100


04 042 044 046 048 05 052 054 056 058120601

C(120601)


119888= 0466 and 120601

119888= 0527 for TP and LP


050100150200250300350400450

Spee

d (120583

mm

s)

000 01 020 030 040 060050

120601

Left-right





0

20

40

60

80

100

120

140

Spee

d (120583

mm

s)

000 010 020 030 040 050 060

120601

Top-bottom







(a) (b)

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

(mV

)

(c)

(d) (e)

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

(mV

)

(f)









5 Conclusions







Acknowledgments


References
























































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of



















NA 541 517 55 535OA 183 162 156 144S 192 321 173 209






04

041

042

043

044

045

046

047

048

049

050

051

052

053

120601

Left-rightTop-down

000

500

1000

1500

2000

2500

3000

3500

P(120601)

()





4 Discussion




042 045 048012345

reen

trie

sN

umbe

r of


051 054

120601

(a)

042 045 048012345

reen

trie

sN

umbe

r of


051 054

120601

(b)


0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

(a) (b)

(c) (d)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)





0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

(a) (b)

(c) (d)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)







0

20

40

60

80

100


04 042 044 046 048 05 052 054 056 058120601

C(120601)


119888= 0466 and 120601

119888= 0527 for TP and LP


050100150200250300350400450

Spee

d (120583

mm

s)

000 01 020 030 040 060050

120601

Left-right





0

20

40

60

80

100

120

140

Spee

d (120583

mm

s)

000 010 020 030 040 050 060

120601

Top-bottom







(a) (b)

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

(mV

)

(c)

(d) (e)

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

(mV

)

(f)









5 Conclusions







Acknowledgments


References
























































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















042 045 048012345

reen

trie

sN

umbe

r of


051 054

120601

(a)

042 045 048012345

reen

trie

sN

umbe

r of


051 054

120601

(b)


0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

(a) (b)

(c) (d)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)





0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

(a) (b)

(c) (d)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)







0

20

40

60

80

100


04 042 044 046 048 05 052 054 056 058120601

C(120601)


119888= 0466 and 120601

119888= 0527 for TP and LP


050100150200250300350400450

Spee

d (120583

mm

s)

000 01 020 030 040 060050

120601

Left-right





0

20

40

60

80

100

120

140

Spee

d (120583

mm

s)

000 010 020 030 040 050 060

120601

Top-bottom







(a) (b)

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

(mV

)

(c)

(d) (e)

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

(mV

)

(f)









5 Conclusions







Acknowledgments


References
























































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

0 016 032 048 064 080 0960

016

032

048

064

080

096

x (cm)

y(c

m)

(a) (b)

(c) (d)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)

0

minus80

(mV

)







0

20

40

60

80

100


04 042 044 046 048 05 052 054 056 058120601

C(120601)


119888= 0466 and 120601

119888= 0527 for TP and LP


050100150200250300350400450

Spee

d (120583

mm

s)

000 01 020 030 040 060050

120601

Left-right





0

20

40

60

80

100

120

140

Spee

d (120583

mm

s)

000 010 020 030 040 050 060

120601

Top-bottom







(a) (b)

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

(mV

)

(c)

(d) (e)

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

(mV

)

(f)









5 Conclusions







Acknowledgments


References
























































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















0

20

40

60

80

100


04 042 044 046 048 05 052 054 056 058120601

C(120601)


119888= 0466 and 120601

119888= 0527 for TP and LP


050100150200250300350400450

Spee

d (120583

mm

s)

000 01 020 030 040 060050

120601

Left-right





0

20

40

60

80

100

120

140

Spee

d (120583

mm

s)

000 010 020 030 040 050 060

120601

Top-bottom







(a) (b)

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

(mV

)

(c)

(d) (e)

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

(mV

)

(f)









5 Conclusions







Acknowledgments


References
























































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















(a) (b)

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

(mV

)

(c)

(d) (e)

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

(mV

)

(f)









5 Conclusions







Acknowledgments


References
























































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of





















Acknowledgments


References
























































of






Disease Markers



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