SIMPLIFIED MATHEMATICAL MODEL for GENERATING ECG SIGNAL and FITTING THE MODEL USING NONLINEAR LEAST SQUARE TECHNIQUE

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1 Procdngs of th Intrnatonal Confrnc on Mchancal Engnrng (ICME) 8- Dcmbr, Dhaa, Bangladsh ICME-RT- SIMPLIFIED MATHEMATICAL MODEL for GENERATING ECG SIGNAL and FITTING THE MODEL USING NONLINEAR LEAST SQUARE TECHNIQUE Md. Abdul Awal, Shh Shanawaz Mostafa and Mohuddn Ahmad Dpartmnt of Bomdcal Engnrng, Khulna Unvrsty of Engnrng & Tchnology, Khulna Bangladsh Dpt. of Elctrcal and Elctronc Engnrng, Khulna Unvrsty of Engnrng & Tchnology, Khulna, Bangladsh ABSTRACT ECG modls ar complx and thr computatonal tm s hgh. In ths papr, w propos a Gaussan wav-basd modl whch can smulat ECG wav as wll as P, Q, R, S and T wavs ndvdually. In addton, ths modl s capabl of smulatng varous nds of practcal phnomna. Th coffcnt of th modl was calculatd by nonlnar last squar tchnqu usng Gauss-Nwton algorthm. In ordr to valuat th ffctvnss of th modl, dffrnt nd tm doman and frquncy doman tchnqus such as PSD and MSC wr usd. Th goodnss of fttng was calculatd usng MSE, NMSE, RMSE, NRMSE and PRD and compard wth ral and modl ECG sgnal. Th lowr valu of ths rror and hghr cross-corrlaton coffcnt of.98 btwn modl and ral ECG ndcats th outstandng prformanc of th modl. Th modl s also succssful n gnratng nosy ECG sgnal. Kywords: ECG Sgnal, Gaussan Wav, Nonlnar Last Squar Tchnqu, Goodnss Of Fttng. INTRODUCTION Th hart s a hollow muscular organ that bats n rhythm to gnrat th forc for pumpng blood through th whol body. Each bats trggrd by a bolctrc sgnals orgnatd at snoatral (SA) nod and sprads throughout th body. And th lctrocardogram (ECG or EKG) s an nvstgatv tool for a wd rang of hart condtons, from mnor to lf thratnng that masur and rcords th lctrcal actvty of th hart n suprb dtal. So ECG sgnal modlng and procssng s th most sgnfcant topcs n bomdcal ngnrng. For modlng of ECG, dffrnt tchnqus hav bn dvlopd n th past. A pol-zro modls of th ECG was rprsntd by [] for fatur xtracton and data comprsson. Anothr rsarch [] rportd that, th pols and zros form clustrs and th clustrs can b rlatd to th consttunt wavs of th ECG modls. Transform-typ mthods l nonlnar transform usng multplcaton bacward dffrnc for dtctng QRS proposd by dffrnt rsarchrs. Howvr, ths typs of modlng cannot provd a drct rprsntaton of th consttunt wavs n th ECG as cardac spcalst ar ndd for mang dagnoss. Chp Away Dcomposton (ChAD) algorthm whch s an tratv mthod for Gaussan paramtr dtrmnaton was usd for dcomposng and rprsntng th ECG modl by [3]. Clfford t al. [4] usd svn Gaussan functons for modlng of ECG by mans of 3D stat-spac modl whch rqur numrcal ntgraton usng a fourth-ordr Rung-Kutta mthod. S. Paravna t al. [5] usd a larg numbr of Gaussan (4 to 33) wth no bas ln drft factor basd on mnmum ban mthod and zro crossng mthod. But fttng ths modl to th ral ECG sgnal, startng and nd pont of any ntrval usng zro crossng mthod s not ffcnt. In addton, growng numbr of Gaussan functons nvolv much tm to run th program. [6] proposd a modl usng Gaussan functon. Howvr thy cannot rprsnt QRS wav ndvdually as wll as t s unabl to ft wth th ral ECG at a sgnfcant lvl. Thy usd doubl dffrntaton of th Gaussan functon whch s tm consumng and nd complx mathmatcal opraton. Th fttng tchnqus wr ncomptnt bcaus thy wr not capabl to ft any ngatv valus n thr modl whch was qut common n ral data. Ths papr propos a Gaussan wav bas modl whch can smulat ECG wav as wll as P, Q, R, S and T wav ndvdually and s vry smpl as compard to arlr mntond modl. Nvrthlss, ECG sgnals ar corruptd by varous nds of noss l othr lctrcal sgnal,, such as () powr lns ntrfrnc [7], () hgh-frquncy lctromyography (EMG) nos, () moton artfacts, (v) mpdanc changs at th sn/lctrod, (v) basln drfts [8], (v) lctrosurgcal ICME RT-

2 nos [9], and (v) wht nos. Thrfor, nosy ECG s mportant as normal ECG to gnrat ralstc ECG sgnal. In addton, nosy ECG can b gnratd by th modl. Th coffcnt of th modl s calculatd scton by scton by nonlnar last squar tchnqu usng gauss-nwton algorthm. Th prformanc of th proposd modl and fttng algorthm ar valuatd by usng ECG from rcordd data. Thrby, for llustratng modl prformanc, goodnss of fttng ar calculatd. Ral data ar pr-procssd for bttr fttng by usng Buttrworth fltrng.fnally; th frquncy-doman analyss of ECG sgnals s dmonstratd. Th rst of ths papr s organzd as follows. Scton ntroducs th proposd modl ECG modl. Dtals mthods of th proposd modl ar prsntd n Scton 3. Scton 4 provds xprmntal rsults and faturs of th modl and concluson and futur wor com n Scton 5.. ECG MODEL ECG sgnal s rsmblng of th combnaton of bll curv l P, Q, R, S and T wavs; t falls toward both sts whch ar on of th charactrstc of Gaussan wav. If P, Q, R, S, T thn Gaussan wav for ach componnt of ECG wav hav followng paramtr: M s hght of curvs pa, t s th cntr poston of th pa and W controls th wdth thn ECG componnts can b wrttn as P wav : M P Q wav : M Q R wav : M t P WP t W R d dt S wav : M S T wav : M T Q Q t T WT M Q t R WR t S WS t W Q Q Smply th gnral quaton (-5) can b wrttn as f P, R, S, T N, SNR ( t) tt w d M M dt Q, d In Eq. (6), dpnd on. dt d d ff R dt dt tt w d dt. and ff S () () (3) (4) (5) Othrws th trm dos not xst. If w prform normal sum of Eq. (6), th ndvdual P, Q, R, S, T componnt of ECG wll b ovrlappd wth (6) ach othr wav. To solv ths problm, ECG componnts ar fttd to thr rght poston usng shftng and zro paddng mthod. N, SNR ( t) s th nos paramtr n th modl. ndcats varous typs of ECG nos, such as wht and color nos, muscl artfact (MA), lctrod movmnt (EM), and basln wandr (BW). SNR ndcats th nput sgnal-to-nos rato. As ral ECG s contamnatd by ths noss, so nos should b tan nto account for mor ralstc modlng. Varous noss ar modld n th followng ssson. (a)synthtc noss Wht nos s a random sgnal wth a flat powr spctral dnsty. It has all frquncy componnts. Flcr nos or color nos s a typ of low-frquncy lctronc nos wth an nvrsly proportonal powr spctral dnsty compard wth th frquncy. Rsstanc fluctuaton s th man rason for flcr nos gnraton and that s why all rsstors has flcr nos. For th currnt study, w hav modld th nos color by a sngl paramtr rprsntng th slop of a spctral dnsty functon that dcrass monotoncally wth frquncy by followng Eq. [] S( f ) (7) f Whr f s th frquncy, s th varanc of th orgnal sgnal and β s th slop; a masur of nos color. Wht nos (β=), pn or flcr nos (β=), and brown nos or th random wal procss (β=), ar thr of th most commonly rfrncd noss.wht nos and color nos ar smulatd havng 3dB nput SNR usng MATLAB. (b) Ral Nos Ral noss ar xtractd from th nos strss tst databas n MIT-BIH []. Low-ampltud muscl nos s common n ECG. Muscl nos s, n contrast to basln wandr and 5/6 Hz ntrfrnc. Mchancal movmnt of rcordng lctrods or sn strtchng rsults n changs n potntal. Du to altraton n th physcal dmnsons of th lctrod-sn half cll thus modfyng cll potntal and sn-lctrod mpdanc []. Th proposd modl s tstd on th ECG data rcordd from BIOPAC data acquston systm [3]. Th nosy sgnal was gnratd by addng BW, EM, and MA artfacts (Nos Strss Tst Databas of MIT-BIH) and wht and color nos wr addd to th clan ECG sgnals []. 3. MATERIALS AND METHODS 3. ECG Th xprmnt s prformd to collct th data for ths rsarch wor. Th subct s a mal of 6 yars old wth no nown cardovascular dsordr. For ECG and masurmnt, rqurd qupmnts ar BIOPAC lctrod lad st (SSL), BIOPAC dsposabl vnyl lctrods (EL53), BIOPAC data acquston unt (MP36) [] wth cabl and powr. For ECG masurmnt, wht lad was placd on rght forarm, ICME RT-

3 rd lad on th lft lg and th blac lad was placd on rght lg as shown n Fg.. Subcts was satd n a char rlaxng and asd to b as stll as possbl to nsur lowr moton artfact and EMG sgnal on th data. Aftr runnng calbraton squnc ECG data was rcordd. rqurd to solv ths problm. Consdr that an ECG functon y=f(x) of a varabl of x tabulatd at valus, whr y =f(x ), y =f(x ) y =(x ). Morovr, assum that th nown analyss form th functon dpndng on paramtrs f x;,,., ) and th st of quaton wll b ( y y y f ( x;,,., ) f ( x ;,,..., ) f ( x ;,,..., ) (8) (a) W hav to solv th quaton to obtan th valu of,,., whch satsfs our modl proprly. At frst an ntal valu s pcd for and dfnd d y f x,,.., ) (9) ( ; And thn th stmaton for th chang d ndd to rduc d to d l dl d l x For =,,., whr,,..., ) ( () Fg. (a)mp36 Bopac (b)placmnt of lad n ECG masurmnt. 3. Prprocssng Th acqurd ECG data ar prprocssd to rmov nos, artfacts, and basln wandr usng Savtzy-Golay Fltrng [3]. To ths nd, two frquncy-slctv fourth-ordr Buttrwort fltrs [4] ar usd: on hgh-pass fltr wth cutoff frquncy at.5hz and on low-pass fltr wth cutoff frquncy at 9 Hz. To supprss th ntrfrncs from th powr ln grd, a notch fltr cntrd around 5 was usd. Agan, ths fltr s mplmntd as a fourth-ordr Buttrworth fltr l [5]. As from th proprts of Gaussan wav t s nown that, th wav do not cross th zro. But P, Q, R, S and T wavs cross th zro ln n th ral ECG sgnal. To solv ths zro crossng problm.5 was addd to ral ECG sgnal bfor calculatng th coffcnts, so that th basln of ral ECG ls n zro ln. 3.3 Nonlnar Fttng For fttng th mathmatcal modl wth th ral world data statstcal hypothss tstng l: tst of normalty of rsduals, ch squar tst, analyss of varanc, last squar tst tc [6] s ndd. As t obsrvd that, ECG modl s nonlnar n th coffcnts. So th nonlnar last squar tchnqus can b th bst choc for ths fttng. Nonlnar modls ar mor dffcult to ft than lnar modls bcaus th coffcnt cannot b stmatd usng smpl tchnqus nstad an tratv approachs Ths lmnt can b wrttn as a matrx of partal drvatvs of A l Thn, d d d l x x x. d d d x x x l () d A d () And th brf quaton s d Ad (3) If by dfnng a=a T A and b=a T dβ W fnd, a d b (4) Thn Eq. (4) s solvd for d usng Gaussan lmnaton tchnqus. Ths offst s appld to α and a nw d s calculatd. By ntractvly applyng ths procdur untl th lmnts of d bcom smallr than dsrd lmt, a soluton s obtand. Th sum of squar rsduals s calculatd by R d. d aftr th fnal ICME 3 RT-

4 Ampltud traton. 3.5 Prformanc Evaluaton Paramtrs 3.5. Cohrnc Th magntud squard cohrnc (MSC) stmat s a functon of frquncy wth valus btwn and that ndcats how wll th modl ECG corrsponds to ral ECG at ach frquncy. Th MSC stmat C xy of th nput sgnals (x and y) usng Wlch's avragd, modfd prodogram mthod [7]. Th MSC s nothng but a functon of th powr spctral dnsts (P xx (f) and P yy (f)) of x and y and th cross powr spctral dnsty (P xy (f)) of x and y. P ( f ) C xy xy P ( f ) P ( f ) (5) xx yy In ths papr; x, y rprsnts th modl and ral ECG sgnal rspctvly and x and y must b th sam lngth Powr spctral dnsty (PSD) PSD rprsnts th strngth of th nrgy as a functon of frquncy [7].In othr words, t shows at whch frquncs varatons ar strong and at whch frquncs varatons ar wa. Enrgy can b obtand wthn a spcfc frquncy rang by ntgratng PSD wthn that frquncy rang. Computaton of PSD s don drctly by th mthod calld FFT or computng autocorrlaton functon and thn transformng t Cross-corrlaton coffcnt If x( b th rcordd or collctd ECG sgnal and x m ( b th ECG sgnal gnratd by th mathmatcal modl, thn cross-corrlaton coffcnt ρ btwn x ( and x m ( s gvn by [8] ][ x ( ] x m m (6) x m Whr dnots th avrag calculatd by summng ovr th obsrvd tm srs, ndxd by n. whr µ x and σ x ar th man and standard dvaton of x(, and m and m ar th man and standard dvaton of (. A valu of ρ rflcts a strong x m corrlaton, ρ mpls a strong antcorrlaton, and ρ ndcats that x( and x m ( ar uncorrlatd. Ths mans that a valu of ρ = suggsts that modl and ral ECG ar dntcal Error Assssmnt: Th Man Squar Error (MSE) s dfnd as [9]. N N n MSE [ x ( n ) x m ( n )] (7) Th normalzd form of MSE s N [ x( xm ( ] n NMSE (8) N n ] Anothr masurmnt s Root Man Squar Error, whch s N N n RMSE [ x ( n ) x m ( n )] (9) Th Normalzd vrson of RMSE s NRMSE N n x N n m ] ( ] Prcnt Root Man squar Dffrnc (PRD) can b dtrmnd by Eq. (6). N () xm( ] n PRD % () N n ] 4. RESULT AND DISCUSSION Th coffcnt of th modl s calculatd by nonlnar last squar tchnqu usng Gauss-Nwton algorthm havng 95% confdnc lvl and th coffcnt ar shown n Tabl. Tabl : Coffcnt to ft th modl wth BIOPAC rcordd ECG ECG Coffcnt Componnt A B t P Q = = R S T Usng th valu of th modl coffcnt, th ral and modl ECG ar qut smlar. Ths s rprsntd n fg Ral ECG Modl ECG Sampls Fg. Comparson btwn ral and modl ECG sgnal ICME 4 RT-

5 MSC PSD To valuat th prformanc of th modl, goodnss of fttng was calculatd usng Eq. (7) ~ Eq. () and th valus ar gvn n Tabl. It s obsrvd that, th valus of dffrnt rrors wr qut low and th valu of cross-corrlaton coffcnt was.95 whch ndcatd th hghly corrlaton btwn th ral and modl ECG. (Slow hart rat), Tachycarda (Fast hart rat) tc. From Fg. 5, brachycarda was smulatd for 5 sc possssng 4 bat. So th bat pr mnut (BPM) was 48. Snus rhythm and tachycarda wr smulatd n th sam way. Tabl : Goodnss of fttng Goodnss of fttng Valu (BIOPAC) MSE.779 NMSE.7477 RMSE.8865 NRMSE.9748 PRD Th modl was not only valuatd n th tm doman but also n th frquncy doman. Fgur 3 and Fg. 4 showd th powr spctral dnsty (PSD) and magntud squar cohrnc (MSC) rspctvly. In Fg 3, th modl and ral ECG ar rasonably smlar and th MSC n fg 4, was xsts n most of th frquncs whch s a bttr ndcaton of th proposd modl Powr spctral dnsty Frquncy (Hz) Fg 3. Comparson of powr spctral dnsty btwn ral and modl ECG sgnal Modl ECG Ral ECG Frquncy(Hz) Fg 4. Magntud squard cohrnc (MSC) btwn ral and modl ECG sgnal. Th modl not only smulat ECG bat but can smulat dffrnt hart condton l Brachycarda Fg 5. ECG smulaton by modl for 5 Sc. For brachycarda wth BPM 48, snus rhythm wth BPM 7, tachycarda wth BPM 8. Anothr fatur of th modl s that, t can gnrat ralstc and smulatd nosy ECG sgnal. ( ) of N, SNR t Eq. (6) ndcats th nos paramtr of th modl. Dffrnt nosy ECG sgnal (dffrnt ) wr smulatd for th nput SNR of 3dB n fg Wht nos Ral muscl artfacts Ral basln wandr Colord nos Ral lctrod movmnts Compost nos Sampl Sampl Fg 6. Dffrnt typs of nosy ECG sgnal. 5. CONCLUSION Th proposd modl s capabl of rplcatng many mportant faturs of th human ECG wav. A numbr of faturs and applcatons of th modl ar: Instad of usng too many paramtrs l othr, th proposd systm can gnrat ECG and capabl of smulatng varous nd of practcal phnomna such as brachycarda, tachycarda. It dos not nd thr dmnsonal stat spacs whch ar dffcult for ralzaton and smulaton. Nosy ECG sgnal can b modld by smply addng a nos paramtr wth th modl. In frquncy doman th ral and modld ECG showd almost sam proprts. A ralstc ECG databas can b cratd by ICME 5 RT-

6 fttng th modl wth ndvdual subct s ECG. Ths can b usd for furthr analyss and for ducaton purpos. By only savng th coffcnt of th modl, ECG data comprsson can b possbl. Howvr, th lowr valu of rror paramtrs l MSE.779 ndcats th ffctvnss of ths smplfd modl and th proposd modl nabls us to nvstgat wav morphology varaton. It can b mprovd for bttr fttng, modl-basd dnosng, comprsson and nural ntwor basd classfcaton tc whch s undr nvstgaton. Fnally, t s hopd that ths smpl modl wll provd an ffcnt tool for tstng and procssng of th ECG sgnals wth dffrnt lvl of nos and/or moton artfact. 7. REFERENCES. Murthy, Ivatur S. N., Rangara, M. R., Udupa, K.J., Goyal, A. K., Homomorphc Analyss and Modlng of ECG Sgnals, IEEE Trans Bomd Eng., vol. BME-6 Issu:6 pp ,Jun Murthy, I.S.N., Prasad, G.S.S.D., Analyss of ECG from pol-zro modls, IEEE Trans Bomd Eng., vol: 39, ssu:7 pp ,July Suppappola, S., Sun, Y., Charamda. S.A., Gaussan puls dcomposton: An ntutv modl of lctrocardogram wavforms, Annals of Bomdcal Engnrng. vol. 5, no. pp. 5-6, Mar-Apr, Clfford, G.D., Vllarrol, M., Modl-Basd Dtrmnaton of QT Intrvals, IEEE Computrs n Cardology, vol.33 (53) pp , Sp Parvanh, S., Pashna, M., Elctrocardogram Synthss Usng a Gaussan Combnaton Modl (GCM), Computrs n Cardology, vol. 34, pp. 6 64, Rchardson, J., Haywood, L.J., Murthy, V.K., and Harvy,G., A mathmatcal modl for ECG wav forms and powr spctra, Mathmatcal Boscncs, vol., ssus 3-4, pp. 3-38, Dc Sornmo, L., and Laguna,P., Bo-Elctrcal Sgnal Procssng n Cardac And Nurologcal Applcatons, Elsvr Acadmc Prss d., Jun 5, Tam, H., and Wbstr, J. G., "Mnmzng lctrod moton artfact by sn abrason", IEEE Trans. Bomd. Eng., vol. 4, pp , Bgg, R., La, D.T.H., Palanswam, M., Computatonal Intllgnc n Bomdcal Engnrng, CRC Prss, 8.. Wornll, W. G. and Oppnhm, A. V., Estmaton of fractal sgnals from nosy masurmnt usng wavlts,ieee Trans. Sgnal Procss., vol. 4, no. 3, pp. 6 63, Mar www. physont.org. Bopac Studnt Lab 3.7, BIOPAC Systm, Inc., Awal, M.A., Mostafa, S.S., and Ahmad, M., Prformanc Analyss of Savtzy-Golay Smoothng Fltr Usng ECG Sgnal, Intrnatonal Journal of Computr and Informaton Tchnology, vol., no., January. 4. Buttrworth, S., On th thory of fltr amplfrs, Exp. Wrlss Eng., vol. 7, pp , Oct Vullngs, R., Vrs, B.D., and Brgmans, J.W.M., An Adaptv Kalman Fltr for ECG Sgnal Enhancmnt, IEEE Transactons on Bomdcal Engnrng, Vol. 58, No. 4,pp 94-3, Aprl. 6. Prss, W.H., Tuolsy, S.A., Vttrlng, W. T., Flannry, B. P., Numrcal rcps n C++, nd d, Cambrdg Unvrsty Prss. 7. Wlch, P. D., Th us of fast Fourr transform for th stmaton of powr spctra: a mthod basd on tm avragng ovr short, modfd prodograms, IEEE Trans. Audo Elctroacoustc, vol. AU-5, pp. 7 73, Jun Clfford, G. D., Azua F.,. McSharry, P. E., Advancd Mthods and Tools for ECG Data Analyss, Artch hous, nc Manandan, M.S., Dandapat, S., Wavlt nrgy basd dagnostc dstorton masur for ECG, Bomdcal Sgnal Procssng and Control, vol., ssu, pp 8-96, Apr MAILING ADDRESS Md. Abdul Awal Dpartmnt of Bomdcal Engnrng, Khulna Unvrsty of Engnrng & Tchnology, Khulna-93, Bangladsh E-mal: awalc4@yahoo.com Shh Shanawaz Mostafa Dpartmnt of Bomdcal Engnrng, Khulna Unvrsty of Engnrng & Tchnology, Khulna-93, Bangladsh E-mal: shawan44@gmal.com Mohuddn Ahmad Dpt. of Elctrcal and Elctronc Engnrng, Khulna Unvrsty of Engnrng & Tchnology, Khulna-93, Bangladsh Emal: ahmad@.ut.ac.bd ICME 6 RT-