Rhythm of the Heart. Megan Levine. Abstract

Transcription

1 Rhyhm of he Hear Megan Levine Asrac An overlook of he hear and how i works as a pump wih an eplanaion of how o uild a mahemaical model using differenial equaions. The Hear. The Hear Bea The hear ea is he only angile, eernal sign we have ha ells us our hear is eaing in a seady rhyhm all of he ime. Each ea represens a pump of lood hrough he hear and ou o oher pars of he ody. This process is driven y a pach of cells, in he upper par of he hear, called he Sinoarial Node. These cells send waves of elecrochemical aciviy over he hear which causes i o conrac and rela over and over again.. Blood Flow The lood flow is coninuous and a cycle like he hear ea cycle. If we sar where unoygenaed, lue lood eners he hear, we are a he vena cava. Blood goes hrough his o ener he righ arium, and hen ino he righ venricle, and hen pumped ou o he lungs hrough he pulmonary areries. The lungs echange oygen for caron dioide and hen he lood is sen ack o he hear hrough he pulmonary veins. These veins ake he lood o he lef arium, hrough o he lef venricle and pumped ou of he hear o he res of he ody hrough he aora. The pump needed o send lood o he lungs is weaker han he pump used o send lood o he res of he ody. Therefore, he righ side of he hear is a low-pressure pump, while he lef side of he hear is a high-pressure pump..3 Equilirium Saes The hear has wo differen saes of eing where i is in equilirium. Diasole is he relaed sae of he hear. I is a sale equilirium poin ecause he hear always reurns o diasole, even afer i sops eaing. Sysole is he conraced sae.

2 .4 Hear Bea Cycle A logical place o sar would e diasole, he sale equilirium. The pacemaker sends a wave of elecrochemical aciviy over he hear. Firs i goes over he aria, causing hem o slowly conrac. This is he firs ligh pump ha fills up he venricles wih lood. This is acually he arium s jo; o fill up he venricle efore he srong pump, so o ge he mos efficien amoun of lood ou o he res of he ody. There is a poin where he slow wave of elecriciy changes o a rapid wave. This goes over he venricles, causing hem o quickly conrac, which in urn pumps lood ou of he hear. The poin of he conracion is he sysole equilirium. Afer ha, here is a rapid relaaion ack o diasole. The cycle repeas iself, slowing down or speeding up in response o eernal or inernal condiions, such as sleeping or running, aniey or relaaion. Derivaion of he Differenial Equaions. Sipulaions. The model mus have an equilirium poin a diasole and a sysole.. The model needs o show a hreshold for riggering he elecrochemical aciviy ino changing from slow o rapid. 3. The model should reflec a rapid reurn o diasole equilirium afer reaching sysole.. Variales represens he lengh of muscle fier. represens he elecrochemical aciviy. T represens he over-all ension of he sysem. This is also he main parameer for he sysem of differenial equaions..3 Hypohesis. The rae of change of muscle fier lengh, ino and ou of conracion, depends on ension, T, and elecrochemical aciviy,.. The chemical conrol,, changes a a rae direcly proporional o muscle fier lengh,.

3 .4 Where o Begin? The sysem of differenial equaions is = f(, ) = g(, ) I has already een deermined ha diasole is an equilirium poin, so i will e denoed as (, ). Sysole will e denoed y he poin (, ). When he sysem is in diasole f(, )=g(, )=. If we linearize aou he equilirium poin we will end up wih hese equaions. = a ( )+a ( ) = a ( )+a ( ) Higher order erms have een omied ecause hey are negligily small near he equilirium. In his equaion a is he parial derivaive of f wih respec o, a is he parial derivaive of f wih respec o, a is he parial derivaive of g wih respec o, anda is he parial derivaive of g wih respec o. The Jacoian mari is ( ) a a J(, ) = a a Diasole is a sale equilirium poin, herefore, he eigenvalues of mari J need o e real, o avoid oscillaory ehavior, and negaive, so i acs like a sink. We also wan o reflec in he equaions hypoenuse, which says ha changes a a rae direcly proporional o. We can herefore say ha a = and a =. Now for he eigenvalues o e oh real and negaive, a and a mus oh e negaive. The model needs o also show a rapid reurn o equilirium, which can e accomplished y making a large and negaive. a also needs o e large and negaive, wih a eing larger relaive o a. To make hings easier o undersand and wrie. I am going o inroduce wo new variales o help represen a and a. a = a ε a = ε a and ε are oh posiive consans, wih epsilon eing small. Now he sysem can e wrien as ε = a( ) ( ) =( ). This sysem is a good model around he diasole equilirium u i is a ad approimaion away from his poin. The local model does no reflec he riggering of elecrochemical aciviy from slow o rapid. Researchers, like E.C. Zeeman, found a modified equaion ha saisfies he necessary riggering, hrough 3

4 rial and error. ε = a( ) ( ) ( ) 3 3 ( ) = I was ale o simplify he equaion for he derivaive of wih hese compuaions. ε = [a( )+( )+( ) 3 +3 ( ) ] = [a a ( + )] = [ a a + ] = [ a a + ] = [ 3 (3 + a)+ +(3 a )] A his poin i is good o define ension o e T =3 + a and so 3 a = = 3 a. These equaions are also rue if we susiue he diasole equilirium poin for he sysole equilirium (, ). The sysem is more simply wrien as ε = ( 3 T+ ) () =. () The sysem is a very good model of he hear ea ecep ha i acs funny around sysole equilirium. The flow, in he phase plane, is going he wrong direcion around sysole. This prolem occurs ecause he sysem was made o represen he iological phenomena around one equilirium poin, diasole. The model needs o e old when o change and wha o change o when i is near sysole. The prolem in he direcion of he flow is ha i is going righ insead of lef, wih respec o he -ais. There is no prolem wih he up or down flow, herefore he only equaion ha needs o e changed is. When he flow is near sysole he equaion of -prime needs o change o =. If we incorporae a sep funcion U. =( )+U( ) When U is equal o one, cancels ou and he equaion ecomes =. 4

5 The -values where U is equal o one is when is eween and,andalso a he same ime 3 T+ >. Acually, his jus means ha < orhe lengh of he muscle fier is decreasing or oherwise pu,conracing. If hese wo aren rue a he same ime hen U can also equal one if >,oherwise pu, he amoun of elecriciy is larger han ha needed o e in sysole. The sep funcion is wrien U =(> )+[(> ) ( < ) ( 3 T+ >)] If he erm inside he parenhesis is rue, ha erm equals one and he oher is zero, and U =so =. If he erms inside he rackes are all rue ha erm equals one, he oher erm is zero, U =and =. A any oher ime U =and =. Now we have he complee sysem of equaions. ε = ( 3 T+ ) (3) =( )+U( ) (4) 3 Modeling Beas To model his sysem I used Mala s ODE suie and found ha ode3s, he siff solver, worked es. The ode funcion file I wroe wen like his, funcion yprime=hear(,y,flag,e,t,,,,) yprime=zeros(,); U=(y()>)+((y()>=)&(y()<=)&(y()^3-T*y()+y()>)); yprime()=(y()-)+u*(-); yprime()=(-/e)*(y()^3-t*y()+y());. To generae graphs of he model I wroe his small program, opions=odese( refine,8); [,y]=ode3s( hear,[,],[;],opions,e,t,_,_,_,_); a=aes( posiion,[.,.4,.8,.5]); plo(y(:,),y(:,)),lael( ),ylael( ) a=aes( posiion,[.,.,.35,.]); plo(,y(:,)),lael( ),ylael( ) a3=aes( posiion,[.55,.,.35,.]); plo(,y(:,)),lael( ),ylael( ) The firs hing I did in he program, was o se he refinemen of he numerical approimaor higher han normal. This gave me a more accurae approimaion. The aes command efore each plo command is how I posiioned he graphs of versus, versus, and versus. When graphing his phase plane, i is desirale o graph he elecrochemical aciviy on he independen aes, ecause i eer represens he flow from diasole o sysole and ack o diasole. This also affecs how I wroe he ode funcion file. The firs equaion of he sysem is. 5

6 Figure : Tension equals four A prolem wih he model is ha i is hard o prove ha here will e a closed rajecory, or limi cycle, ha he graph goes o. If he ension is se oo high, he hear will never e ale o make i ou of diasole, u if i is oo low, he conracion ino sysole will e slow, and no give a hard enough pump o send lood o all pars of he ody. A model wih good resuls is presened in Figure where I se T =4and(, )as( 4, ), and (, )as(4, ). The soluion o he sysem in Figure has an iniial condiion of (, ) If we change he iniial condiion o somehing ouside of he rajecory cycle,like (5, ), i will sill approach he limi cycle. If I se he ension oo high, T = 5, he sysem will no e ale o leave diasole far enough o reach sysole. Figure 3 shows his siuaion wih an iniial condiion a (, ). 6

7 Figure : Iniial Condiion a (5, ) 7

8 Figure 3: Tension is equal o 5 8

9 Figure 4: Tension equals zero Anoher model o look a is when he ension is oo low. Figure 4 has he ension se as T =. In his scenario he ension is so low i reaches sysole u canno leave i. 3. Conclusion This is a good eginning for a model of he hear ea. Also, he processes involved are good raining for modeling oher iological phenomena in he fuure. There is one prolem ha I have noiced aou his model. The hear consiss of wo pumps. The righ hand side is a low-pressure pump ha sends lood o he lungs. The high-pressure pump is on he lef-side of he hear and i sends lood o he res of he ody. These wo pumps do no ea simulaneously u one righ afer he oher, and ogeher hey make one ea of he hear. The model does no represen wo pumps of differing srenghs. This is why he graph of he elecro-chemical aciviy versus ime has he same ampliude for every wave of elecriciy. I would look more like an elecrocardiagram if he wo pumps were oh represened in he model. Over ime and wih research a eer model will e found, hopefully I ll have somehing o do wih i. 9

10 References [] Dany, J.M.A., Compuing Applicaions o Differenial Equaions Resin Pulishing 985 [] Hoppensead, F.C., Peskin, C.S., Mahemaics in Medicine and he Life Sciences Springer-Verlag 99 [3] Jones, D.S, Sleeman, B.D., Differenial Equaions and Mahemaical Biiology George Allen & Unwin 983 [4] Srogaz, Seven H., Nonlinear Dynamics and Chaos Addison-Wesley 994