JOY: The Journal of Yoga Summer 2008, Volume 7, Number 2

Transcription

1 JOY: The Journal o Yoga Summer 008, Volume 7, Number The Mahemaical modeling o Pranic Body Saria*, VKKaiyar**, PPradhan * *Dearmen o Mahemaics & Saisics Gurukul kangri Univerisiy, aridwar, Uarakhand India **Dearmen o Mahemaics Indian Insiue o Technology Roorkee, aridwar, Uarakhand India sariamah@gmailcom Absrac In Yogic anaomy, each o us has nine bodies oher han he hysical one we are wellacquained wih One o hose bodies is called Pranic body ere by Pranic body is o be hough as he breah which we all inake In his aer we develoed a mahemaical model or dieren breahing aern or unseady-sae equaions All hese equaions are ime deenden under consan meabolic rae A comarmen model o breah uncion rom lungs o issues is discussed and we have also discussed he analyical soluion o breah exressions The Variaion o he concenraion o oxygen in hear, lungs, and in body or normal breah is shown in ig (), ig (), and in ig () Key words: Anaomy, seady-sae, meabolic rae Inroducion Pranayama is an imoran, ye lile known ar o Yoga Is echniques have been raciced or cenuries by arden sudens o yoga in remoe ashrams, and have been resened or us hrough many generaions boh in racice and in hand wrien books

2 This ar and science o yogic breahing was almos comleely unknown o he conman man like many oher ancien Indian ars The modeling o Pranic body discussed by Brain Bergen (006) by ug seady sae condiion o he governing equaion [] and also discusses he behavior o lung, hear, body cells and breahes uncion or one minue breah In his aer we have develoed a comarmen model o breah uncion rom lungs o issues and we discussed he analyical soluion o breah exressions The main assumion or he irs model we consruced ha he oxygen eners and exis by he lungs wih breah uncion R (), wih uni volume/ime and hen ranserred o he hear/bloodsream a a rae ha deends on he amoun available in he lungs and he amoun already in he blood Since i is ossible ha here is sill oxygen le in he lungs (indeed here is a minimal volume o abou L, and he air ha ges caugh in he bronchial ree a he end o he inhale will be he irs o be exhaled [5]) The change in he concenraion o oxygen in he lungs due o exhaling is also considered We assume ha he low rom lungs o he blood deends on he concenraion already in he blood ce oxygen binds o he hemoglobin in he blood In realiy, his is a non- linear relaionshi [5], bu or simliciy, i is assumed ha i is direcly roorionae o he dierence in concenraions o he wo chambers So we assumed ha oxygen lows rom he blood o he cells o he body and Vice-Versa Finally, he oxygen is consumed by he body a a rae roorionae o he amoun available and he meabolic rae M []

3 Mahemaical Model & Equaions () () () (4) (5) Lungs R () ear/blood sream Tissue/body cells } { ) ( ) ( ) ( ; ) ( ) ( Q Q A R where MB B d db B L d d RQ L L V R d dl

4 Variables and Parameers L concenraion o oxygen in lungs concenraion o oxygen in hear/blood B concenraion o oxygen in body/cells M meabolic rae how much oxygen is being used by he body R breahe uncion Inhale R>0 Exhale R<0 Maximum rae o ranser o molecules rom he lungs o he bloodsream V Maximum rae o ranser o molecules rom he bloodsream o he body cells Maximum lung volume Period o breah I is also he breahing eriod Q < 4 L/min, amoun o oxygen All he concenraion o make sense, L 0, 0, B 0 Analyical Soluions o he Equaions Solving he above simulaneous dierenial equaion(-) and we ge he values o L, B, and such as B A Q A V ( M ) (6) 4

5 ( ) ( ) [ ] ( ) cos M A R Q M RQ A Q (7) Where cos ( ) R V M And he concenraion o oxygen in lungs ( ) ( ) [ ] [ ] ( ) B Q A Q A L ( δ β α Where he consan symbols used as: (8) cos 4 cos 4 5 cos cos 6 ( ) R Q AQ M cos 7 ( )( ) A V M cos 8 5

6 9 RQ ( M ) α A ( M ) Q β ( ) [ A Q 9 ] A R Q( M ) cos δ ( ) ( ) A ( M ) R cos V V The consan values o he included arameers such as 4L/min, 600ml, 07/s, and 00/s, Q, V,, M are 0/s, Analysis I he uncion R does no deend on ime, hen he sysem becomes auonomous, as in he case, where one is orever inhaling or exhaling When R0, Q0 and R>0 So, as execed, he concenraion o oxygen in he body goes o 0 I one is always exhaling More inereg is he case or R>0 The breah uncion R is linearly deends uon he concenraion o oxygen The basic resuls obained rom he equaions are: R B redicaed B acual Resuls In order o generae he ollowing resuls, we used he ollowing arameers, V600ml, 0/sec, 07/sec, M00/sec For he normal breah uncion, R was calculaed rom he idal volume (abou 500ml) and he imum lung volume was used and he breah rae is deined as 0sec inhale, 0sec susension, 0 sec exhale 6

7 We observed he uncions L,, and B (Concenraion o oxygen in lungs, hear and body or he normal breah in ig Comare i o he one minue breah uncion, in ig Though i aears ha he one minue breah hasn a chance o reach equilibrium Aer only one minue, i seems similar o he normal breah in value, exce here is some smoohing in he amoun o oxygen in he lungs The breah o ire aern, however, rums he oher wo The breah o ire reaches a non-oscillaory equilibrium For he Pranic body, rom he ig () i is clear ha he concenraion o oxygen in lungs (when we inhale), irs i will increase coninuously, aer a ime eriod i will become consan ie he sae o sauraion The condiion is ulilled also or he exhalaion ie when we release oxygen rom our body irs i will decrease hen aer some ime eriod his is coninuous, his sae is also called sae o sauraion From ig (), in he body cells he oxygen alls eriodically and becomes consan or one minue In ig (), iniially he concenraion o oxygen in lungs is very low, as soon as ime increases concenraion increases u o a imum value and raidly alls down o he saring oin Aer ha concenraion becomes aroximaely consan during one minue ime inerval in dee breahing osiion Similarly he concenraion o oxygen in body increases iniially and becomes aroximaely consan The concenraion o hear alls down iniially and becomes aroximaely consan during one minue ime inerval in dee breahing osiion 7

8 Fig (): Normal breahing simulaion or one minue (X-axis shows ime, Y-axis shows L, B, uncions) 8

9 0 08 L,,B ime Fig (): one minue breahe simulaion 9

10 breahe uncion ime in minues Fig (): Comarison o hree breahing aerns (normal, one minue, Breah o ire) over minues 0

11 Reerences ) Bhajan, Y (kundalini Research Insiue 005) The Aquarian Teacher KRI Inernaional Yoga Teacher Training Level ) Peskin, C, oensesd, F (Sringeer 00) Modeling and Simulaion in Medicine and he Lie Sciences second ed ) Brain Breger, An Advenure in Modeling: The ranic Body 4) Edelsein-Keshe, L (Siam 005) Mahemaical Models in Biology classes ediion 5) Planzer, R Basal Meabolic Rae h://wwwbioaccom 6) Waler, L, ow Does Weaher Aec air Pressure? 008 JOY: The Journal o Yoga wwwjournaloyogaorg