Estimation of Diffusion Coefficient in Gas Exchange Process with in Human Respiration Via an Inverse Problem

Transcription

1 Ausralian Journal o Basic and Applied Sciences 5(): ISS Esimaion o Diusion Coeicien in Gas Exchange Process wih in Human Respiraion Via an Inverse Problem M Ebrahimi Deparmen o Mahemaics Karaj Branch Islamic Azad Universiy Karaj Iran Absrac: This paper is inended o provide a sochasic approach involving he combined use o he Feynman-Kac ormula and Mone Carlo mehod as a soluion algorihm or esimaing he imedependen eecive diusiviy in a one-dimensional parabolic inverse problem The inverse problem is purposed o design a mahemaical model or he gas-diusion process wihin he alveolar region o he human's lung The model depends on a represenaive physical propery o he alveolar region ermed he eecive diusiviy In he presen sudy he uncional orm o he eecive diusiviy is unknown a priori The unknown eecive diusiviy is approximaed by he polynomial orm To modiy he coeiciens o he polynomial orm o he unknown eecive diusiviy we inroduce a deerminisic opimizaion mehod based on leas squares minimizaion A numerical es is perormed in order o show he eiciency and accuracy o he presen work Key words: Gas-exchange process Human respiraion Inverse problem Feynman-Kac ormula Mone Carlo mehod ITRODUCTIO In he presen work a speciic sochasic combined algorihm is used or obaining he soluion o inverse diusion equaion as par o a parabolic inverse problem ha arise in gas-diusion process wihin he alveolar region o he lung during human respiraion This algorihm uses Feynman-Kac ormula o represen he soluion o he parabolic inverse problem a a poin as he expeced value o uncionals o Brownian moion rajecories sared a he poin o ineres We are ineresed o have soluion o a parabolic inverse problem wihou any need o discreize he problem domain For many problems described by parial dierenial equaions (PDEs) such soluions are delivered by he so-called Feynman-Kac ormulas (Csaki E 993; Modese 006; Budaev BV and DB Bogy 003) The lieraure reviews showed ha E Csaki (993) have applied Feynman- Kac ormula o solve iniial-boundary value problems In (Csaki E 993) a discree Feynman-Kac ormula has employed or linear parabolic PDEs wih zero boundary condiions Modese e al (006) have used a kind o nonlinear Feynman-Kac ormula o give a probabilisic inerpreaion o he soluions o parabolic quasi-linear PDEs Budaev and Bogy (003) presened a probabilisic approach o sysems o PDEs on he basis o he wellknown Feynman-Kac ormula To dae various mehods have been developed or he analysis o he parabolic inverse problems involving he esimaion o boundary condiion or diusion coeicien rom measuremen inside he maerial (Shidar A 009; Wang J and Zabaras 004; Shidar A 007; Shidar A 006; Shidar A 006; Farnoosh R and M Ebrahimi 00; Dehghan M 005) Shidar e al (009) have applied an algorihm based on conjugae gradien mehod o esimae he unknown ime dependen mel deph during laser maerial processing in liquid phase In his aricle he deerminaion o he mel deph is reaed as a onedimensional ransien inverse hea conducion problem umerical procedure shows a good agreemen wih experimenal and analyical resuls In (Wang J and Zabaras 004) Wang and Zabaras have used a Bayesian inerence approach o he inverse hea conducion problem Their work is based on Mone Carlo mehod and experimen resuls show a good esimaion on he linear inverse problems in wo dimensions Shidar e al (007) have used an accurae and sable numerical algorihm based on inie dierence mehod o solve an inverse parabolic problem in one dimension To our bes knowledge he problem o gas-exchange in human respiraory sysem wih unknown ime dependen diusion coeicien has no been sudied Furhermore according o laes inormaion rom he research works i is believed ha he soluion o inverse problem based on sochasic algorihm included he Feynman-Kac ormula has been invesigaed or he irs ime in he presen sudy Descripion o he Problem: Formulaion o he direc and inverse problem is given as ollows: I) Direc Problem: The mahemaical ormulaion o a one dimensional linear parabolic problem is given as ollows: Corresponding Auhor: M Ebrahimi Deparmen o Mahemaics Karaj Branch Islamic Azad Universiy Karaj Iran moebrahimi@kiauacir 333

2 Aus J Basic & Appl Sci 5(): u a( uxx 0 x 0 u( x0) ( x) 0 x u( 0 ( 0 u( ( 0 () () (3) (4) where is he inal ime or measuremens The direc problem considered here is concerned wih he deerminaion o he medium parial pressure when he ime-dependen diusion coeicien a ( he iniial condiion (x) and he boundary condiions ( and ( are known coninuous uncions Theorem : The problem ()-(4) has a unique soluion i (x) ( and ( are coninuous uncions and a ( 0 (Cannon JR 984) II) Inverse Problem: For he inverse problem he diusion coeicien a ( is regarded as being unknown In addiion an overspeciied condiion is also considered available To esimae he unknown coeicien a ( he addiional inormaion o measuremens on he boundary x x 0 x is required Le he parial pressure measuremens be denoed by aken a x x over he ime period u x h( 0 (5) ( The addiional condiion is perormed based on simulaing numerically he gas-diusion process during a single-breah lung-diusing capaciy es rouinely perormed in clinical seings (Kulish V 006) I is eviden ha or an unknown uncion a ( he problem ()-(4) is under-deermined and we are orced o impose addiional inormaion (5) o provide a unique soluion pair ( u a( ) o he inverse problem ()-(5) We noe ha he measured parial pressure u( x h( should conain measuremen errors Thereore he inverse problem can be saed as ollows: by uilizing he above-menioned measured daa esimae he unknown uncion a ( over he enire space and ime domain I is worh noing ha a ( 0 (Cannon JR 984) Cerain ypes o physical problems can be modeled by ()-(5) The coeicien a ( can represen physical quaniies or example he conduciviy o a medium The exisence and uniqueness o he soluions o his problem and also some more applicaions are discussed in (Dehghan M 005; Cannon JR 984) The numerical soluion o he problem ()-(5) has been discussed by several auhors For example i can be ound ha he numerical based on several inie dierence schemes proposed by Dehghan (005) applied o he above menioned parabolic inverse problem ()-(5) The resuls show ha he accuracy o his work are very reasonable Overview o he Soluion Algorihm: The applicaion o he presen mehod o ind he soluion o problem ()-(5) can be divided ino he ollowing seps: Sep Remove he Time Dependen Diusion Coeicien: We uilize he ransormaion a( a( y) dy 0 0 o reduce he equaion () o ha involving he diusion equaion wih consan diusion coeicien Since 334

3 Aus J Basic & Appl Sci 5(): ( a( 0 0 hen here exiss a unique uncion ( ) such ha ( ( )) 0 ( ) ( ( ) 0 and ( ) ( ( )) (a( ( )) (a( ) 0 ( ) Le U ( x ) u( x ( )) hen U ( x ) u ( x ( )) ( ) u ( x ( ))( a( ( ))) u xx ( x ( )) U xx ( x ) Consequenly o obain he represenaion or u ( x we subsiue ( ino he represenaion or U ( x ) Thereore he problem ()-(5) becomes U U xx 0 x 0 (6) U ( x0) ( x) 0 x U 0 ) ( ( )) 0 T ( U ( ) ( ( )) 0 T (7) (8) (9) U ( x ) h( ( )) 0 (0) T where T ) ( Sep Feynman-Kac Formula: In his sep we begin wih a mehod o solve PDEs based on he represenaion o poin soluions o he PDEs as expeced values o uncionals o sochasic processes deined by he Feynman-Kac ormula The paricular sochasic processes ha arise in he Feynman-Kac ormula are soluions o speciic sochasic dierenial equaions deined by he characerisics o he dierenial operaor in he PDE The Feynman-Kac ormula is applicable o a wide class o linear iniial and iniial-boundary value problems or ellipic and parabolic PDEs On he basis o he well-known Feynman-Kac ormula providing explici probabilisic represenaion or he soluion o problem (6)-(0) Theorem (The Feynman-Kac ormula): Le : R R be a coninuous diereniable uncion wih compac suppor in R and W( x ) : R R R is a uncion wih coninuous derivaives up o order and wih respec o and x respecively a) Pu U x ) E[ ( X )] () ( 335

4 Aus J Basic & Appl Sci 5(): hen U ( x ) saisies he problem (6)-(9) b) Moreover i W ( x ) is bounded on K R or each compac K R and W solves (6)-(9) hen W ( x ) U ( x ) given by () For proo o he Theorem we reer o (Oksendal B 998) ow we use he Feynman-Kac ormula o obain he soluion U ( x ) o he direc problem (6)-(9) given previously by using an approximaed a ˆ( or he exac a ( Thereore U ( x ) is given by () For some cases he expecaion value o random variable Y ( X ) can be calculae wih mahemaical sowares such as Mahemaica 7 exacly In ac based on wha he iniial condiion u( x0) ( x) is we can obain he soluion o direc problem ()-(4) exacly or approximaely For example when we consider ( x) x he exac soluion o E [ ( X ] can be compue simply For oher cases such as ( x) sinx he Mone Carlo mehod is employed o solve he inegral ( x y) E[ ( X )] (sin y) exp( ) dy Sep 3 Mone Carlo Simulaion o Esimae E[ ( X ] : The idea o Mone Carlo simulaion is o draw an idenical independenly disribued se o samples ( i) { X } rom a arge prior probabiliy densiy uncion i p ( X ) These samples can be used o approximae he arge densiy wih he ollowing empirical poin-mass uncion: p ( X ) ( i ) ( X ) X i where i ) ( X ) ( X expecaion o any uncion o denoes he dela-dirac mass locaed a X by is mean as ollows: (i) X Consequenly one can approximae he E ( i) [ ( X )] ( X i ) By he srong law o large numbers E ( ) converges o E ( ) ie lim E [ ( X )] E[ ( X )] R ( X ) p( X ) dx Sep 4 Opimizaion Technique: In his work he polynomial orm is proposed or he unknown uncion a ( beore perorming he inverse calculaion Thereore a ( can be approximaed as he ollowing orms: ) A polynomial wih degree m and real coeiciens: a ˆ( ) c c m 0 c m ) A rigonomeric polynomial wih degree m and real coeiciens: m a ˆ( c0 c j cos( j c j m j j sin( j where { c0 c cm} are consans which remain o be deermined simulaneously The unknown coeiciens c c c } can be deermined in such a way ha he ollowing uncional is minimized: { 0 m T cal 0 c cm ) U ( x : c0 c cm ) h( 0 J ( c 336

5 Aus J Basic & Appl Sci 5(): Here U ( x c0 c cm ) are he calculaed parial pressures These quaniies are deermined rom he soluion o he direc problem given previously by using an approximaed aˆ ( or he exac a ( The esimaed values o c j j m are deermined unil he value o J ( c0 c cm) is minimum The compuaional procedure or esimaing unknown coeiciens c j are described as ollows: Consider he ollowing deerminisic opimizaion problem min J( C) J( C ) J m C DR () where J (C) is real-valued bounded uncion deined on a closed bounded domain D R m and C The uncion J (C) C ( c0 c cm) I is assumed ha J achieved is minimum value a a unique poin may have many local minimum in D bu only one global minimum When J(C) and D have some aracive properies or insance J (C) is a diereniable concave uncion and D is a convex region hen a local maximum is also a global maximum and problem () can be solved explicily by mahemaical programming mehods (Rubinsein RY 98) The ormula () reveals ha he paper in ac inds he bes i o he unknown a ( by he mehod o leas squares I he problem canno be solved explicily hen numerical mehods based on random sampling in paricular Mone Carlo mehods can be applied (Farnoosh R and M Ebrahimi 00) umerical Experimens: In his secion we are going o demonsrae some numerical resuls or deermining ( u a( ) in he inverse problem ()-(5) All he compuaions are perormed on he PC However o urher demonsraing he accuracy and eiciency o his mehod he presen problem is invesigaed and an example is illusraed Thereore he ollowing example is considered and he soluion is obained Example: In a clinical seing we have he se o experimenal daa u a( u xx 0 x 0 (3) u( x0) sin( x) 0 x (4) u( u( 0 0 u( 05 h( 0 (5) (6) (7) The diusion equaion (3) wrien or he parial pressure o gas process in he represenaive elemenary volume (REV) when a ( is he ime dependen eecive diusion coeicien o he alveolar region Selsuiciency o he REV allows us o impose he boundary condiion o he REV being impermeable o he gas in quesion ha is o require he mass lux hrough he REV boundary o be zero u ( 0 0 Also he red blood cells (RBC) disribued by random wihin he REV are model led as inernal sinks o he gas and consequenly can be viewed as anoher boundary condiion o he gas parial pressure being always zero a he RBC s sies u ( 0 [] The iniial condiion o he parial pressure is u( x0) sin( x) The exac soluion o he problem (3)-(7) is U ( x exp a( y) dy Sin( x) 0 and 337

6 Aus J Basic & Appl Sci 5(): h( a( h( While h ( is posiive coninuously diereniable h ( is negaive and h ( 0) or example h( Figure is presened o show he plo o error beween a ˆ( and a ( when he polynomial wih degree m 3 is used o approximae a ( Figure is presened o show he error or he medium parial pressure u ( x beween he exac soluion o he problem (3)-(7) and numerical resuls ha obained by using he algorihm o he presen work We also employed he uncion esimaor aˆ ( c cos( c sin in presen algorihm and conclude he Figures 34 and 5 Figure 3 show he error or ime dependen diusion coeicien a ( and Figure 4 show he error or he medium parial pressure u ( x Figure 5 is perorm o show he plo o error beween a ˆ( and a ( when 0 Error Fig : Resuls or error beween ( 3 a and aˆ ( c c c c when 0 3 Fig : Resuls or error beween exac ( x orm is used u and presen numerical resuls wih when polynomial 338

7 Aus J Basic & Appl Sci 5(): Error Fig 3: Resuls or error beween a ( and a( c cos( c sin when ˆ Fig 4: Resuls or error beween exac ( x polynomial orm is used u and presen numerical resuls wih when rigonomeric Error Fig 5: Resuls or error beween ( a and a( c cos( c sin when 0 ˆ 339

8 Aus J Basic & Appl Sci 5(): Conclusions and Fuure Direcions: ) The presen sudy successully applies he sochasic mehod involving he Feynman Kac ormula o a onedimensional parabolic inverse problem ) From he illusraed example i can be seen ha he proposed sochasic mehod is eicien and accurae o esimae he unknown eecive diusiviy in a one-dimensional parabolic inverse problem 3) The resuls presened here sugges ha he synhesis o he Feynman-Kac ormula provides a promising probabilisic approach o parabolic inverse problem o he heory o gas ranser The advanages o his approach include bu are no limied o versailiy he possibiliy o compuing he uncions o ineres a isolaed poins wihou compuing hem on massive meshes and he opporuniy o having simple scalable implemenaions wih pracically unlimied capabiliy or parallel processing 4) Here we explored he basic ideas o he Feynman-Kac ormula in conjuncion wih Mone Carlo mehods o a gas low problem In uure papers we plan o exend he approach o nonlinear inverse problems and linear inverse problems wih high dimensions REFERECES Budaev BV and DB Bogy 003 Probabilisic approach o he Lame equaions o linear elasosaics Inernaional Journal o Solids and Srucures 40: Cannon JR 984 The One-Dimensional Hea Equaion Addison-Wesley Menlo Park Caliornia Csaki E 993 A discree Feynman-Kac ormula Journal o Saisical Planning and Inerence 34: Dehghan M 005 Ideniicaion o a ime-dependen coeicien in a parial dierenial equaion subjec o an exra measuremen umerical Mehods or Parial Dierenial Equaions (3): 6-6 Farnoosh R and M Ebrahimi 00 Mone Carlo simulaion via a numerical algorihm or solving a nonlinear inverse problem Communicaion in onlinear Science and umerical Simulaion 5: J ocedal S J Wrigh umerical Opimizaion Second Ediion Springer Science+Business Media ew York (006) Kulish V 006 Human Respiraion: Anaomy and Physiology Mahemaical Modeling umerical Simulaion and Applicaions WIT PRESS UK Modese Y Ouknine and A Sulem 006 Regulariy and represenaion o viscosiy soluions o parial dierenial equaions via backward sochasic dierenial equaions Sochasic Processes and heir Applicaions 6: Oksendal B 998 Sochasic dierenial equaions Fih ediion Springer-Verlag Berlin Heidelberg Rubinsein RY 98 Simulaion and he Mone Carlo mehod Wiley ew York Shidar A GR Karamali and J Damirchi 006 An inverse hea conducion problem wih a nonlinear source erm onlinear Analysis 65: 65-6 Shidar A J Damirchi and P Reihani 007 An sable numerical algorihm or ideniying he soluion o an inverse problem Applied Mahemaics and Compuaion 90: 3-36 Shidar A M Alinejadmorad and M Garshasbi 009 A numerical procedure or esimaion o he mel deph in laser maerial processing Opics and Laser Technology 4: Shidar A R Pourgholi and MEbrahimi 006 A numerical mehod or solving o a nonlinear inverse diusion problem Compuers and Mahemaics wih Applicaions 5: Wang J and Zabaras 004 A Bayesian inerence approach o he inverse hea conducion problem Inernaional Journal o Hea and Mass Transer 47: