DETERMINING OF THE NEONATAL THERMAL MODEL PARAMETERS USING INVERSE THERMAL ANALYSIS

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1 11t World Congress on Computatonal Means (WCCM XI) 5t European Conferene on Computatonal Means (ECCM ) 6t European Conferene on Computatonal Flud Dynams (ECFD I) E. Oñate. Olver and A. Huerta (Eds) DEERMIIG OF HE EOAAL HERMAL MODEL PARAMEERS USIG IERSE HERMAL AALYSIS OAA LASZCZYK * AA MACZKO WOCIECH WALAS AD ADRZE. OWAK * * Insttute of ermal enology Slesan Unversty of enology Konarsego Glwe Poland e-mal: oanna.laszzy@polsl.pl; andrze..nowa@polsl.pl Medal Centre n Opole Department for Cldren Intensve erapy Al. Wtosa Opole Poland Key Words: Compartment neonatal termal model Pennes oeat equaton lood perfuson nverse termal analyss. Astrat. In ts paper omputatonal model of te numeral termal model wtn te neonatal ody ale of ontrollng te ran oolng proess s presented. e man am of te analyss was to fnd te proper parameters of te proess. e smplfed geometr model of neworn's ody onsstng of te numer of ompartments was ult usng Desgn Modeler. e negourng ompartments exange eat y onduton troug tssues ut also nterat wt ea oter troug flowng lood. o smulate te natural eat transfer proesses troug te uman tssues te Pennes oeat equaton was mplemented nto Ansys Fluent usng UDF (User Defned Funton) apalty. Model parameters are determned usng nverse termal analyss. 1 IRODUCIO e pernatal ypox-sem enepalopaty s a farly frequent prolem w te neonatologsts ave to ope wt. Untl reently tere was not any metod to remedy ts dsease. At present te terapeut mld ypoterma of te nfant's ran s more and more frequently used [3 5]. Su ypotermal terapy generally stops apoptoss of neurons. However to ounterat te sde effets of volent modfatons of te temperature durng te ypotermal terapy te deep ody temperature sould e ept at te onstant and seure level. In order to perform te treatment n a proper way a dstruton of te temperature nsde te neonatal ody (espeally nsde te ran) durng te wole terapy as to e nown. Sne t was already proved [5] tat temperature level of o C s seure durng ran oolng at te egnnng of te terapy te deep ody temperature s dereased to tat level. In te ase of neonate sufferng from te ypoxa s temperature s usually dereasng spontaneously 0.5 K per our tus te deep ody temperature at te ertan moment s

2 oanna Laszzy Anna Mazo Woe Walas and Andrze. owa approprate to start terapy. If not s temperature as to e redued utlsng oolng elmet or oter deve. ext durng te terapy te deep ody temperature s mantaned at te onstant level. Sne te most vulnerale organ s te ran te temperature of te salp s ept y oolng elmet at te lowered level equal to aout 4 o C. Smultaneously n order to mantan te deep ody temperature at te seure level te sn of te rest of te neonatal ody as to e eated y te radant warmer. e terapy s arred out for 7 ours and ten te deep ody temperature s gradually nreased to te normal level (aout 37 o C). It s very mportant to guarantee tat temperature augments not faster tan 0.5 K per our wat entals many enormous dffultes. In ts wor te attempt s made to develop a numeral termal model wtn te neonatal ody ale of ontrollng te ran oolng proess. e man am of te analyss was to fnd te proper parameters of te proess. e smplfed geometr model of neworn's ody onsstng of te numer of ompartments [1 3] was ult usng Desgn Modeler. As te sape of te ead an ellpsod was used. e forms of ne and trun were smplfed to ellpt ylnders. e lms were modelled as ylnders. Insde te ead te salp sull and ran were separated. e ran was surrounded y te ran ererospnal flud. e remanng part of te ead was modelled as a spae ouped y te ar. e ne onssted of te spne musle and sn. e spne was lengtened to te torso. Interor of te trun was dvded nto lungs and vsera w were surrounded y one and sn. Lms were splt nto tree tssues: one musle and sn. e ult geometr model of te alf of te neworn's ody and te more detaled snapsot of te ult geometr model of te ead are presented n Fgure 1. Fgure 1: e ult geometr model of te neonate s ody (left) and of te ead (rgt) e negourng ompartments exange eat y onduton troug tssues ut also nterat wt ea oter troug flowng lood. o smulate te natural eat transfer proesses troug te uman tssues te Pennes oeat equaton [4] was mplemented nto Ansys Fluent [] usng UDF (User Defned Funton) apalty. GOERIG EQUAIOS e analyss of eat transfer proesses durng ran oolng sould generally e arred out as a transent one wat eventually an lead to te steady-state. However t sould e noted tat te eat apaty of te lood wtn one numeral ell s small omparng wt te eat apaty of te tssue and as a onsequene equatons for arteral and ven lood temperature an e formulated as steady-state ut for te onseutve tme steps. As already mentoned t s assumed tat transent eat transfer n uman tssues s

3 oanna Laszzy Anna Mazo Woe Walas and Andrze. owa governed y Pennes oeat equaton [4]: t t t t ( ) q t t t a t met t were susrpts () t () and () a refer to tssue lood and artery respetvely. Symol stands for temperature (n K) t W/(mK) s a tssue termal ondutvty ρ g/m 3 ts densty wle /(gk) s a spef eat. Quantty W/m 3 represents a metaolsm and s an amount of energy w s generated wtn a unt tme and a unt volume of a tssue. Blood perfuson s mared y 1/s tme s depted y t s and s a wegtng fator varyng etween 0 and 1. Ovously for steady-state one an wrte: ( ) q t t t a t met t 0 () e lood perfuson rate and te metaol eat rate are depended on te tssue temperature and tey an e alulated n te followng way: ( t 0 )/ 10 3 (3) t t as q met t met t as ( t 3 )/ 10 (4) 0 q were W/m 3 1/s stand for te metaol eat rate and te lood perfuson rate n te referene temperature 0 K equal to K. As already mentoned te geometr model s dvded nto setors: a ead a fae a ne a soulder a forearm a left palm a leg a left foot a torax and an adomen. Ea setor dentfed y ndex () as ts own arteral lood temperature a (n K) ven lood temperature v (n K) and lood mass flow g/s. In te analyss t s assumed tat mass flows of te artery and te ven loods are equal. Ea setor s also dvded nto approprate numer of tssues dentfed y ndex () wle ndex () n te derved equatons refers to a partular numeral ell. In te developed model t s assumed tat te rulatory system ontans te mxng spot alled te entral lood pool gaterng te ven lood from all setors and dstrutng te arteral lood to ea setor. e temperature of te entral lood pool p K s ommon for te wole ody and s dentfed wt te retal temperature of te neworn. Let s nsder now a sngle segment dentfed y ndex w s reevng from te lood pool arteral lood avng temperature a and sendng to lood pool a ven lood avng temperature v. s stuaton s sematally sown n Fgure. (1) Fgure : e ult geometr model of te neonate s ody (left) and of te ead (rgt) 3

4 oanna Laszzy Anna Mazo Woe Walas and Andrze. owa e temperature of te arteral lood leavng te lood pool s equalled to p and te temperature of te arteral lood reang te segment s equalled to a. e temperature of te ven lood leavng te segment s equalled to v wlst te temperature of te ven lood reang te lood pool s equalled to pv. Wle te arteral lood flows from te lood pool to te segment and n te same tme te ven lood flows from te segment to te lood pool. erefore te amount of eat equalled to s exanged etween te arteral and te ven vessel and an e desred as: Q m a p a (5) Q m v pv v (6) were stands for te arteral lood mass flow rate reang te segment wlst stands for te ven lood mass flow rate leavng te segment. e eat flow an e also expressed as Q a v (7) were W/(m K) stands for a ounter-urrent eat exange oeffent for te segment and taes nto onsderaton te arrangement of te rulatory system. Connetng equatons (5) - (7) and assumng tat te followng relatonsp an e reeved: a (8) m m m a v v o determne te temperature of te arteral and ven lood te ea segment was dvded nto tssues. In te model te followng 6 tssues ave een dstngused: sn musle ran vsera lungs and one. ey are dentfed y ndex (). e lood mass flow for te segment s related to te lood perfuson rate for te partular type of tssue y te followng relatonsp: m m m 1 p were represents te total amount of all types of tssue of te segment stands for te lood perfuson rate for te tssue n te segment s te lood volume flow rate n te segment wlst stands for te volume of te tssue n te segment and s te sum of te volumes of all numeral ells n te tssue n te segment : 1 were represents te total amount of numeral ells n te tssue n te segment. us s eventually alulated y te followng relatonsp: 1 (9) (10) (11) 4

5 oanna Laszzy Anna Mazo Woe Walas and Andrze. owa 5 p v a (1) e eat transfer aounts for te dfferenes of te arteral and lood temperatures as dependent on te dstane from te lood pool. Sne te neonate s ody w s generally small omparng to an adult's ody t s assumed ere tat te value of s generally small (almost equal to 0) for ea segment. It s assumed tat te ven lood temperature leavng a gven numeral ell n tssue n a segment s equalled to te temperature of te onsdered numeral ell. Addtonally te energy of te ven lood leavng te tssue n a segment s equalled to te total energy of te ven lood leavng all numeral ells n te tssue as well as tat te eat apaty and densty of lood s ommon for wole ody te temperature of te ven lood s expressed y: t v (13) In te entral lood pool te lood from ea segment s mxed. After some smple algera manpulatons [6] one an otan te equaton to determne lood pool temperature w s extenson of result gven y Fala et. al. [1]: K K t p m m m m 1 1 (14) e lood mass flow for te segment s related to te lood perfuson rate for te partular type of tssue y te followng relatonsp: Ovously for steady-state one an wrte: 3 IERSE AALYSIS Despte te adopted geometral and matematal smplfatons te nverse analyss was used to retreve te needed parameter values of te eat transfer model le lood perfuson rate used n equaton (3) nternal eat generaton used n equaton (4) as well as te eat transfer oeffent on te sn. Generally te metod onssts of tunng te temperature feld of te numeral model to te temperature feld measured on te real oet

6 oanna Laszzy Anna Mazo Woe Walas and Andrze. owa y angng te set of parameters values. As te oetve funton Δ te standard leastsquare funton was appled. e oetve funton onssted of te sum of squares of devatons etween temperature predted y te matematal model and temperature U measured at te seleted loatons on te nfant s sn w ntrodung te vetors of temperatures values at onsdered loatons an e wrtten as follows: U U mn (15) were supersrpt ndates matrx transposton. e measurements were olleted from 35 spots on te surfae of te sn and onsderng te deep ody temperature (retal temperature n ase of a real ay and lood pool temperature n ase of numeral smulaton). e loaton of tose 35 spots on te surfae of te model as t s sematally sown n Fgure 3 as well as te temperature feld U nludng te lood pool temperature (te retal temperature n te real ay) was onsulted wt te neonatologsts and approved y tem. Fgure 3: e dstruton of ponts on te surfae of te sn wt measured (nown) temperature (left) and exemplary IR measurements of sn temperatures otaned from neonatologsts (rgt) Mnmzaton of te oetve funton Δ s arred out mang use te senstvty oeffents Z determned numerally: Z Y Y * * (16) Y Y were Y stands for te seleted parameter (any of 5 lood perfuson rates any of 5 nternal eat generatons or eat transfer oeffent) affetng temperature at seleted loaton. In order to mae te prolem over-determned te numer of measurements was more tan tree tmes ger tan te numer of estmated parameters. s sould generally elp n redung te nfluene of expermental errors on results. Quanttes Y* and * stand for te ntal guess of unnown model parameters. Colletng all te parameters values nto a vetor Y all senstvty oeffents nto te senstvty matrx Z and expandng temperature nto a aylor seres (ut after frst-order dervatves): * 6

7 oanna Laszzy Anna Mazo Woe Walas and Andrze. owa * * Z Y Y (17) were Y* stands for te vetor of ntal guess of unnown model parameters. ese model parameters were taen from te lterature. In many ases te values of tese model parameters ave een adopted dretly from data avalale only for adults. e vetor of * temperatures s te result of usng derved model usng te vetor Y* of model parameters. e fnal soluton of optmsaton prolem (15) an now e otaned from te followng system of equatons [7]: * * Z Z Y Z U Z Z Y (17) were Y* stands for te vetor of ntal guess of unnown model parameters. ese model parameters were taen from te lterature. In many ases te values of tese model parameters ave een adopted dretly from data avalale only for adults. 4 UMERICAL EXAMPLE e ntal temperature of te wole neworn ody was assumed to e aout 34 o C. Besdes t was presumed tat te naed neonate lad on te eatng mattress and was surrounded y te ar. us on te ottom part of te sn te eat flux was estmated to e 0 W/m (nsulaton oundary ondton). Furtermore to te top part of te sn te onveton wt te eat transfer oeffent equalled to = 5 W/m K was presred. e ament temperature was assumed to e 5 o C. en te ay s treated y te ypotermal terapy w lasts 7 ours. Smlarly to a real terapy t was assumed tat on te ay's ead a oolng elmet was fxed wlst te rest of te ody was rradated y te radant warmer. e oolng elmet was mtated y te onvetve oundary ondton (t a = 10.8 o C = 19.5 W/m K). e radant warmer and te free onveton to te ament ar were smulated y te mxed oundary ondton (onveton: t a = 5 o C = 5 W/m K radaton: = 100 W/m ). Many smulatons onfrmed tat te proper temperature dstruton durng te terapy an e aeved usng nverse termal analyss. e example temperature felds at te egnnng and durng te terapy are dsplayed n Fgure 4: Fgure 4: e temperature dstruton (n o C) on te sn surfae (a) and on te entral ross seton (d) at te egnnng (a) and at te end of te oolng terapy (d). 7

8 oanna Laszzy Anna Mazo Woe Walas and Andrze. owa 5 COCLUSIOS e presented model of te ypotermal terapy nludes a geometr model of neonate a omputer program desrng te eat transfer n te uman ody and te fle n Fluent n w te proess of te terapy s preset. For model te most requred are values of parameters ot for te terapy and for te eat transfer n te neonate s ody. Su parameters an e reeved usng te nverse analyss presented n ts wor. unng te model to te data from te realty one an otan te set of proper and reasonale parameters. Addtonally t sould e stressed tat values of lood perfuson rate and eat generaton depend on te tssue temperature so tey are qute unversal. It an e assumed tat f te set of parameters fts to one expermental data and s valdated wt anoter data from experment te fnal set of parameters an e appled to test anoter stuatons e.g. test varous settngs or ondtons le temporary ange of ament temperature or anges of metaol eat. Summng up te presented model s very unversal. e results of te nverse analyss sowed tat te predtng of values of mportant parameters of oeat transfer s possle ut t stll requres some tests and mprovements. e dfferene etween te measured/assumed and te alulated temperatures usng a new set of model parameters an stll e oserved. It ould e valuale to nlude te wegtng oeffents n te optmzaton proedure e.g. te retal temperature requres te ger value of wegtng oeffent tat te rest of measurements. Besdes te optmzaton proedure an e mproved y omnaton wt onstrants taen from te lterature. everteless t an e stated tat otaned results are valuale from te sentf and pratal pont of vew. e fnal values of parameters ontrollng te neonatal ran oolng proess are extremely valuale from tenal and sentf pont of vew. ACKOWLEDGEME e prnpal autor of ts paper s very grateful for fnanal support otaned wtn te frame of te grant o enttled e analyss of te ran oolng proess funded y te atonal Centre for Resear (Poland). e last o-autor s very grateful for fnanal support otaned wtn te frame of statutory resear fund of te Faulty of Energy and Envronmental Engneerng te Slesan Unversty of enology. REFERECES [1] Fala D. Lomas K.. Storer M. A omputer model of uman termoregulaton for a wde range of envronmental ondtons: e passve system Appl Pysol ol. 87 Issue 5 pp [] Ansys Fluent 1.1 n Woren User's Gude ASYS In. (009) [3] Cenguang D. Lang Z. Huan W. Coolng and rewarmng for ran sema or nury: eoretal analyss Annals of Bomedal Engneerng 31 pp [4] Pennes H.H. Analyss of ssue and Arteral Blood emperatures n te Restng Human Forearm Appl Pysol 1 pp [5] WalasW. Hypoterma of ead - g ane for neonates wt pernatal ypoxsem enepalopaty (n Pols) WCM.opole.pl ol (010). [6] Laszzy. Mazo A. Walas W. owa A.. Inverse termal analyss of te neonatal 8

9 oanna Laszzy Anna Mazo Woe Walas and Andrze. owa ran oolng proess Internatonal ournal of umeral Metods for Heat and Flud Flow ol. 4 n press. [7] Kurpsz K. owa A.. Inverse ermal Prolems Internatonal Seres on Computatonal Engneerng. Comp. Me. Pulatons