VALIDATION OF THE MATHEMATICAL MODEL FOR PATIENTS USED IN GENERAL ANESTHESIA

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1 VALIDATION OF THE MATHEMATICAL MODEL FOR PATIENTS USED IN GENERAL ANESTHESIA Diego F. Sendoya-Losada, Faiber Robayo Beancour and José Salgado Parón Deparmen of Elecronic Engineering, Faculy of Engineering, Surcolombiana Universiy, Neiva, Huila, Colombia ABSTRACT In his work he validiy of paien used for predicion in a -based conroller during he clinical rials is examined. The ime consan of he pharmacodynamic and he ime delay inroduced by he monior are varied in order o deermine which one of hese parameers has a greaer influence on he oupu of simulaed. Firs he ime consan is changed and no ime delay is considered in order o observe is effec on he simulaed signal, hen he ime consan is considered fixed and he ime delay value is changed. The resuls show ha he ime delay has a greaer influence on he simulaed han he ime consan. Therefore, in he predicion used by he -based conroller is very imporan o have a good esimaion of he ime delay because if ime delay is sub-esimaed, he conrol acion is useless. Keywords: aneshesia conrol, pharmacodynamic, pharmacokineic, ime consan, ime delay.. INTRODUCTION Adequae aneshesia can be defined as a reversible pharmacological sae where he paien's muscle relaxaion, analgesia and hypnosis are guaraneed. Aneshesiologiss adminiser drugs and adjus several medical devices o achieve such goals and o compensae for he effec of surgical manipulaion while mainaining he vial funcions of he paien. One of he devices used by clinicians o assess he deph of aneshesia is he Bispecral Index () monior, which uses elecroencephalographic (EEG) signals (closely relaed o he level of consciousness of he paien) in order o derive a monoonous measure of deph of aneshesia in a range from o (see Figure-). equals o means ha he paien does no have cerebral aciviy and equals o denoes ha he paien is awake and conscious. When he paien is in Inensive Care Uni (ICU), he desired arge is 5 and mus remain beween 4 and 6. In order o develop a -based conroller for he drug adminisraion during aneshesia in ICU, a paien is necessary. If a -based conroller is projeced for a precise adminisraion of drugs, he used in predicion becomes of vial imporance for simulaion and conrol. The used for predicion should no be oo complex, in order no o ake oo much compuaional ime. On he oher hand, i mus represen he dynamics of he paien as good as possible, in response o he specific drug considered (in his case Propofol). 2. MATERIALS AND METHODS The relaionship beween he Propofol infusion rae and is effec can be described by pharmacokineic () and pharmacodynamic () s. describes he disribuion of Propofol in he paien body and describes he relaionship beween Propofol blood and is clinical effec. DRUG INFUSION (inpu) Cl2 Cl3 V2 MUSCLE x2 k2 k2 INTRAVASCULAR BLOOD k3 V3 k3 FAT V x x3 EFECT SITE COMPARTMENT xe ke k Clearance: Cl Nonlinear Drug/Effec Relaion Real Time Monioring (oupu) ke Figure-. Index range guidelines. Figure-2. Muli-comparmenal. 655

2 In he pas, differen paien s have been developed for Propofol; however one popular is he Propofol hree-comparmenal Schnider (Schnider e al., 998). This can be described by a hreecomparmenal (see upper par Figure-2) and i is represened in space saes by he following equaions: x k + k + k k k x [ x ] = [ k k ] [ x ] + [ ] u x k k x = = [ ] [ ] where denoes he amoun of drug in he cenral comparmen (blood) and is unis are milligrams (mg). The peripheral comparmens he drug exchange beween he blood and he ohers body issues. The amoun of drug in hese comparmens is denoed by (muscle issue) and (fa mass), respecively. The consans, for, denoe he drug ransfer rae from he h ohe h comparmen. The consan is he consan rae for he process ha irreversibly removes Propofol from he cenral and peripheral comparmens; is unis are min -. Finally, is he Propofol infusion rae in he cenral comparmen (blood) and represens he inpu of he sysem, is unis are milligrams per second (mg/s) and i is limied o 3.33 mg/s by he pump. The oupu of he is he plasma of Propofol ( ), which is calculaed dividing by, he cenral comparmen volume. Similarly, he s in he oher comparmens are obained dividing and by and, respecively (Ionescu e al., 28). In his case, he unis for he comparmen volume are liers (l) and hese values are given by: =. =.. = The consans are obained as follows: where (clearance) is he rae a which a subsance is removed from he body by he kidneys and is unis are liers per minue (l/min). These values are obained as follows: C l =. +. weigh. lbm +. heigh =. +. =. wih (lean body mass): lbm A =. weigh weigh heigh lbm A =. weigh weigh heigh As can be observed in he above equaions, some of he values depend on he mass (in kilogram), heigh (in cenimeer), age (in years), and gender of he paiens. Regarding he (see lower par Figure- 2), an addiional hypoheical effec comparmen was proposed in order o represen he lag beween he plasma of Propofol and he paien response o he drug. The effec sie comparmen receives Propofol from he cenral comparmen by a firs-order : = + where and are consans and is he amoun of Propofol in he effec comparmen. If he effec comparmen is supposed very small compared o he oher comparmens, hen will be a very small fracion of (Shafer e al., 998). The apparen of Propofol in he effec sie comparmen can be calculaed once is known, because his will characerize he emporal effecs of equilibraion beween he plasma of Propofol and is corresponding effec. Thus, he equaion is ofen used as: = [ ] where is he effec sie comparmen of Propofol and he value of is.456 min -. Therefore, he effec sie comparmen can be represened as follows: C () p Plasma s k e Figure-3. Effec-sie comparmen. In his, / represens he ime consan of he sysem ( ) and is value is =. seconds for all of he paiens. The measured can be relaed o he effec-sie by he empirical saic bu ime-varying nonlinear relaionship (Bailey and Haddad, 25), called also he Sigmoid Hill Curve: = 5 e () C Effec is he value when he paien is awake, by convenion, a value of is ypically assigned; is he maximum effec ha can be achieved by he infusion of Propofol; 5 is he Propofol a half 656

3 maximum effec and represens he paien sensiiviy o he drug; and deermines he degree of nonlineariy (seepness of he curve) (Haddad e al., 23).Thus, he combinaion of he and s allows a complee ling of he paien. The parameers involved in he are known because hey depend of he biomeric values for each paien. The group of ineres corresponds o paiens o whom he Propofol infusion rae was adminisraed by he nurse. The deailed biomeric values of he paiens in his group are shown in Table-. Paien Table-. Biomeric values of he paiens. Age (years) Lengh (cm) Weigh (Kg) Gender M M F M M M F M M M The average age of he group is 65. years wih a sandard deviaion of ± 6.94 years. The average lengh of he paiens is 7.9 cm wih a sandard deviaion of 9.88 cm. The average weigh of he paiens is 87. Kg wih a sandard deviaion of 2.53 Kg. Regarding he, he value of is also known, bu no he parameers of he Sigmoid Hill Curve. Since hese parameers are unknown for each paien some nominal values have been used for he simulaions. The nominal value for 5 is 2.5µg/ml and for is 3.. The oher wo parameers of he Hill curve, and are considered equal o he value of. 3. RESULTS AND DISCUSSIONS The purpose of his secion is o examine he validiy of paien used for predicion in a based conroller during he clinical rials. The ime consan of he and he ime delay inroduced by he monior were varied in order o deermine which one of hese parameers has a greaer influence on he oupu of simulaed. In order o achieve his, he recorded signals in he clinical rials were used as inpu of a simulaor. Firs he ime consan is changed and no ime delay is considered in order o observe is effec on he simulaed signal, hen he ime consan is considered fixed and he ime delay value is changed. The simulaor developed uses he Propofol infusion rae as inpu signal o he, which uses he biomeric values of each paien o produce he Propofol plasma ; his signal goes o, which has he ime consan of he sysem ( =. seconds for all of he paiens), and he resul is he Propofol effec sie ; his signal is applied o he Sigmoid Hill, which uses nominal values o produce he signal. Someimes he signal qualiy of measured is no very good and a ime delay beween he signal displayed by he monior and he real of he paien appears. This ime delay ( ) is included in he simulaor. In his way, he used in he simulaor of he paien can be represened by he block diagram in Figure-4. Daa of paien were used in his case. In order o appreciae in a beer way and o compare he real and he simulaed signals, he real signal was filered using a 3 rd order low pass filer and a cuoff frequency of 2.5 mhz o eliminae he noise. Propofol and filered signals during a cerain ime inerval are presened in Figure-5. In his case, i can be observed ha when he Propofol level increases he level sars o decrease, indicaing an inverse proporionaliy beween hese signals. Propofol rae u () Plasma C () pprop Effec C () eprop Sigmoid Hill () ( d ) e s d 3rd order s order Time delay Figure-4. Simulaor block diagram. 657

4 Propofol (mg/s) Nominal +2%.5 +5% +8% Figure-5. Propofol (lef) and filered (righ) signals- Paien. 3. Changes in he ime consan Iniially, Propofol signal is applied o he - of he simulaor geing as oupu he effec sie, signal. Propofol rae u () 3rd order Plasma C pprop () s order Figure-6. - s. Effec C eprop () The ime consan of he nominal is =. seconds. The value of his consan is increased by 2%, 5% and 8%, his is s, s and 5.26 s respecively; geing he values shown as follows. Figure-7. when is increased. I can be observed ha when he value of he ime consan is increased he response speed of he sysem is reduced. When he value of he ime consan is increased by 8% he ime peak of he response shows a delay of seconds compared o he nominal sysem response. Subsequenly, he value of he ime consan is decreased by 2%, 5% and 8% and he values presened bellow are obained Nominal -2% -5% -8% Figure-8. when is decreased. When he value of he ime consan is decreased, he response speed of he sysem increases. If he value of he ime consan is decreased by 8% he ime peak of he response is seconds smaller han of he nominal sysem. Secondly, Propofol signal was used as inpu of he simulaor wihou aking ino accoun any ime delay (Figure-9). The ime consan of he nominal is increased and decreased by 2%, 5% and 8% obaining he values shown in Figure-. 658

5 Propofol rae u () Plasma C pprop () Effec C eprop () Sigmoid Hill () 3rd order s order Figure-9. Simulaor wihou ime delay Filered Nominal 2 +2% +5% +8% Filered Nominal 2-2% -5% -8% Figure-. when is increased (lef) and decreased (righ). The increase or decrease of he ime consan only influences he speed of sysem response. When he ime consan is increased or decreased by 8%, he ime peak of he response shows a shif of only seconds compared o he nominal sysem response. Furhermore, he rise and fall slope of he nominal case (wih =. seconds) is closer o he real han he slope in he cases when he ime consan was varied. Furhermore, i can be seen ha when he Propofol signal presens peak values from zero o values higher han.5 mg/s (Figure- - lef), he level falls rapidly (Figure- - righ), and he rise slope of he signal is differen ha he fall slope.this occurs because he sysem never reaches he seady sae and he rise or fall speed depends of he condiions in which he simulaed sysem is when a high value of Propofol is applied. To verify his, a Propofol es signal (Figure-2 - lef) was used in he simulaor as inpu. The oupu of he - is presened in Figure-2 (righ). The saic characerisic response (Figure-3) shows an apparen hyseresis because he sysem never reaches he seady sae. However, when posiive seps are applied (Figure-4 - lef) he dynamic response of he - is very similar o he case when negaive seps are applied (Figure-4 - righ). I can be observed ha he apparen hyseresis occurs because he sysem never reaches he seady sae and he rise or fall speed depends on he condiions in which he simulaed sysem is when a high value of Propofol is applied. Propofol (mg/s) Filered 4 Nominal -2% 3-5% -8% Figure-. Peak of Propofol (lef) and reducion of (righ). 659

6 Propofol (mg/s) Prop.4 mg/s Prop.8 mg/s Prop mg/s Prop.6 mg/s Prop mg/s Prop 4 mg/s Prop mg/s Prop.6 mg/s Prop mg/s Prop.8 mg/s Prop.4 mg/s Prop. mg/s Figure-2. Propofol es signal (lef) and response (righ) Propofol (mg/s) Figure-3. Saic response for es signal Figure-4. - s dynamic response for posiive (lef) and negaive (righ) seps. 3.2 Changes in he ime delay The increase or decrease he ime consan influences he speed of sysem response. However, a shif beween he real and simulaed signals can be clearly observed. In his secion, he ime consan of he sysem is considered fixed ( =. seconds) and he ime delay value is changed. Iniially, a ime delay of 8 samples (8 seconds) is used in he simulaor (Figure-5). The comparison beween he filered real signal and signal from he simulaor is presened in Figure-6. The slope of he simulaed signal is he same as ha of he real, bu he delay of 8 samples is no sufficien ye o ensure ha he peaks of he signals occur in he same momens of ime. Subsequenly, a ime delay of 5 samples (5 seconds) is used in he simulaor. The wo signals (real and simulaed) are presened in Figure-7. 66

7 Propofol rae u () Plasma C () pprop Effec C () eprop Sigmoid Hill () ( d ) e s d 3rd order s order Time delay Figure-5. Simulaor wih ime delay because if ime delay is sub-esimaed, he conrol acion is useless. The 4 h order can be used in predicion if a correc ime delay is esimaed. The esimaion procedure of ime delay will be presened in a fuure work. ACKNOWLEDGEMENT This work has been funded by IWT TBM projec number 6776 suppored by he Flemish Insiue for Innovaion hrough Science and Technology. 2 Filered Delay=8 s Figure-6. Comparison beween real and simulaed (delay 8 s). When a ime delay of 5 samples is used, he response of he simulaed is very close o he acual paien response. In his manner, he changes in ime delay, when he ime consan of he sysem remains fixed in =. seconds, allows obaining a response of he simulaor very close o he acual paien response. The bes approximaion is obained for = samples Filered Delay=5 s Figure-7. Comparison beween real and simulaed (delay 5 s). The ime delay has a greaer influence on he simulaed han he ime consan. Therefore, in he predicion used by he -based conroller is very imporan o have a good esimaion of he ime delay REFERENCES Bailey J. M., Haddad W. M. 25. Drug dosing conrol in clinical pharmacology, IEEE Conrol Sys. Mag. 25: Chilcoa R. 98, A review of he conrol of deph of anaeshesia. Transacions of he Insiue of Measuremen and Conrol. 2(): Cullen D., Eger E., Sevens W., Smih N., Cromwell T., Cullen B., Gregory G., Bahlman S., Dolan W., Soeling R., Fourcade H Clinical signs of aneshesia, Aneshesiology. 36(): Derighei M Feedback Conrol in Anaeshesia, PhD hesis, Swiss Federal Insiue of Technology (ETH), Zürich, Swizerland. Goldmann L Informaion-processing under general aneshesia: a review. Journal of he Royal Sociey of Medicine. 8: Haddad W. M., Hayakawa T., Bailey J. M. 23 Nonlinear adapive conrol for inensive care uni sedaion and operaing room hypnosis. American Conrol Conference. 2: Ionescu C., De Keyser R., Torrico B. C., De Sme T., Sruys M., Normey-Rico J. E. 28. Robus Predicive Conrol Sraegy Applied for Propofol Dosing using as a Conrolled Variable during Anaeshesia. IEEE Transacions on Biomedical Engineering. 55(3): Linkens D. A., Hacisalihzade S. S. 99. Compuer conrol sysems and pharmacological drug adminisraion: a survey, Journal of Medical Engineering & Technology. 4(2):

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