Technical Report, SFB 475: Komplexitätsreduktion in Multivariaten Datenstrukturen, Universität Dortmund, No. 1998,04

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1 econstor Der Open-Access-Publkatonsserver der ZBW Lebnz-Inforatonszentru Wrtschaft The Open Access Publcaton Server of the ZBW Lebnz Inforaton Centre for Econocs Becka, Mchael Workng Paper Coplexty-reducton by frst-order approxaton of non-lnear knetcs Techncal Report, SFB 475: Koplextätsredukton n Multvaraten Datenstrukturen, Unverstät Dortund, No. 1998,04 Provded n Cooperaton wth: Collaboratve Research Center 'Reducton of Coplexty n Multvarate Data Structures' (SFB 475), Unversty of Dortund Suggested Ctaton: Becka, Mchael (1998) : Coplexty-reducton by frst-order approxaton of non-lnear knetcs, Techncal Report, SFB 475: Koplextätsredukton n Multvaraten Datenstrukturen, Unverstät Dortund, No. 1998,04 Ths Verson s avalable at: Nutzungsbedngungen: De ZBW räut Ihnen als Nutzern/Nutzer das unentgeltlche, räulch unbeschränkte und zetlch auf de Dauer des Schutzrechts beschränkte enfache Recht en, das ausgewählte Werk Rahen der unter nachzulesenden vollständgen Nutzungsbedngungen zu vervelfältgen, t denen de Nutzern/der Nutzer sch durch de erste Nutzung enverstanden erklärt. Ters of use: The ZBW grants you, the user, the non-exclusve rght to use the selected work free of charge, terrtorally unrestrcted and wthn the te lt of the ter of the property rghts accordng to the ters specfed at By the frst use of the selected work the user agrees and declares to coply wth these ters of use. zbw Lebnz-Inforatonszentru Wrtschaft Lebnz Inforaton Centre for Econocs

2 Coplexty-reducton by frst-order approxaton of non-lnear knetcs M. Becka Departent of Statstcs, Unversty of Dortund, D Dortund, Gerany Abstract: Ecologcal, toxcologcal, and pharacologcal research s often concerned wth the answer to the queston of how a substance s processed wthn a bologcal syste. The exact knowledge of the correspondng knetc pattern fors the bass for a useful answer. In order to dentfy non-lnear knetcs, a frstorder approxaton ethod s proposed for coplexty-reducton. A sulaton study s presented to nvestgate the error of the approxaton n case of a sple Mchaels-Menten knetc process. The proposed ethod shows to gve useful results whch allow to characterze the underlyng knetc pattern. Furtherore t could be shown that n sulatng knetc processes the appled nuercal ethods ay perfor wth consderable nuercal nstabltes. Key Words: Copartental odel, Dynac process, Frst-order approxaton, Mchaels-Menten knetc, Non-lnear knetc, Nuercal ntegraton 1. Introducton A ajor part of ecologcal, toxcologcal, and pharacologcal research s concerned wth the answer to the queston of how a substance s processed wthn a bologcal syste. Wth qualtatvely ncreasng technology, however, the requred qualty of the answers has been ncreased n the last years as well, and wll contnue to ncrease. Where rsk assessent forerly has been based on ncdences and anal results n cobnaton wth haphazardly chosen safety factors, t becoes ore and ore echanstcally based now. In carcnogeness, for nstance, odels are consdered, where on a olecular bass DNA-daage by adducts after exposton to a substance and DNA-repar echanss for concurrent dynac systes, precedng a potental carcnogenc cascade (Hoel et al. 1983). The developent of a tuor then depends on the repar potental of the speces n contrast to the DNA-daage potental of the substance under vew. The relevant 1

3 echanstc characterstcs are nvestgated and odeled for the target speces n order to get access to ore scentfcally based threshold-values and low-dose extrapolatons. In the developent process of new drugs, the detaled knowledge of ther dynac and knetc characterstcs n dfferent patent populatons becoes econocally ore and ore urgent. New drugs wll have to be optally desgned regardng effcacy and adverse sde effects n the ndvdual patent. The above entoned probles are naturally concerned wth hghly coplex answers and approprate statstcal tools have to be appled, respectvely, to be establshed. Characterzng processes of transport or transforaton wthn dynac systes n ters of knetcs s not a trval task. Ths proble was exeplfed by a re-analyss of n vvo knetc data fro Golka et al. (1989) usng a two-copartent odel and nonlnear regresson. The developent and applcaton of further statstcal ethods for ths purpose offer a eans to account for the coon dscrepances found n the dfferent analyses of toxcoknetc odels (Becka et al.(1992), Csanády and Flser (1993), and Becka (1993)). 2. Copartent odels and knetc pattern The flow of ateral n dynac systes, as for exaple n bologcal unts can be odeled by copartents. The ter copartent here usually eans a possbly fcttous area wthn the syste detached fro others by possbly fcttous boundares where the ateral under vew s dstrbuted hoogeneously. Usng copartent-odels, an nterest s gven to the aount of ateral beng exchanged between the copartents n the course of te resultng n specal ateral profles for each copartent. There are any cases where the aount of ateral can not be observed n all of the areas so that observatons are often only avalable fro a subset of copartents. To nvestgate dynac systes t s usually necessary to dsturb the natural steady state of the unts, whch s usually done by nput of ateral n one or ore of the copartents under vew. The nput then ay be contnuous, however constant over te, or a bolus. 2

4 A typcal characterstc n the context of nvestgatng knetc patterns s that all nforaton have to be derved fro observng dynac processes over te. In contrast to the tedependency of an observed process, the knetc pattern descrbes the law behnd or, n other words, the velocty of the process. Therefore, the frst oblgatory step s to extract the velocty characterstcs out of the observed course. A closer look at the underlyng data structure s necessary to reveal that the statstcal odelng process ndeed s no trval task. The nterestng queston s how a process works at dfferent concentratons of a substrate, whereas the observable data consst of the dynac reacton of ths process to a sngle dsturbance. To answer the frst queston then t s necessary to analyze ultple dfferent dsturbances. In practce, t s often assued that all knetcs of a dynac syste are of frst-order whch eans that the veloctes of the partcpatng processes are lnear functons of the aount of correspondng substrates. Ths assupton allows for a qute easy atheatcal handlng wth easy to obtan results. However, alost all bochecal processes are catalyzed or nfluenced by enzyes or protens where the velocty of a processes depends on the aount of substrate n a non-lnear way, the nuber of bndng stes, the affnty, and so on. In ths case the acceleraton of the process,.e. the change of the velocty n connecton wth the aount of substrate becoes of nterest. Referrng to a frst-order process, the acceleraton s a constant. However, dsturbng a specal process n changng the aount of substrate n one copartent ay result n a hgh acceleraton at low concentraton of substrate untl an optal condton s reached, followed by a decrease of acceleraton wth further ncreasng concentraton. Such characterstcs ay be due to occuped bndng stes and queung syptos. The prevously descrbed stuaton reflects the well known saturable Mchaels-Menten-type knetcs. At low concentraton of substrate, there also ay be a phase of low acceleraton due to few bndng stes and a low chance of coplex-bndng. Ths stuaton then reflects sgodal-type knetcs. 3. Statstcal odelng and reducton of coplexty 3

5 Let us nvestgate a dynac syste where one partal process conssts of the aount of ateral beng transferred fro one specal copartent to another copartent j n the course of te. The cobnaton of all the partcpatng processes results n a ateral profle for each copartent. In order to nvestgate the dynac characterstcs, we wll dsturb the natural steady state of ths syste by nput of ateral n one or ore copartents. Denotng %a the aount of nput whch ay be contnuous, however, constant over te, the transport reactons of the dynac syste to ths nput are used to analyze the nherent processes. Denotng Y ( t) the aount of ateral n copartent at te t, the transport fro copartent to copartent j n the course of te s usually descrbed by the dfferental equaton dy ( t) = f ω, dt ( Y() t ) j j where f j denotes the functonal descrpton of a so-called knetc process and ω j s a pathdependent vector of unknown paraeters. Furtherore, t s usually assued that ths knetc s a functon of the aount of ateral Y ( t) present n copartent at te t. If the dynac syste s odeled by, say, n copartents, t wll be assued that the correspondng copartent-odel s knd of a regular one n so far that 1.) there s no copartent n the syste where the concentraton-te course sply reflects those of another copartent n a lnear way, n ters: there are no copartents, j {,,..., n} soe constants a and b, 12 wth j and Y ( t) = a + b Y ( t) for all t 0 and 2.) each copartent n the syste can receve ateral after soe te, n ters: Y t a) (, % > 0 for all { 12 n},,..., and te-lack τ wth 0 < τ t, and 3.) the dynac syste onotonous responds to nput of ateral, n ters: a% < a% * ples Y t, a ~ Y t, a ~ * 12,,..., n and t 0. ( ) ( ) for all { } Regardng one specal copartent, t wll be addtonally assued that the change n the aount of ateral ( ) Y t at te t s due to the su of the related knetc processes,.e. j 4

6 dy ( t) dt n n = f ( Y ( t) ) f Y ( t) j ωj, + j ω j, j j= 1 j= 1 j j ( ). In practce, t s often assued that all knetcs of a dynac copartent-odel are of frstorder whch ples ( ) ( ) ( ) f ω, Y t = k Y t and j j j dy ( t) dt n = k Y( t) + k Y ( t) j= 1 j j= 1 j j n j j wth constant rate paraeters k j for, j {,,..., n} 12 wth j. Usually the odel then yelds a syste of lnear dfferental equatons whch can be solved nuercally or analytcally for the correspondng ateral profles Y( t), = 1, 2,..., n. There are any cases where the aount of ateral can not be observed n all of the copartents so that observatons are only avalable fro a subset of copartents. If these observatons stll allow to dentfy all of the rate paraeters, the rate constants ay be estated by least squares usng the soluton of the syste of lnear dfferental equatons, however, accountng for heterogeneous and auto-correlated errors. If the knetcs of one or ore of the partal processes fro copartent to copartent j are not known exactly, t wll be necessary to nvestgate dfferent nput values a% 1 < a% 2 < L < a% to produce dfferent profles Yl, ( t), l = 1, 2, K,, for characterzng the knetcs as functons of the aount of ateral. The nput nto the syste through one sngle copartent ay be contnuous, however constant over te, or a bolus, as n the above notaton. The a then s to characterze the correspondng knetc relatonshps f ( Y) ω,, j, frst. Note that the requred characterzaton does not depend on t, whch j j s related to the duraton of the specal experent. To elnate the nfluence of experental te t s useful to look for a te-ndependent characterzaton of Y ( t) knetcs are taken nto account where ( ω,, ( )) fj j Y l t fj *( ωj, Y, l ( t) ): = Y () t s onotone and contnuous as a functon of the concentraton Y for concentraton te-course Y ( t) l, l,. If only those j and t τ, the l, ay be characterzed by the axu value Y,ax whch 5

7 ay be defned as the value at the begnnng or the end of the experent, f the local axu does not exst. Ths can be done because of the onotone relatonshps, where a~ < a~ ples 1 2 and, consequently, ( ) Y( t a ) Y t, a ~, ~ 1 2 Y a ~ Y a ~,ax ( ) ( ) 1,ax 2 yeldng ( ω,ax( 1) ) ω,ax( 2) ( ) f *, Y a ~ f *, Y a ~. j j j j Approxatng, therefore, (, ) f *, Y ( t) ω by ( ) j j l ( ω,ax ) = ( ) f *, Y a~ : f * a ~ j j l j l a syste wth frst-order knetcs can be used n a frst step, where dy ( t) dt n ( ) ( ) ( ) ( ) j l j = f * a ~ Y t + f * a ~ Y t j= 1 j l j= 1 j j n for each experent l, l = 1, 2, K,. If the observed ateral profles allow for dentfyng the rate paraeters of ths frst-order syste, the so defned f ( a ) j * ~ wll be estated by a sple least squares technque accountng for heterogeneous errors, yeldng estates f $ * ( ~ j al ) for, j = 1, 2, K, n, j and each experent l { 12,, K, } l. Ths lnear approxaton characterstcally results n a systeatc auto-correlated error-structure. If the Y a~ can not be observed drectly for = 1, 2, K, n, they wll axu concentratons ( ),ax be replaced by estates Y ( a ) l $ ~,ax l calculated by usng the estated ateral profles of the frst-order syste. In an exploratve anner, the followng step s to analyze the relatonshp between the $ * ~ estates fj ( al ) and the correspondng axu aounts of ateral Y ( a ) l = 1, 2,, K, and for all unknown knetc processes fj ( j Y ),ax ω,,, j {,,, n} ~ for 12 K, j, whch ay requre approprate standardzaton of the estated rate paraeters frst. In the case of a true frst-order knetc process wth ( ) ( ω,, ), ( ) f Y t = k Y t ths yelds j j l j l l 6

8 ( ω ( )) = so that the f $ * ( ~ j al ) should not depend on Y,ax( al) f *, Y, t k j j l j ~, nstead beng constant except soe nter- and ntra-experental varablty. Ths hypothess can be tested $ * ~ usng, for exaple, a nonparaetrc test of ndependence of fj ( al ) and Y ( a ),ax ~ for l = 1, 2, K,. If there s a certan structure of dependence, the correspondng shape ay be analyzed accordng to $ [ j * ( a~ l )] E f = ( ω, ( ~,ax )) Y ( a~ ) f Y a j j l,ax l, l = 1, 2, K,. (1) Dependng on the functonal descrpton of the knetc process f j and usng the axu aounts of ateral Y ( a~ ), the estates f $ * ( a ~ ),ax l j l n odel (1) ay not be assued to be unbased. Based on a two-copartental odel and assung frst-order knetcs, ths approach usng the acceleraton nforaton was appled to nvestgate the knetc processes of propylene nhaled by Sprague-Dawley rats (Becka and Urfer, 1996). l 7

9 4. Frst-order approxaton of a Mchaels-Menten knetc process - a sulaton study 4.1 Systeatc error and qualty of a frst-order approxaton To characterze the systeatc error and the qualty of a frst-order approxaton, a onedensonal dynac Mchaels-Menten type process was sulated (Fgure 1) and analyzed assung a frst-order knetc (Fgure 2) C O N C E N T R A T I O N TIME Fgure 1: Concentraton-te curve based on a Mchaels-Menten knetc wth v ax =600 unts/te unt, k =1100 unts, and an ntal concentraton of unts. 8

10 L N ( C O N C E N T R A T I O N ) TIME Fgure 2: Concentraton-te curve of Fgure 1 on a logarthc scale (sold lne) wth the frst-order approxaton (dotted lne). 600 y ax =1100 unts y ax =10000 unts V E L O C I T Y CONCENTRATION Fgure 3: Mchaels-Menten knetc wth v ax =600 unts/te unt, k =1100 unts (sold lne), and exact frstorder approxatons based on axu concentratons of 1100 and unts (dashed lnes). 9

11 Fgure 3 shows a Mchaels-Menten knetc and an exact lnear approxaton,.e. a leastsquares approxatons based on dscrete ponts of the Mchaels-Menten curve tself up to 1100 unts, respectvely, unts concentraton. Snce the lnear approxaton of the observed logarthzed concentraton-te curve (Fgure 2) s based on copletely dfferent data, the resultng approxaton ust not necessarly be the sae. Snce the velocty of a Mchaels-Menten process s descrbed by we have the acceleraton ( ) f y = v ax y ( k + y) ( ) df y dy = v ax k ( k + y) ( ) = f y. 2 If we are askng for the concentraton at whch the acceleraton of the process s c we get ( ) f y = c v ax k 2 ( k + y) = c y = v ax k c k 0 < Snce we have f ( y) 0< ϑ 1. We then get vax k v we ay choose the constant c so that c = ϑ ax wth k y = v ax k c k = vax k v ϑ k ax 2 k 1 k = k = 1 k. ϑ ϑ In order to characterze the curvature of the knetc we ay ask at whch concentraton the actual acceleraton of the process s 1/4 of the axu acceleraton n the orgn,.e. v k ax. If there s a strong curvature the correspondng concentraton would be uch saller than f the knetc would be qute flat. Accordng to the forulas derved before, the pont at whch the actual acceleraton of the process s 1/4 of the axu acceleraton n the orgn s gven wth concentraton k ndependent fro the axu velocty of the process v ax. 10

12 We ay consder how uch the ntal concentraton of the process devates fro ths pont n ters of y 0 k. Fro Fgure 3 we ay expect that the qualty of the approxaton ncreases as y 0 k decreases. Another rearkable feature of the frst-order approxaton s the resultng systeatc errorpattern as depcted n Fgure R E S I D U A L TIME Fgure 4: Resduals of frst-order approxaton on orgnal scale (estated nus observed). 11

13 4.2 Nuercal Soluton of the Dfferental Equaton and Processng of the Data All calculatons were perfored usng SAS, verson 6.12 TS level 0020, on an Intel 200 MHz MMX workstaton under Wndow NT, verson 4.0. Gven an ntal concentraton of y 0 unts the chosen Mchaels-Menten knetc process defned by ( ) vax y( t) = k + y() t dy t dt was ntegrated usng Adas-Bashforth s 4-step ethod wth controlled step-wdth startng wth an ntal step of 10-9 and keepng the local error of dscretzaton between 10-8 and 10-6, respectvely, wth dfferent constant step-wdths n the range of 10-2 up to For calculatng the startng values, the classcal Runge-Kutta ethod of order 4 and step-wdth 10-9 was appled. The dfferental equaton was ntegrated unless 99.9% of the ntal aount had been elnated. Fro the resultng concentraton-te curve 20 equdstant data-ponts were selected and used to ft the frst-order approxaton wth knowledge of the actual ntal concentraton paraeter-cobnatons were nvestgated, where the axu velocty v ax and the Mchaels-Menten constant k ranged fro 1 to 1000 unts per te unt, respectvely unts. For each paraeter-cobnaton the ntal concentratons 10, 100, 1000, and unts were consdered. Only those runs where analyzed further where the nuercal ntegraton took ore than 1000 steps so that n total runs were nvestgated. 12

14 R O O T - M S E Y0/KM Fgure 5: Relaton of root-mse and y 0 /k (95.8% of all runs wth controlled step-wdth). All sulatons revealed consderable nuercal nstabltes n the tals,.e. yelded hgher or lower concentraton values n the tals than actually would be present. Ths defcency becae obvous n the subsequent data-processng. Fgure 5 shows the root ean squared error of the lnear approxaton (logarthzed concentraton scale) n relaton to the rato y 0 k up to a value of 100. The two curves actually reveal two dfferent nstablty processes. Wth saller constant step-wdths ths defcency could not be corrected and the pattern reaned the sae, however, the nuber of obvously dfferent nstablty processes ncreased. Also applyng the classcal Runge-Kutta ethod of order 4 could not prove ths nstablty characterstcs. 13

15 Step-Wdth V ax Process (Te Unts) Duraton of No. of Integraton Steps Root-MSE of Frst-Order Approxaton Estated Velocty Constant Estated Intal Concentraton Controlled Controlled Table 1: Nuercal nstablty n ntegratng a Mchaels-Menten equaton wth k =21 unts and ntal concentraton of 1000 unts applyng step-wdth control. 4.3 Results The analyzed data were consdered to be non-norally dstrbuted wth systeatc errorstructure defned by the nstablty processes of the nuercal ntegraton. The estated (negatve) frst-order velocty constants f $ * ( y0 ) were copared wth v ( k + y ) forula 1 of secton 3) usng ther dfferences ax o (see for D= $ v * ax f ( y0) + k y 0 and ther ratos ( y ) f ( y ) ( k y ) f$ * $ 0 * R = = vax k y Snce the Mchaels-Menten knetc s strct onotonous ncreasng n y and because we are usng all concentraton data y 0 for calculatons we have v ax. 14

16 $ v f * ( y ) ax 0 k y 0. The results are gven n Tables 2 and 3. y 0 STEP-WIDTH N MEAN SD MIN Q 25 MEDIAN Q 75 MAX controlled controlled controlled Overall Table 2: Dfference D between estated frst-order velocty constants f $ * ( y0 ) and v ( k + y ) ax o. In 1166 cases (2.8%) the absolute dfference D showed to be 1 and n 7735 cases (18.9%) the absolute dfference was

17 y 0 STEP-WIDTH N MEAN SD MIN Q 25 MEDIAN Q 75 MAX controlled controlled controlled Overall Table 3: Rato R of estated frst-order velocty constants f $ * ( y0 ) and v ( k + y ) ax o. In cases (45.1%) the relatve dfference R showed to be 20%. Of these 7295 (17.8%) had an absolute dfference of 0.2 and 1139 (2.8%) an absolute dfference of 1. 16

18 5. Dscusson and outlook Due to the observed nuercal nstabltes, sulatons of knetc processes usng nuercal ntegraton have to be appled very carefully wth regard to used procedures and the knetc processes under vew. Sulatng coplete concentraton-te courses usng nuercal ethods see to be no useful tool for analyss purposes snce the sulated concentratonte curve ay not necessarly represent the actual course very well. Analyzng coplete concentraton-te courses for a sngle knetc process as presented n ths sulaton anly refer to re-analyses of hstorcal data, where knetc experents were perfored for any reasons and where the data were analyzed ore or less under the assupton of frst-order knetcs. In ths stuaton the presented approach showed to work qute well, although for the sulatons the underlyng error-dstrbuton (.e. the nuercal nstablty) was not exactly dentfed. Only about 3% of the sulatons yelded crtcal results where a noncrtcal result was defned as an absolute devaton of <1 fro the expected value n addton to a relatve devaton of <20%. It ay be assued that under approxate norally dstrbuted errors the presented procedure would yeld even better results. Analyzng a sngle non-lnear knetc process, the descrbed approach s based on easy to perfor calculatons. In experents purely desgned to dentfy sngle knetc processes, concentraton-te courses would not be observed untl the syste returns to ts steady-state. Snce the concentraton changes over te the velocty of the process wll also change, f the underlyng knetcs are not of frst-order. The ost portant dynac characterstcs concerned wth the dsturbance of the syste ay, therefore, be seen n a short te wndow after the syste was dsturbed. However, analyzng the nherent knetc processes of dynac systes sultaneously wll always requre to nvestgate coplete concentraton-te courses n order to reach axu veloctes n all nvolved copartents. How the proposed procedure works n ths stuaton reans to be elucdated. Studes and experents desgned for knetc n vvo assessents are usually based on collectons of ndvdual subjects. One or ore nterestng response varables are easured at 17

19 several selected tes leadng to so-called te-dependent response curves. The selected ndvdual subjects are assued to be a representatve saple of a populaton of ndvduals, all of whose responses are related to te n a bascally slar way but wth soe varaton fro ndvdual to ndvdual. It s portant to consder the easureent process on the ndvdual response curves and also to take nto account the varaton of these ndvdual response curves around a ean populaton curve. The easured data structure s deterned by the nuber k of ndvduals sapled fro the populaton and the nuber n of observatons ade on ndvdual ( =1,2,...,k ). The observed response of the -th ndvdual at te t j wll be denoted by y j ( =1,2,...,k ; j=1,2,...,n ). We assue that the relatonshp between the response y j and the te of easureent t j takes the for y = f ( θ, t ) + e, = 1, 2,..., k ; j = 1, 2,..., n, j j j where f descrbes the expected value of the response at the dfferent tes t j for a gven paraeter vector θ of ndvdual and e j s a zero-ean easureent error assued ndependent fro observaton to observaton wth constant varance. For a saple of ndvduals fro the sae populaton the sae for of functonal relatonshp f s expected. The paraeter vector θ whch deternes the precse shape of the curve for each ndvdual s assued to vary across the populaton n the anner of a rando saple. Several statstcal ethods have been proposed for the nference of knetc paraeters n odels that cobne ndvdual and populaton coponents, however, there s only sparse lterature regardng the dentfcaton of the underlyng functonal relatonshp tself. Whatever procedure s followed up to now, a substantal aount of dagnostc and cross valdatng checkng has to be perfored to nvestgate the approprateness of the choce of response curves, error varance fors, dstrbutonal assuptons, and approxatons nvoked n the nference procedure. The presented approach usng a frst-order approxaton ay help to reduce the coplexty for ths purpose. 18

20 Acknowledgent Ths work has been supported by the SFB 475 of the DFG. References Bates, D.M. and Watts, D.G. (1988): Nonlnear Regresson Analyss and Its Applcatons. John Wley & Sons. Becka, M. (1993): Statstcal analyss of toxcoknetc data by nonlnear regresson: Reply. Arch. Toxcol. 67, pp Becka, M. and Urfer, W. (1996): Statstcal aspects of nhalaton toxcoknetcs. Envronental and Ecologcal Statstcs 3(1), pp Becka, M., Bolt, H.M., and Urfer, W. (1992): Statstcal analyss of toxcoknetc data by nonlnear regresson (exaple: nhalaton pharacoknetcs of propylene). Arch. Toxcol. 66, pp Becka, M., Bolt, H.M., and Urfer, W. (1993): Statstcal evaluaton of toxcoknetc data. Envronetrcs 4(3), Csanády, G.A. and Flser, J.G. (1993): Statstcal analyss of toxcoknetc data by nonlnear regresson. Arch. Toxcol. 67, pp Flser, J.G. (1992): The closed chaber technque - uptake, endogenous producton, excreton, steady-state knetcs and rates of etabols of gases and vapours. Arch. Toxcol. 66 : pp Golka, K., Peter, H., Denk, B., and Flser, J.G. (1989): Pharacoknetcs of propylene and ts reactve etabolte propylene oxde n Sprague-Dawley rats. Arch. Toxcol. Suppl. 13: pp Hoel, D.G., Kaplan, N.L., and Anderson, M.W. (1983): Iplcaton of nonlnear knetcs on rsk estaton n carcnogeness. Scence 219, pp

21 Mats, J.H. and Wehrly, T.E. (1994): Copartental odels of ecologcal and envronental systes. In Envronental Statstcs, G.P. Patl and C.R. Rao, eds. Handbook of Statstcs, Vol. 12, Elsever, Asterda, pp Seber, G.A.F. and Wld, C.J. (1989): Nonlnear Regresson. Wley, New York. Urfer, W. and Becka, M. (1996): Exploratory and odel-based nference n toxcoknetcs. In: B.J.T. Morgan (Ed.): Statstcs n Toxcology, Oxford Unversty Press, pp