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Application of Neural Networks to Population Pharmacokinetic Data Analysis

HSIAO-HUI CHOWX, KRISTIN M. TOLLE, DENISE J. ROE, VICTOR ELSBERRY, AND HSINCHUN CHEN

Received September 24, 1996, from the Department of Pharmacy Practice and Science, Department of Management InformationSystems, and Arizona Cancer Center, University of Arizona, Tucson, AZ 85721. Accepted for publication March 24, 1997X.

Abstract 0 This research examined the applicability of using a neuralnetwork approach to analyze population pharmacokinetic data. Such datawere collected retrospectively from pediatric patients who had receivedtobramycin for the treatment of bacterial infection. The informationcollected included patient-related demographic variables (age, weight,gender, and other underlying illness), the individuals dosing regimens(dose and dosing interval), time of blood drawn, and the resultingtobramycin concentration. Neural networks were trained with thisinformation to capture the relationships between the plasma tobramycinlevels and the following factors: patient-related demographic factors,dosing regimens, and time of blood drawn. The data were also analyzedusing a standard population pharmacokinetic modeling program, NON-MEM. The observed vs predicted concentration relationships obtainedfrom the neural network approach were similar to those from NONMEM.

The residuals of the predictions from neural network analyses showed apositive correlation with that from NONMEM. Average absolute errorswere 33.9 and 37.3% for neural networks and 39.9% for NONMEM.Average prediction errors were found to be 2.59 and 5.01% for neuralnetworks and 17.7% for NONMEM. We concluded that neural networkswere capable of capturing the relationships between plasma drug levelsand patient-related prognostic factors from routinely collected sparse within-patient pharmacokinetic data. Neural networks can therefore beconsidered to have potential to become a useful analytical tool forpopulation pharmacokinetic data analysis.

Introduction

In clinical practice, collection of s pars e w ithin-patientconcentration-t im e data along w ith the clinically relevantprognostic factors and the dosing history are routinely con-ducted as pa rt of thera peut ic monitoring for a variety of dru gs.P opulation pharm acokinetic data analys is techniques havebeen developed to allow analysis of such dat a to gain furt herinsight into population pharmacokinetic characteristics andthe effects of different demographic factors on the behaviorof a d r u g. I t h a s b ee n r e a li ze d t h a t , s in ce on ly a fe wconcentration measurements need to be collected from eachpatient for analysis, collection of this type of data could beeasily incorporated into all phases of the drug developmentprocess to maximize the amount of knowledge that could bediscovered.

In recent years, a number of population modeling programs

have become available.1

A m ong them , the N O N M EM pro-gr a m w a s t h e fi rs t d eve lop ed a n d h a s b ee n u s ed m os textensively to analyze actual clinical pharmacokinetic data.Building a population pharm a cokinetic m odel requires un-derstan ding and selection of various mat hema tical/stat isticalmodels that include a pha rmacokinetic structure m odel relat-ing dose, sampling time, and pharmacokinetic parameters toplasma drug levels, regression models for relationships be-tw een patient characteris tics and the pharm acokinetic pa-ram eters, a population model for intersu bject variability, and

a variance m odel for random res idual variation in the data(intrapatient variability).2,3 This can be a t edious an d tim e-

consuming process.

N e u r a l n e t w or k s a r e com p u t a t i on a l s ys t e m s w h ich a r e

developed to simulate the neurological processing abilities of

biological systems a nd a re kn own for their ada ptive learning

an d self-organ izationa l capa bilities. Application of th is tech-

nique t o biopharm a ceutical data analys is has received con-

siderable attention recently.4-7 Brier et al .8 have exam ined

the us e of neura l n etw orks for population pharm a cokinetic

data analys is an d concluded tha t neural n etw orks a nd N O N -

MEM provided comparable predictions of plasma drug con-

centrat ions . The s trength of neural netw orks is that t hey do

not assum e a specific model. Instea d, they learn to establish

the input and output relationships from the data provided to

them . This greatly simplifies the m odeling work in volved intra ditional population pharma cokinetic data ana lysis. In this

s tudy, w e further explored the applicability of the neural

network approach to population ph arm acokinetic data ana ly-

sis to capture the relationship between patient-related prog-

nostic factors a nd plas m a drug levels , us ing data com piled

from our clinical practice s ite. A com paris on w as m ade

betw een the res ults from the neural netw ork approach and

thos e obtained us ing th e N O N M EM program .

Methods

D a t a C o m p i l a t i o nsThe da ta were obtained r etrospectively from

the clinical dosing services of the Department of Pediatrics of theUniversity Medical Center in Tu cson, AZ. Pat ient-related informa tion

was collected from th e char ts of 101 pediatric patients who, between

1983 and 1992, received multiple doses of tobramycin infusion for

the t reat ment of bacterial infection. This informa tion included age

weight, gender, other underlying illness (cystic fibrosis or cancer),

dose, dosing interval, time of blood drawn, and the resulting serum

tobramycin concentr ation. Blood samples were collected a t t wo time

points (roughly at the peak and trough) after the fourth or the fifth

dose of infusion. Steady-state sh ould have been at tained before th e

collection of the blood samples.

Some of the patients had received more than one dosing regimen

of t ob r a m yci n t r e a t m e n t d u r in g on e h os p it a l s t a y a n d a s iz a bl e

n u m b er of p a t ie n t s h a d r e ce iv ed t ob r a m yci n t r e a t m e n t d u r in g

separat e i nfect i on episodes. Thi s resul t ed i n t he avai labi li t y of

multiple measurements of tobramycin levels in these patients. There

was, t herefore, a t ot al of 311 pai rs of peak and t rough t obramycinconcentra tion dat a which provided 622 t obram ycin concentr ation

measurement s.

NONMEMsThe entire data collection was analyzed using NON-

MEM version IV, level 2.1. Since steady-stat e levels should ha ve been

attained after the fourth or the fifth dose, a one-compartment model

wit h i nt ravenous i nfusi on and first -order el imi nat i on was used

Various regression models relating different combinations of clinical

fact ors (i.e., age, weight , gender, and i ll ness) t o pharma coki net ic

para meters were examined, and t he model that provided the sma llest

objective function was selected. Various sta tistical models describing

the between-patient and within-patient variability were also tested.

The final models used in the NONMEM program are shown belowX Abstract published in Advance ACS Abstracts, May 15, 1997.

S0022-3549(96)00401-7 CCC: $14.00840 / Journal of Pharmaceutical Sciences 1997, American Chemical Society andVol. 86, No. 7, July 1997 American Pharmaceutical Association


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wher e for child i, C ij ) predicted concentration at measurement j, Koi) infusion rat e, Ti ) infusion time (0.5 h), i ) dosing interval, tij )elapsed time at measu rement j after end of infusion, CLi ) clearance,and Vd i ) volume of distribution.

where TVCL is the typical value of clearance; TVVd is the typicalvalue of volume of distribution; a nd s are model paramet ers.

where s are t he i nt eri ndi vi dual random effect vari abl es and t hevariances ofs are model paramet ers.

where C is the observed concentration and C is the concentrationpredicted from th e pha rma cokinetic structure model, is the intrain-di vi dual random effect vari abl e, and t he vari ance of i s a modelparamet er.

B eca u s e of t h e n a t u r e of t h e p r og r a m , d i ffe r e n t t i m e e ve n t sincluding time of initiation of the treatment, time of initiation of eachinfusion, time of the end of each infusion, and time of blood drawnw er e p r ov id e d t o t h e p r og r a m . T h es e t i m es w er e con v er t e d t oaccumulated values counting from the initiation of the tobramycintrea tmen t for an y given hospital sta y. The tobramycin concentr at ionswere converted to logarithm ic values. The dat a derived within one

hospitalization for any given patient were entered into the programas a set of data . After t he final models were selected an d converged,predicted t obram ycin concentra tions were obtained from NONMEM.

N e u r a l N e t w o r k ssA backpropagation algorithm implemented inC language (running on a Pentium-based PC, 133 MHz) was devel-oped. The specifi c al gori t hm adopt ed i n t hi s research has beendescribed previously.6,9 Figure 1 illustrates a schematic representa-tion of the neural network used to predict drug concentration frominforma tion on clinical prognostic factors, dosing regimens, a ndelapsed time after dosing. This network ha d seven input var iables(i.e., age, gender, illness, weight, dose, dosing interval, and time ofblood drawn). The time of blood dra wn was pr esented in two differentfashions, one as accumulated t ime counting from th e initiation of thetobramycin t reatment (test I) and the other as t ime after the initiationof the fourt h or the fifth infusion within one dosing interval (test II).The va lues of age, weight, dose, dosing int erval, or t ime of blood drawnwere normal i zed l i nearly t o range bet ween 0 and 1. Gender and

illness var iables had values of 0 or 1, with male)

0, female)

1 a n doncology ) 0, cystic fibrosis ) 1, respectively. The logarith m of thet obramyci n concent rat i on was used as out put val ues, whi ch werenormalized according to t he following formu la t o result in outputvalues ranging from 0 to 1.

Fl exi ble select i on of net work paramet ers such as t he number of hidden units, learning rat e, and moment um factor was allowed. Afterexperi ment at i on, ei ght and seven hi dden nodes were select ed t o

provi de t he i nt erconnect i ons bet ween i nput and out put for neuralnetworks test I a nd test II, respectively. As shown in Figure 1, eachnode i n a l ayer i s connect ed t o every node i n t he next l ayer. Eachinterconnection between nodes is a ssociated with a weighting factor

Each of t he 622 t obramycin measurement s al ong wi t h i t s corre-sponding clinical information, dosing regimens, and time of bloodd r a wn w a s t r e a te d a s a n e w in p u t a n d o u t pu t d a t a p a ir . D a t aassoci at ed wi t h t he same pat i ent were arranged i n adj acent orderwhen t hey were used t o t rai n t h e net work. Of t he 622 set s of inputand out put dat a, t he fi rst t wo-t hi rds of t he dat a (415 set s of dat a)were from pat i ent s who recei ved t hei r f i rst t obramyci n t reat mentbefore 1989 and were used as a t raining data set. The last one-thirdof the data (207 sets of data) were from patients who received their

fi rst t obramyci n t reat ment aft er 1989 and were used as t he t uni ngset. Because of subsequent hospitalizations, some of the pat ients inthe training data set had had tobramycin plasma levels collected after1989. Furt herm ore, no apparent t i me dependent di fferences i n t hepatient population could be observed to h ave r esulted in significantdifferences in these two data sets.

T h e n e t w o r k s w e r e t r a i n e d w i t h t h e i n p u t-output (prognosticfactors and dosing regimens vs tobramycin concentrations) data pairsi n t he t ra i ning set . The net work st ar t ed out wit h a random set oconnect ion weight s. Each pa i r of dat a went t hr ough t wo st ages oact ivat i on: a forward pass and a backward pass. The forward passi n vol ve d p r e se n t in g a s a m p le i n pu t t o t h e n e t wor k a n d l et t i n ga c t iv a t ion fl ow u n t i l t h e ou t p u t l a ye r w a s r e a ch e d . D u r in g t h ebackward pass, t he net works act ual out put from t h e forward pass(predi ct ed concent rat i ons) was compared wit h t he t arget out put(observed concentrations), and errors were computed for the outputuni t s. The weight s connect ed t o t he out put uni t s were adj ust ed in

order to reduce those errors. The error estimates of the output un itswere then used to derive error estimates for the units in the hiddenl ayers. Fi nal ly, errors were propagat ed back t o t he connect ionsstemming from the input units. The delta ru le9 was used as the errorcorrection formula to adjust the connection weights each time thenetwork saw a new input-output pa ir. A complete round of forwar d-backward passes and wei ght adj ust ment s usi ng al l i nput-outputpairs in the tr aining data set is called an epoch. A backpropagationn e t wor k n e e ds t o l ea r n f r om t h e s a m e d a t a s e t t h r o u gh a fe whundredsssometimes even thousandssof epochs in order to gra duallyrefine its connection weights.

In t hi s work, aft er each epoch of t rai ning and adj ust ment of t heconnect ion wei ght s, t h e t rai ned n et work was evaluat ed wit h t het uni ng dat a. The errors of t he out put val ues from t h e t uni ng dat awere calculated an d reported. These errors were not propagated backfor connection weight a djustment s. Convergence of training (termi-nat i on of t rai ni ng) was det ermi ned when t he net work showed a

minimum sum of square error in predicting the tobramycin concen-t rat i ons i n t he t uni ng dat a set .

S t a t i s t i c a l A n a l y s i ssThe observed and predicted logarithm oftobramycin concentrations were transformed back to actual concen-trations and then used to evaluate and compare the performances oft h e n e u r a l n e t wor k t e s t s a n d of N O N ME M . M ea n s qu a r e d e r r o r(MSE), mean prediction error (ME), and their corresponding 95%confidence intervals were estimated according to that described bySheiner and Beal.10 MSE and ME were used as measures of precisionand bias, respectively. Precision an d bias were also evaluat ed relativeto the observed tobramycin concentra tions using percent absoluteerr or (%AE) and percent pr ediction err or (%PE), respe ctively. Thesepercent errors were est i mat ed according t o t h e foll owing equa-

Figure 1sSchematic representation of the neural network used in estimatingplasma tobramycin levels from demographic factors and dosing regimens.

the ph arma cokinetic structure model

C ij )koi(1 - e

-(CL i/Vdi)Ti)

CL i(1 - e-(CL i/Vd i)i)

e-(CL i/Vd i)tij

the regression model

TVCL ) 1 ln(weight) + 3 a ge

TVVd ) 2 ln(weight)

the population model

CL i ) TVCL e1

Vd i ) TVVd e2

the variance model

ln Cij ) ln C ij + ij

normalized output value )

[ln(observed concn) - ln(smallest observed concn)]

[ln(largest observed concn) - ln(smallest observed concn)]

Journal of Pharmaceutical Sciences / 841Vol. 86, No. 7, July 1997


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tions:

St at i st i cal measures of t he performances of t he neural net worktests and NONMEM were compared using a paired t-t est . A p value


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Figure 3 demonstrates t he relationship between th e residu-als of the predictions (estimated from predicted concentration- observed concentra tion) from NONMEM versus t hat fromneural n etw ork analys es. M os t of the res iduals w ere foundto ra nge from -4 t o 4 . I t a p p e a r s t h a t t h e r e s id u a l s of t h epredictions of the neural netw ork tes ts s how ed a pos itivecorr elation with th ose of NONMEM, with a h igher corr elationfound betw een neural netw ork tes t II and N O N M EM (cor-rela tion of coefficient , r, of 0.82).

Table 2 summarizes the statistical measures of the perfor-m a n c e of n e u r a l n e t w or k t e s t s a n d N O NM E M. W h en a c-cum ulated tim es w ere us ed, neural netw ork tes t I res ulted

in significantly higher MSE tha n NONMEM analysis. With-ou t t h e u s e o f a c cu m u l a t e d t i m e s , n e u r a l n e t w or k t e s t I Iprovided precision of th e predicted concentr at ions compara bleto that w ith N O N M EM analysis . W hen the precis ion w asmeasu red relative t o the observed concentra tion (i.e., %AE),a better average precision was seen with neural network testII th an with NONMEM (33.9% vs 39.9%). NONMEM resultss h ow ed a s m a ll er M E t h a n t h a t of t h e n e u r a l n e t wor k analys es. H owever, w hen the bias w as m eas ured relative tothe observed concentration (i.e., %PE), neural network testII exhibited only a 2.59% m ean prediction error, w hereasNONMEM had a 17.3% mean prediction error.

Discussion

T h is w or k s h ow ed t h a t n e u r a l n e t wor k s a r e ca p a b le ofcaptur ing the relations hips betw een plas m a dr ug levels andthe following var iables, patient -related prognostic factors,dos ing regim ens , a nd tim e of blood draw n from routinelycollected s pars e w ithin-patient pharm a cokinetic data. Thisres earch furth er docum ents the applicability of neural net-w orks to population pharm a cokinetic data analys is . I t als o

s how ed that proper pres entation of the tim e factor in theneura l network appr oach contr ibuted to improved concentra -tion predictions.

Because pharmacokinetic processes are recognized to behighly time-dependent events, the analysis of plasma concen-trat ion data alw ays requires cons ideration of the relativetimes of collection of blood samples and administration of thedose. Previous related work by Brier et al.8 could not considerthis factor s ince the actual t im ing of the peak and troughs am ples had not been recorded. In the current res earch, thetime of blood dra wn was originally presented to the networka s e la p s ed t i m es a ccu m u l a t e d fr om t h e i n it i a t ion of t h ether apy within one hospitalization (test I). This was done tom atch the data input into the N O N MEM program . Becaus eof t h e n a t u r e of t h e N O N ME M p r og r a m , i n for m a t i on ondifferent time events including time of the initiation of thetobramycin treatment, time of the initiation or the terminationof each infusion, and time of blood drawn was provided to theprogram. These times were converted to accumu lated valuescount ing from the initiat ion of the t hera py. However, inter-nally w ithin the N O N M EM program m the concentration-time profile was analyzed within each dosing interval, i.e. ,the blood sampling times were converted back to the valueswithin each dosing interval. In th e neural network approachw h e n t h e a c c u m u l a t e d t i m e v a l u e s w e r e p r e s e n t e d t o t h enet work, t he differences between t he t imes of th e pair of bloods am ples collected w ithin one dos ing interval becam e verys m all w hen com pared to the total accum ulated t im es. Thenetwork therefore attempted to capture the global relationshipov er t h e t ot a l a ccu m u l a t e d t i m e, w h ich r e s u lt e d i n l es ss atis factory perform ance of the es tim ation of tobram ycin

concentra tions within a dosing int erval. When the nonaccu-m u l a t e d t i m e s w e r e u s e d a s t h e i n p u t d a t a ( t e s t I I ) , t h eperformance of the network improved, resulting in smallererrors in the predicted concentrations than thos e from theN O N M EM program an d neura l netw ork tes t I . Addition ofinput variables w hich could account for the accum ulatednatu re of the tim e factor (e.g., th e t otal num ber of dos esreceived) could improve the performance of neural networktes t I . Com paris on of the tw o neural netw ork tes ts s ugges tsthat the w ay th at the t im e factor is pres ented to the netw orkis import an t to the net works perform an ce on time-dependentevents.

Figure 3sPlots of residuals of the prediction from the neural network approachversus that from the NONMEM program. Residuals were calculated as thedifference between the predicted and observed concentrations. Neural networktest I used accumulated times within one hospitalization, whereas test II wastrained with unaccumulated times within one dosing interval. Solid line representsthe linear regression line of residuals of the two methods [y) 0.670 + 0.683x,r2 ) 0.44 for the plot of neural network test I vs NONMEM (a) and y) 0.378

+ 0.776x, r2 ) 0.67 for the plot of neural network test II vs NONMEM (b)] anddashed line represents the line of identity. Residual concentrations are inmicrograms/milliliter.

Table 2sStatistical Measures of the Performance of Neural Networks andthe NONMEM Program

Neural Network

NONMEM Test I Test II

Mean squared error 1.88 2.38b 1.78c

(MSE) (1.39, 2.37)a (1.81, 2.96) (1.30, 2.26)Mean prediction error 0.077 0.62b 0.32b,c

(ME) (0.031, 0.18) (0.73, 0.51) (0.42, 0.22)% mean absolute error 39.9 37.3 33.9b,c

(mean AE) (36.4, 43.3) (34.6, 40.1) (31.2, 36.6)

% mean prediction error 17.3 5.01b 2.59b,c(mean PE) (12.8, 21.7) (9.04, 0.98) (1.23, 6.42)

a Data in parentheses represent the 95% confidence interval of the meanb Significantly different from NONMEM analysis, p< 0.05. c Significantly differenfrom neural network test I, p < 0.05.



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I n ou r n e u r a l n e t wor k a n a l ys is , e x p er i m en t a t i on w a sperformed t o define thr ee different par amet ers associated withthe t raining of a backpropagation n eural netw ork: learningrate, hidden units , and m om entum . Although th e s electionof t h e s e p a r a m e t e r s cou l d s om e t im e s b e e m pi r ica l , i t i srecognized that each of these choices could have a significanteffect on net work performa nce. The num ber of hidden un itsrefers to the number of nodes which are placed between thei n pu t a n d ou t p u t l a ye r s t o p r ov id e t h e i n t er con n e ct i on sbetw een the input inform at ion a nd output values . A largenum ber of hidden units can m ap a com plex input-output

relationship, but a network will consequently require morecomputation time and may suffer in its ability to generalizefor an un known data set. Learning rate and momentum, withvalues th at could ra nge from 0 to 1, are netw ork para m etersaffecting the magnitu de of weight adjustmen t between input ,hidden, and output layers. The following simplified relation-s hip pres ented by Erb 11 points out the effects of thes e tw oparam eters on the w eight adjus tm ent:

T h e h igh e r t h e l ea r n in g r a t e , t h e fa s t er t h e le a r n in gprocess. However, if the lear ning r at e is too high, oscillationin the weight change can prevent convergence to an optimal

s ol u t ion . M om e n t u m i s a p a r a m e t e r w h i ch t a k e s t h e l a stweight change into consideration when making new weightch a n g es . T h e h i gh e r t h e m o m en t u m v a lu e , t h e f a s t er t h eneural n etw orks tr aining pace. But, s im ilar t o the im pactfrom a high learning ra te, high momentu m results in fluctua -tion of new weight changes t hat can a lso impede convergenceto an optim al s olution. W e have exam ined the perform anceof t h e n e t wor k w h en t h e n u m be r of h id de n u n it s w a sincreased from 1 to 60 for different learning rates (0.05-0.60)a n d m om e n t u m fa ct or s (0-0 .9 0). As i n ou r p r e vi ou s r e -search,6 w e found th at a m oderate learning rate w ithout theus e of a m om entum factor provided the optim um netw ork.Furt hermore, minimal errors were obtained when t he n etworkcontained s even to eight un its in t he h idden layer.

O ve r -t r a i n in g of n e u r a l n e t wor k s h a s oft e n cr e a t e d a nundes irable effect on the generalization proces s . U pon re-peated t raining, the system continu es to minimize the differ-ences between the predicted and observed concentrations ofthe tra ining data. If training is continued, the m ean s quaree r r or m a y e ve n t u a l ly b e l ow er t h a n t h a t fr om N O NM E Mana lysis. However, the network actually memorizes th e inputand output relationships of the data from the tr aining set andeventua lly loses its capa bility to predict unkn own dat a. Weadopted a method suggested by Rumelhart 12 to incorporate at u n i n g (or v a li da t i on ) d a t a s et t o p r e ve n t g en e r a li za t i onproblems. This technique also had been su ccessfully adoptedin our previous w ork to im prove the predictive pow er of atrained netw ork for unknow n data. 6 In th e current applica-tion, this als o provided a fair com paris on w ith the res ultsobta ined from t he NONMEM a nalysis. After ea ch epoch oftraining, the trained netw ork w as evaluated w ith t he tun ing

data . The sum of square error was estimated. The epoch thatgave the low es t s um of s quare error for the tuning s et w ass el ect e d a s t h e op t im u m e p och a n d t h e t r a in i n g w a stermina ted. The final net work with t he optimized connectionweight s was used t o estimate t obram ycin concentr at ions. Withthis early terminat ion of the tr aining, the mean squar ed errorand average percent of abs olute err or from neura l netw orktes t II w ere com parable to or s m aller than thos e from theNONMEM program.

Because tobramycin is extensively cleared by the kidney,it w ould be expected that creatinine clearance w ould s erveas a major covariate in the population pharmacokinetic data

an alysis. However, since serum creat inine levels had not beencompiled from these pat ients, creatinine cleara nce could n otbe estimated and used as a clinical variable in our analyses.

Previous work by Brier et al.8 included both serum creatininea n d cr e a t i n in e cl ea r a n c e a s t h e i n p u t v a r i a b le s i n t h e a p -

plication of neural netw orks to prediction of gentam ycinconcentrations . Creatinine clearance als o was u s ed as th e

covariable for systemic clearance in their NONMEM analysisof t h e g e n t a m yci n d a t a . I t w a s f ou n d t h a t N O NM E M a n d

neural networks exhibited average absolute errors of 18.6%and 16.5% in pr edicting th e peak gent amycin concentra tions

of a set of leave-out data, respectively, and 59% and 48.3% inpredicting the trough gentam ycin concentra tions, respectively.In our res earch, the perform an ces of the t w o m ethods w ere

not evaluated s eparat ely for the peak a nd trough concentra -tions becaus e the blood s a m ples w ere not dr aw n exactly at

those time points. The average absolute err or on th e overalld a t a w a s 3 9. 9% f r om N O NM E M a n d 3 3.9 % f r om n e u r a l

network t est II, despite th e lack of informa tion on creatininecleara nce. Furt hermore, the residuals of the pr edictions from

neura l network test II were highly correlat ed with t hose fromNONMEM. Consistency in residuals observed between dif-ferent prediction methods tends to suggest that the residuals

are likely to be attributed to errors in data recording or lackof certain common covariables. This observat ion suggests t hat

not having included information on creatinine clearance in

this com parative s tudy s hould not have bias ed the perfor-mance of either th e neura l network a nalyses or t he NONMEManalysis.

The m ajor s trength of the us e of neural netw orks in the

ana lysis of sparse within-patient pharm acokinetic data is t hatthey do not assu me a specific model. Instea d, neural net workslearn from the data to es tablis h the input-output relation-s h ip . T h is e li m in a t e s t h e n e ed t o d e a l w i t h t h e com p le xnonlinear mixed effect models r equired for th e population

phar macokinetic modeling approach. With th e incorporationof a hidden layer and fully connected units between layers,neural networks are capable of capturing complex nonlinear

relations hips and polynom ial s urfaces , including third orhigher order terms as well as cross-product terms correspond-ing to intera ctions between clinical factors. From th e stand-

point of modeling a nd theoretical scientists, the major advan-tage of the n eural n etworks could tur n int o its major weakn ess

Without a specific underlying model, neura l networks will notbe capable of providing population values of pharmacokineticparameters and the within-patient and between-patient vari-

ability of these par ameter s. However, even in th e traditionalpopulation phar macokinetic modeling approach, insu fficientinform at ion from the data m ay prevent accurat e es tim ation

o f s o m e o f t h e s e p a r a m e t e r s a n d q u e s t i o n s r e g a r d i n g t h emodelsreliability could still exist. Becau se the ult imat e goals

of population pha rma cokinetic an alysis during t he process ofdrug development a re to identify potent ial prognostic factorsw hich could affect the res ulting plas m a drug levels and to

recognize a special patient population which will handle thedrug differently, incorporating other statistical analyses in

determinin g th e effects of demograph ic factors on t he plasm adrug levels will provide practical information n eeded durin gthe process of drug development.

In conclusion, th is work sh owed th at neur al net works were

capable of capturing the relationships between patient-relatedfactors and plasma drug levels from routinely collected sparse

within-patient pha rmacokinetic data. This furth er documentsthe applicability of neural netw orks to population pharm a -cokinetic data ana lysis. Our ongoing work in volves incorpora-

tion of a graphical interface and w hat-if analys is into theneural netw orks that w e have developed on P ent ium -bas edP Cs . W ith thes e additional features , w e hope to develop a

new weight change ) lear ning r at e error +

momentum last weight change

844 / Journal of Pharmaceutical SciencesVol. 86, No. 7, July 1997


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flexible, robust, and easy-to-use system wh ich is appr opriatefor population pharmacokinetic data analysis.

References and Notes1 . Y u h , L .; B ea l , S .; D a vi di a n , M .; H a r r i s on , F . ; H e s t e r , A. ;

Kowalski, K.; Vonesh, E.; Wolfinger, R. Biometrics 1994 , 50 ,566-575.

2. Sheiner, L. B.; Ludden, T. M. Annu . R ev. Pharm acol. T oxicol.1992, 32 , 185-209.

3. Whiting, B.; Kelman, A. W.; Grevel, J. Clin. Pharmacokinet.1986, 11 , 387-401.

4. 4.Veng-Pederson, P.; Modi, N. B. J. Pharmacokinet. Biopharm.1992, 20 , 397-412.5. 5.Hussa in, A. S.; Johnson, R. D.; Vachhar ajani, N. N.; Ritschel,

W. A. Ph a rm. R es. 1993, 10 , 466-469.6. Chow, H.; Chen, H.; Ng, T.; Myrdal, P .; Yalkowsky, S. H. J .

Chem. Inf. Comput. Sci. 1995, 35 , 723-728.7. Gobburu, J. V. S.; Chen, E. P . J. Ph a rm. S ci . 1996, 85 , 505-

510.

8. Brier, E. M.; Zurada, J. M.; Aronoff, G. R. Ph a rm. R es. 199512 , 406-412.

9. Rumel hart , D. E.; Hi nt on, G. E.; Wil li ams, R. J . In ParallelDistributed Processing ; Rumelhart , D. E., McClelland, J . L., thePDP Resear ch Group, Eds.; MIT Press: Cambridge, MA, 1986pp 45-76 .

10. Sheiner, L. B.; Beal, S. L. J. Pharmacokinet. Biopharm. 19819, 503-512.

1 1. E r b , R. J . Ph a rm. R es. 1993, 10 , 165-170.

12. Rumelha rt, D. E.; Widrow, B.; Lehr, M. A. Commun. ACM 199437, 87-92.

Acknowledgments

Thi s proj ect was support ed i n part by a gran t from t he Nat i onalScience Foundation (IRI9525790).

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