research article parameter estimation of population pharmacokinetic models...

Research ArticleParameter Estimation of Population PharmacokineticModels with Stochastic Differential Equations Implementationof an Estimation Algorithm

Fang-Rong Yan12 Ping Zhang12 Jun-Lin Liu12 Yu-Xi Tao12

Xiao Lin12 Tao Lu12 and Jin-Guan Lin123

1 State Key Laboratory of Natural Medicines China Pharmaceutical University Nanjing 210009 China2Department of Mathematics China Pharmaceutical University Nanjing 210009 China3Department of Mathematics Southeast University Nanjing 210096 China

Correspondence should be addressed to Fang-Rong Yan fryan163com

Received 19 May 2014 Revised 20 September 2014 Accepted 28 September 2014 Published 10 November 2014

Academic Editor Steve Su

Copyright copy 2014 Fang-Rong Yan et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Population pharmacokinetic (PPK) models play a pivotal role in quantitative pharmacology study which are classically analyzedby nonlinear mixed-effects models based on ordinary differential equations This paper describes the implementation of SDEsin population pharmacokinetic models where parameters are estimated by a novel approximation of likelihood function Thisapproximation is constructed by combining the MCMC method used in nonlinear mixed-effects modeling with the extendedKalman filter used in SDEmodelsThe analysis and simulation results show that the performance of the approximation of likelihoodfunction for mixed-effects SDEs model and analysis of population pharmacokinetic data is reliable The results suggest that theproposed method is feasible for the analysis of population pharmacokinetic data

1 Introduction

Population pharmacokinetic (PPK)models play a pivotal rolein quantitative pharmacology studyThey combine the classicpharmacokinetic (PK) analysis with population statisticalmodels whichmakes it possible to investigate the determinedfactors of drug concentration in patients group quantitativelyto guide the adjustment of administration and to providecomprehensive quantitative information for a more rationaland effective clinical administration regimen In recent yearswith the trend that model based drug development (MBDD)is promoted and strongly advocated by the FDA the popu-lation pharmacokinetic theories and practice researches havebeen greatly developed

Nonlinear mixed-effects modeling has been proven to bea kind of widely used and effective tool for describing thepharmacokinetic (PK) and pharmacodynamic (PD) proper-ties of drug [1 2] However traditionally the most commonlyused nonlinear mixed-effects PPK models are based on

ordinary differential equations (ODEs) supplemented by amodel for the interindividual variations in the structuralmodel parameters and a model for the variation of theresiduals and those two models are assumed to be inde-pendent so that the residuals are determinate However thepharmacokinetic process will be affected by much stochasticvolatility which cannot be described by the determinateerrors so the preceding assumption is not appropriate [3]

Stochastic differential equations (SDEs) are a kind ofmodelingmethod developed recently which could be used todeal with such kind of questions In nonlinear mixed-effectsmodeling with SDEs the differences between individualvalues and observations are explained by three fundamentallydifferent types of noise (1) the dynamic noise which entersthrough the dynamics of the system and may arise fromstructural model misspecification or unpredictable randombehavior of underlying process (2) random effect whichdescribes the interindividual variations (3) themeasurementnoise which represents the uncorrelated part of the residual

Hindawi Publishing CorporationJournal of Probability and StatisticsVolume 2014 Article ID 836518 8 pageshttpdxdoiorg1011552014836518

2 Journal of Probability and Statistics

variability that may be due to assay error [4] The simulationresults show that the introduction of the SDEmay contributeto the better estimates of the interindividual variations andstructural parameters [5]

The first paper encouraging the introduction of randomfluctuations in PKPD was published by DrsquoArgenio et al[6ndash8] The authors underlined that both deterministic andstochastic component contribute to PKPDmodelsHoweverestimating parameters in nonlinearmixed-effectsmodel withSDE is a difficult problem and not straightforward exceptfor simple cases Naturally likelihood inference would bea feasible approach but the transition densities are rarelyknown thus explicit likelihood function is usually hard toget Actually there are hardly any theories for SDE modelsat present Kristensen NR proposed maximum likelihoodmethod and maximized a posterior to estimate the param-eters in PK models with SDE but this method only focusedon single subject modeling where no interindividual variancecomponents were estimated [9] Overgaard RV suggestedapplying the Kalman filter to approximate the likelihoodfunction for a SDE model with a nonlinear drift term and aconstant diffusion term [10] Tornoe CW used this algorithmto estimate SDEs in NONMEM [11] but the NONMEMimplementation cannot be used to form Kalman smoothingestimates which is an important feature of the SDE approachwhere all data is used to give optimal estimates at each sam-pling point Sophie Donnet proposed a Bayesian inferenceto analyze growth curves using mixed-effects models basedon stochastic differential equations and obtained good results[12] Struthers and McLeish applied Bayesian method to themulticenter AIDS cohort study [13]

Inspired by this the present work describes an approx-imation of likelihood function of nonlinear mixed-effectspharmacokinetic model which is constructed by combin-ing the extended Kalman filter with the MCMC methodThe parameters in the nonlinear mixed-effects models areestimated based on stochastic differential equation and thewhole process is implemented in Matlab Section 2 presentsthe classical nonlinear mixed model and the mixed modeldefined by SDEs In Section 3 the algorithm of nonlinearmixed-effects model with SDEs is proposed In Section 4example with simulated data and a case study are describedConclusion and discussion are listed in Section 5

2 Models and Notation

Nonlinear mixed-effects models can be regarded as hierar-chical models where the variability in concentrationeffectis split into intraindividual variability described by the first-stage distribution and interindividual variability describedby the second-stage distribution This section describes thenotation for nonlinear mixed-effects models used in thepresent paper

21 Nonlinear Mixed-Effects Model Based on Ordinary Dif-ferential Equation Let 119910 = (119910

119894)1le119894le119899

= (119910119894119895)1le119894le1198991le119895le119899119894

119894 =1 119899 119895 = 0 119899

119894 be the true observations where 119910

119894119895is

measurement for individual 119894 at time 119905119894119895 119899 is the number of

individuals and 119899119894is the number of measurements for indi-

vidual 119894 Traditional nonlinear mixed-effect models usuallymodel this process by nonlinear mixed-effects model basedon ordinary differential equations Formally the classicalnonlinear mixed-effects model is written as

120597119909

120597119905= 119892 (119909 120601 119905)

119910119894119895= 119891 (119909 (119905

119894119895) 120601119894) + 120576119894119895 120576

119894119895sim119894119894119889119873(0 120590

2

120576)

120601119894sim 119873 (120583Ω)

(1)

where 119891 is a possibly nonlinear function and Φ = (120601119894)1le119894le119899

is the individual parameter vectors 120601119894is assumed to be

independently and identically normally distributed withexpectation 120583 and varianceΩ 120576

119894119895are the measurement errors

and are also assumed to be independently and identicallynormally distributed with null mean and variance 1205902

120576

22 Nonlinear Mixed-Effects Model Based on Stochastic Dif-ferential Equation While the introduction of SDEs doesnot change the fundamental hierarchical structure theydo change the entities in the first-stage density and theconstruction Under the framework of ordinary differentialequations noise is only introduced through themeasurementequation see (1) This allows the measurement noise term toabsorb the whole error due to model miss-specification ortrue random fluctuations of the states and hence may ignorethe correlated residuals In order to consider the correlatedresiduals a stochastic process is added to the state spacemodel such that the nonlinear mixed-effects model based onSDEs can be written as

119889119909119905(120601119894) = 119892 (119909

119905 120601119894 119905) 119889119905 + Γ (119909

119905 120601119894 1205902) 119889119882119905

119909 (119905 = 0) = 1199090 (120601)

119910119894119895= 119891 (119909 (119905

119894119895) 120601119894 119905119894119895) + 120576119894119895 120576

119894119895sim119894119894119889119873(0 120590

2

120576)

120601119894sim 119873 (120583Ω)

(2)

where Γ(119909119905 120601 1205742) 119889119882

119905is called the diffusion term and

describes the stochastic part of the system 119882119905is a stan-

dard Wiener process defined by 1199081199052minus 1199081199051

sim 119873(0 |1199052minus

1199051|119868) 119892(119909

119905 120601 119905) 119889119905 is called the drift term and describes the

deterministic part The stochastic dynamics of the system isdefined by the drift and diffusion terms together

In nonlinear mixed-effects model based on SDEs (see(2)) the total variance is divided into three fundamentallydifferent noises the interindividual variability Ω describingthe individual difference the system noise 120590

2 reflectingthe random fluctuations around the corresponding dynamicmodel and the measurement noise 1205902

120576representing the

uncorrelated residuals originating from measurement assayor sampling errors

Journal of Probability and Statistics 3

3 Algorithm of the NonlinearMixed-Effects Model with SDEs

To solve the nonlinear mixed-effects model with SDEswe calculate the approximation of likelihood function con-structed by combining the extended Kalman filter with theBayesian inferencesThedetails are described inwhat follows

31 Extended Kalman Filter We use the extended Kalmanfilter to calculate the one-step predictions and the one-step predicted variances for a stochastic differential equationwith additive diffusion and measurement noise The Kalmanfilter is a recursive estimator which means that only theestimated state from the previous time step and the currentmeasurement are needed to compute the estimate for thecurrent state The algebra presented in the following is allperformed on the individual level and the 119894 index referring tothe individual had been dropped for simplicity The generalintraindividual model can be written as

119889119909 = 119892 (119909 120601 119905) 119889119905 + 120590119908119889119882

119910119895= 119891 (119909 (119905

119895) 120601) + 120576

119895

(3)

where 119909 is the vector of state variables 119910119895is the vector

of measurements at time 119905119895 120576119895are the associated normally

distributed measurement errors with covariance matrix sumand 120590

119908119889119882 is the system noise

The state of the filter is represented by two variables asfollows

(1) 119909119896|119896

is a posteriori state estimate at time 119896 givenobservations up to and including time 119896

(2) 119875119896|119896

is a posterior error covariance matrix

And we need to initiate the extended Kalman filter (EKF)with a prediction of the initial state 119909

1|0and a prediction of

the covariance of the initial state 1198751|0

From this point the EKF is most often conceptualized astwo distinct phases ldquopredictrdquo and ldquoupdaterdquoThe predict phaseuses the state estimate from the previous time step to producean estimate of the state at the current time step which isachieved by

119910119895|119895minus1

= 119891 (119909119895|119895minus1

120601)

119877119895|119895minus1

= 119862119895119875119895|119895minus1

119862119879

119895+ sum119895|119895minus1

(4)

The update phase uses the actual measurement to updateour state prediction and variance This is performed by theupdate equations that is

119909119895|119895= 119909119895|119895minus1

+ 119870119895(119910119895minus 119910119895|119895minus1

)

119875119895|119895= 119875119895|119895minus1

minus 119870119895119877119895|119895minus1

119870119879

119895

119870119895= 119875119895|119895minus1

119862119879119877minus1

119895|119895minus1

(5)

where119870119895is the Kalman gain

The final step in the recursive algorithm is to predictthe state and the state variance at the time of the followingmeasurement This is performed by solving the predictionequations that is

119889119909119905|119895

119889119905= 119892 (119909

119905|119895 119905 120601)

119889119875119905|119895

119889119905= 119860119905119875119905|119895+ 119875119905|119895119860119879

119905+ 120590119908120590119879

119908

(6)

where 119860119905= (120597119892120597119909)|

119909=119909119905|119895minus1

So the algorithm of extended Kalman filtering proceedsas follows

(1) Given parameters and initial prediction 120601 1199091|0

and1198751|0

(2) For 119895 = 1 to 119899119894do

(3) Use (4) to compute 119910119895|119895minus1

and 119895|119895minus1

(4) Use (5) to compute the Kalman Gain119870119895

(5) Use (5) to compute updates 119909119895|119895

and 119875119895|119895

(6) Use (6) to compute 119909(119895+1)|119895

and (119895+1)|119895

(7) end for(8) Return (for all 119895) 120576

119895= 119910119895minus 119910119895|119895minus1

and 119877119895|119895minus1

After the prediction of the state value at the followingmeasurement we start again with predictions of the actualmeasurements until all the one-step prediction 119910

119895|119895minus1and all

the one-step prediction variance 119877119895|119895minus1

have been calculated

32 The Likelihood Function for the Nonlinear Mixed-EffectsModel with SDEs In the present work we will use a quasi-likelihood method that is a method that uses the Gaussianapproximation so we assume that the conditional densitiesare well approximated by Gaussian densities The calculationof the conditional densities for the intraindividual model isfacilitated by the extended Kalman filter (EKF)

The EKF approximates the conditional densities withGaussian distributionsThe conditional densities describe thedistribution of the following measurement on the conditionof all the previous measurements so that the mean of thedistribution is identical to the prediction of the followingmeasurement that is the one-step prediction 119910

119894(119895|119895minus1) Like-

wise the covariance of the conditional densitywill be the one-step prediction covariance 119877

119894(119895|119895minus1) We have thus completely

described the approximate Gaussian conditional densities bythe conditional mean and covariance which are

119910119894(119895|119895minus1)

= 119864 (119910119894119895| 119884119894(119895minus1)

sdot)

119877119894(119895|119895minus1)

= 119881 (119910119894119895| 119884119894(119895minus1)

sdot)

(7)

The one-step prediction error 120576119894119895is given by

120576119894119895= 119910119894119895minus 119910119894(119895119895minus1)

isin 119873(0 119877119894(119895119895minus1)

) (8)


Using the notation above the Gaussian approximation ofthe first-stage distribution density function can be written as

1199011(119884119894119899119894| sdot) asymp

119899119894

prod119895=1

exp (minus (12) 120576119879119894119895119877119879119894(119895119895minus1)

120576119894119895)

radic100381610038161003816100381610038162120587119877119894(119895|119895minus1)

10038161003816100381610038161003816

(9)

The second-stage density can be written as 1199012(120601119894| Ω)

which is included in the same way as for ordinary differentialequations This gives us the full nonlinear mixed-effectslikelihood function

119871 (120601sum 120590119908 Ω) prop

119873

prod119894=1

int1199011(119884119894119899119894| 120601119894sum 120590

119908) 1199012(120601119894| Ω) 119889120601

119894

=

119873

prod119894=1

int exp (119897119894) 119889120601119894

(10)

where

119897119894= minus

1

2

119899119894

sum119895=1

(120576119879

119894119895119877minus1

119894(119895|119895minus1)120576119894119895+ log 100381610038161003816100381610038162120587119877119894(119895|119895minus1)

10038161003816100381610038161003816)

minus1

2120601119879

119894Ωminus1120601119894minus1

2log |2120587Ω|

(11)

is the approximate of a posterior log-likelihood function forthe random effects of the 119894th individual It is observed that thelikelihood function is based on the one-step prediction error

33 The Parameter Estimation for the Nonlinear Mixed-EffectsModel with SDEs As usual for nonlinearmixed-effects mod-els the likelihood function cannot be solved analyticallyApproximations therefore have to be made in order toestimate the parameters and Bayesian inferential method isconsideredThe Bayesian approach allows prior distributionsto be incorporated with the likelihood function to evaluatethe posterior distribution of the population parameters inthe SDE model Thus the first step is the choice of the priordistributions Usual diffuse prior distributions can be chosenbut the resulting posterior distributions may not be properTherefore we propose to use standard prior distributionssuggested by de la Cruz-Mesıa and Marshall [14]

120583119896sim 119873(119898

prior119896

Vprior119896

) 119896 = 1 119901

Ωminus1sim 119882(119877 119901 + 1)

1

1205902sim Γ (120572

prior120590

120573prior120590

)

(12)

where 119882 and Γ are the Wishart distributions and Gammadistributions respectively We select a uniform prior on 1205742In practice the specification of hyper parameters119898prior

119896 Vprior119896

119877 120572prior120590

and 120573prior120590

may be very difficult therefore we choosethe noninformative priors for the hyperparameters

For the ODE model (see (1)) Gibbs sampling algorithmshave been proposed in the literature [15] These algorithmswill not be detailed here For the SDE model we proposeto use a Gibbs algorithm In the case when the posterior

distributions of the parameters have no explicit form wepropose to approximate it using the Metropolis-Hastingsrandom walks The convergence of the Metropolis-Hastingsalgorithm is ensured by the theorem proposed byMengersenand Tweedie [16]

Given the starting value 120579(0) at iteration 119905 = 0 thealgorithm proceeds as follows

Metropolis-Hastings random-walks algorithmFor (119905 = 1 119879) repeat

(1) Draw 120579lowast from the proposed distribution 119902(sdot | 120579119905minus1)

(2) Compute the ratio 120572(120579lowast 120579119905minus1) =

min1 120587(120579lowast)119902(120579lowast 120579119905minus1)120587(120579119905minus1)119902(120579119905minus1 120579lowast)

(3) Set

120579119905=

120579lowast with probability 120572120579119905minus1 with probability 1 minus 120572

(13)

Repeat the procedures until it converges then we throwthe front unstable Markov chains and take the mean ofthe following stable Markov chain as the estimators ofparameters

4 Numerical Studies

In this section we introduce the algorithm to computethe estimator of the parameters in the population phar-macokinetic model The model structure is a linear two-compartment model with elimination from the central com-partment We use the simulation and an application to thereal data to illustrate the reliability of the algorithm

41 Simulation Study In the simulation we simulate 100datasets consisting of 119899 = 10 individuals and 119899

119894=

10 measurements from the following linear model whichis proposed with independent Brownian motions on eachequation and diffusion coefficients [17]

1198891199091= (119896211199092minus 119896121199091minus 119896101199091) 119889119905 + 120590

1119889119908

1198891199092= (119896121199091minus 119896211199092) 119889119905 + 120590

2119889119908

119910 = 1199091+ 120576 120576 sim 119873 (0 120590

2

120576)

120601 = (119896101198961211989621) sim 119873 (120583Ω)

120583 = [

[

12058310

12058312

12058321

]

]

Ω = [

[

119908210

0 0

0 119908212

0

0 0 119908221

]

]

(14)

where 1199091is the concentration of drug in the central com-

partment 1199092is the concentration of drug in the peripheral

compartment 11989610 11989612 and 119896

21are the rate constant 119910 is the

measurement 120576 is themeasurement error 120590120576is the coefficient

of variation for the measurement error 119908 is a standardWiener process and 120590

1and 120590

2are the magnitude of the

system noise


Table 1 Mean estimates and 95 credibility interval in brackets obtained from the SDE mixed-effects models and ODE models on 500simulated datasets

Parameter True value SDE estimation(confidence interval)

ODE estimation(confidence interval)

12058310

02 01999(01999 02006)

00984(minus06448 08416)

12058312

05 04755(047503 05213)

05339(04089 06589)

12058321

025 02501(02499 02501)

01815(minus00303 03932)

11990810

001 00173(00063 00467) mdash

11990812

01 01482(00176 01812) mdash

11990821

002 00174(00035 00402) mdash

120590120576

02 01488(01343 02441) mdash

1205901

01 00793(00402 01091) mdash

1205902

01 01087(00767 01515) mdash

For all simulations the sampling time period of thesimulation interval was 5 The initial values for popula-tion parameters were 120583

119879 = [02 05 025] diag(Ω) =

(0012 012 0022) 1205902120576

= 004 1205901= 01 120590

2= 01 On

the simulated datasets parameters were estimated usingalgorithm described previously We chose 10000 iterationsto make sure of the convergence of the MCMC algorithmEstimates were obtained as the expectation of the parameterposterior distribution of the last 5000 simulated trajectoriesof the SDE generated during the MCMC algorithmThe truevalues and estimators are reported in Table 1 where the 95credibility intervals are given in brackets The results arepresented in Figure 1 Figure 2 shows the basic goodness-of-fit graph for simulated data From the results we can concludethat the parameters are well estimated

The results of SDE and ODE as presented in Table 1demonstrate that the SDE estimation worksmuch better thanthe ODE estimationThe estimation of SDE is much closer tothe true value and the 95 confidence interval is smaller thanthe ODE method Only the 120583

12has a relative rational result

and the other two parameters are not well estimated in ODEsimulationThe result gives an obvious comparison of the twomethods on parameters estimation and it is believed that theSDE does better

When the time points turned out to be less than 10maybe 6 or even less the SDE method does not perhapsperform well enough Sparse data cannot satisfy the mod-eling need so the parameter estimation may be unreliableConsidering the sparse data data augmentation for instancelocal polynomial is a good tool dealing with sparse data inpharmacokinetics The local polynomial data augmentationbased on limited data inserts data into the adjacent data toget more data points so as to satisfy the statistical need [18]

Resp

onse

Time

Simulation data

Simulation data

Predicted dataConfidence interval

10

9

8

7

6

5

4

3

2

1

0

0 05 1 15 2 25 3 35 4 45 5

Figure 1 Simulation data are given by the circle and prediction dataare given by the asterisk linked in a lineThe 95 credibility intervalsare given by the dotted line

Then the SDEmodels are applied to the augmented data andexpected estimation results can be reached

42 Application to C-Peptide Data The method was usedto model the C-peptide data [19] C-peptide also known aslinker peptide is the secretion of islet 120573 cells and releasedto capillary along with insulin It reflects the capacity of


Table 2 The parameters estimators of C-peptide concentrations

Parameter 11989610

11989612

11989621

11990810

11990812

11990821

120590120576

1205901

1205902

Estimator 00778 02445 06667 01092 01044 00233 01352 02090 00469

DV

pred0

10

9

8

7

6

5

4

3

1

2

0

5 10

(a)

ipred

DV

10

9

8

7

6

5

4

3

1

2

0 5 10

0

(b)

Figure 2 Basic goodness-of-fit graph for simulated data (a) Plot of observed versus population predicted value and (b) observed versusindividual predicted value The solid lines are the lines of identity

islet 120573 cells to synthesize and release insulin so it is usefulin the diagnosis of islet cell tumor In this chapter weapplied the proposed method to in vivo metabolism dataof C-peptide In 14 normal humans a bolus (average massof about 50000 pmol) of biosynthetic CP was intravenouslyadministered In order to avoid the confounding effect ofendogenously secreted CP CP pancreatic was suppressedthrough a somatostatin infusion (started two hours before thebolus administration and thereafter continued throughoutthe experiment) Blood samples were collected at min 23 4 5 6 7 8 9 10 11 14 17 20 25 30 35 40 45 5055 60 70 80 90 100 110 120 140 160 and 180 and C-P plasma concentration was measured For simplicity wechoose the plasma concentration data at min 2ndash11 of sevensubjects with same initial dose and time interval and establishamixed-effectmodel based onODE to analyze the data Two-compartment models are recommended in some literature todescribe this process

So the CP plasma concentration data is modeled withtwo-compartment model for intravenous administration andparameters are estimated by Kalman filter and Bayesianinference method The results are listed in Table 2

Figure 3 describes the CP concentrations data (given bycircle) and prediction data (given by asterisk)

Figure 4(a) describes the relationship between observedconcentration and population predictions Figure 4(b)describes the relationship between observed concentrationand individual predictions

Resp

onse

Time

15

10

5

0

0 2 4 6 8 10 12

Figure 3 C-peptide concentrations data are given by the circle andprediction data are given by the asterisk

In Figure 4 ldquopredrdquo represents population predictionsldquoipredrdquo represents individual predictions and ldquoDVrdquo repre-sents latent variables (observations) Figures 3 and 4 showa good linear relationship between observations and predic-tions of CP concentrations which illustrates that the actualdata are well fitted and parameters are accurately estimated


pred

DV

12

11

10

9

8

7

6

5

4

3

4 6 8 10 12

(a)

ipred

DV

12

11

10

9

8

7

6

5

4

3

2

0 5 10 15

(b)

Figure 4 (a) Relationship between observed concentration and population predictions (b) Relationship between observed concentrationand individual predictions

5 Conclusion and Discussion

The Kalman filter also known as linear quadratic estimationis an algorithm that uses a series of measurements observedover time containing random variations and other inaccura-cies and produces estimates of unknown variables that tendto be more precise than those based on a single measurementaloneTheKalman filter is a recursive estimator whichmeansthat only the estimated state from the previous time step andthe current measurement are needed to compute the estimatefor the current state The Kalman filter has been widely usedin many fields Extended Kalman filter is an extension toKalman filter which gradually becomes the standardmethodto deal with the parameter estimation in nonlinear system

In the present work we apply extended Kalman filter topopulation pharmacokinetics model based on SDEWe bringthe model to a more generic situation by changing singlestatus equation to status equation set To overcome the diffi-culty of getting the likelihood function for complexmodel weuse extended Kalman filter method to get the approximationlikelihood function and then estimate parameter values byBayesian inference but there are still manymore problems instatistical inference of stochastic differential equations worthour deeper study An interesting area for future research isthe exploration of the model with covariate Moreover theextension of this work to multidimensional SDEs would alsobe an interesting direction

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported in part by the project NSFC(Program nos 11171065 and 81130068) and the FundamentalResearch Funds for the Central Universities (Program nosJKPZ2013015 and JKQZ2013026)

References

[1] L B Sheiner and J-L Steimer ldquoPharmacokineticpharmaco-dynamic modeling in drug developmentrdquo Annual Review ofPharmacology and Toxicology vol 40 pp 67ndash95 2000

[2] L Aarons ldquoPharmacokinetic and pharmacodynamicmodellingin drug developmentrdquo Statistical Methods in Medical Researchvol 8 no 3 pp 181ndash182 1999

[3] M O Karlsson E N Jonsson C G Wiltse and J R WadeldquoAssumption testing in population pharmacokinetic modelsillustrated with an analysis of moxonidine data from congestiveheart failure patientsrdquo Journal of Pharmacokinetics and Biophar-maceutics vol 26 no 2 pp 207ndash246 1998

[4] R Krishna Applications of Pharmacokinetic Principles in DrugDevelopment Springer New York NY USA 2004

[5] M O Karlsson S L Beal and L B Sheiner ldquoThree new resid-ual error models for population PKPD analysesrdquo Journal ofPharmacokinetics and Biopharmaceutics vol 23 no 6 pp 651ndash672 1995

[6] D Z DrsquoArgenio and K Park ldquoUncertain pharmacokineticpharmacodynamic systems design estimation and controlrdquoControl Engineering Practice vol 5 no 12 pp 1707ndash1716 1997

[7] M Ramanathan ldquoA method for estimating pharmacokineticrisks of concentration-dependent drug interactions from pre-clinical datardquo Drug Metabolism and Disposition vol 27 no 12pp 1479ndash1487 1999


[8] M Ramanathan ldquoAn application of Itorsquos Lemma in populationpharmacokinetics and pharmacodynamicsrdquo PharmaceuticalResearch vol 16 no 4 pp 584ndash586 1999

[9] N R Kristensen H Madsen and S B Joslashrgensen ldquoParameterestimation in stochastic grey-box modelsrdquo Automatica vol 40no 2 pp 225ndash237 2004

[10] R V Overgaard N Jonsson C W Tornoslashe and H MadsenldquoNon-linear mixed-effects models with stochastic differentialequations implementation of an estimation algorithmrdquo Journalof Pharmacokinetics and Pharmacodynamics vol 32 no 1 pp85ndash107 2005

[11] C W Tornoslashe R V Overgaard H Agersoslash H A Nielsen HMadsen and E N Jonsson ldquoStochastic differential equations inNONMEM implementation application and comparison withordinary differential equationsrdquo Pharmaceutical Research vol22 no 8 pp 1247ndash1258 2005

[12] S Donnet J-L Foulley and A Samson ldquoBayesian analysisof growth curves using mixed models defined by stochasticdifferential equationsrdquo Biometrics vol 66 no 3 pp 733ndash7412010

[13] C A Struthers andD LMcLeish ldquoA particular diffusionmodelfor incomplete longitudinal data application to the multicenterAIDS cohort studyrdquoBiostatistics vol 12 no 3 pp 493ndash505 2011

[14] R de la Cruz-Mesıa and G Marshall ldquoNon-linear randomeffects models with continuous time autoregressive errors aBayesian approachrdquo Statistics in Medicine vol 25 no 9 pp1471ndash1484 2006

[15] B P Carlin and T A Louis Bayes and Empirical Bayes Methodsfor Data Analysis Chapman amp HallCRC 2000

[16] K L Mengersen and R L Tweedie ldquoRates of convergence ofthe hastings andmetropolis algorithmsrdquoTheAnnals of Statisticsvol 24 no 1 pp 101ndash121 1996

[17] C-A Cuenod B Favetto V Genon-Catalot Y Rozenholc andA Samson ldquoParameter estimation and change-point detectionfrom dynamic contrast enhanced MRI data using stochasticdifferential equationsrdquoMathematical Biosciences vol 233 no 1pp 68ndash76 2011

[18] T Yuxi ldquoData augmentation for sparse data in population phar-macokinetics based on local polynomial regressionrdquo ChineseJournal of New Drugs vol 22 no 12 pp 1361ndash1366 2013

[19] E T Shapiro H Tillil A H Rubenstein and K S PolonskyldquoPeripheral insulin parallels changes in insulin secretion moreclosely than C-peptide after bolus intravenous glucose admin-istrationrdquo The Journal of Clinical Endocrinology amp Metabolismvol 67 no 5 pp 1094ndash1099 1988

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


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OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


variability that may be due to assay error [4] The simulationresults show that the introduction of the SDEmay contributeto the better estimates of the interindividual variations andstructural parameters [5]

The first paper encouraging the introduction of randomfluctuations in PKPD was published by DrsquoArgenio et al[6ndash8] The authors underlined that both deterministic andstochastic component contribute to PKPDmodelsHoweverestimating parameters in nonlinearmixed-effectsmodel withSDE is a difficult problem and not straightforward exceptfor simple cases Naturally likelihood inference would bea feasible approach but the transition densities are rarelyknown thus explicit likelihood function is usually hard toget Actually there are hardly any theories for SDE modelsat present Kristensen NR proposed maximum likelihoodmethod and maximized a posterior to estimate the param-eters in PK models with SDE but this method only focusedon single subject modeling where no interindividual variancecomponents were estimated [9] Overgaard RV suggestedapplying the Kalman filter to approximate the likelihoodfunction for a SDE model with a nonlinear drift term and aconstant diffusion term [10] Tornoe CW used this algorithmto estimate SDEs in NONMEM [11] but the NONMEMimplementation cannot be used to form Kalman smoothingestimates which is an important feature of the SDE approachwhere all data is used to give optimal estimates at each sam-pling point Sophie Donnet proposed a Bayesian inferenceto analyze growth curves using mixed-effects models basedon stochastic differential equations and obtained good results[12] Struthers and McLeish applied Bayesian method to themulticenter AIDS cohort study [13]

Inspired by this the present work describes an approx-imation of likelihood function of nonlinear mixed-effectspharmacokinetic model which is constructed by combin-ing the extended Kalman filter with the MCMC methodThe parameters in the nonlinear mixed-effects models areestimated based on stochastic differential equation and thewhole process is implemented in Matlab Section 2 presentsthe classical nonlinear mixed model and the mixed modeldefined by SDEs In Section 3 the algorithm of nonlinearmixed-effects model with SDEs is proposed In Section 4example with simulated data and a case study are describedConclusion and discussion are listed in Section 5

2 Models and Notation

Nonlinear mixed-effects models can be regarded as hierar-chical models where the variability in concentrationeffectis split into intraindividual variability described by the first-stage distribution and interindividual variability describedby the second-stage distribution This section describes thenotation for nonlinear mixed-effects models used in thepresent paper

21 Nonlinear Mixed-Effects Model Based on Ordinary Dif-ferential Equation Let 119910 = (119910

119894)1le119894le119899

= (119910119894119895)1le119894le1198991le119895le119899119894

119894 =1 119899 119895 = 0 119899

119894 be the true observations where 119910

119894119895is

measurement for individual 119894 at time 119905119894119895 119899 is the number of

individuals and 119899119894is the number of measurements for indi-

vidual 119894 Traditional nonlinear mixed-effect models usuallymodel this process by nonlinear mixed-effects model basedon ordinary differential equations Formally the classicalnonlinear mixed-effects model is written as

120597119909

120597119905= 119892 (119909 120601 119905)

119910119894119895= 119891 (119909 (119905

119894119895) 120601119894) + 120576119894119895 120576

119894119895sim119894119894119889119873(0 120590

2

120576)

120601119894sim 119873 (120583Ω)

(1)

where 119891 is a possibly nonlinear function and Φ = (120601119894)1le119894le119899

is the individual parameter vectors 120601119894is assumed to be

independently and identically normally distributed withexpectation 120583 and varianceΩ 120576

119894119895are the measurement errors

and are also assumed to be independently and identicallynormally distributed with null mean and variance 1205902

120576

22 Nonlinear Mixed-Effects Model Based on Stochastic Dif-ferential Equation While the introduction of SDEs doesnot change the fundamental hierarchical structure theydo change the entities in the first-stage density and theconstruction Under the framework of ordinary differentialequations noise is only introduced through themeasurementequation see (1) This allows the measurement noise term toabsorb the whole error due to model miss-specification ortrue random fluctuations of the states and hence may ignorethe correlated residuals In order to consider the correlatedresiduals a stochastic process is added to the state spacemodel such that the nonlinear mixed-effects model based onSDEs can be written as

119889119909119905(120601119894) = 119892 (119909

119905 120601119894 119905) 119889119905 + Γ (119909

119905 120601119894 1205902) 119889119882119905

119909 (119905 = 0) = 1199090 (120601)

119910119894119895= 119891 (119909 (119905

119894119895) 120601119894 119905119894119895) + 120576119894119895 120576

119894119895sim119894119894119889119873(0 120590

2

120576)

120601119894sim 119873 (120583Ω)

(2)

where Γ(119909119905 120601 1205742) 119889119882

119905is called the diffusion term and

describes the stochastic part of the system 119882119905is a stan-

dard Wiener process defined by 1199081199052minus 1199081199051

sim 119873(0 |1199052minus

1199051|119868) 119892(119909

119905 120601 119905) 119889119905 is called the drift term and describes the

deterministic part The stochastic dynamics of the system isdefined by the drift and diffusion terms together

In nonlinear mixed-effects model based on SDEs (see(2)) the total variance is divided into three fundamentallydifferent noises the interindividual variability Ω describingthe individual difference the system noise 120590

2 reflectingthe random fluctuations around the corresponding dynamicmodel and the measurement noise 1205902

120576representing the

uncorrelated residuals originating from measurement assayor sampling errors





119889119909 = 119892 (119909 120601 119905) 119889119905 + 120590119908119889119882

119910119895= 119891 (119909 (119905

119895) 120601) + 120576

119895

(3)






(1) 119909119896|119896


(2) 119875119896|119896






119910119895|119895minus1

= 119891 (119909119895|119895minus1

120601)

119877119895|119895minus1

= 119862119895119875119895|119895minus1

119862119879

119895+ sum119895|119895minus1

(4)


119909119895|119895= 119909119895|119895minus1

+ 119870119895(119910119895minus 119910119895|119895minus1

)

119875119895|119895= 119875119895|119895minus1

minus 119870119895119877119895|119895minus1

119870119879

119895

119870119895= 119875119895|119895minus1

119862119879119877minus1

119895|119895minus1

(5)



119889119909119905|119895

119889119905= 119892 (119909

119905|119895 119905 120601)

119889119875119905|119895

119889119905= 119860119905119875119905|119895+ 119875119905|119895119860119879

119905+ 120590119908120590119879

119908

(6)

where 119860119905= (120597119892120597119909)|

119909=119909119905|119895minus1



and1198751|0

(2) For 119895 = 1 to 119899119894do


and 119895|119895minus1



and 119875119895|119895

(6) Use (6) to compute 119909(119895+1)|119895

and (119895+1)|119895


119895= 119910119895minus 119910119895|119895minus1

and 119877119895|119895minus1







119894(119895|119895minus1) Like-




119910119894(119895|119895minus1)

= 119864 (119910119894119895| 119884119894(119895minus1)

sdot)

119877119894(119895|119895minus1)

= 119881 (119910119894119895| 119884119894(119895minus1)

sdot)

(7)


120576119894119895= 119910119894119895minus 119910119894(119895119895minus1)

isin 119873(0 119877119894(119895119895minus1)

) (8)



1199011(119884119894119899119894| sdot) asymp

119899119894

prod119895=1

exp (minus (12) 120576119879119894119895119877119879119894(119895119895minus1)

120576119894119895)

radic100381610038161003816100381610038162120587119877119894(119895|119895minus1)

10038161003816100381610038161003816

(9)



119871 (120601sum 120590119908 Ω) prop

119873

prod119894=1

int1199011(119884119894119899119894| 120601119894sum 120590

119908) 1199012(120601119894| Ω) 119889120601

119894

=

119873

prod119894=1

int exp (119897119894) 119889120601119894

(10)

where

119897119894= minus

1

2

119899119894

sum119895=1

(120576119879

119894119895119877minus1

119894(119895|119895minus1)120576119894119895+ log 100381610038161003816100381610038162120587119877119894(119895|119895minus1)

10038161003816100381610038161003816)

minus1

2120601119879

119894Ωminus1120601119894minus1

2log |2120587Ω|

(11)



120583119896sim 119873(119898

prior119896

Vprior119896

) 119896 = 1 119901

Ωminus1sim 119882(119877 119901 + 1)

1

1205902sim Γ (120572

prior120590

120573prior120590

)

(12)


119896 Vprior119896

119877 120572prior120590

and 120573prior120590









(3) Set

120579119905=


(13)


4 Numerical Studies



119894=


1198891199091= (119896211199092minus 119896121199091minus 119896101199091) 119889119905 + 120590

1119889119908

1198891199092= (119896121199091minus 119896211199092) 119889119905 + 120590

2119889119908

119910 = 1199091+ 120576 120576 sim 119873 (0 120590

2

120576)

120601 = (119896101198961211989621) sim 119873 (120583Ω)

120583 = [

[

12058310

12058312

12058321

]

]

Ω = [

[

119908210

0 0

0 119908212

0

0 0 119908221

]

]

(14)



compartment 11989610 11989612 and 119896




1and 120590


system noise





12058310

02 01999(01999 02006)

00984(minus06448 08416)

12058312

05 04755(047503 05213)

05339(04089 06589)

12058321

025 02501(02499 02501)

01815(minus00303 03932)

11990810

001 00173(00063 00467) mdash

11990812

01 01482(00176 01812) mdash

11990821

002 00174(00035 00402) mdash

120590120576

02 01488(01343 02441) mdash

1205901

01 00793(00402 01091) mdash

1205902

01 01087(00767 01515) mdash


119879 = [02 05 025] diag(Ω) =

(0012 012 0022) 1205902120576

= 004 1205901= 01 120590

2= 01 On






Resp

onse

Time

Simulation data

Simulation data


10

9

8

7

6

5

4

3

2

1

0

0 05 1 15 2 25 3 35 4 45 5






Parameter 11989610

11989612

11989621

11990810

11990812

11990821

120590120576

1205901

1205902

Estimator 00778 02445 06667 01092 01044 00233 01352 02090 00469

DV

pred0

10

9

8

7

6

5

4

3

1

2

0

5 10

(a)

ipred

DV

10

9

8

7

6

5

4

3

1

2

0 5 10

0

(b)






Resp

onse

Time

15

10

5

0

0 2 4 6 8 10 12




pred

DV

12

11

10

9

8

7

6

5

4

3

4 6 8 10 12

(a)

ipred

DV

12

11

10

9

8

7

6

5

4

3

2

0 5 10 15

(b)







Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119889119909 = 119892 (119909 120601 119905) 119889119905 + 120590119908119889119882

119910119895= 119891 (119909 (119905

119895) 120601) + 120576

119895

(3)






(1) 119909119896|119896


(2) 119875119896|119896






119910119895|119895minus1

= 119891 (119909119895|119895minus1

120601)

119877119895|119895minus1

= 119862119895119875119895|119895minus1

119862119879

119895+ sum119895|119895minus1

(4)


119909119895|119895= 119909119895|119895minus1

+ 119870119895(119910119895minus 119910119895|119895minus1

)

119875119895|119895= 119875119895|119895minus1

minus 119870119895119877119895|119895minus1

119870119879

119895

119870119895= 119875119895|119895minus1

119862119879119877minus1

119895|119895minus1

(5)



119889119909119905|119895

119889119905= 119892 (119909

119905|119895 119905 120601)

119889119875119905|119895

119889119905= 119860119905119875119905|119895+ 119875119905|119895119860119879

119905+ 120590119908120590119879

119908

(6)

where 119860119905= (120597119892120597119909)|

119909=119909119905|119895minus1



and1198751|0

(2) For 119895 = 1 to 119899119894do


and 119895|119895minus1



and 119875119895|119895

(6) Use (6) to compute 119909(119895+1)|119895

and (119895+1)|119895


119895= 119910119895minus 119910119895|119895minus1

and 119877119895|119895minus1







119894(119895|119895minus1) Like-




119910119894(119895|119895minus1)

= 119864 (119910119894119895| 119884119894(119895minus1)

sdot)

119877119894(119895|119895minus1)

= 119881 (119910119894119895| 119884119894(119895minus1)

sdot)

(7)


120576119894119895= 119910119894119895minus 119910119894(119895119895minus1)

isin 119873(0 119877119894(119895119895minus1)

) (8)



1199011(119884119894119899119894| sdot) asymp

119899119894

prod119895=1

exp (minus (12) 120576119879119894119895119877119879119894(119895119895minus1)

120576119894119895)

radic100381610038161003816100381610038162120587119877119894(119895|119895minus1)

10038161003816100381610038161003816

(9)



119871 (120601sum 120590119908 Ω) prop

119873

prod119894=1

int1199011(119884119894119899119894| 120601119894sum 120590

119908) 1199012(120601119894| Ω) 119889120601

119894

=

119873

prod119894=1

int exp (119897119894) 119889120601119894

(10)

where

119897119894= minus

1

2

119899119894

sum119895=1

(120576119879

119894119895119877minus1

119894(119895|119895minus1)120576119894119895+ log 100381610038161003816100381610038162120587119877119894(119895|119895minus1)

10038161003816100381610038161003816)

minus1

2120601119879

119894Ωminus1120601119894minus1

2log |2120587Ω|

(11)



120583119896sim 119873(119898

prior119896

Vprior119896

) 119896 = 1 119901

Ωminus1sim 119882(119877 119901 + 1)

1

1205902sim Γ (120572

prior120590

120573prior120590

)

(12)


119896 Vprior119896

119877 120572prior120590

and 120573prior120590









(3) Set

120579119905=


(13)


4 Numerical Studies



119894=


1198891199091= (119896211199092minus 119896121199091minus 119896101199091) 119889119905 + 120590

1119889119908

1198891199092= (119896121199091minus 119896211199092) 119889119905 + 120590

2119889119908

119910 = 1199091+ 120576 120576 sim 119873 (0 120590

2

120576)

120601 = (119896101198961211989621) sim 119873 (120583Ω)

120583 = [

[

12058310

12058312

12058321

]

]

Ω = [

[

119908210

0 0

0 119908212

0

0 0 119908221

]

]

(14)



compartment 11989610 11989612 and 119896




1and 120590


system noise





12058310

02 01999(01999 02006)

00984(minus06448 08416)

12058312

05 04755(047503 05213)

05339(04089 06589)

12058321

025 02501(02499 02501)

01815(minus00303 03932)

11990810

001 00173(00063 00467) mdash

11990812

01 01482(00176 01812) mdash

11990821

002 00174(00035 00402) mdash

120590120576

02 01488(01343 02441) mdash

1205901

01 00793(00402 01091) mdash

1205902

01 01087(00767 01515) mdash


119879 = [02 05 025] diag(Ω) =

(0012 012 0022) 1205902120576

= 004 1205901= 01 120590

2= 01 On






Resp

onse

Time

Simulation data

Simulation data


10

9

8

7

6

5

4

3

2

1

0

0 05 1 15 2 25 3 35 4 45 5






Parameter 11989610

11989612

11989621

11990810

11990812

11990821

120590120576

1205901

1205902

Estimator 00778 02445 06667 01092 01044 00233 01352 02090 00469

DV

pred0

10

9

8

7

6

5

4

3

1

2

0

5 10

(a)

ipred

DV

10

9

8

7

6

5

4

3

1

2

0 5 10

0

(b)






Resp

onse

Time

15

10

5

0

0 2 4 6 8 10 12




pred

DV

12

11

10

9

8

7

6

5

4

3

4 6 8 10 12

(a)

ipred

DV

12

11

10

9

8

7

6

5

4

3

2

0 5 10 15

(b)







Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1199011(119884119894119899119894| sdot) asymp

119899119894

prod119895=1

exp (minus (12) 120576119879119894119895119877119879119894(119895119895minus1)

120576119894119895)

radic100381610038161003816100381610038162120587119877119894(119895|119895minus1)

10038161003816100381610038161003816

(9)



119871 (120601sum 120590119908 Ω) prop

119873

prod119894=1

int1199011(119884119894119899119894| 120601119894sum 120590

119908) 1199012(120601119894| Ω) 119889120601

119894

=

119873

prod119894=1

int exp (119897119894) 119889120601119894

(10)

where

119897119894= minus

1

2

119899119894

sum119895=1

(120576119879

119894119895119877minus1

119894(119895|119895minus1)120576119894119895+ log 100381610038161003816100381610038162120587119877119894(119895|119895minus1)

10038161003816100381610038161003816)

minus1

2120601119879

119894Ωminus1120601119894minus1

2log |2120587Ω|

(11)



120583119896sim 119873(119898

prior119896

Vprior119896

) 119896 = 1 119901

Ωminus1sim 119882(119877 119901 + 1)

1

1205902sim Γ (120572

prior120590

120573prior120590

)

(12)


119896 Vprior119896

119877 120572prior120590

and 120573prior120590









(3) Set

120579119905=


(13)


4 Numerical Studies



119894=


1198891199091= (119896211199092minus 119896121199091minus 119896101199091) 119889119905 + 120590

1119889119908

1198891199092= (119896121199091minus 119896211199092) 119889119905 + 120590

2119889119908

119910 = 1199091+ 120576 120576 sim 119873 (0 120590

2

120576)

120601 = (119896101198961211989621) sim 119873 (120583Ω)

120583 = [

[

12058310

12058312

12058321

]

]

Ω = [

[

119908210

0 0

0 119908212

0

0 0 119908221

]

]

(14)



compartment 11989610 11989612 and 119896




1and 120590


system noise





12058310

02 01999(01999 02006)

00984(minus06448 08416)

12058312

05 04755(047503 05213)

05339(04089 06589)

12058321

025 02501(02499 02501)

01815(minus00303 03932)

11990810

001 00173(00063 00467) mdash

11990812

01 01482(00176 01812) mdash

11990821

002 00174(00035 00402) mdash

120590120576

02 01488(01343 02441) mdash

1205901

01 00793(00402 01091) mdash

1205902

01 01087(00767 01515) mdash


119879 = [02 05 025] diag(Ω) =

(0012 012 0022) 1205902120576

= 004 1205901= 01 120590

2= 01 On






Resp

onse

Time

Simulation data

Simulation data


10

9

8

7

6

5

4

3

2

1

0

0 05 1 15 2 25 3 35 4 45 5






Parameter 11989610

11989612

11989621

11990810

11990812

11990821

120590120576

1205901

1205902

Estimator 00778 02445 06667 01092 01044 00233 01352 02090 00469

DV

pred0

10

9

8

7

6

5

4

3

1

2

0

5 10

(a)

ipred

DV

10

9

8

7

6

5

4

3

1

2

0 5 10

0

(b)






Resp

onse

Time

15

10

5

0

0 2 4 6 8 10 12




pred

DV

12

11

10

9

8

7

6

5

4

3

4 6 8 10 12

(a)

ipred

DV

12

11

10

9

8

7

6

5

4

3

2

0 5 10 15

(b)







Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













12058310

02 01999(01999 02006)

00984(minus06448 08416)

12058312

05 04755(047503 05213)

05339(04089 06589)

12058321

025 02501(02499 02501)

01815(minus00303 03932)

11990810

001 00173(00063 00467) mdash

11990812

01 01482(00176 01812) mdash

11990821

002 00174(00035 00402) mdash

120590120576

02 01488(01343 02441) mdash

1205901

01 00793(00402 01091) mdash

1205902

01 01087(00767 01515) mdash


119879 = [02 05 025] diag(Ω) =

(0012 012 0022) 1205902120576

= 004 1205901= 01 120590

2= 01 On






Resp

onse

Time

Simulation data

Simulation data


10

9

8

7

6

5

4

3

2

1

0

0 05 1 15 2 25 3 35 4 45 5






Parameter 11989610

11989612

11989621

11990810

11990812

11990821

120590120576

1205901

1205902

Estimator 00778 02445 06667 01092 01044 00233 01352 02090 00469

DV

pred0

10

9

8

7

6

5

4

3

1

2

0

5 10

(a)

ipred

DV

10

9

8

7

6

5

4

3

1

2

0 5 10

0

(b)






Resp

onse

Time

15

10

5

0

0 2 4 6 8 10 12




pred

DV

12

11

10

9

8

7

6

5

4

3

4 6 8 10 12

(a)

ipred

DV

12

11

10

9

8

7

6

5

4

3

2

0 5 10 15

(b)







Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Parameter 11989610

11989612

11989621

11990810

11990812

11990821

120590120576

1205901

1205902

Estimator 00778 02445 06667 01092 01044 00233 01352 02090 00469

DV

pred0

10

9

8

7

6

5

4

3

1

2

0

5 10

(a)

ipred

DV

10

9

8

7

6

5

4

3

1

2

0 5 10

0

(b)






Resp

onse

Time

15

10

5

0

0 2 4 6 8 10 12




pred

DV

12

11

10

9

8

7

6

5

4

3

4 6 8 10 12

(a)

ipred

DV

12

11

10

9

8

7

6

5

4

3

2

0 5 10 15

(b)







Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










pred

DV

12

11

10

9

8

7

6

5

4

3

4 6 8 10 12

(a)

ipred

DV

12

11

10

9

8

7

6

5

4

3

2

0 5 10 15

(b)







Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article parameter estimation of population pharmacokinetic models...

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