Development of a population pharmacokinetic model describing enterohepatic circulation based on total concentrations of mycophenolic acid and its
inactive glucuronide metabolite
Rebecca Irene Kongsvik
Thesis submitted for the degree of Master in Pharmacy
45 credits
Department of Pharmacy
Faculty of Mathematics and Natural Sciences UNIVERSITY OF OSLO
May 2021
Development of a population pharmacokinetic model describing enterohepatic circulation based on total concentrations of mycophenolic acid and its inactive
glucuronide metabolite.
Rebecca Irene Kongsvik
Thesis submitted for the degree of Master in Pharmacy
45 credits
Department of Pharmacy
Faculty of Mathematics and Natural Sciences UNIVERSITY OF OSLO
May 2021 Supervisor
Anders Åsberg, professor dr. cand. Scient
© Rebecca Irene Kongsvik 2021
Sammendrag
Bakgrunn: Det kan være utfordrende å individualisere dosering av legemidler som har en stor intra- og interindividuell variabilitet i kinetikken og et smalt terapeutisk vindu.
Mykofenolsyre (MPA) brukes som en sentral del i den immunsuppresive behandlingen av organtransplanterte, og har en komplisert kinetikk og et relativt smalt terapeutisk vindu. MPA gjennomgår enterohepatisk resirkulering (EHC), det vil si at det gjennomgår flere
absorpsjonsfaser, som bidrar til den totale eksponeringen av legemidlet. MPA er assosiert med variasjon både innad i populasjonen, men også på individnivå. Terapeutisk
legemiddelmonitorering (TDM) er derfor et nyttig verktøy for å individualisere doseringen.
Standard metoder for TDM er måling av MPA trough konsentrasjon og ved noen tilfeller komplettert med en «mini-AUC», det vil si AUC estimert ut fra konsentrasjoner målt for eksempel før og 0.5 og 2 timer etter dose der den totale AUC er estimert ut fra en standard algoritme. Ved å bruke farmakokinetisk populasjonsmodellering kan kinetikken beskrives mer nøyaktig for hvert individ. Målet med denne oppgaven var å utvikle en farmakokinetisk populasjonsmodell som beskriver enterohepatisk resirkulering av MPA hos nyretransplanterte pasienter. Modellen er basert på observerte konsentrasjoner for MPA og den inaktive
metabolitten 7-O-MPA-glukuronide (MPAG).
Metode: En farmakokinetisk populasjonsmodell ble utviklet med rike pasientdata fra tre ulike kliniske studier utført på nyretransplanterte pasienter. Alle pasientene som ble inkludert i analysene brukte mykofenolat mofetil, takrolimus og prednisolon som immunsuppresiv behandling. Modellen ble utviklet i R-pakken Pmetrics med en ikke-parametrisk adaptiv gridmetode. Validering ble utført på modellen med et utviklingsdatasett og et
valideringsdatasett.
Resultat: Den utviklede modellen predikerte EHC på et individnivå når flere konsentrasjoner i et doseintervall var kjent. Modellen predikerte individuelle konsentrasjoner, men viste svak prediksjonsevne på populasjonsnivå, noe som skyldes et begrenset datamateriale, mangel på inklusjon av kovariater, samt stor variasjon innad pasientgruppen. RMSE (root mean square error) for populasjonsmodellen var 17,5% for MPA og 6.4 % MPAG for utviklingsdatasettet og 52 % for MPA og 23.7 % for valideringsdatasettet.
Konklusjon: Den farmakokinetiske populasjonsmodellen som ble utviklet beskriver en enterohepatisk resirkulering for MPA nøyaktig. Modellen har potensiale men trenger et større pasientgrunnlag for å kunne utvikles videre.
Abstract
Background: Individualized treatment with drugs that has a large intra- and inter-individual variability in addition to a narrow therapeutic window is challenging. Mycophenolic acid (MPA) is used as a central part in the immunosuppressive treatment of organ transplanted patients. It has a complicated kinetics and a relatively narrow therapeutic window. MPA undergoes enterohepatic cycling (EHC), which means it undergoes multiple absorption phases that contributes to its total systemic exposure. MPA show high variability both within the population as well as on an individual level. Therapeutic drug monitoring (TDM) is therefore mandatory for individualizing drug dosing. The common methods for TDM are sampling of MPA trough concentration (C0),and in some cases it is complemented with “miniAUC”, which is AUC estimated with an algorithm from concentrations sampled for example predose, 0.5 hours and 2 hours after administering the dose. The total AUC is calculated using a
standard algorithm. By using pharmacokinetic population modelling, MPA kinetics may be better described on a more individual level. The goal with this thesis is to develop a
pharmacokinetic population model describing the EHC of MPA, and the primary metabolite MPAG with kidney transplanted patients.
Method: A pharmacokinetic population model was developed with rich patient data from three different clinical studies performed on kidney transplanted patients. All patients that were included in the analyses used mycophenolate mofetil, tacrolimus and prednisolone as immunosuppressive maintenance treatment. The model was developed in the R package Pmetrics, with a non-parametric adaptive grid method. Both development and external validation was done on the model with a development dataset and a validation dataset.
Results: The final model was able to predict EHC on an individual level when several
observed concentrations within the dose interval were known. The model predicted individual concentrations, but displayed a poor population prediction ability due to the limited data set, and absent inclusion of covariates, and a large between subject variability. RMSE (root mean square error) was reported at 17.5 % for MPA and 6.4 % MPAG for the development dataset, and 52.0 % for MPA and 23.7 % for MPAG the validation dataset.
Conclusion: The pharmacokinetic population model that was developed described
enterohepatic cycling of MPA accurately. With an increased amount of patient profiles, the model has potential of further development.
Forord:
Denne oppgaven ble utført ved seksjon for farmakologi og farmasøytisk biovitenskap, Farmasøytisk institutt, Universitetet i Oslo i perioden august 2020, til mai 2021, med tilknytning til Avdeling for Transplantasjonsmedisin, OUS-Rikshospitalet HF.
Først og fremst må jeg takke Anders Åsberg for din veiledning og engasjement rundt denne masteroppgaven. Det har vært svært inspirerende å ha deg som veileder. Takk for at du alltid har vært tilgjengelig til å svare på spørsmål og hjelpe med Pmetrics når ting satt seg fast. Takk til Markus Herberg Hovd for masse gode tips og svar underveis.
Ellers må jeg takke hele gjengen på nyrelabben. Det har vært en fryd å bli kjent med dere, og jeg kommer til å savne dere alle sammen.
Takk til min kjære Torstein Holtlien Blø for korrekturlesing, datakyndig hjelp og alt annet du har stilt opp med gjennom studietiden min. Malin O. Syversen og Thao Minh Nguyen, studietiden har vært helt fantastisk sammen med dere. Takk til øvrig familie for all støtte gjennom mine fem år som farmasistudent.
Oslo, mai 2021.
Rebecca Irene Kongsvik
TABLE OF CONTENTS
ABBREVIATIONS ... 1
1.0 INTRODUCTION ... 2
1.1 KIDNEY TRANSPLANTATION ... 2
1.2 MYCOPHENOLIC ACID ... 3
1.2.1 MECHANISM OF ACTION ... 3
1.2.2 ADVERSE EFFECTS ... 3
1.2.3 PHARMACOKINETICS OF MYCOPHENOLIC ACID ... 4
1.3 PHARMACOKINETIC PARAMETER DETERMINATION ... 5
1.3.1 NON-COMPARTMENTAL ANALYSIS ... 5
1.4 POPULATION MODELLING ... 7
1.4.1 MODELLING SOFTWARE ... 8
1.4.2 ASSESSMENTS IN MODEL DEVELOPMENT ... 11
1.5 CLINICAL UTILITY VALUE ... 13
1.6 AIM ... 14
2.0 METHOD ... 15
2.1 PATIENT AND DATA ... 15
2.1.1 MODEL DEVELOPMENT ... 15
2.1.2 MODEL VALIDATION ... 15
2.2 BIOANALYSIS ... 16
2.3 MODELLING USING PMETRICS ... 16
2.4 MODEL DEVELOPMENT ... 18
2.4.1 STRUCTURAL MODEL ... 18
2.4.2 ENTEROHEPATIC MODELING ... 20
2.4.3 STEADY STATE ... 21
2.5 ABSORPTION AND REABSORPTION ... 21
2.5.1 DELAY OF ABSORPTION ... 21
2.5.2 DELAY OF REABSORPTION ... 22
2.5.3 THE HEAVISIDE STEP FUNCTION ... 22
2.6 MODEL PARAMETERS ... 24
2.7 COVARIATES ... 25
2.8 MODEL VALIDATION PLAN ... 26
2.8.1 DEVELOPMENT VALIDATION ... 26
2.8.2 EXTERNAL VALIDATION ... 26
3.0 RESULTS ... 27
3.1 PATIENTS ... 27
4.3 NPAG WORKFLOW ... 46
4.4 EXTERNAL VALIDATION ... 47
4.5 LIMITATIONS ... 48
4.6 MODEL PERSPECTIVES ... 49
4.7 FUTURE PERSPECTIVES ... 51
5.0 CONCLUSIONS ... 53
6.0 REFERENCES: ... 54
7.0 ATTACHMENTS ... 57
ABBREVIATIONS
-2LL 2*log likelihood value
AIC Akaike Information criteria
AcMPAG Acyl MPAG
AUC Area under the plasma concentration-time curve
CSV Comma separated values
EHC Enterohepatic circulation
GIT Gut intestinal tractus
IMPDH Inosine monophosphate dehydrogenases
LLOQ Lower limit for quantitation
LSS Limited sampling strategy
MEP Mean error of prediction
MMF Mycophenolate mofetil
MPA Mycophenolic acid
MPAG MPA 7-O-glucuronide
NCA Non-compartment analysis
NPAG Non-parametric adaptive grid
OBJ Minimum value of OFV
OFV Objective function value
OP-plot Observed versus predicted concentration plot
PopPK Population pharmacokinetics
RMSE Root mean square error
SD Standard deviation
SSM Model sum of squares
SST Total sum of squares
TAC Tacrolimus
TDM Therapeutic drug monitoring
UGT Uridine 5`diphosphoglucoronosyltransferase
1.0 INTRODUCTION
1.1 KIDNEY TRANSPLANTATION
Solid organ transplantation of a kidney is a common treatment for patients with end-stage renal failure [1, 2]. The first successful human to human transplant was performed in 1954, were a recipient received a kidney from his identical twin brother [3]. In 1956, the first clinical organ transplant was performed in Norway; a kidney from an allogenic donor was transplanted to a patient with end-stage renal decease. The patient survived for 30 days, and the cause of death was due to a cardiac arrest during a wound revision. The
immunosuppressive treatment consisted of total body irradiation and corticosteroids [4].
An immune response is activated when the immune system identifies the transplanted organ as foreign. The activation is divided into two phases that will lead to a destruction of the graft [5].
Phase 1 include the innate immune system and is rapidly activated with an early onset.
Triggers that cause an activation of the system is related to tissue injury that is sustained by an organ retrieval or cell isolation. Macrophages, neutrophiles and dendritic cells mediates the activation. If the response is powerful enough, phase 2 is activated. Phase 2 is mediated by lymphoid cells, such as T and B lymphocytes. The response has a slower onset and occurs after the innate response. Phase 2 may result in a graft destruction if not prevented [5, 6]. To prevent the immune system from rejecting a transplanted organ, the recipient is treated with lifelong immunosuppressive therapy which gives the patients a higher survival rate [1, 4, 7].
Immunosuppressive therapy has evolved significantly during the past 70 years. From the use of total body irradiation as a base in immunosuppressive treatment, until the
immunosuppressive derivate azathioprine was introduced in the 1960s and used as treatment combined with the steroid prednisone [3]. The one-year rejection rate was high; reported from 40-60 %. At the time, organ transplantation was considered an experimental treatment due to poor outcomes. In 1978, cyclosporine, a fungal derivate that works as a calcineurin inhibitor,
mycophenolic acid was introduced in combination with corticosteroids. Mycophenolic acid (MPA) is an antiproliferative agent and is frequently used as an important part of modern immunosuppressive protocols [3, 8].
1.2 MYCOPHENOLIC ACID
Mycophenolic acid (MPA) was first discovered in 1893 by an Italian physician, Bartolomeo Gosio. The drug was first intended as an antibacterial agent, but because of the adverse effects it had on immune cells it was abandoned [9]. In 1969, the potent effect on lymphocytes was discovered and made MPA a candidate for immunosuppressive treatment. Due to poor bioavailability of the active agent, an ester prodrug was developed [10]. In addition to kidney transplant therapy, MPA is used in immunosuppressant treatment of liver and heart
transplantation, as well as some autoimmune disorders [9].
As of today, oral administration of MPA is available as the ester prodrug mycophenolate mofetil (MMF; CellCeptâ) and enteric-coated mycophenolate sodium (EC-MPS; Myforticâ) [11]. The present thesis will only focus on MMF.
1.2.1 MECHANISM OF ACTION
The key effect of MPA is a selective non-competitive inhibition of inosine monophosphate dehydrogenases (IMPDH). IMPDH is a rate-limited enzyme that is dependent in the synthesis of guanosine nucleotides. The inhibition results in an interference with the de novo pathway of purine synthesis and DNA replication. T and B cells are highly dependent on the IMPDH pathway, and when inhibited, it produces a cytostatic effect on the lymphocytes. Unbound MPA is the pharmacological active moiety and is responsible for the inhibition of IMPDH.
[12, 13].
1.2.2 ADVERSE EFFECTS
In addition to the immunosuppressive effect on T and B cells, adverse effects are reported on other cell types showing high turn-around, like bone marrow and enterocytes in the gut intestinal tractus (GIT). The most clinically significant adverse effects of MPA are related to the GIT and presented as vomiting, diarrhea, nausea and abdominal pain. In addition, the immunosuppressive properties of MMF increase the risk of infections significantly [14, 15].
1.2.3 PHARMACOKINETICS OF MYCOPHENOLIC ACID
Orally administered MMF, is rapidly absorbed and hydrolyzed by carboxylesterases present in the gut wall, to active MPA [16]. MMF displays a high oral bioavailability, approximately 80-90 % [11]. When present in the plasma, MPA is highly bound to the plasma protein albumin, 97-99 %. Further, MPA is metabolized by uridine 5´diphospho-
glucuronosyltransferase (UGT) system into the inactive metabolite MPA 7-O-glucuronide (MPAG) and the active acyl-MPAG (AcMPAG). The UGTs is present in the liver, the intestines and the kidneys, and the metabolization of MPA is extensive. MPAG present in the systemic circulation is also highly bound to plasma protein albumin, around 87 %. MPAG undergoes an active transport from the liver hepatocytes into the bile and is circulated back to the small intestines, as described in figure 1.1. Glucuronides present in the intestinal flora convert the inactive MPAG back to MPA, which may undergo reabsorption back to the systemic circulation [17, 18]. This process is called enterohepatic circulation (EHC). The reabsorption can be seen as a second plasma peak in an area under the plasma concentration- time curve (AUC) for MPA. The second peak is usually present around 6-12 hours after oral administration, and contributes significantly to the systemic plasma exposure, around 30-40 % of the total exposure [11].
EHC is a process in the liver were biliary acids are circulated from the blood via the liver, the bile, the intestine and back to the systemic circulation (Figure 1.1) [19]. Bile release flow determines the timing of the EHC cycle and is regulated by the activity of the Sphincter of Oddi, which refers to a smooth muscle located in the junction of the bile duct and duodenum.
A decrease in the resistance of the muscle facilitates bile flow into the duodenum. Sight, smell and ingestion of food may enhance the flow in addition to the basal biliary flow, which leads to a possible sporadic excretion of bile and complex pharmacokinetics of drugs subjected to EHC [20-22]. Studies of the gall bladder excretion shows a wide variability within the individual and within a population, which makes it challenging to predict dosing of MPA for each individual [23].
Figure 1.1: Administrated MMF is rapidly converted to MPA and circulated to the liver, were the UGT system metabolize MPA to MPAG. MPAG undergoes enterohepatic cycling through the gallbladder and is circulated back to the duodenum. In the gut, MPAG is deconjugated to MPA and may be reabsorbed to central circulation or eliminated through feces. MPAG circulating in central circulation is eliminated through renal elimination. MPAG may undergo several EHC during a dose interval.
1.3 PHARMACOKINETIC PARAMETER DETERMINATION 1.3.1 NON-COMPARTMENTAL ANALYSIS
A fundamental approach to analyse the kinetics of a drug is by using the non-compartment analysis (NCA). NCA calculates AUC based on several plasma concentration samples over a period of time. When creating an AUC of a drug, lines between time-concentration points are drawn and the concentration curve is divided into several trapezes, as described in figure 1.2.
By this approach, AUC can easily be estimated and pharmacokinetic parameters of the drug such as clearance, elimination half-life, Tmax and Cmax can be generated without any assumption of the drugs pharmacokinetic properties [27].
Figure 1.2: AUC measurements with the trapezoidal approach. A(n) represent a trapeze compartment between two blood samples. The trapezes are summarized, and the total AUC exposure is measured.
To generate a complete AUC of a drug, repeated sampling is required to cover the complete absorption and elimination profile of the drug within one half-life. For a drug administered once a day, this requires a sample time frame of 24 hours. In the first hours of sampling, multiple blood samples are required to be able to describe a thorough AUC of the drug [27].
Trough concentration (C0) sampling strategy depends on measuring the concentration of a drug present in blood before the next dose is administered, as illustrated in figure 1.3. C0 is commonly used to adjust drug doses, because of its simplicity, considering only one blood sample is required. However, C0 correlates poorly with the total exposure for a drug with more complex kinetics and a different approach is necessary [28, 29].
Figure 1.3: Drug measurements indicating steady state, were C0 is the concentration at time before a new dose is administered (predose). The red dots on the x-axis indicates the time interval between drug administration. The therapeutic range is between the toxic range and subtherapeutic range. The therapeutic range is presented as the red arrow.
Blood sampling throughout the dose range of 12 hours may be challenging. An abbreviated approach of creating an AUC for MPA is by measuring the blood concentration at three specific times; before administration, 0.5 hours and 2 hours after administration and the total AUC is estimated by a regression algorithm. The regression algorithm is highly dependable on accurate time point of sampling [29, 30].
AUC can be estimated by a limited number of blood samples for future patients based on a population model. The population model has to contain an adequate amount of patient profiles to include possible variability within the population. The limited sampling strategy (LSS) uses few blood samples and is then correlated with predicted AUC created by the model. Using Bayesian estimates, AUC is predicted by estimating the most likely AUC values based on former PK-profiles of similar populations [31, 32].
1.4 POPULATION MODELLING
Studies of population pharmacokinetics aim to identify and quantify sources of variability in drug concentration in a chosen patient population. Modelling is a mathematical method to describe how the body handles a drug. Clinically, a successful model may be used to optimize
drug therapy in individuals by predicting relevant individual pharmacokinetic parameters [25]. Population pharmacokinetic (PopPK) modelling is a commonly used approach to describe pharmacokinetic data [33].
Kinetic data can be modelled based on a compartment approached model. The model
describes the distribution between compartments, and is structured into several compartments that describes transport rate and drug concentrations within a given compartment. An
absorption compartment is related to the amount of drug in the gut before absorption. A central compartment describes the concentration in the central circulation and is often related to observed concentrations. The model may contain several central compartments based on measurements of different components, such as the active and inactive metabolites. In a two- or multi-compartment model, a peripheral compartment is included to describe distribution.
This compartment is not observable and is dependent on a manipulation process. The peripheral compartment will represent the amount of drug that is not present in the central compartments [34, 35].
1.4.1 MODELLING SOFTWARE
Pmetrics, derived from the term “pharmacometrics”, is a library for R, were R and Rstudio is a free software environment for statistical computing and graphics [36-38].
The non-parametric approach is considered a superior method for detecting unexpected sub- populations or outliers because it estimates the discrete distribution of the known parameters.
The non-parametric method does not approximate the likelihood function as the parametric method but obtain the most likely representation of the entire population parameter
distribution, as illustrated in figure 1.4 [34, 39].
Figure 1.4: Illustration of a parametric and non-parametric population model. The left model shows a parametric distribution (unimodal), which have a constant form and are described from statistical parameters, such as mean. The right model describes a non-parametric distribution (multimodal) and consist of the entire population parameter distribution.
Modified from Jelliffe et al. [39].
In Pmetrics, a model can either be executed parametric or non-parametric. The parametric approach operates with mean and standard deviation. The distribution is determined by an assumed function. Commonly the distribution is log-normal or normal, as shown in figure 1.4.
The adaptive grid is a method were Pmetrics will detect the maximum likelihood of a vector within a large grid. A simplified approach to describe this method is shown in figure 1.5. A two-dimension fixed grid G0 containing a number of support points, X. The adaptive grid method will search for support points with the highest likelihood and the points with very low probability will be rejected. In the process a new grid G1 is created, containing the support points from G0 with the highest likelihood. To expand the G1, the support points is adjoined with similar support points Xn, with X at its centre. Pmetrics will further search for the highest likelihood support points and deleting low probability points. The process will lead to the creation a new grid G2. G2 will replace G1, and the iteration process will continue to increase the likelihood of the grid. When the increase is essentially zero, convergence is reached [40].
Figure 1.5: Describing the adaptive grid in a two-dimension grid. G0is the starting fixed grid, containing a number of support points, X. The support points with the highest likelihood, marked red, is adjoined with similar support points, Xn, creating a new expanded grid, G1. The support points with low probability, marked black in the left figure, is rejected.G1 is replaced with a new grid, G2 and the iteration process is continued until convergence is reached.
The algorithm used in Pmetrics is the non-parametric adaptive grid (NPAG) and aim to detect the vector of parameters that calculates the highest maximum likelihood of a population distribution. Because the approach is non-parametric, all distributions within the dataset is included in the estimate. The method will make a non-parametric population mixed-effect model of discrete support points [36].
The mixed-effect refer to random effects and fixed effects. The random effects include model parameters such as clearance (Cl) and distribution volume (Vd) that are estimated within the population. The fixed effects are an error model calculated by the standard deviation (SD) from the dataset. The SD is estimated by the following equation;
𝑆𝐷 = 𝐶0 + 𝐶1 ∙ [𝑜𝑏𝑠] + 𝐶2 ∙ [𝑜𝑏𝑠]0+ 𝐶3 ∙ [𝑜𝑏𝑠]2
𝐸𝑟𝑟𝑜𝑟 = 𝑆𝐷 ∙ 𝑔𝑎𝑚𝑚𝑎
𝐸𝑟𝑟𝑜𝑟 = 8𝑆𝐷0∙ 𝑙𝑎𝑚𝑏𝑑𝑎0
The gamma or lambda value is terms to capture extra process noise related to the observation, including mis-specified dosing and observation times. The value of lamda or gamma is a starting value related to C0 and is prespecified in the model file. Gamma is a multiplicative model, while lambda is an additive model [38].
1.4.2 ASSESSMENTS IN MODEL DEVELOPMENT
An observed versus predicted plot (OP-plot) is a graphic interpretation of an observed and predicted value. Observed values from the dataset in the input file is plotted against values that are predicted by the model at the respective time-point and creates a plot similar to the plot shown in figure 1.6.
Figure 1.6: Population (left) and individual (right) observation versus prediction plot for a NPAG run conducted in Pmetrics. A residual is the distance between a data point (red dots) and the regression line (stapled line).
The OP-plot shows how well the estimates predicted by model fit the observed values by calculating the residual error;
𝑅𝑒𝑠𝑖𝑑𝑢𝑎𝑙 𝑒𝑟𝑟𝑜𝑟 = (𝑦AB − 𝑦B)
𝑦AB is the predicted value, and 𝑦B is the observed value for each individual i in the dataset.
Ideally the error is zero, and the aim is for the residual error to be as close to zero as possible.
The coefficient of the determination ratio, R2 is calculated by establishing the ratio between model sum of squares (SSM) and total sum of squares (SST). The R2 range from 0 to 1, were the aim to reach as close to 1 as possible. 𝑦E is the mean observation. R2 is described by the equation;
𝑅0 = 𝑆𝑆𝑀
𝑆𝑆𝑇 = ∑(𝑦AB− 𝑦E)0
∑(𝑦AB− 𝑦E)0
Imprecision is random variability about a target. Root mean squared error (RMSE) of
prediction is the measure of how far away the model predictions are from the observed values and is described with the equation;
𝑅𝑀𝑆𝐸 = I∑(𝑦AB − 𝑦B)0 𝑛
Mean error of prediction (MEP) shows whether the model over- or underestimate. A positive value indicates overestimation while a negative value indicates underestimation.
𝑀𝐸𝑃 = ∑(𝑦AB − 𝑦B)0 𝑛
Akaike Information criteria (AIC) is used as a tool when comparing models in a development process. The value of AIC in itself has no absolute meaning, but when comparing between different models with the same structure, the lowest value of AIC indicates a better fit with the observed data, corrected for number of parameters in the model. AIC is based on the
Were np is the total number of parameters in the model. OBJ is the minimum value of OFV [33].
1.5 CLINICAL UTILITY VALUE
TDM is based on measurements of the concentration of drugs in blood/plasma. Metabolites and active drug can be measured, and the concentration can be used to optimize treatments for each individual. For the method to be clinically useful, there must be a relationship between the plasma concentration of the drug and clinical effect. TDM is commonly used for drugs with a narrow therapeutic window [41, 42]. The therapeutic index window indicates the margin between the therapeutic dose and the toxic dose. [41]. MPA is a suitable drug for TDM, because of its narrow therapeutic window and its variability both intra-individual and within a population.
To describe the relationship between MPA exposure and acute rejection in kidney
transplanted patients, several studies has concluded that the AUC0-12h is the best and suitable form to establish the exposure within the therapeutic range of 30 to 60 mg*h/L [31, 43]. A comprehensive review article was recent established were the MPA exposure was
investigated. In the review, 36 publications were identified and used in the full-text article [44]. A strong relationship has been displayed between acute rejection and AUC0-12h, which is favoured compared to trough concentrations (C0) of MPA in TDM. For many therapeutic drugs, a trough concentration is well corelated with AUC, but for MPA, the relationship is less precise. This can be explained with the highly pharmacokinetic/pharmacodynamic
(PKPD) characteristic of MPA, such as EHC, were C0 of MPA will not be able to detect EHC within the individual and patient population [44, 45].
PopPK models can be used in therapeutic drug monitoring (TDM) to accurately describe the relationship between dosage and blood concentration. Clinically, population models can be used as tools to individualize an individual patient dose for optimizing of the clinical response to a drug. By developing Bayesian estimates and combining these with LSS, a patient specific dosage regimen can be optimized with only a few samples within a dose interval [31, 43].
1.6 AIM
The aim of this thesis is to develop and validate a PopPK model with a special focus on describing enterohepatic recycling of MPA physiologically, for the ultimate use to obtain Bayesian estimates to be used to individualize MPA therapy in adult kidney transplanted recipients.
2.0 METHOD
2.1 PATIENT AND DATA 2.1.1 MODEL DEVELOPMENT
Pharmacokinetic (PK) data on MPA and MPAG in renal transplanted recipients was obtained from the DagNattTac-study [46] and MicrobioTac_MPA-study (ongoing clinical trial,
NCT04207177). Only PK-profiles obtained from patients on steady-state MMF therapy were included in the development and validation processes. The patients had similar drug therapy regimen, with tacrolimus and prednisolone as additional maintenance immunosuppressive therapy, and similar covariates were available from the two studies.
The DagNattTac-study was a prospective, open, nonrandomized PK study and was performed at the National Transplant Center in Norway, Oslo University Hospital – Rikshospitalet. The duration of the study was from December 2015 until May 2017, and included renal transplant recipients older then 18 years 12-15 months after transplantation [46]. From this study, 9 PK- profiles were included in the model development. The patients were fasting at the day of investigations and include data from 12-hour sampling were available.
The MicrobioTac_MPA-study is an exploratory, open, prospective parallel cohort study and is conducted at the center as the DagNattTac-study. The primary objective of the study is to investigate the association between microbiome diversity and 12-hour mycophenolate
kinetics. Patient data was collected between 3-8 weeks and one year after transplantation. The study is ongoing, and the sampling was done in parallel with the model development.
Data collected from fall of 2019 until March of 2021 were used. MPA sampling were performed over a 12-hour period after administration of the morning oral dose of MMF.
2.1.2 MODEL VALIDATION
To validate the popPK model, patient data from the AdvaOmega-study was used [47]. This study was performed at the National Transplant Center in Norway, Oslo University Hospital – Rikshospitalet [47]. The study contributed with 22 PK-profiles for external validation of the model. The patients were fasting, and cotreated with tacrolimus and prednisolone. Sampling was performed for 8 hours after the morning dose of MMF.
2.2 BIOANALYSIS
The bioanalytical analysis of MPA and MPAG total plasma concentrations were performed at Farmasøytisk institutt, University of Oslo. The analyses were conducted by a validated
UPLC-MS/MS method for quantification of both MPA and MPAG. The plasma samples were deproteinized by a precipitation reagent (95 % acetonitrile and 5 % methanol) and added 100 ng/mL internal standard solution for MPA (MPA-d3) and 1 µg/mL for MPAG (MPAG-d3).
The supernatant was added a mobile phase of 50 mM ammonium acetate with 5 % acetonitrile.
In the UPLC-MS system, chromatographic separation of MPA and MPAG from internal standard, were done with a T3 column (Acquity UPLC HSS T3 1,8 µm 1 x 150 mm) with a T3 pre column (Acquity UPLC HSS T3 1,8 µm 2,1 x 150 mm). Mobile phase A and B consisted of 50 mM ammonium acetate with 5% acetonitrile and 100 % acetonitrile with a flow rate of 0.400 mL/min.
Lower limit of quantitation (LLOQ) was reported to 0.2 mg/L for MPA and 2.0 mg/L for MPAG which is accepted values lower than 5. The variation coefficient (CV%) for MPA was reported to less than 8.7 %. Mean precision of MPA was between 88.9 % and 112 %. CV%
reported for MPAG was less than 13 % and mean precision valued between 86.4 % and 104.6
% [48].
2.3 MODELLING USING PMETRICS
To be able to run Pmetrics, two different files are required in the run directory; a data file (.csv) and a model file (.txt) to be able to run. The NPAG run is executed by the argument NPrun() command. Pmetrics will create a summary report and several data objects which can be loaded into R, using PMload() for further data analyses.
The input file consists of a spreadsheet “matrix” format, and is designed for input of multiple
Figure 2.1: Section of the input file created in Excel.
The model file is formatted as a text file and Pmetrics will transform the model file into a Fortran-model. The codes made in the model file must follow the Fortran syntax. The model file contains the description of the structural model and describe how the data in the input file is included in the model. The model structure must follow a certain structural format,
described in the Pmetrics manual. In addition to information about the structural model, the model file also contains information about the range of primary parameters, covariates and an error model. To run, the model file must at least contain primary variables, outputs and error [38].
A thorough description of the format of the model and input file is available in the Pmetrics manual at the “Laboratory of Applied Pharmacokinetics and Bioinformatics” website [38].
2.4 MODEL DEVELOPMENT
The drug input value in the csv-file corresponds with the administered dose of MPA. Several output equations can be specified for one input value [38]. The MPA-model to be developed has two output equations related to MMF dose administered; the observed concentrations of MPA and MPAG and since it is calculating based on mass-movement of the MPAG output normalized to a “MPA-weight” by the molecular weight ratio of MPA:MPAG.
Several of the recipient in the studies contributed with more than one 12-hour PK-profile of MPA and MPAG, obtained at different time after transplantation. In the modelling these data were handled as pseudo id’s, i.e. adding ID of n.1, n.2 or n.3 indicating which PK profile the observed concentrations belonged to in order for Pmetrics to be able to identify specific parameter estimates for each investigation, i.e. Pmetrics treat the data as from separate individuals. Time after transplantation were in addition specified as a covariate.
Because MicrobioTac_MPA is an ongoing study, not all data were available during the early stages in the development phase. In the beginning, the development dataset included 12 PK- profiles collected from the Dag&Natt-study. Later in the process, MicrobioTac_MPA-data were included in addition.
2.4.1 STRUCTURAL MODEL
The model development was based on a four compartment model created by a former student [49]. The model structure of the basis is shown in figure 2.2 and worked as the base model in the development phase. The base model only included observed concentrations of MPA and had to be revised to also include MPAG observations.
Figure 2.2: A four compartment model used as base for development of the structural compartment model. The base model consist of an absorption compartment, central
compartment of MPA, bile compartment and a separate gut reabsorption compartment. K is the rate constant, and is divided into absorption and reabsorption (Kabs/Kreabs), metabolism (Km), and elimination (Kel). Transition of bile was described as the K rate constant, Kbile.
The base model was expanded by adding a second central compartment related to the observed concentrations of MPAG, and tested in a NPAG run. Further development of the model included incorporation of a peripheral compartment connected to the central
compartment of MPA and to include “bile-acid compartment”, creating the six-compartment model (Figure 2.3).
Next step was to assess different ways to include a lag-time in the absorption process both for the dose administered as well as during the EHC process. The standard lag-time function implemented in Pmetrics was used as a reference and a Heaviside step function was tested on the transition rate constant of absorption (Ka1) and EHC-reabsorption (Ka2). Two separate lag times were tested. The final structure is presented in figure 2.3.
Figure 2.3: Final model structure describing the pharmacokinetics of MPA and MPAG. K is the rate constant of the drug transfer between compartments. A compartment consists of an amount of drug and the volume that the drug is distributed in, within the given compartment.
Table 2.3 gives a description of the rate constants included in the model.
2.4.2 ENTEROHEPATIC MODELING
EHC is dependent on the transition of K56 between the bile compartment and the gut
reabsorption compartment. An early approach to determine when the transition of bile release occurred, was by introducing selective execution, were the estimated time parameter, time of gallbladder emptying (TGB) was introduced;
𝑇 < 𝑇𝐺𝐵 𝑡ℎ𝑎𝑛 𝐾56 ≠ 0
approach was tested by adding a second estimated parameter, gall interval (GI). A time interval from minimum to maximum time was introduced;
𝑇𝐺𝐵 < 𝑇 < 𝑇𝐺𝐵 + 𝐺𝐼 𝑡ℎ𝑎𝑛 𝐾56 ≠ 0
T at lower values of TGB and T at higher values than TGB + GI, the transition of K56 is equal to zero and EHC will not occur. The model was able to control the transfer opening and closing, and EHC were seen as a second peak in the AUC-plots. The specified interval was able to predict one cycle of EHC.
2.4.3 STEADY STATE
Initial conditions were included in the model to describe the amount of MPA and MPAG in respectively compartment before administration of the dose from which the subsequent concentrations were determined. IC1 and IC2 is respectively the measured trough
concentration of MPA (compartment 2) and MPAG (compartment 4), and is presented as covariates in the model file, available in attachment 1. The initial amount in respective compartment is hence the measured concentration multiplied with the estimated volume of each compartment. For compartment 3 the initial amount was described by the ratio of the rate constants between compartment 2 and 3 (K23/K32) multiplied with the amount in compartment 2. Amount in compartment 5 is not possible to relate to any of the
measurements at predose and different strategies were applied with a primary variable to describe the amount present. Compartment 6 was set to 0 at each dosing.
2.5 ABSORPTION AND REABSORPTION 2.5.1 DELAY OF ABSORPTION
The delay in absorption following oral ingestion was coded into the model by introducing a lag time for each absorption. The delayed transfer is described as a lag time for the primary variable Ka1 and will determine when the transfer of the observation can start. To describe the transition of absorption in a realistic physiologic approach, a Heaviside function with a primary variable describing a lambda value was included in the function in secondary variables. The function include the delay caused by tlag, and the lambda function creating a sigmoidal transition between compartments.
2.5.2 DELAY OF REABSORPTION
The transition of reabsorption was described in a separate Heaviside step function, including a second lag time stating when the reabsorption occurs. Reabsorption is represented as the primary variable, Ka2. The transition of drug is in addition described with a lambda function.
The Heaviside step function is described further.
2.5.3 THE HEAVISIDE STEP FUNCTION
The Heaviside step function, also called the unit step function, is commonly used in engineering problems since it enables an easy representation of functions that appear for a limited period of time. The function is given by the equation;
𝐻(𝑡) = 0 𝑓𝑜𝑟 𝑡 < 0 𝑎𝑛𝑑 𝐻(𝑡) = 1 𝑓𝑜𝑟 𝑡 ≥ 0
In the model, the step function was included in absorption from compartment 1 to 2, and reabsorption from compartment 6 and back to the central compartment 2. The function appears during an interval [a,b] and vanishes outside this interval. The step function makes it possible to formalize the transfer within the interval. The Heaviside step function has no derivate at 0, and is not continuous [50].
0
Figure 2.4 illustrates the Heaviside step function with the inclusion of the lambda function.
Including a delay stated by a lag time, the lambda, displayed as a green and a blue function, illustrate the transition gradually depending on the value of lambda. At lower values of lambda, the transition is described as a sigmoidal function, while at higher values, the function is presented as a steeper line, presenting a more rapid transition.
Transformed into the model, the Heaviside step function is coded in Fortran:
𝐾𝑎] ∙ (1
2^1 + 𝐴𝑇𝐴𝑁(𝐿𝐴𝑀 (𝑇 − 𝑇𝐿𝐴𝐺]))2 𝜋 b)
The parameter n is the given state of absorption or reabsorption. Ka is the absorption or reabsorption rate constant of MPA. ATAN is the Fortran syntax computing for arctangent of x.
LAM is the function of lambda which affects the transition of absorption and reabsorption.
Tlag is the respected lag time for absorption defined as a primary variable.
2.6 MODEL PARAMETERS
The discrete support point distribution after a completed NPAG run is visualized for each primary parameter in a marginal plot as shown in figure 2.5 below. During model
development these plots will give an indication of if the boundaries of a primary variable should be expanded or narrowed in order to optimize the model.
Figure 2.5: Marginal plots for GI and TGB. A marginal plot is created for each primary variable after a completed NPAG run. The red lines represent the areas in which the
likelihood is detected. GI is accordance with its upper limit of 5, and the boundary should be expanded to higher values.
Altering the boundaries of one parameter will result in an altered likelihood for all other variables. The process to detect the most likely boundary range for each of the primary variables is a time-consuming process. To assist the initiation of this process, reported literature values were used as initial conditions of the respective boundaries.
2.7 COVARIATES
The covariates available in the analysis are presented in table 2.1. The plan was to test covariates in the final structural model. This could lead to an improvement of the population prediction of the model. A standard inclusion/exclusion strategy would have been applied and identification of potentially relevant covariates would have seen screened for by using the PMstep()-function in Pmetrics. Using the PMstep function, P-values for the linear
regression coefficients for each subject covariates versus Bayesian posterior parameter values is reported and is used in the covariate analysis.
A covariate is included into the model by a graphic interpretation of the correlation between a covariate and the model parameter. The model parameter is tested in the equations;
𝑀𝑜𝑑𝑒𝑙 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 = 𝑒𝑠𝑡𝑖𝑚𝑎𝑡𝑒𝑑 𝑚𝑜𝑑𝑒𝑙 𝑝𝑎𝑟𝑎𝑚𝑡𝑒𝑟 ∙ 𝑐𝑜𝑣𝑎𝑟𝑖𝑎𝑡𝑒
𝑀𝑜𝑑𝑒𝑙 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 =𝑒𝑠𝑡𝑖𝑚𝑎𝑡𝑒𝑑 𝑚𝑜𝑑𝑒𝑙 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝑐𝑜𝑣𝑎𝑟𝑖𝑎𝑡𝑒
Decreased in population %RMSE and values as well as a positive impact on the individual time-concentration plots is the main diagnostic criterion for evaluating each covariate.
Table 2.1: Description of the different covariates in the model and input file #COV.
#COV Description
TAC Tacrolimus dose (mg)
WHT Body weight (kg)
HGT Body height (cm)
PRED Prednisolone dose (mg)
*BSA Body surface areal (m2)
BMI Body mass index (kg/m2)
M1K0 Sex
TX Time after transplantation (days)
CREAT Creatinine (µmol/L)
AGE Age (years)
* Mosteller formula
2.8 MODEL VALIDATION PLAN 2.8.1 DEVELOPMENT VALIDATION
Internal validation was performed by comparing the models through the development period.
Decreased values of RMSE and AIC weighted the most in the validation process. In addition, time-concentration plots for each individual were investigated and values of bias and
imprecision were taken into consideration when determining the overall outcome of a possible model.
2.8.2 EXTERNAL VALIDATION
Patient data from the AdvaOmega-study were used to test how well the model predicts a second population that is separate from the data used in the development phase. The final model was applied to the external dataset specifying zero cycles to create Bayesian posteriors.
RMSE and individual concentration-time curves were investigated for each of the profiles.
3.0 RESULTS
3.1 PATIENTS
The demographic and patient characteristic data are presented in table 3.1. In all, 338
concentration values of both MPA and MPAG from 18 patients were used to develop the PK- model, and 222 values of both MPA and MPAG from 11 patients were used for external validation of the final model.
Table 3.1: Patient characteristic and demographic data for model development and model validation.
Development data Mean (range)
Validation data Mean (range)
Patients 18 11
PK-profiles 22 22
MPA samples 338 222
MPAG samples 338 222
C0 TAC (mg) 6.5 (3.6, 8.9) 6.5 (3.6 , 11.0)
Age, years 56 (29, 70) 55 (32, 70)
Height, cm 177 (165, 190) 178 (165, 185)
Weight, kg 87 (52, 109) 90 (65, 123)
Sex (male/female) 14/4 9/2
MMF dose, mg/day 1500 (1500, 1500) 1500 (1500, 1500) Prednisolone, dose mg/day 12 (5, 20) 6 (5, 15)
TAC dose, mg/day 8 (2, 10) 8 (3, 10)
P-creatinine, µmol/L 118 (65, 154) 107 (73, 187)
Time post transplantation, days 89 (13, 383) 416 (389, 452) Body mass index (kg/m2) 27.7 (19.1, 36.0) 28.7 (21.4, 36.0)
*Body surface area (m2) 2.06 (1.54, 2.30) 2.10 (1.73, 2.45) C0: Predose concentration. MMF: Mycophenolate mofetil. TAC: Tacrolimus.
MPA: Mycophenolic acid. MPAG: MPA O 7-glucuronide. *Mosteller formula.
3.2 MODEL DEVELOPMENT
Table 3.2 describes the summary of the development process. The base model was restructured to include MPAG. Model 1 includes a second central compartment with
additional primary variables to describe the transfer. AIC is not comparable between the base and model 1 due to alterations in the data set used for development. %RMSE increased for model 1 compared to the base due to the inclusion of a second compartment which affected the likelihood of the primary variables.
Table 3.2: Selection of runs throughout the development process showing the results of individual observed versus predicted concentrations. %RMSE reported for MPA.
Model %RMSE R2 Bias
(mg*h/L)
Imprecision (mg*h/L)
AIC Description
BASE 21.3 0.953a -0.42a 13.7a 1243 MPA base model.
1 54.67 0.561a/0.795b -0.19a/-0.74b 0.42a/0.87b 1243 BASE + Central compartment of MPAG
2 48.33 0.702a/0.906b -0.2a/-0.14b 0.58a/0.74b 1171 Peripheral compartment for MPA
3 36.35 0.793a/0.93b -0.30a/-0.16b 0.68a/0.66b 1098 Inclusion of the Heaviside unit step function and Tlag2.
4 18.49 0.867a/0.961b -0.51a/-0.10b 0.86a/2.88b 850 TGB<T<TGB+GI
5 12.73 0.967a/0.96b -0.001a/-0.06b 0.83a/1.04b 749 Primary variable of amount present in bile.
6 15.5 0.952a/0.956b -0.16a/-0.06b 4.63a/0.98b 2149 Inclusion of data from the MicrobioTac-MPA_study.
7 17.51 0.938a/0.958b -0.02a/0.01b 1.7a/0.89b 2199 Lambda error model alteration L = 0.6
MPAa MPAGb
Model 2 included a peripheral compartment for MPA creating a six-compartment structure compared to model 1 with five compartments. The %RMSE and AIC values decreased when the boundaries of the primary variables were adjusted to fit the alteration. Additionally, two primary variables were added to describe peripheral transition.
Model 3 included the Heaviside step function to improve the transition of absorption of the primary variable Ka1. In addition, a second lag time was defined to establish the second peak related to EHC. Compared to model 2, inclusion of the Heaviside function decreased both
%RMSE and AIC.
Model 4 was altered by including a second time parameter; GI, to determine the duration of the rate transfer K56. The improvement of the model is displayed in figure 3.1, comparing the time-concentration plots from model 3 to the alteration done in model 4, in ID 25.1, 26.1 and 27.1.
Figure 3.1: Time-concentration plots of three PK-profiles during model development, the three profiles on the upper represent results from model 3. The profiles under describing EHC by model 4. Triangles are related to observed concentrations, while continuous lines are related to predicted concentrations. MPA is related to black triangles and lines, while MPAG
included a primary variable (A5) describing the initial conditions in the bile compartment as presented in the figure, in which at time zero, the compartment contains an amount. The improvement was seen as a decreased value of %RMSE and AIC.
Figure 3.2: Black lines and triangles are related to MPA, red is related to MPAG. The green continuous line represent the concentration of MPAG represents in the bile compartment (compartment 5) per hour, while the blue line represent the concentration of MPA present in the gut reabsorption compartment (compartment 6) per hour. The output equation of bile and gut is rescaled 3 times.
The increase in AIC of model 6 compared to model 5 is related to the inclusion of the
additional data from 9 patients obtained from the MicrobioTac_MPA-study. The dataset were expanded from 9 PK-profiles to 22 PK-profiles, which had an impact on the overall outcome of the parameters.
The model 7 run was started with a lower initial lambda value (L=0.6) in the error function compared to the other models throughout the development period. The previous models run with an initial lambda value of 10, indicating a higher level of noise in the observed
concentrations. The lower value of lambda increased the value of AIC and RMSE of the model. However, less fluctuation of MPAG were seen in individual time-concentration plots.
The primary variables included in the final model is presented in table 3.3. The boundaries of the rate transition variables (Kn) were in early stages tested in a wider range (0.1 – 10). The values of GI and TGI were tested in several runs to detect the best fit for the stated time interval. The boundaries of each variable were tested at lower and higher values. The volume V1 and V2 were tested in a wide range at early stages (1-80). The range were further decreased to lower values due to results in the marginal plots.
Table 3.3: Description of the final model parameters included in the model file and figure 2.3.
Primary variables: Description:
Ka1 Absorption rate constant of MPA (h-1) Ka2 Reabsorption rate constant for MPAG (h-1)
K23 Transfer rate constant of MPA to peripheral compartment (h-1) K32 Transfer rate constant of MPA from peripheral compartment (h-1) K24 Metabolism rate constant of MPA to MPAG (h-1)
K40 Renal excretion rate constant of MPAG (h-1)
K45 Transfer rate constant of MPAG to biliary compartment (h-1)
K56 Transfer rate constant of gallbladder emptying of MPAG to gut (h-1) K60 Feces excretion rate constant of MPA/MPAG (h-1)
V1 Volume of central compartment of MPA (L) V2 Volume of central compartment of MPAG (L)
A5 Drug present in bile compartment before administered dose (mg).
Tlag1 Lag time for absorption (h) Tlag2 Lag time for reabsorption (h)
TGB Time of bile release after dosing (h) GI Gallbladder emptying interval (h)
LAM Lambda, used for the Heaviside step function
FA0 Bioavailability of MPA
value of 100. Transition of K56 appeared within the time interval stated in secondary variables (#SEC) shown in figure 3.4. The statements of EHC predicts one cycle of
circulation. Differential equations headlined as #DIF in figure 3.4 were used to describe the transition of drug through the various compartments.
#SEC
&IF(T<TGB) K56 = 0.D0
&IF(T>TGB + GI) K56 = 0.D0
#DIF
XP(1) = - K12 × X(1)
XP(2) = K12 × X(1) + K32 × X(3) + K62 × X(6) – K24 × X(2) – K23 × X(2) XP(3) = K23 × X(2) – K32 × X(3)
XP(4) = K24 × X(2) – K40 × X(4) – K45 × X(4) XP(5) = K45 × X(4) – K56 × X(5)
XP(6) = K56 × X(5) – K62 × X(6) – K60 × X(6)
Figure 3.3: Statements of bile release related to the primary variable K56, stated as secondary variables (#SEC) in the model file. Differential equations (#DIF) describing enterohepatic cycling in the model file. XP(n) is the notation of dX(n)/dt, where n is the given compartment number corresponding to the structural model. X(n) is the amount of drug within the compartment. K56 is the parameter responsible for EHC. K12 represent the absorption rate transfer Ka1, while K62 represent the reabsorption rate transfer Ka2.
TGB: Time of gallbladder emptying, GI: gallbladder interval, 0.D0: Value of 0.00000.
Some of the individual time-concentration profiles used for model development showed potential of more than one peak related to EHC. In figure 3.4, time-concentration plots of ID 404.1, 23.1 and 24.1 is presented for observed concentrations of MPA, and in addition, time- concentration plots of MPA and MPAG illustrating how the model predicts profiles with several peaks during the 12-hour sampling interval. The second and third peak is seen around 5 hours for 23.1 and 24.1, and 6 hours for 404.1. A third peak can be seen around 8 hours for 24.1, and at 10 hours after administration for 404.1 and 23.1. This were seen in several of the time-concentration plots, only a selection of three profiles is presented in the figure 3.4.
Figure 3.4: Time-concentration plots of observed MPA (upper) and observed and predicted MPA and MPAG (lower) showing potential of more than one EHC present during 12 hours for ID 404.1, 23.1, 24.1.
3.3 FINAL MODEL
The final model converged after 8481 cycles. %RMSE were reported to 17.51 % for MPA and 6.42 % for MPAG and AIC of 2199. Attachment 1 present the complete model file for the final model. The model included all primary parameters listed in table 3.4. Final lambda value was reported at 0.66.
Table 3.4: Population parameters value summaries of the final model.
Parameter MEAN SD CV% MEDIAN
Ka1 8.36 5.03 60.10 8.07
Ka2 7.95 8.26 103.82 3.19
LAM 14.08 13.45 95.51 6.54
K23 6.60 2.55 38.56 6.28
K32 0.96 0.82 85.94 0.45
K24 2.11 1.02 48.56 1.71
K40 0.07 0.06 89.76 0.04
K45 0.22 0.09 38.35 0.22
*K56 100
K60 0.41 0.61 149.50 0.1
V1 8.31 5.38 64.81 6.24
V2 7.12 3.49 49.06 7.28
FA0 0.78 0.20 25.80 0.76
A5 50.94 33.52 65.80 52.68
GI 3.32 1.56 46.84 3.58
TGB 5.46 1.53 28.06 5.60
Tlag1 1.16 0.80 69.22 0.92
Tlag2 7.32 1.54 21.02 7.52
* Fixed value
SD – Standard derivation. CV% - confidence interval
Figure 3.5 shows the OP-plot of MPA and MPAG. The population plots of both MPA and MPAG displays a poor outcome of R2 (MPA=0.196, MPAG = 0.477) and high values of imprecision (MPA=163, MPAG= 9.96).
The individual plot of MPA and MPAG presented better results of R2, MPA = 0.938 and MPAG = 0.958. Imprecision of the individual plots for MPA = 1.7, and MPAG = 0.887, and bias (MPA = -0.02, MPAG = 0.01). The %RMSE of the final model was reported to 17.51 % for MPA and 6.42 % for MPAG.
5 10 15 20
5101520
Population Predicted
Observed
R−squared = 0.196 Inter = 1.94 (95%CI 1.35 to 2.54) Slope = 0.423 (95%CI 0.328 to 0.518) Bias = 0.856
Imprecision = 163
5 10 15 20
5101520
Individual Posterior Predicted
Observed
R−squared = 0.938
Inter = −0.0282 (95%CI −0.175 to 0.119) Slope = 1.01 (95%CI 0.979 to 1.04) Bias = −0.0236
Imprecision = 1.7
20 30 40 50 60
2030405060
Population Predicted
Observed
R−squared = 0.477 Inter = −3.9 (95%CI −8.19 to 0.389) Slope = 1.33 (95%CI 1.18 to 1.49) Bias = −1.99
Imprecision = 9.96
10 20 30 40 50 60
102030405060
Individual Posterior Predicted
Observed
R−squared = 0.958
Inter = 0.0238 (95%CI −0.761 to 0.809) Slope = 1 (95%CI 0.977 to 1.02) Bias = 0.0105
Imprecision = 0.887
MPA MPA
MPAG MPAG
Residual plot in figure 3.6 shows the distribution of weighted residuals (WRES) for the individual predictions by predicted concentrations and time after dose, respectively, for the final model. The residuals are distributed heterogeneously with outliers at higher
concentrations, both at negative and positive values. The distribution of WRES compared to time shows several outliers at the early time course. The model tends to overpredict the concentrations, due to the mean prediction error value of 0.0032 mg/L. Both overprediction and underprediction tends to occur at higher concentrations, and at lower time courses.
Figure 3.6: Residual plot of weighted residuals (predicted – observed) versus predicted concentrations (left) and weighted residuals versus time (right).
Individual time-concentration plots for a selection of PK-profiles are presented in figure 3.7.
The final model was able to predict EHC presented as a second peak for several of the profiles. The complete set is presented in attachment 2. Rapid absorption and reabsorption were seen in several of the plots.
Fluctuation of MPAG concentrations, which is presented as red lines and triangles, were seen in several of the profiles, such as for ID 404.1, 23.1 and 28.1 presented in attachment 2.
AUC calculated for MPA and MPAG. Significant differences were seen in ID 404.1, comparing the predicted value of MPAG to the observed value, were the increase in the predicted AUC0-12 is calculated at 15.6 %.
Table 3.5: Observed and predicted AUC of MPA and MPAG is displayed in table 3.4.
Observed concentrations of MPA and MPAG is calculated by the trapezoidal rule using the makeNPA() function in Pmetrics. Individually predicted values of MPA and MPAG are used for the calculation of AUC with the trapezoidal rule with the function makeAUC(). Values reported in mg*h/L.
MPA MPAG
#ID OBS AUC0-12
mg*h/L
PRED AUC0-12
mg*h/L
OBS AUC0-12
mg*h/L
PRED AUC0-12
mg*h/L
2.1 35.29 37.17 376.35 383.07
23.1 52.00 60.47 454.80 468.07
24.1 33.09 33.68 274.18 279.19
25.1 24.68 25.94 190.46 198.94
26.1 39.11 43.24 225.80 234.42
27.1 41.52 42.77 367.38 380.40
28.1 28.95 30.86 283.98 293.17
29.1 39.05 42.25 310.23 315.70
30.1 42.59 43.07 269.64 275.65
31.1 42.45 41.12 429.09 439.17
401.1 49.50 54.53 385.63 445.01
401.2 43.50 48.61 373.15 337.30
402.1 62.33 63.51 349.41 369.54
403.1 48.18 47.80 494.97 499.32
403.2 52.21 52.24 462.48 464.44
404.1 65.65 67.40 490.36 509.85
405.1 59.33 62.45 564.16 555.62
405.2 38.02 40.28 428.24 436.51
407.1 48.95 48.98 375.48 399.84
407.2 32.60 38.76 286.35 314.37
408.1 33.22 34.56 333.73 357.82
411.1 58.63 64.13 327.31 361.97
MEAN 44.13 46.54 366.05 378.15
SD 11.11 11.63 93.66 93.05
MIN 24.68 25.94 190.46 198.94
MAX 65.65 67.40 564.16 555.62
OBS: Observed, PRED: predicted, AUC: Area under the plasma concentration-time curve.
MPA: Mycophenolic acid, MPAG: 7-O-MPA glucuronide, SD: Standard deviation.
3.4 EXTERNAL VALIDATION
Data from the AdvaOmega-study were run without cycling of the model. Mean prediction error was reported at -0.91 mg/L, %RMSE at 52.3 % for MPA and 23.7 % for MPAG.
Figure 3.8 displays the OP-plots. The population OP-plot of both MPA and MPAG displayed poor prediction abilities, with R2 of 0.369 for MPA and 0.64 for MPAG. Individual R2 value was 0.695 for MPA and 0.755 for MPAG. The values of R2 for external validation were increased significantly compared to values of the final model and an increase in %RMSE for MPA and MPAG. In addition, the p-value reported was increased, which means the model prediction abilities was worse for the external validation.
Figure 3.8: OP-plot of the final model for MPA (top), and MPAG (under). The left shows the population OP-plot, while the right shows the individual OP-plot.
0 5 10 15 20 25 30 35
05101520253035
Population Predicted
Observed
R−squared = 0.369 Inter = 0.511 (95%CI −0.631 to 1.65) Slope = 0.953 (95%CI 0.778 to 1.13) Bias = 0
Imprecision = 0
0 5 10 15 20 25 30 35
05101520253035
Individual Posterior Predicted
Observed
R−squared = 0.695 Inter = 0.738 (95%CI 0.113 to 1.36) Slope = 1.03 (95%CI 0.939 to 1.13) Bias = 0
Imprecision = 0
20 40 60 80
20406080
Population Predicted
Observed
R−squared = 0.64
Inter = −7.01 (95%CI −11.4 to −2.58) Slope = 1.66 (95%CI 1.48 to 1.83) Bias = 0
Imprecision = 0
20 40 60 80
20406080
Individual Posterior Predicted
Observed
R−squared = 0.755 Inter = −5.21 (95%CI −8.46 to −1.96) Slope = 1.27 (95%CI 1.17 to 1.37) Bias = 0
Imprecision = 0
MPA
MPAG MPA
MPAG