dose-response-time data analysis: an underexploited trinityanalysis of the connection between dose,...

1521-0081/71/1/89–122$35.00 https://doi.org/10.1124/pr.118.015750PHARMACOLOGICAL REVIEWS Pharmacol Rev 71:89–122, January 2019Copyright © 2018 by The Author(s)This is an open access article distributed under the CC BY-NC Attribution 4.0 International license.

ASSOCIATE EDITOR: GUNNAR SCHULTE

Dose-Response-Time Data Analysis: AnUnderexploited Trinity

Johan Gabrielsson, Robert Andersson, Mats Jirstrand, and Stephan Hjorth

Division of Pharmacology and Toxicology, Department of Biomedical Sciences and Veterinary Public Health, Swedish University ofAgricultural Sciences, Uppsala, Sweden (J.G.); Fraunhofer-Chalmers Centre, Gothenburg, Sweden (R.A., M.J.); Pharmacilitator AB,

Vallda, Sweden (S.H.); and Department of Molecular and Clinical Medicine, Institute of Medicine, Sahlgrenska Academy at GothenburgUniversity, Gothenburg, Sweden (S.H.)

Abstract. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90I. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90II. Methods and Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

A. What Can Be Learned from Response-Time Data Patterns? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92B. How Is the Biophase Typically Defined?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93C. The Drug “Mechanism” Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94D. Structure of the Pharmacodynamic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95E. Response-Time Patterns Impact Model Structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

III. Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96A. Case Study 1: Dose-Response-Time Analysis of Nociceptive Response . . . . . . . . . . . . . . . . . . . . . 96

1. Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 962. Models, Equations, and Exploratory Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 963. Results and Conclusions with Respect to Dose-Response-Time Analysis. . . . . . . . . . . . . . . . 994. Pharmacodynamic Interpretation and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

B. Case Study 2: Dose-Response-Time Analysis of Locomotor Stimulation. . . . . . . . . . . . . . . . . . . . 991. Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 992. Models, Equations, and Exploratory Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 993. Results and Conclusions with Respect to Dose-Response-Time Analysis. . . . . . . . . . . . . . . . 1014. Pharmacodynamic Interpretation and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

C. Case Study 3: Dose-Response-Time Analysis of Functional Adaptation . . . . . . . . . . . . . . . . . . . . 1021. Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1022. Models, Equations, and Exploratory Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1023. Results and Conclusions with Respect to Dose-Response-Time Analysis. . . . . . . . . . . . . . . . 1034. Pharmacodynamic Interpretation and Comment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

D. Case Study 4: Dose-Response-Time Analysis of Antipsychotic Effects (Brief PsychiatricRating Scale Scores) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1031. Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1032. Models, Equations, and Exploratory Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1043. Results and Conclusions with Respect to Dose-Response-Time Analysis. . . . . . . . . . . . . . . . 1054. Pharmacodynamic Interpretation and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

E. Case Study 5: Dose-Response-Time Analysis of Bacterial Count . . . . . . . . . . . . . . . . . . . . . . . . . . 1061. Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1062. Models, Equations, and Exploratory Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063. Results and Conclusions with Respect to Dose-Response-Time Analysis. . . . . . . . . . . . . . . . 106

Address correspondence to: Johan Gabrielsson, Division of Pharmacology and Toxicology, Department of Biomedical Sciencesand Veterinary Public Health, Swedish University of Agricultural Sciences, Box 7028, SE-750 07 Uppsala, Sweden. E-mail: [email protected]

This work was funded in part through the Marie Curie Seventh Framework Programme People Initial Training Networks EuropeanIndustrial Doctorate Project [Innovative Modeling for Pharmacological Advances through Collaborative Training Grant No. 316736] and bythe Swedish Foundation for Strategic Research [(both to R.A. and M.J.)].

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4. Pharmacodynamic Interpretation and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107F. Case Study 6: Dose-Response-Time Analysis of Cortisol–Adrenocorticotropic Hormone

Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1071. Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1072. Models, Equations, and Exploratory Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1073. Results and Conclusions with Respect to Dose-Response-Time Analysis. . . . . . . . . . . . . . . . 1074. Pharmacodynamic Interpretation and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

G. Case Study 7: Dose-Response-Time Analysis of Miotic Effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1081. Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082. Models, Equations, and Exploratory Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1093. Results and Conclusions with Respect to Dose-Response-Time Analysis. . . . . . . . . . . . . . . . 1094. Pharmacodynamic Interpretation and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

H. Case Study 8: Meta-Analysis of Dose-Response-Time Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1101. Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1102. Results and Conclusions with Respect to Dose-Response-Time Analysis. . . . . . . . . . . . . . . . 1103. Pharmacodynamic Interpretation and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

I. Overall Conclusions of the Case Studies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111IV. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

A. Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112B. What Do Different Doses, Routes, and Rates of Input Add? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112C. Drug Delivery by Means of Iontophoresis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119D. Permutations of Smolen’s Model: The Kinetic-dynamic K-PD Rate of Change Model. . . . . . . 120E. General Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

Abstract——The most common approach to in vivopharmacokinetic and pharmacodynamic analysesinvolves sequential analysis of the plasma concen-tration- and response-time data, such that theplasma kinetic model provides an independent func-tion, driving the dynamics. However, in situationswhen plasma sampling may jeopardize the effectmeasurements or is scarce, nonexistent, or unlinkedto the effect (e.g., in intensive care units, pediatricor frail elderly populations, or drug discovery),focusing on the response-time course alone may bean adequate alternative for pharmacodynamic analyses.Response-timedata inherentlycontainuseful informationabout the turnover characteristics of response (targetturnover rate, half-life of response), as well as the drug’sbiophase kinetics (biophase availability, absorptionhalf-life, and disposition half-life) pharmacodynamicproperties (potency,efficacy).Theuseofpharmacodynamictime-response data circumvents the need for a

direct assay method for the drug and has theadditional advantage of being applicable to casesof local drug administration close to its intendedtargets in the immediate vicinity of target, or whentarget precedes systemic plasma concentrations.This review exemplifies the potential of biophasefunctions in pharmacodynamic analyses in bothpreclinical and clinical studies, with the purposeof characterizing response data and optimizingsubsequent study protocols. This article illustratescrucial determinants to the success of modelingdose-response-time (DRT) data, such as the doseselection, repeated dosing, and different inputrates and routes. Finally, a literature search was alsoperformed to gauge how frequently this technique hasbeen applied in preclinical and clinical studies. Thisreview highlights situations in which DRT should becarefully scrutinized and discusses future perspectivesof the field.

I. Introduction

Pharmacological time-series data are typically rich ininformation. However, response-time series are oftencompared with plasma or blood exposure prior to anyanalysis of the connection between dose, response, andtime data per se. Dose-response-time (DRT) data anal-ysis is therefore an underexplored approach for quan-tification of the onset, intensity, and duration of a

pharmacological response when information about thedrug concentration is scarce or even totally lacking(Levy, 1971; Smolen, 1971a,b, 1976a,b, 1978; Smolenet al., 1972; Smolen and Weigand, 1973). Usually,plasma or tissue concentrations of drug are used asthe “driver” of the pharmacological response. However,plasma concentrations may be of marginal valuein situations with difficult or unethical sampling (e.g.,pediatric or frail elderly populations), when due to local

ABBREVIATIONS: ACTH, adrenocorticotropic hormone; AUC, area under the curve; BPRS, Brief Psychiatric Rating Scale; CTC,circulating tumor cell; CV, coefficient of variation; DRT, dose-response-time; FFA, free fatty acid; ID50, 50% of the maximum drug-inducedinhibitory effect of the inhibitory drug mechanism function; K-PD, kinetic-dynamic model; NiAc, nicotinic acid; PTH, parathyroid hormone;SD50, biophase amount at 50% of maximum drug-induced effect.

90 Gabrielsson et al.

administration exposure at the pharmacological targetprecedes plasma concentrations (e.g., dermal or pulmo-nary applications, iontophoretic techniques), or in thecase of extreme or unusual concentration-time profiles(e.g., the presence of active or interactive metabolites,mixture kinetics, or oligonucleotides). If the (pharma-cological) biomarker response follows a time series, thattime series normally contains useful information abouttarget behavior [e.g., target turnover rates and half-lives (t1/2)] and drug properties [e.g., ED50 values, rateconstants]. In some cases, the time course of drugmolecules in the biophase (i.e., immediately adjacentto the target) may also reveal itself in the onset,intensity, and duration of response when the input rateto the biophase becomes the rate-limiting step. Thebiophase kinetics frequently differ from the plasmakinetics of a drug, particularly for peripherally placedtargets and for slowly developing responses. Ideally, thein vivo drug response-time course is made up of threefundamental components: 1) the time course of drugmolecules at the target (with plasma concentrationsoften used as a proxy and driver of the “drug mecha-nism” function), 2) the concentration-response relationship

that contains potency and efficacy of the drug, and 3) theturnover of the target (or drug-target complex) as such.Thus, the underlying assumption in DRT analysis is thatthe time course of a drug at its target can be represented bya biophase function—in the absence of actual biophaseconcentrationsof thedrug. Is it thenpossible todeconvolutethe biophase time course based on information about dose,rate, and route of administration coupled to the pharma-codynamic response-time courses? Figure 1 schematicallyshows the differences between the plasma concentration-time (exposure)–driven pharmacological exploration andan analysis using biophase-driven response-time courses.

The exact driver of a pharmacological response willalways be an approximation, independently of whether itis the concentration in plasma, in a tissue, or in a hypothet-ical biophase. However, it is important to remember thatupon repeated dosing, the concentrations in plasma, tissue,and biophase are at equilibrium. This is not to say that theconcentrations in the various matrices have to be equal, buttheir relative proportions are constant at equilibrium. Tofacilitate analysis of DRT data, pivotal parts of the pharma-cological time course are summarized and compared withcertain pharmacodynamic parameters in Table 1.

Fig. 1. Schematic illustration of exposure-driven response models (upper row) and biophase-driven response (bottom row). The plasma concentrationis typically the (proxy) driver of the pharmacological model in an exposure (C)–driven model, in contrast to the DRT model in which a biophasecompartment model serves as a surrogate for drug exposure. In DRT analyses, information about dose, route, and rate are pivotal for a stringentanalysis. The structure of the hypothetical biophase model is assumed to be extracted from the time course(s) of biologic response-time course as such.The underlying assumption about pharmacological response-time data is that some kind of kinetic information is embedded into them, which then maybe approximated by the biophase model. In turn, this represents the potential time course (shape) of the total amount of drug in the body that isexpected to “drive” the measured pharmacological response. The gray shaded areas are inferred from experimental concentration-time data (upperrow) or dosing information (bottom row) coupled to response-time courses. The response-time courses in the DRT analysis provide the extra “kinetic”information to the model that concentration-driven pharmacodynamic models do not need. PO, per oral; SC, subcutaneous.

Dose-Response-Time Data Analysis 91

This review aims to introduce the basic concepts ofDRT analysis, anchor the concepts onto real-life casestudies highlighting typical data patterns and designsof pharmacological studies in which DRT data analyseshave proven to be useful, and finally, compile relevantliterature about DRT data analysis.The basic concepts of DRT analysis are introduced

and illustrated by eight previously published casestudies, each of which exemplifies different diseaseareas. The raw data for case studies 1–7 are availablein a spreadsheet format (Gabrielsson and Weiner,2010). The eighth case study is aimed at demonstratinghow data from several studies in a drug discoveryproject can be merged and analyzed simultaneously(Andersson et al., 2017). This latter meta-analysisutilizes data from multiple dose provocations withnicotinic acid (NiAc; dose, route, rate, and mode), aswell as two intertwined biomarkers (plasma fatty acids

and insulin) in a large population of control and diseasemodel rats. Finally, we searched the literature forstudies in which DRT data analyses have been success-fully performed. The result of this exploration istabulated and collected in the list of references, whichmay then serve as a repository of relevant DRTliterature.

II. Methods and Models

A. What Can Be Learned from Response-TimeData Patterns?

What may be extracted from time-series assessmentsof pharmacological responses? Figure 2 shows sometypical features to look out for when analyzing DRT dataafter intravenous and subcutaneous dosing (Gabrielssonand Hjorth, 2016). These features include the follow-ing (to note, letters in parentheses refer to their

TABLE 1Overview of in vivo drug response phases, features of data, parameters, and case studies

Phase Features in Data Parameter Case Study

A: baseline Stable, variable, oscillating, or drifting(e.g., due to disease)

R0, kin, kout 1, 2–8

B: time delay Time delay between expected plasma peakconcentration and peak of response

kout, K, Ka 1–6

C: onset of action Concave or convex rise of response (onset of action);overshoot during onset

kout, K, Ka, number of transitcompartments, ktol

1–7

D: peak shift Peak shift or not; a shift suggests a nonlinearstimulation/inhibition of action

Smax/Imax, SD50/ID50, n 1–6, 8

E: response maximumor minimum

May indicate a nonlinear drug action or physiologiclimit; the same with an inhibitory drug action

Smax/Imax, SD50/ID50, n,physiologic limit

1–5, 7, 8

F: return to baseline Different dose routes may reveal absorptionrate–limited elimination of drug from biophase;rebound during washout

kout, K, Ka, ktol 1–3, 5–8

Fig. 2. Typical response-time courses obtained after different doses and routes of administration (3 and 10 mg i.v., and 10, 50, and 100 mg s.c.). Lettersdenote the following: A, baseline; B, time delay between expected peak concentration in plasma and observed peak response; C, concave onset of action;D, peak shifts in the response-time courses with increasing doses; E, saturation at top doses; and F, decline of response is slower (absorption rate–limited elimination from biophase) after subcutaneous dosingcompared with intravenously at equal doses. The black bar in the lower-left corner showsthe expected peak time of drug in plasma after an intravenous dose or oral solution. See also Table 1 for a suggestion of parameterizations. IV,intraveonous; SC, subcutaneous.


corresponding letters in Fig. 2): baseline level (A; madeup from the ratio of turnover rate to fractional turnoverrate of response; i.e., decay rate of response), time delaybetween expected plasma concentration peak and peakof response (B; may be due to indirect pharmacologicalaction), onset of action (C; delayed onset may indicate acascade of events upstream prior to biomarker compart-ment), peak shifts with increasing doses (D; indication ofsaturable stimulation), saturation of response (E; due tononlinear drug mechanism or physiologic limit), andpostpeak decline of response (F; the 10 mg intravenousand subcutaneous time courses in Fig. 2 differ due toabsorption rate–limited elimination after subcutaneousadministration). Each portion and phase of the curveshas its own story to tell with respect to drug properties,system properties, and biophase kinetics (Table 1).

B. How Is the Biophase Typically Defined?

A requirement for a robust description of the biophasemodel is that dose, rate, route, and mode of adminis-tration are known. The dose information combined withresponse-time courses bracket the biophase structure(Fig. 1, bottom row). If only response-time data areknown, the individual contribution of input from drugadministration and of biophase function cannot beseparated. The biophase is a replacement of anypossible time course of drug that “drives” the pharma-cological effect and may be viewed as a first-generationeffect-compartment approach. The biophase functionrepresents the time course of drug in the body deconvo-luted from response-time data necessary to drive theonset, intensity, and duration of the pharmacologicalresponse. The more remote and inaccessible the target(e.g., nuclear receptors) or the slower the turnover ofresponse, often the slower the rate of its presentationand loss and the longer the action of drug at the targetsite. The transfer of the drug molecule to and from thetarget as well as the affinity for the target are elements(drug properties) separated from the turnover proper-ties of the target as such and accompanying postrecep-tor events (biologic properties). It should be noted thatdrug and biologic properties jointly impact andmake upthe onset, intensity, and duration of a pharmacologicalresponse. Since the pharmacological response is aconsequence of the two, it is reasonable to assume thatthe time course of response also contains some kind ofkinetic information, which is governed by the biophasecompartment. The biophase compartments in Fig. 3 areassumed to mimic that part of the pharmacological timecourse. The biophase kinetics will, to a varying extent,depend on the dose, the input rate (or rate of absorptionof drug), and the route of administration, let alone thequality of response data.The input/output of the biophase is assumed to make

up all absorption and disposition processes that a drugmay undergo until it hits the pharmacological target.Even though a drug was given as a bolus dose into

plasma, the rise and fall of the pharmacological re-sponse may reveal a deviating pattern if the target iswell separated from the plasma compartment and or thebiophase exposure or response develops slowly. Thebiophase is meant to capture the rate-limiting step inthe absorption and disposition of drug causing thenecessary shape of exposure time at target.

The mathematical functions of the biophase amount(Ab) can be expressed as integrated solutions (eq. 1; top,bolus input; middle, first-order input/output; bottom,zero-order rate input):8>>>>>><>>>>>>:

Ab ¼ Div ×�e2K × t�

Ab ¼ Ka × F × Dev

ðKa 2KÞhe2K × ðt2 tlagÞ2 e2Ka × ðt2 tlagÞi

Ab ¼ Rin

K

�12 e2K × Tinf

�× e2K × t9

ð1Þ

Div is the intravenous bolus dose,K is the elimination rateconstant, Ka is the absorption rate constant, F* is thebiophase availability, Dev is the extravascular dose, tlag isthe lag time, and Rin is the rate of infusion. The biophaseavailability is the fraction of the dose that reaches thebiophase relative to an intravenous dose, as approximatedfrom pharmacological response data. This new parameteris different from the bioavailability,which is the fraction ofthe dose reaching the systemic circulation intact uponextravascular dosing. The biophase availability may begreater than unity (1) for a compound with pharmacolog-ically active metabolites or saturable action.

The bolus, first-order input, and zero-order input canalso be given as solutions of differential equations of alinear first-order system (eq. 2; second line from top,bolus input; second line from bottom, first-order inpu-t/output; and bottom, zero-order rate input):8>>>>>>>>>>>><

>>>>>>>>>>>>:

dAb

dt¼ Input rate2 output rate

dAb

dt¼ 2K × Ab

dAb

dt¼ ka × F × Dev × e2Ka × t 2K × Ab

dAb

dt¼ Input rate2K × Ab

ð2Þ

For nonlinear elimination, the three input modes(bolus, first-order input, and zero-order input) becomethe following (eq. 3):8>>>>>>>><

>>>>>>>>:

dAb

dt¼ 2

Vmax × Ab

Km þ Ab

dAb

dt¼ Ka × F × Dev × e2Ka × t 2

Vmax × Ab

Km þ Ab

dAb

dt¼ Input2

Vmax × Ab

Km þ Ab

ð3Þ


For saturable input and output, the differential equa-tions become the following (eq. 4):8>>><

>>>:

dDev

dt¼ 2

Ka;max × Dev

Ka;m þDev

dAb

dt¼ F ×

Ka;max × Dev

Ka;m þDev2

Vmax × Ab

Km þ Ab

ð4Þ

where Dev is the extravascular dose variable at theapplication site, Ka,max is the maximum absorption rate,Ka,m is theMichaelis–Menten constant (amount) at thehalf-maximal absorption rate, and F* is the biophase availabil-ity. Again, the biophase availability F* is the fraction of thedose that reaches the biophase relative to intravenousdosing and is derived from pharmacological response-timedata. Note that the biophase availability is not synonymouswith bioavailability obtained from intravenous and oraldosingdata.Thebiophaseavailabilitymaybecontaminatedby active or interactive metabolites, antagonistic or syner-gistic action, feedback, and so forth and is a parameterderived from response-time data as such only.Even though all of a bolus dose or a fraction (biophase

availability) of an extravascular dose reaches the

hypothetical biophase, only a small amount of drug istypically likely to reach its biologic target. The re-mainder will distribute nonspecifically to other bodyregions. The shape of the hypothetical biophase func-tion is deconvoluted from the response-time data. As aresult, it is not rational to assume that a hypotheticalplasma time course will result from dividing the timecourse of the biophase by the volume of distribution. Wetherefore discourage any transformation ofAb to plasmaconcentration-time courses by means of the volume ofdistribution (Vd). However, converting the biophaseamount at 50% of maximum drug induced effect(SD50) or 50% of the maximum drug-induced inhibitoryeffect of the inhibitory drug mechanism function (ID50)to EC50 or IC50 with volume is still feasible whenoperating under equilibrium conditions. The biophasestill represents the body, but the shape of the functionmimics the target site.

C. The Drug “Mechanism” Function

The interface between the biophase kinetics and thepharmacological model is the drugmechanism function.The biophase kinetics drives a linear (eq. 5) or nonlinear

Fig. 3. Schematic illustration of different biophase models used for DRT analysis. Numbers indicate the following: (1) Biophase approximated by bolusadministration and first-order elimination. (2) First-order input/output of the biophase. (3) Constant (zero-order) rate input followed by washout. (4)Bolus plus Michaelis–Menten (saturable) elimination. (5) Saturable absorption coupled to first-order elimination (note the peak shift). (6) Saturableabsorption-elimination from biophase. (7) Bolus on top of endogenous ligand production. Baseline shows endogenous ligand concentration. (8) Step-down input to biophase simulating longer half-lives for rapidly eliminated compounds (e.g., NiAc). (9) Biphasic decline in the biophase after a bolusdose. TRN, turnover.


(eq. 6) drug mechanism function either as a stimulatoryprocess [parameterized with the proportionality con-stant in the linear drug mechanism function (a) or theefficacy parameter of the drug mechanism function(Smax), the SD50 value, and the Hill exponent (nH)] oras an inhibitory process [parameterized with the pro-portionality constant in the linear (inhibitory) drugmechanism function (b) or the Imax, ID50, and nH].�

SðAbÞ ¼ 1þ a × AbIðAbÞ ¼ 12 b × Ab

ð5Þ

The nonlinear stimulatory and inhibitory drug mecha-nism functions are as follows:8>>><

>>>:SðAbÞ ¼ 1þ Smax × A

nHb

SDnH50 þ AnH

b

IðAbÞ ¼ 12Imax × A

nHb

IDnH50 þ AnH

b

ð6Þ

Logarithmic and exponential drug mechanism func-tions may also be applied, provided that data supportsuch an approach. The pharmacological response isdriven by the drug mechanism function, which convertsdrug input (typically concentration or amount) to apharmacological action.

D. Structure of the Pharmacodynamic Model

When the equilibrium between the biophase and theobserved pharmacological response is rapid, the bio-phase function can be used to directly drive an analyt-ical function of the response E. Equation 7 shows astimulatory (upper row) and an inhibitory (bottom row)efficacy parameter of the Hill equation (Emax) modeldriven by the biophase amount Ab.8>>><

>>>:E ¼ E0 þ

Emax × AnHb

EDnH50 þ AnH

b

E ¼ E0 2Imax × A

nHb

IDnH50 þ AnH

b

ð7Þ

The model structure includes a baseline value E0, themaximum effect Emax or Imax, the dose at half-maximaleffect ED50 or ID50, and the sigmoidicity factor nH. Inthis case, the pharmacological response model is some-what similar to the previously presented drug mecha-nism function mentioned above.When a drug acts on factors responsible for the

buildup or loss of a pharmacological response, thekind of turnover model shown in Fig. 4 is typicallyapplied. The drug either stimulates or inhibits theproduction of response [zero-order turnover rate (kin)]or the loss term of response [first-order fractionalturnover rate (kout)]. The drug directly impacts factorscontrolling the responseR such as kin and kout. The drug

indirectly, via kin and kout, affects the pharmacologicalresponse R as such.

The drug mechanism function [S(Ab) or I(Ab)] is thenincorporated into one of the response production or losssystems (see, Fig. 4) as follows:8>>>>>>>>>>>><

>>>>>>>>>>>>:

dRdt

¼ kin × SðAbÞ2 kout × R

dRdt

¼ kin 2 kout × SðAbÞ × RdRdt

¼ kin × IðAbÞ2 kout × R

dRdt

¼ kin 2 kout × IðAbÞ × R

ð8Þ

Equation 8 shows the four basic turnover systems withstimulation or inhibition of either turnover rate kin, oron the fractional turnover rate kout. These models maybe adjusted depending on the particular pharmacolog-ical target and system studied. The onset of responsemay reflect a series of transduction compartments tocapture a concave buildup of response. Other systemsdisplay oscillating time courses and may also requirediurnal variations in turnover rate. A third may involveone or more amplification steps (Gabrielsson et al.,2000; Gabrielsson and Weiner, 2010). A more detailedaccount of biophase structure, drug mechanism, andturnover systems will be given for each case study.

E. Response-Time Patterns Impact Model Structure

We previously described “pattern recognition” as agraphical tool for delineating the structure of a phar-macological model (Gabrielsson and Hjorth, 2016).That article also discussed practical experimental de-sign to highlight some pivotal guidelines. In brief,information on the baseline is central to model build-ing. Significant information is also contained in theoverall shape of the curve: potential time delays be-tween plasma exposure and a pharmacological re-sponse, peak shifts in the response-time course withincreasing doses (indicative of whether there is a linearor nonlinear drug mechanism function at play), andsaturation at the maximum or minimum of response,which in combination with peak shifts is a strongindicator of the nonlinear nature of the drugmechanismfunction. This generic approach is likewise applicable

Fig. 4. Conceptual diagram of the turnover model (indirect responsemodel) where drug either impacts the production of response kin or theloss of response kout. The drug acts on factors controlling the level ofresponse R. The red arrows show the rise or decrease in response as aconsequence of drug action.


to the DRT analysis we describe here. Figure 5 showsfour common patterns in response-time curves and theirtentative models (and case studies).

III. Case Studies

Eight case studies of previously published data arecompiled and presented here to demonstrate the prosand cons of DRT analysis. This analysis contains eightcase study datasets that were in part originally pub-lished in Pharmacokinetic and PharmacodynamicData Analysis: Concepts and Applications (first tofifth editions) by Gabrielsson and Weiner (2010, 2016)in 1994–2016 or in Gabrielsson et al. (2000) but havenot been easily accessible to a larger audience; theyare therefore included due to their availability andfunctionality. The experimental data of case studies 1–7are available electronically as models, command files,and project files via https://dipublish.com/pkpd/. Dataon cortisol–adrenocorticotropic hormone (ACTH) dy-namics (Urquhart and Li, 1968), locomotor activity(van Rossum and van Koppen, 1968), and Brief Psychi-atric Rating Scale (BPRS) scores (Lewander et al., 1990)were scanned from the literature. Case study 8 is asummary of Kroon et al. (2017). Some of the figures anddatasets have been used for teaching and at variousconferences. The datasets were selected to illustrateresponse-time pattern and design feature points ofview, rather than to compare models and drugs within

specific therapeutic classes. Table 2 gives an overview ofthe background on each case study.

A. Case Study 1: Dose-Response-Time Analysis ofNociceptive Response

1. Background. This case study comprises phar-macological data obtained using two different routesof administration (intravenous and subcutaneous)to model and estimate pertinent screening parame-ters such as SD50, biophase availability, and turn-over characteristics of the (analgetic) response, suchas half-life. The underlying assumption is that thepharmacodynamic data convey information about thekinetics of the drug in the biophase (rate constants,biophase availability), which can be obtained simul-taneously from DRT data. Since both intravenousand subcutaneous administration modes were ap-plied, a new parameter (biophase availability) couldbe estimated.

A potential analgesic test compound was given at twodose levels (3 and 10 mg) via the intravenous route andat three dose levels (10, 50, and 100 mg) subcutaneously.The time (in seconds) to respond to a noxious stimulus(laser beam challenge in the tail-flick test) was de-termined at different times after dosing. Response-timedata were obtained from each dose level after an acutedose (Fig. 6).

2. Models, Equations, and Exploratory Analysis.The underlying biophase-coupled turnover model isshown in Fig. 6 (right). The biophase behaves as a

Fig. 5. Commonly encountered response-time courses and their corresponding potential turnover models. (A) Standard convex onset of action afterstimulation of turnover rate kin after two doses (see also case study 1). (B) Concave onset of effect as a consequence of an early upstream stimulation ofturnover rate at two doses followed by one or more transduction processes prior to the final response-generating compartment (see case study 2 onlocomotor activity). (C) Diurnal variation in the turnover rate, resulting in an oscillating (baseline and) response-time course. The dashed oscillatorycurve shows response after inhibitory drug intervention. (D) Feedback-regulated response. A period of drug-induced stimulation of response is followedby rebound (response-time course below the predose baseline level). The red horizontal bar in plot (D) shows the drug exposure period. The small redbars in the upper row plots show the expected peak time of drug in plasma (see also case studies 1–3 and 6).


https://dipublish.com/pkpd/

TABLE

2Ove

rview

ofinpu

t,da

tafeatur

es,mod

el,pa

rameters,

observations

,an

dreferenc

esforthecase

stud

ies

No.

Cas

eStudy

Inpu

tFea

turesof

Data

Mod

elParam

eter

Referen

ce

1Tailflick

i.v.(3

and10

mg),s.c.

(10,

50,a

nd10

0mg)

Bas

eline,

peak

shift,sa

turation

,ab

sorp

tion

rate–limited

duration

ofresp

onse

Bolus

andfirst-orde

rinpu

t/ou

tput

biop

hase

coup

ledto

aturn

over

resp

onse

mod

el

F,Ka,K,k i

n,k o

ut,

SD

50,S

max,

nH

Gab

rielsson

etal.(200

0)

2Locom

otor

activity

stim

ulation

s.c.

(3.12an

d5.82

mg/kg

)Noba

seline

,pe

aksh

ift,zero-ord

errise

andde

clineof

resp

onse,

saturation

ofbu

ild-up

andloss

First-ord

erinpu

t/ou

tput

biop

hase

coup

ledto

aturn

over

resp

onse

mod

el

K’,k o

ut,k M

,SD

50,S

max,

nH

vanRossu

man

dva

nKop

pen(196

8)

3Fatty

acid

resp

onse

Multiplei.v

.infusion

spluswas

hou

tBas

eline,

saturation

,rebo

und,

toleranc

eZe

ro-ord

erinpu

tan

dfirst-orde

rloss

biop

hase

mod

elcoup

led

toaturn

over

feed

back

mod

el

K,R0,k

out,k t

ol,

ID50,nH

Gab

rielsson

and

Peletier(200

8),

Isak

sson

etal.

(200

9)4

Psych

osis

score

(BPRS)

120–

600mgda

ily

p.o.

dosing

Bas

eline,

timeto

PD

stea

dystate,

subcateg

orizingresp

onde

rsCon

stan

tbiop

hase

expo

sure

coup

ledto

aturn

over

mod

elk i

n,k o

ut,I m

ax

Lew

ande

ret

al.

(199

0),Gab

rielsson

andWeine

r,(201

6)5

Bacterial

grow

than

dkill

Multiple

i.v.do

ses

Bas

eline,

grow

thlimit,sa

turation

First-ord

ergrow

than

dsecond

-ord

erkillrate

k g,k k

Gab

rielsson

and

Weiner

(201

0)6

Cortisolreleas

edu

ring

andafterex

posu

reto

ACTH

Exp

erim

entalmod

elof

ACTH

expo

sure

driving

thereleas

erate

ofcortisol

Bas

eline,

rebo

und,

toleranc

eturn

over,en

doge

nous

agon

ist,

squa

rewav

eof

ACTH

Squ

are-wav

eex

posu

reto

the

endo

genou

sag

onistACTH

coup

ledto

aturn

over

feed

back

mod

elof

cortisol

releas

erate

k in,k o

ut,SD

50,

Smax,

nH

Urquhartan

dLi(196

8),

Gab

rielsson

and

Weiner

(201

0)

7Mioticresp

onse

inthecatey

eThr

eedo

ses(0.1,1.0,

and10

mglatanop

rost)

give

ntopically

Bas

eline,

partialinhibition

,sa

turation

,rapideq

uilibrium

mod

elIn

hibitory

Emaxmod

elKa,K,t

lag,

ID50,I m

ax,

nH

Smolen

(197

1a),

Gab

rielsson

etal.(200

0)8

Meta-an

alysis

offattyacid

andinsu

lin

resp

onse

tomultiple

NiA

cpr

ovocations

(NLME

analysis)

Multiple

i.v.an

ds.c.

doses,

rates,

andmod

esof

NiA

cpluswas

hou

t

Bas

eline,

saturation

,rebo

und,

toleranc

eturn

over,multiple

biom

arke

rs,multipledr

ugpr

ovocations

,en

doge

nous

agon

ist,inpu

t-stim

ulated

was

hout

profile

Satur

able

inpu

t-ou

tput

biop

hase

mod

elof

endo

geno

usag

onists

coup

ledto

aturn

over

feed

back

mod

elof

FFA

andinsu

linin

normal

and

diseas

edan

imals

KaM,V

max,

K,k

in,k o

ut,

SD

50,S

max,

nH

Isak

sson

etal.

(200

9),And

ersson

etal.(201

6,20

17)

NLME,n

onlinea

rmixed

effects;

PD,p

harmacod

ynam

ic.


one-compartment bolus model for intravenous data(eq. 9, top row) and as a first-order input-output modelfor subcutaneous dosing (eq. 9, bottom row). Thebiophase model then drives the stimulatory drugmechanism function (eq. 10), acting on a series oftransit compartments that capture the slight concaveonset of response in both intravenous and subcutea-nous data (eq. 11). Drug action occurs via stimulationof the production of response (i.e., analgesia, resultingin a delayed reaction time to noxious stimuli). A zero-order input and first-order input/output model governsthe turnover of the response.The amount of drug in the biophase compartment

after intravenous and subcutaneous dosing is modeledwith first-order kinetics assuming the volume is unity(eq. 9):8><

>:Ab ¼ Div ×

�e2K × t�

Ab ¼ Ka × F × Dsc

ðKa 2KÞhe2K × ðt2 tlagÞ2 e2Ka × ðt2 tlagÞi ð9Þ

Div and Dsc denote the intravenous and subcutaneousdoses, K is the elimination rate constant, Ka is theabsorption rate constant, and F* and tlag are thebiophase availability and lag time, respectively. Notethat there are no volume terms in eq. 10 for the biophaseamount expressions. The stimulatory drug mechanismfunction driver is as follows:

SðAbÞ ¼ 1þ Smax × AnHb

SDnH50 þ AnH

bð10Þ

Smax, SD50, and nH denote the maximum drug-inducedeffect of the drug mechanism function, the dose gener-ating 50% of maximum drug-induced response, and thesigmoidicity parameter, respectively. The turnoverfunction of the pharmacological response with stimula-tion of turnover rate kin of R1 is written as a series oftransit compartments, which capture the concave onsetof response.8>>>>>>><

>>>>>>>:

dR1

dt¼ kin × SðAbÞ2Kout × R1

dR2

dt¼ kout × R1 2Kout × R2

dRobs

dt¼ kout × R2 2Kout × Robs

ð11Þ

kin and kout are the turnover rate and the fractionalturnover rate constants, respectively. The maximumdrug-induced change D is obtained from eq. 12:�

D ¼ Rss 2R0 ¼ R0 × ð1þ EmaxÞ2R0 ¼ R0 × Emax10 ¼ R0 × Emax ¼ 5 × 2

ð12Þ

where D is derived from the absolute difference be-tween baseline R0 and Rmax. An approximate estimateof biophase availability F* can be obtained by areaunder the curve AUCEsc/AUCEiv for the 10 mg intrave-nous and subcutaneous doses, which gives approxi-mately 100%. To note, when noncompartmentalassessment of the biophase availability is used, theareas under the baseline have to be subtracted fromeach total AUCE.

Fig. 6. (Left) Observed (symbols) and model-predicted (lines) response (tail flick) vs. time of drug X after different doses given by the intravenous (red)and subcutaneous (blue) routes. The subcutaneous doses were 10, 50, and 100 mg, and the intravenous doses were 3 and 10 mg. Note the slow terminaldecline of the 10 mg subcutaneous dose compared with the 10 mg intravenous dose, suggesting absorption rate–limited elimination. (Right) Schematicillustration of the biophase (top) and the turnover response (bottom) models. The drug in the biophase stimulates the production of the antinociceptiveresponse. Also note that we have added a transit compartment to mimic the concave buildup of response. Ka denotes the first-order input rate constant,Ab is the biophase amount, k is the first-order elimination rate constant, S(Ab) is the stimulatory “drug mechanism” function, kin is the turnover rate,R is the measured (biomarker) response, and kout is the fractional turnover rate. IV, intraveonous; SC, subcutaneous.


Initial estimates of the kout parameter (0.3 h21) werederived from the initial slope of the upswing of theresponse-time curve after the largest dose. The initialestimate of the kin parameter was derived from kout andthe corresponding ratio of kin/kout. ID50 was obtained bysimulating the model with a number of realistic valuesof SD50 within the 0–10 mg range.3. Results and Conclusions with Respect to Dose-

Response-Time Analysis. A high correlation betweenobserved and estimated response-time data (Fig. 6), aswell as well defined parameters with good precision(Table 3), is obtained by simultaneously fitting eqs. 9–11 to all five response-time courses.Comparison of the intravenous and subcutaneous

10 mg DRT data suggested absorption rate–limitedelimination via the subcutaneous route. Thus, in spiteof a higher initial response of the 10 mg intravenousdose, the time course declined faster than the corre-sponding response-time course of a 10 mg subcutaneousdose. The biophase availability was 86%. The simulatedbiophase (dose) response relationship is shown in Fig. 7.4. Pharmacodynamic Interpretation and Comments.

Depending on the drug target involved, analgetic drugactions may involve central (spinal, supraspinal) aswell as peripheral components (e.g., Sawynok andLiu, 2014). For the particular case described above, itis assumed that one of these compartments is thedominant antinociceptive biophase for compound X.The tail-flick response represents a classic model ofacute thermal pain sensitivity considered to involvecentral (spinal/supraspinal) pathways. It might besuggested that in this case (regardless of whether thebiophase target sites are fully occupied, and in theabsence of further information on test compound prop-erties), maximal analgesia via this particular target bycompound X is attained. Moreover, the observed timecourse may represent a buildup of biophase exposure toreach this level and/or contributions from the actualbiologic mechanism underlying the analgetic responsemonitored. In this regard, the protracted response curveseen with subcutaneous versus intravenous adminis-tration may suggest that compound X has propertiesthat may limit absorption and potentially also penetra-tion across blood-brain/spinal fluid barriers, thus influ-encing the biophase levels. This case thus illustrates thevalue of the integrated use of dose, response, and time

data to arrive at biophase estimates with plausibleprecision.

B. Case Study 2: Dose-Response-Time Analysis ofLocomotor Stimulation

1. Background. Response-time data on rat loco-motor activity after intraperitoneal administration ofdexamphetamine at two dose levels were obtainedand digitized from van Rossum and van Koppen(1968). These data suit the purpose of DRT modeling,since the resolution is high, with adequate granular-ity in both the rise and decline in response, a clearpeak shift with dose, and more than one dose level(assuming first-order input) used. A slight concavitywas observed in the onset of response, followed by anapparently linear rise during the first 30 minutesafter dosing. The initial rise in locomotor responsesuperimposed for both doses and was followed by anabrupt change from rise to decline. The response tothe higher dose peaked later than the response to thelower dose. The slope of the postpeak linear decline ofresponse was independent of dose (3.12 and 5.62mg/kg) or route of administration (intraperitoneal orintramuscular application; see van Rossum and vanKoppen, 1968) (Fig. 8).

2. Models, Equations, and Exploratory Analysis.Figure 8 (right) shows the suggested biophase compart-mentmodels of this analysis. They also demonstrate the“duality of models” when two competing models areequally good representatives of the pharmacology. Notethat the drug stimulatory action is assumed to act onthe production of response.

Two biophase models were fitted: one approximatingthe input to the biophase as a bolus, and the otherapplying a first-order input (eq. 13), resulting in thefunctions where Ab denotes the biophase amount aftersubcutaneous dosing. The biophase availability F* wasset equal to unity since no comparative data fromintravenous dosing were available.

Fig. 7. Simulated response (tail flick) vs. biophase amount (dose) ofcompound X. The baseline (vehicle treatment) response is about 5 secondsand the maximum drug-induced response is 10 seconds, resulting in a15-second response maximum in these settings.

TABLE 3Initial and final parameter estimates and their corresponding

precision (CV%)

Parameter Initial Estimate Final Estimate CV%

Ka (min21) 0.01 0.018 8K (min21) 0.05 0.063 19F* 1 0.86 11kout (min21) 0.3 0.13 8nH 2 2.0 14SD50 (mg) 1–10 1.8 22Smax 2 2.0 5


(AbðtÞ ¼ D × e2K 9 × t

AbðtÞ ¼ Fp × D × K 9 × t × e2K :9t ð13Þ

The stimulatory drug mechanism function is shown ineq. 14:

8>>>>>>>>>><>>>>>>>>>>:

SðAbÞ ¼Smax × A

nHb

SDnH50 þ AnH

b

SðAbð0ÞÞ ¼ 0

SðAbÞ ¼ 1þ Smax × AnHb

SDnH50 þ AnH

b

SðAbð0ÞÞ ¼ 1

ð14Þ

where Smax, SD50, and nH correspond to the maximumdrug-induced efficacy, potency, and Hill exponent, re-spectively. Note that the drug mechanism function (eq.14, upper two rows) lacks the constant term (1) typicallyfound in these functions (eq. 14, bottom two rows) whenno baseline information is available. This is because thelocomotor activity is zero or negligible in the absence ofdrug in this experiment.The drug mechanism function is then incorporated

into the systems equation describing the pharmacolog-ical response (eq. 15):

dRdt

¼ SðAbÞ2 koutðRÞ × R ð15Þ

where kin has been removed since the baseline value oflocomotor activity is zero. In light of the linear decay ofresponse (Fig. 8), the loss of response will be modeled as

a saturable term kout(R) with a fractional turnover ratekout as a function of R (eq. 16):

koutðRÞ ¼ kout;max ×1

kM þ Rð16Þ

The terminal slope of the response-time course whenR . . kM is then written as shown in eq. 17:

dRdt

¼ 2 kout;max � 30U=min ð17Þ

where kout,max is approximately 30 response units perminute. Similarly, the slope of the rise in responseequals the following (eq. 18):

dRdt

¼ Smax 2 kout;max � 90U=min ð18Þ

where Smax can be approximated as 90 + 30 =120 U/min, provided both processes are saturated. Thepharmacological response will reach an equilibriumstate if the rate of production is less than the maximalrate of loss; that is, when the first term in eq. 15 issmaller than the maximal upper bound of the secondterm [i.e., if S(Ab), kout,max]. The equilibrium biophaseamount versus response relationship becomes the fol-lowing (eq. 19):

RSSðAbÞ ¼ kM ×SðAbÞ

kout;max 2SðAbÞ ð19Þ

The data are rich since they contain high-resolutionresponse-time courses at two dose levels. It appears thatthe dexamphetamine-locomotor activity scores are a

Fig. 8. (Left) Rat locomotor activity score (counts per minute) time data after dexamphetamine is given intravenously at 3.12 (blue symbols) or 5.62mg/kg doses (red symbols) (van Rossum and van Koppen, 1968). Note the apparently linear and parallel decline in response over time independently ofdose and the peak shift to the right with increasing dose. A slight concave onset of action may also be discernible, particularly with the lower of the twodoses. No saturation manifested as a plateau (flat horizontal) response-time region can be seen with the highest dose. Dashed lines represent the bolusinput biophase model, and the solid lines represent the first-order input/output model. (Right) Conceptual schematic model of the DRT data. K’ denotesthe first-order input/output biophase rate constant and Smax, SD50, and n are the pharmacodynamic parameters of maximum efficacy, biophase amountat 50% of maximum efficacy, and the Hill-parameter, respectively. The constant term (1) in the definition of S(Ab) in eq. 14 could be eliminated becauseno baseline activity was observed in the data. Note the saturable loss of response given by the fractional turnover rate, kout(R) = kout,max/(kM + R). Graysolid lines represent simulated plasma time courses of dexamphetamine at the two doses given.


linear relation of the time postpeak independent of dose.Equation 16 is therefore suggested as a reasonableapproximation of the zero-order decline of response-time data. The model is simultaneously fit to bothresponse-time courses only changing the dose betweenthe two datasets. Dose-normalized areas increased withdose, which indicates some kind of saturation in the lossof response and/or saturable stimulation function suchas eq. 16. This is also supported by the peak shift in theresponse-time courses.3. Results and Conclusions with Respect to Dose-

Response-Time Analysis. The regression lines of thetwo biophase models (eq. 13) coupled to the pharmaco-logical model (eqs. 14 and 15) of the data are shown inFig. 8. The final parameter estimates are listed inTable 4, together with their precision [coefficient ofvariation (CV%)].4. Pharmacodynamic Interpretation and Comments.

Normally, locomotor activity in rodents is dependent onseveral factors, including novelty of the environmentversus habituation and phase of the day-light cycle(rodents being nocturnal). In the locomotor stimulationcase described above, the rats are likely to have beenpreadapted to the mobility meter boxes in the lightphase of the 24-hour cycle to minimize exploratoryactivity. From the modeling perspective, this simplifiesanalysis of the data because baseline may be set to zero,both prior to and after the period of drug challenge. Theturnover rate kin was therefore eliminated from eq. 15,and the drugmechanism function could be condensed. Aclear peak shift (time shift in tmax values) was observedin response-time data with increasing doses of dexam-phetamine. The linear decay of the locomotor responsetoward baselinemay be contingent on the elimination ofdexamphetamine from the biophase, but it may addi-tionally involve desensitization of central dopamineneuronal and/or receptor mechanisms. It is clear fromthe simulated predicted plasma concentration-timecourses (superimposed solid gray lines on experimentaland model-predicted response-time courses in Fig. 8)that a biophase compartment is needed to capture time-course differences between plasma exposure and loco-motor activity after intraperitoneal dosing. Evidently,

plasma concentrations do not directly drive the phar-macological response, and access to brain target levelsaligned to the plasma time course of dexamphetaminewould have been a useful addition to the locomotoractivity data to verify the biophase structure. Note thatthe model-predicted kM value is very low (0.001 re-sponse units), which therefore makes the decline oflocomotor activity a zero-order process at the presentdose levels, whereas the plasma concentrations follow afirst-order disappearance with a drug half-life of about90 minutes. Information about the shape of the drugtime course in the biophase in relation to plasma andlocomotor activity levels could therefore potentially giveclues to underlying contributory mechanisms andevents. Although it is beyond the scope of this review,it would also be interesting to see what further insightson pharmacodynamic similarities and differences ofdrugs able to stimulate motor activity (e.g., amphet-amine, cocaine, ethanol, caffeine, etc.) might be gainedfrom a crosscomparison DRT analysis based onmultipledose levels, high-resolution behavioral responses, andcorresponding plasma (or, ideally, biophase) data ofsuch agents. This said, it should be recalled that severalof the aforementioned agents possess dose- (concentra-tion-) and time-dependent psychostimulant as well asdepressant properties and therefore may be challeng-ing to compare or rank by means of a DRT approach(see Fig. 9). Drugs with well known biphasic exposure-response relationships include ethanol, caffeine, andmorphine (see Dafters and Taggart, 1992; Calabrese,2008), where the extent of excitatory and depres-sant actions depend on the drug in question, dose, andtime after administration. DRT analysis may also beof limited value for the assessment of safety margins,where establishing systemic exposure to unbounddrug concentrations should instead be the method ofchoice (Gabrielsson andHjorth, 2012, 2018; Gabrielssonet al., 2018).

Fig. 9. (Left) Schematic illustration of the principles of a biphasic dose-response relationship. (Right) Example showing the complexity ofcomparing two compounds with a slight shift between their dose-responserelationships. Even though both have similar shapes, one compound maydemonstrate an excitatory (stimulant) response at the same dose as theother compound demonstrates an inhibitory (depressant) response. Inthis particular situation, DRT analysis may fail to characterize theirpharmacodynamic properties. Exposure-response analysis would there-fore be feasible option and is a better alternative.

TABLE 4Final parameter estimates of first-order and bolus models, their relative

S.D. (CV%), and their objective function value (WSSR)

Parameter

First-OrderBiophase Model

Bolus BiophaseModel

Estimate CV% Estimate CV%

Smax 249 1 190 7K’ (first-order) (h21) K (bolus) (h21) 5.96 4 2.54 6SD50 (mg) 1.02 4 1.43 8nH 1.63 5 2.05 18kout, max (h21) 30.1 4 32.9 4kM (h21) 0.001 9 0.001 1WSSR 768 640

WSSR, weighted sum of squared residuals.


C. Case Study 3: Dose-Response-Time Analysis ofFunctional Adaptation

1. Background. This case study is aimed atdemonstrating DRT analysis containing functionaladaptation (Isaksson et al., 2009) of a metabolicresponse to drug (NiAc) treatment. The biophasekinetics, ke, will be estimated simultaneously withsystem parameters (kout and ktol) and drug param-eters (ID50 and n). The mechanism of action isvia inhibition of the turnover rate [I(Ab) · kin]. Fullinhibition of response is assumed with a sufficientlyhigh dose. Data are obtained after six consecutiveintravenous infusions of NiAc (plasma half-lifeof 1 to 2 minutes) followed by washout (Fig. 10).The plasma concentration-time course of NiAc willtherefore have the shape of a square wave due to theshort half-life.The proposed model is shown in Fig. 10 (right). The

model contains two parallel turnover models con-nected via ktol · R and kin · R0/M serving as input tothe moderator compartment (M) and R, respectively.The ratio of ktol to kout, also called k, gives an ideaabout the speed of tolerance development and thepossibility of observing rebound. This system isshown to be applicable to many diverse situationswith drug tolerance and rebound (Gabrielsson andPeletier, 2008).2. Models, Equations, and Exploratory Analysis.

The multiple zero-order input steps and first-order lossof the biophase compartment are expressed as adifferential equation (eq. 20):

dAb

dt¼ Input rate2K × Ab ð20Þ

Input rate, K, and Ab are the escalating zero-orderinfusion rates of NiAc, the elimination rate constant,and the biophase amount, respectively. The biophasefunction is then driving the drug mechanism functionvia an inhibitory process (Imax, ID50, and nH), where theImax parameter is set to a constant value of 0.95 (eq. 21).

IðAbÞ ¼ 12Imax × A

nHb

IDnH50 þ AnH

bð21Þ

The biophase and drug mechanism functions are thenincorporated into one of the turnover systems of aresponse model slightly different from the one pre-sented in Isaksson et al. (2009), as shown in eq. 22:

dRdt

¼ kin × IðAbÞ × R0

M2 kout × R ð22Þ

where kin is the turnover rate, M is the moderatorcompartment, kout is the fractional turnover rate, andI(A) is the drug inhibitory function. The action ofM onRwas assumed to occur via inhibition of production ofR, rather than via stimulation of loss of R, as theformer is more attractive from both amechanistic andan energy-conserving point of view. The R0/M ratiodenotes the baseline-normalized feedback of themoderator M. The turnover of the moderator M isshown in eq. 23:

dMdt

¼ ktol × R2 ktol × M ð23Þ

Imax is assumed to be 1, which means 100% inhibitionfor a very high dose. Input is then governed by themultiple consecutive intravenous infusion regimens.

Fig. 10. (Left) Observed (filled symbols) and model-predicted (solid line) response vs. time data after a multiple intravenous infusion regimen of NiAc,followed by washout (Isaksson et al., 2009). The infusion pump stopped for a brief period at about 90 minutes until the next syringe had been loaded.During this event, there is a rapid return of response toward and past the baseline. The six yellow bars represent six different infusion rates (0.008,0.016, 0.032, 0.064, 0.128, and 0.256 U/min). (Right) Schematic illustration of the proposed feedback model. Solid lines denote flows and dashed linesdenote control steps. Ab and ke denote the biophase amount and elimination rate constant out of the biophase, respectively. kin denotes the turnoverrate, I(Ab) is the drug mechanism function, R0 is the baseline response, M is the moderator compartment, and kout and ktol are the fractional turnoverrate and fractional turnover rate of the moderator compartment, respectively. FFA is the free fatty acid response biomarker R. The moderatorcompartment M represents all lumped feedback mechanisms in fatty acid regulation (e.g., insulin).


The model parameters of eqs. 20–23 are the baselinevalue R0, the fractional turnover rate of response kout,the rate constant of tolerance development ktol, and thebiophase kinetic rate constant ke. ID50 is the amount inthe system resulting in 50% of Rmax. The n parameterallows more flexibility in the inhibitory function. Thebaseline parameterR0 can be approximated to 0.6 units.The ktol parameter governs the rebound between

200 minutes (at the end of infusion) and 240 minutes(return of response to baseline). If we assume that fourhalf-lives of ktol (t1/2, ktol) have elapsed during the periodfrom 240 to 200minutes (i.e., 40minutes), then the half-life is about 10 minutes. The first-order ktol parameterthen becomes ln(2)/10, which is about 0.07 min21.The overall goal is to explain the complex behavior of

the biomarker R as a simple biophase amount-responserelationship. At pharmacodynamic steady state andwith inhibition of kout, the response becomes thefollowing (eq. 24):

RSS ¼(

R0 × I�Ab;SS

�R0 ×

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiI�Ab;SS

�q ð24Þ

3. Results and Conclusions with Respect to Dose-Response-Time Analysis. High consistency was foundbetween the observed and model-predicted data (Fig.10, left). Table 5 contains initial and final parameterestimates and their corresponding CV% values. Notethat ktol (0.058 min21) is 10 times smaller than kout(0.59 min21). The k parameter is defined as ktol/kout,which suggests that it takes 10 times longer to developthe tolerance equilibrium than the response equilib-rium. The size of k (,1) also suggests a substantialrebound effect, provided that the kinetics of the testcompound does not confound the rebound development.Figure 11 shows two simulations of eq. 24, represent-

ing a tolerant and nontolerant system, respectively, atequilibrium. We have by means of a modeling approach(eqs. 20–23; Fig. 10) condensed the complexity into arelatively simple dose-response relationship, displayedin Fig. 11. A tolerant system requires more drug to givethe same response at equilibrium. In this case example,the tolerant system also shows a reduced ability tosuppress fatty acids to the same low level (0.05 mM) asseen in the nontolerant condition. This is revealed bythe upward shift of the red dose-response curve.

Data showed a baseline value prior to test compoundadministration. An almost complete inhibition of re-sponse was seen with the highest dose, which suggeststhat more extensive blockage of response is possible in anontolerant system. Data also demonstrated adapta-tion with rebound and a limited ability to suppress thelevel of free fatty acids (FFAs) in this condition (Fig. 11).

4. Pharmacodynamic Interpretation and Comment.Adaptational phenomena in physiology and pharmacol-ogy are commonplace and need to be accounted for in alldrug treatment contexts involving repeated dosing(chronic indications). Needless to say, the degree ofadaptational and compensatory alterations variesbroadly across physiologic response, disease, target,and drug class conditions. In the case example describedabove, the nontolerant system responded to NiAc with amarked shutdown of the FFA response. That said, itshould be noted that FFA concentrations were nottotally suppressed; this suggests that at least for thetype of rapid-acting drug target intervention repre-sented by NiAc, other physiologic mechanisms willsustain a minimum FFA concentration limit. In turn,the upward shift of this minimum seen in the tolerantsystemmay indicate adjustment in the sensitivity of theprimary NiAc target sites in its intended biophaseand/or recruitment of other buffering/compensatorymechanisms to ascertain upholding of “safe”FFA levels.

D. Case Study 4: Dose-Response-Time Analysis ofAntipsychotic Effects (Brief Psychiatric RatingScale Scores)

1. Background. This case study highlights the as-sessment of the antipsychotic action of remoxipride inpatients with schizophrenia (Lewander et al., 1990).The antipsychotic efficacy of the dopamine receptor–blocking agent remoxipride was established in a seriesof clinical studies, and analysis suggested three impor-tant aspects of its effects in acute-phase schizophrenia.First, 25%–30% of patients did not respond. Second, theremoxipride treatment response appeared to affect both

Fig. 11. Response vs. NiAc concentration (Ab,ss/Vss) relationship atequilibrium. Curves are normalized to the same baseline value. The graycurve shows a system without feedback, and the red curve shows a systemwith feedback. The same dose results in less effect in a tolerant system atequilibrium as compared to a nontolerant system.

TABLE 5Initial and final model parameter estimates and their corresponding

precision (CV%)


R0 (mM) 0.6 0.58 3kout (min21) 0.1 0.59 36ktol (min21) 0.07 0.058 25nH 1 1.40 7ID50 (mmol) 0.1 0.13 26K (min21) 0.2 0.22 13


positive and negative symptoms of the disorder to asimilar degree. Third, the onset of antipsychotic actionin responders seemed to be unusually rapid (i.e., within1 to 2 weeks). Patients with schizophrenia in an acutephase of the illness received remoxipride (dose range,120–600 mg/d) and were repeatedly scored for symptomseverity across a time period of at least 6 weeks. Theobserved mean BPRS time courses in four treatmentresponse categories are shown in Fig. 12.The patients were divided into four categories. Non-

responders were defined as having a ,25% reduction,low responders as a 25%–49% reduction, mediumresponders as a 50%–74% reduction, and high re-sponders as a .75% reduction in total BPRS scoresfrom baseline to the last available rating.The turnover model of the antipsychotic response-

time data is given in eq. 25. This is therefore an exampleof evaluating the effect of a drug via usage of a diseasemodel, with a decrease in BPRS score being indicative ofa reduction in disease severity. Drug exposure andresponder category are embedded in the Imax parameterfor each responder category.2. Models, Equations, and Exploratory Analysis.

Exposure to remoxipride was assumed to be constantacross the whole period of BPRS score measurements.The half-life of remoxipride is about 4 to 5 hours inplasma, compared with the half-life of BPRS score (inweeks). The analysis focused on modeling BPRS re-sponder category (Fig. 12) as a function of time. Theactual mean doses of the drug used in the studies in thelast 4- to 6-week period ranged from 226 to 556 mg/day(typically approximately 400 mg/day; Lewander et al.,1990), but “dose” was not used as a covariate forseparating the responder categories.The rate of change of response dR/dt (Fig. 12, right)

can be described by eq. 25:

8>><>>:

dRdt

¼ kin × IðAbÞ2Kout × R

dRdt

¼ kout × ðR0 × IðAbÞ2RÞð25Þ

where kin, kout, R0, and I(Ab) are the turnover rate,fractional turnover rate, baseline biophase response,and inhibitory drug mechanism function (eq. 26), re-spectively. Equation 25 will, under constant drugexposure, describe the drug action in Fig. 12. The timeto dynamic steady state, assuming a constant biophaseexposure to remoxipride, Abss, will be governed by kout.It is reasonable to assume that the exposure can be heldconstant across two consecutive dosing intervals. Evi-dently, the time course of the response (half-life of 1 to2 weeks) is not directly correlated to the plasma half-lifeof the drug, which is much shorter (4 to 5 hours). It wastherefore decided to model the inhibitory effect of drugaccording to the following expression:

IðAbÞ ¼ 12 Imax ð26Þ

where Imax is the rating category of responders (nonre-sponders and low,medium, and high responders). In theabsence of drug, I(Ab) = 1 and the baseline level ofresponse becomes (eq. 27)

R0 ¼ kinkout

ð27Þ

In the presence of drug, the level of response becomes(eq. 28)

RSS ¼ kinkout

× ð12 ImaxÞ ¼ R0 × ð12 ImaxÞ ð28Þ

where Rss is the minimum steady-state response. Thelower the value of Rss, the better the drug effect. If we

Fig. 12. (Left) Observed (symbols) response (BPRS score, medians) vs. time in four treatment response categories defined according to the percentagechange at the last available rating (Lewander et al., 1990). Data were digitized and slightly adapted, in that the last available rating data point was setto 12 weeks in our analysis. The experimental data are connected by straight lines. (Right) The conceptual model with its parameters turnover rate kin,fractional turnover rate kout, and drug mechanism function I(Ab). R denotes the BPRS score.


had full response in that Imax equaled unity (1), thenRss

would equal 0. The difference betweenRss (Rmin) andR0

is given by eq. 29.8>>><>>>:

DR ¼ R0 2RSS ¼ kinkout

2kinkout

× ð12 ImaxÞ ¼

kinkout

× Imax ¼ R0 × Imax

ð29Þ

From the intercept of the response-time curve with theeffect axis (Fig. 12), the baseline value R0 (36.5 units)can be obtained. R0 is the ratio of kin/kout, if we assumeeq. 27 to be an appropriate description of the drug-freestate. The kout parameter is obtained from the log-linearslope of the downswing of the highest responder (.75%)curve (eq. 30):

Kout ¼ lnð36:5Þ2 lnð10:3Þ02 3

� 0:42wk2 1 ð30Þ

The kin parameter can then be calculated (15.4 U/wk)from the baseline value (36.5 units) and kout (0.42wk21).Imax was estimated according to eq. 26 to be 0.8 for thehighest responder (.75%) group. That value was usedas the starting value for Imax in all groups.

3. Results and Conclusions with Respect to Dose-Response-Time Analysis. Figure 13 displays observeddata together with model-predicted curves of the turn-over model. Table 6 gives the final parameter esti-mates of the four patient categories together with theparameter precision. Note that the parameter pre-cision was high in general, except for the nonre-sponder category. The half-life of response rangedbetween 1.1 weeks (25%–49% and .75% respondercategories) and 1.7 weeks (50%–75% responder cate-gory). The Imax values were 0.41, 0.65, and 0.87 for the25%–49%, 50%–74%, and.75% responder categories,respectively (Table 6).

The quantitative category-response-time analysisapproach enabled us to assess the actual respondervalues (Imax) as well as the half-life of response for eachcategory. Since the half-life of response was about aweek, we know that 90% response should have beenestablished within 3 to 4 weeks (i.e., three to fourhalf-lives). The half-life of response [ln(2)/kout] clearlydemonstrates the adjustment of the dopaminergicsystem to drug intervention. It follows in this case thatwhether a patient responds to the remoxipride therapymay be possible to establish within 1 to 2 weeks from

Fig. 13. Observed (symbols) and model-predicted (lines) response (BPRS score) vs. time in four treatment response categories defined according to thepercentage change at the last available rating (Lewander et al., 1990). Dotted lines represent the dynamic half-life (BPRS scores) in the .75%responder category: approximately 1 week.

TABLE 6Final parameter estimates of the four patient categories

Data are presented as the estimate 6 CV%.

Parameter Nonresponders(,25% Reduction)

Responders

25%–49% Reduction 50%–74% Reduction .75% Reduction

Imax 0.13 6 19 0.41 6 2 0.65 6 4 0.87 6 2kin (scores/wk) 45 6 60 22 6 7 15 6 20 24 6 9kout (wk21) 1.2 6 60 0.61 6 6 0.40 6 10 0.66 6 8


the start of drug treatment. Therefore, the modelinglesson to be learned from this case study is that frequent(two or three times daily) dosing may not necessarily beneeded for a response such as an antipsychotic effectwhen the half-life of the response is much greater thanthe plasma half-life of the drug, unless there are safetyissues related to drug intake. The time to dynamicsteady state is unaffected by dose and/or dose frequency.In other words, pharmacokinetic reasoning cannotstand alone but has to be integrated with dynamics inthe decision-making process.4. Pharmacodynamic Interpretation and Comments.

Ideally, we would have liked to see a return to baselineafter the cessation of drug treatment, as this may havegiven an indication of possible rebound or adaptation.Although withdrawal of a drug and therefore reduction(and possibly obliteration) of the response may beinformation rich in terms of determining the potencyof the drug (EC50/IC50), drug-free periods may jeopar-dize patient health and should therefore be avoided(certainly in difficult psychiatric afflictions). Althoughsuch a design may thus be considered when developinga new drug for schizophrenia, its application is compli-cated by the typical chronic and progressing nature ofthis illness, along with the fact that a significantnumber of patients may have developed supersensitivedopamine (and possibly other) target mechanisms as asequel to having been previously exposed long term toone ormore antipsychotic drugs. Careful selection of thede novo patients may be envisaged as a potential meansto circumvent said complications, and thereby gathermore complete DRT (coupled to plasma drug exposuredata) information from clinical studies to the benefit offuture antipsychotic drug development. From a strictlytherapeutic point of view, the data also underline themarked heterogeneity encountered among patients di-agnosed with schizophrenia. Identification of underly-ing reasons for this may pave the way for better, moreindividualized treatments. Incidentally, the analysis ofthis case study suggests the intriguing possibility that itmight be possible to identify responders and nonre-sponders already in the treatment initiation phase.Further attesting to this prospect, comparable datawere found with haloperidol in the same study(Lewander et al., 1990) as well as with other agents ina number of other literature reports (e.g., Levine andLeucht, 2010; Marques et al., 2011; Stauffer et al.,2011), all of which display very similar patterns withrespect to antipsychotic drug treatment onset andresponse trajectories in patient subgroups. DRT mod-eling across these studies indicates that responders(including “high,” “medium,” and “low” responders)display a half-life of 1 to 2 weeks for symptom improve-ment and 4–6 weeks to pharmacodynamic steady state,and it also suggests early spotting of patients withpoor or no response to (antidopaminergic) treatment.That said, any comparisons of efficacies, potencies,

and/or safety-related issues between compounds, spe-cies, or studies must be based on plasma exposurerather than dose, as pharmacokinetic properties such asbioavailability, clearance, and half-life also need to befactored in. Further stratified DRT analyses of re-sponses to other psychiatric drug treatment classeswould be of great interest toward future individualizedpharmacotherapy approaches and strategies.

E. Case Study 5: Dose-Response-Time Analysis ofBacterial Count

1. Background. This case study highlights somecomplexities in DRT modeling of bacterial growth/killdata. Antibacterial compound X was being developed(Gabrielsson and Weiner, 2010). To establish its po-tency in a resistant bacterial strain, a 10,000-U dose ofbacteria was injected into the bloodstream of five groupsof Wistar rats. A dose of 0 (vehicle control), 1.5, 2, 4, or6 mg of the antibiotic was given to each of the groups.Blood was drawn at selected time points for bacterialcount (Fig. 14).

2. Models, Equations, and Exploratory Analysis.The bacterial growth/kill model is shown in Fig. 14(right). The vehicle dose was included to explore thenatural bacterial growth, kg, in the absence of antibiotic.The model was extended to incorporate an uppergrowth-limiting factor, 1 – N/Nmax (eq. 31):

dNdt

¼ kg × N ×�12

NNmax

2 kk × f ðAbÞ × N ð31Þ

whereNmax is themaximum value of the bacterial countin the system (defined by the bacterial growth endpointin the vehicle-treated group). When the number ofbacteria N approaches the steady-state level Nmax, thefactor 1 –N/Nmax approaches zero and bacterial growthis temporarily stopped. During the early cell kill phase,the decline in bacterial count becomes

dNdt

¼ 2 kk × f ðAbÞ × N ð32Þ

Figure 15 shows the results from fitting the extendedgrowth/kill model to the bacterial count versus timedataset. Note that the starting values at time 0 were notequal for the different groups of animals (see Table 7,N01–05). From the growth of bacteria in the vehiclegroup, the steady-state value of growth/kill can beestimated to about 25–30,000 bacteria.

3. Results and Conclusions with Respect to Dose-Response-Time Analysis. The proposed DRT modelnicely fits the five (0, 1.5, 2, 4, and 6 mg) colony-forming unit time courses in Fig. 14, and Table 7 showsthe final parameter estimates and their CV%. DRTanalysis lent itself to assessment of the growth and killparameters. This model may now be used for optimizing


the next study on repeated dosing with an alternativedosing regimen.4. Pharmacodynamic Interpretation and Comments.

Bacterial antibiotic resistance development is a wellknown, global problem in need of novel drugs andsolutions. Analysis of the DRT data in the studydescribed in this case example gave robust parameterestimates to assess growth and kill rates upon exposureto the antibacterial agent X. The data in Fig. 14 suggestthat bacterial (re)growth was kept at bay for approxi-mately 6 hours after administration of the drug, afterwhich growth exceeded the kill rate. Thus, from atreatment perspective, it could be suggested that asecond dose of antibiotic given within the 6-hour timeframe should be considered tomore effectively eradicatethe infection. The data derived from the DRTmodel andanalysis therefore had significant impact on the onwarddesign of drug treatment protocols, and potentially alsofor the optimization of pharmaceutical formulations fortherapeutic use.

F. Case Study 6: Dose-Response-Time Analysis ofCortisol–Adrenocorticotropic Hormone Action

1. Background. Data were scanned from an articleby Urquhart and Li (1968), who studied the dynamics ofcortisol secretion of the perfused canine adrenal glandin situ upon stimulation by ACTH. The dogs wereacutely hypophysectomized to eliminate endogenousACTH. The cortisol secretion rate (response) wasexpressed as the product of the venous blood flow andthe concentration of cortisol in adrenal venous blood(Fig. 16). The proposed model captures feedback regu-lation at constant drug exposure. This dataset thereforeviolates the previous assumption about time invariantsystems.2. Models, Equations, and Exploratory Analysis.

The proposed feedbackmodel is shown in Fig. 16 (right).Let us assume that the rise and fall of the cortisol

responseR shown in Fig. 16 (left) can be modeled by thefollowing simple feedback model (eq. 33):

dRdt

¼ SðAbÞ × kin 2 kout × M ð33Þ

S(Ab), which represents the ACTH drug mechanismfunction, stimulates the production of R, which isthen counterbalanced by means of the endogenousmodulator that we denote as M. Note that the loss ofR is indirectly governed by means of M. M is in turngoverned by R and the rate constant for developmentof tolerance ktol, which can be written as shown ineq. 34:

dMdt

¼ ktol × R2 ktol × M ð34Þ

The ktol parameter was selected to govern both pro-duction and elimination ofM in this particular example.However, this may not always be the case where datacontain more information about the different rateprocesses. When R increases, the production of modu-lator M is stimulated and then increases. The increasein M counterbalances R by increasing the rate of loss ofR. It is assumed that the rates in and out of M aregoverned by the first-order rate constant for tolerancektol.

The ACTH drug mechanism function S(Ab) is writtenas follows (eq. 35):

SðAbÞ ¼ ½Ab�n ð35Þ

where the ACTH concentration is either 1mU/ml (unity)or 2 mU/ml, and n is an amplification parameterallowing a disproportional rise in S(Ab) with a doublingof ACTH.

3. Results and Conclusions with Respect to Dose-Response-Time Analysis. The regression of eqs. 33–35is compatible with data as shown in Fig. 16. The model

Fig. 14. (Left) Observed (symbols) and predicted (lines) response (bacterial count, CFU) vs. time after administration of 1.5, 2, 4, and 6 mg of a newantibiotic. Note the large range of counts (more than three orders of magnitude). (Right) First-order bacterial growth coupled to the second-orderbacterial kill model. The first-order growth kg × N × (1 2 N/Nmax) is saturable by means of the (1 2 N/Nmax) term, where kg, N, and Nmax denotethe first-order growth rate constant, the number of bacteria, and the maximum bacterial count, respectively. The bacterial kill process is approximatedby –kk f(Ab) × N, where kk, N, and Ab denote the first-order kill rate constant, the number of bacteria, and the plasma concentration of drug. CFU,colony-forming unit.


is made up of two parallel turnover models that areinterconnected, in that the response compartmentdrives the buildup of the moderator and the change ofthe moderator is fed back to the response compartmentvia the loss term of the latter. This mechanism makesup the feedback process that governs the intertwinedcortisol-ACTH system. Data show the predose baseline,the initial rapid rise resulting in an overshoot in cortisolrelease when the ACTH level rises, pharmacodynamicsteady-state and the post-exposure rebound. Cortisollevels then decline to a new steady state, which uponlowering of the ACTH exposure plunges to a level below(rebound) the original baseline at about 2. The reboundthen oscillates back to baseline at the end of theexperimental period. The kout and ktol parameters areof the same magnitude (0.16–0.18 min21), which corre-sponds to a half-life of cortisol release of about 3minutes(Table 8).The biophase exposure to cortisol was assumed,

but not confirmed, to be 1 mU/ml at baseline, approx-imately doubled during the experiment, and thenreturned back to baseline exposure during washout.This experimental setup allowed us to use a square-wave biophase function for driving the pharmacody-namic response. Data contained information about themodel rate constants kin, kout, and ktol and the ampli-fication factor n. All parameters were estimated withhigh precision.4. Pharmacodynamic Interpretation and Comments.

The interdependent ACTH-cortisol secretion system isan important and integral part of the hypothalamic-pituitary-adrenal axis, responding to stressful condi-tions and challenges. The studies of Urquhart and Li(1968) used hypophysectomized dogs to maintain con-trol of the ACTH exposure in their perfusion experi-ments. Interestingly, the overshoot observed in thiscase study appeared dependent both on dose (exposure)and rate of ACTH rise in the arterial blood during

perfusion (Urquhart and Li, 1968). The underlyingmechanism(s) was suggested to involve changes incortisol hydroxylation, and/or other unknown actionsof ACTH on steroidogenesis, which is also related to themoderator M in our analysis. We attempted to rean-alyze data with either an extended model (according toAhlström et al., 2011) or an alternative tolerance model(Urquhart and Li, 1968; used in the original analysis).However, neither of these approaches was successful infully capturing the oscillations. It appears that addi-tional discrete individual data are required to estimateaccurate and precise parameters that may account foroscillatory behavior (and tolerance development) in thiscase.

G. Case Study 7: Dose-Response-Time Analysis ofMiotic Effects

1. Background. Miotic response-time data are anexample of the use of DRT analysis when there are nosupportive drug concentrations. When we use localapplications (e.g., eye drops for drug action in glau-coma), plasma exposure to drug is “downstream” ofthe biophase containing the pharmacological targetand thus cannot serve as a driver of the pharmaco-logical response. This case study therefore illustratesthe feasibility of using DRT data in an early pre-clinical setting in the absence of systemic exposuredata.

The miotic response in the cat eye after application oflatanoprost (Gabrielsson et al., 2000; Gabrielsson andWeiner, 2016), an ester prodrug analog of prostaglandinF2a, is assumed tomirror an interaction with prostenoidF receptors in the smooth muscle of the iris. Thus, in ascreening program to find drugs for the treatment ofglaucoma, the miotic response in the cat eye was one ofthe models used to evaluate potential drug candidates.Domestic cats were specially trained to receive eyedrops and to allow measurement of the pupillary re-sponse. Six animals were used in each of three dosegroups. The horizontal diameter of the pupil wasmeasured. The precision of the measurements of thepupil diameter was 1 mm (10%). Figure 17 shows theobserved response-time data, and eq. 36 is the biophasemodel fit to the data.

TABLE 7Initial and final parameter estimates and their CV%

k values are given in h21, and N values are given in colony-forming units.


kg (h21) 0.10 2Ke (h

21) 1.0 0.77 2kk (h21) 1.0 2Nmax 37,000 6N01 11,000 3N02 11,000 3N03 12,000 4N04 13,000 5N05 11,000 3

Fig. 15. Semilogarithmic plot of observed (symbols) and model-predicted(lines) response-time data on how to estimate the growth kg and kill-ratekk parameters.


2. Models, Equations, and Exploratory Analysis.The kinetics of latanoprost in the biophase was as-sumed to be described by a first-order input/outputmodel, including a lag-time:

Ab ¼ Ka × F × Dev

ðKa 2KÞhe2K × ðt2 tlagÞ2 e2Ka × ðt2 tlagÞi ð36Þ

Ab is the drug amount in the biophase, Dev is the actualextravascular dose applied on the cornea,Ka is the first-order input rate constant, K is the first-order elimina-tion rate constant (assuming Ka . . K), and tlag is thelag time during the input of drug into the effectcompartment. We also assume that the biophase avail-ability and volume of the biophase compartment are setequal to unity. The biophase function (eq. 36) is thendirectly driving the response (eq. 37):

E ¼ E0 2Imax × A

nHb

IDnH50 þ AnH

bð37Þ

including the baseline value of the contralateral controleye E0, the maximum effect Imax, the dose at half-maximal effect ID50, and the sigmoidicity factor nH. Ka,K, tlag, Imax, ID50, and nH were then estimated bysimultaneously fitting eqs. 36 and 37 to the meanvalues of the observed effect-time data obtained fromeach dose (Fig. 18).3. Results and Conclusions with Respect to Dose-

Response-Time Analysis. The treatment data presentedhere exemplify the use of a biomarker (response)when there are no drug concentration data for kinetic/dynamic analyses. We assumed a first-order input/output

kinetic model that drives the dynamics. The kinetic anddynamic models were then fit simultaneously to thedata to estimate Ka, K, tlag, Imax, ID50, and n (Table 9).The simulated biophase amount versus miotic responseis shown in Fig. 18.

4. Pharmacodynamic Interpretation and Comments.This analysis example shows that a well designedexperiment that includes only response-time data maynonetheless lend itself to estimation of the underlyingkinetic processes without any additional concentration-time measurements. It also shows that absorption doesnot have to be instantaneous to enable estimation ofbiophase kinetics and dynamics. It has been suggestedthat the cornea acts like a slow-release depot to theanterior segment when latanoprost is topically admin-istered into the eye (Russo et al., 2008), possibly addingto the lag-time effect introduced by the ester conversionof latanoprost to its active species in the eye. The resultsfrom a DRT exercise could be used for more refinedrecommendations regarding dose, dosing interval, andconcentration-response sampling times in future (pre)-clinical studies. In this particular case and based onhuman data (Sjöquist and Stjernschantz, 2002), oncethe drug reaches the systemic circulation, its break-down is very rapid (t1/2 of 17 minutes); thus, this further

TABLE 8Final parameter estimates

Data are presented as the estimate 6 CV%.

Parameter kin kout ktol n

concentration/min min21 min21

Estimate 0.32 6 6 0.16 6 5 0.18 6 14 0.78 6 6

Fig. 16. (Left) Cortisol release rate vs. time in the perfused canine adrenal gland in situ upon stimulation by ACTH (gray bar). Experimental data(solid circles) and the model-predicted curve (solid line) after a doubling of cortisol exposure. The drug provocation is modeled by means of a square-wave function that changes from 1 mU/ml (20–60 minutes) to 2 mU/ml (60–122 minutes), and then back to 1 mU/ml at 122 minutes. (Right) Schematicpresentation of the conceptual model used for fitting experimental data of cortisol release rate as a function of ACTH exposure. R denotes the cortisolrelease rate, M is the moderator, kin is the turnover rate constant, kout is the fractional turnover rate constant, and ktol is the fractional turnover rateconstant of the moderator compartment. The stimulatory drug mechanism function is given as a square-wave function with a jump of ACTH exposurefrom 1 to 2 mU/ml at 60 minutes and then an equally abrupt change back to 1 mU/ml at 122 minutes. PD, pharmacodynamic.


complicates concentration-time–only approaches tomodeling. Notably, as the application of latanoprost isin the immediate vicinity of the target biophase, plasmaexposure is downstream of the miotic response andtherefore not the driver in this situation. Moreover, thefact that the effect precedes (rather than follows) theappearance of drug in the circulation, combined with aprolonged response duration versus a very short plasmahalf-life of the drug, would make any attempt tocorrelate the miosis readout to plasma concentrationextremely difficult.

H. Case Study 8: Meta-Analysis of Dose-Response-Time Data

1. Background. The standard pharmacological ap-proach of activating a target around the clock often failsbecause of time-dependent loss of drug efficacy andpostdosing rebound above predose baseline levels. Thiscase study presents data that originally explored theidea that a more comprehensive understanding of therelationship between plasma exposure and a pharma-cological response combined with knowledge of thephysiologic regulation of metabolism (i.e., simulta-neously analyzed considering the effect of key bio-markers) can be used to mitigate these barriers,allowing the invention of new pharmacostrategies

(Andersson et al., 2017; Kroon et al., 2017). The meta-analysis aimed at investigating in what way our in-ference about the system is changed when all drugexposure data are removed. Is it still possible to deriveinformation about pivotal target (FFAs) properties byconducting a DRT data analysis? If not, in what way dothe results fail to give insight about system behavior?

This DRT analysis contained the following informa-tion: 1) multiple response-time courses of two differentbiomarkers (FFA and insulin) under acute and semi-chronic NiAc treatments, conducted in both leanSprague-Dawley and obese Zucker rats; 2) time delays(between known plasma exposure and biomarker re-sponse), feedback patterns (e.g., overshoot and re-bound), and two distinct patterns of systemicadaptation prevail (one as a result of insulin control inlean animals, and another pattern due to drug re-sistance in obese animals); and 3) utilization of a priorexposure-driven analysis, which provided a good oppor-tunity for comparisons between DRT- and exposure-driven analyses. Table 10 contains information aboutthe experimental setup, animals used, administrationroutes, and study duration, which were all part of themeta-analysis.

2. Results and Conclusions with Respect to Dose-Response-Time Analysis. The meta-analysis of theNiAc-FFA-insulin data challenges several of the pre-viously established constraints to DRT analysis:namely, linear dynamics, stationary system (toleran-ce/adaptation not allowed), instantaneous equilibriumbetween plasma and biophase (response), linear plasmakinetics, and so forth. The analysis may also provide a

Fig. 17. (Top) Observed (symbols) and model-predicted (lines) response-time data for latanoprost at the 0.1, 1.0, and 10 mg dose levels in the cateye. (Bottom) Simulated corresponding biophase time courses using finalparameter estimates from the regression of the pharmacodynamic modelto response-time data in the top panel. The thick dashed red lineschematically shows the time course of test compound after intravenousand topical (tmax 10 minutes) dosing onto the cornea, with a plasma half-life of ,10 minutes in rabbits and cynomolgus monkeys (Sjöquist et al.,1999).

Fig. 18. Observed (symbols) and predicted (solid lines, eq. 37) mioticeffect (in millimeters) vs. biophase amount after three different doses oflatanoprost (filled red circles, 0.1 mg; blue squares, 1.0 mg; and graycircles, 10 mg). The observed effects are mean values with a coefficient ofvariation of approximately 30% for each observation.

TABLE 9Final parameter estimates

Data are presented as the estimate 6 CV%. Half-lives of Ka and K are given.

Parameter t1/2Ka t1/2K tlag Imax ID50 N

min mm mg min

Estimate 35 6 30 140 6 7 26 6 5 7.2 6 2 0.17 6 10 1.3 6 7


framework to conduct meta-analyses of both preclinicaland clinical studies when exposure data are scarce orlacking. To our knowledge, multiple mutually interde-pendent biomarkers confounded by dynamic feedback(e.g., with FFA and insulin, in this case) have not yetbeen analyzed by means of a DRT approach.The results were generally well defined system

parameters, except for an unacceptably low precisionfor some of them. By and large, the traditional exposureresponse and the DRT approaches, however, gavesimilar results (Table 11). It is also interesting to notethat parameter estimates of a very wide numericalrange were obtained in the analysis, which probablymade the differential equation system rigid. The half-lives ranged from a few seconds to more than 100 hours,which is not surprising in a highly regulated metabolicsystem such as the FFA-insulin interplay. The outcomemay help optimize the next study design to increaseprecision in pivotal parameters. We are aware of thefact that the DRT analysis resulted in some poorlydefined parameters (high CV%), which makes sensein light of particulars of the dataset. Specifically,targeted time ranges will be of greater importance inthe design of subsequent studies. However, note thegreat discrepancy between the half-lives of exposure-driven tolerance versus DRT-driven tolerance in thelean Sprague-Dawley group. The former suggests a longhalf-life of almost 70 hours, whereas the latter analysisshowed a half-life of slightly less than an hour.3. Pharmacodynamic Interpretation and Comments.

The results illustrate the value of applying a quantita-tive approach to accompany comprehensive physiologicand kinetic-dynamic understanding. A well defineddosing regimen, rate, extent, and timing of drug in-tervention were designed sequentially across severalstudies. This resulted in suppression of tolerance andrebound and, most importantly, in profound improve-ments of the metabolic profile of a preclinical diseasemodel (Andersson et al., 2017; Kroon et al., 2017). Inturn, this information may provide vital input toprojects aimed at discovery and development of noveldrugs for metabolic disorder. As noted by Kroon et al.

(2017), the possibility remains to be investigated whetherthe response patterns observed with different NiAcadministration protocols are exclusive to this agent ormay be avoided using other drugs also targeting theHM74/GPR109A receptor. Whether antilipolysis viaother targets would display similar response propertiesto the NiAc HM74/GPR109A receptor interaction is alsoclearly a worthwhile task for future study. Regardless,this case study is an example of the importance ofoptimal design of dosing regimens in the treatment ofmetabolic disorder. Furthermore, the data also empha-size the importance of drug testing in a disease model(obese Zucker rats) compared with control conditions(lean Sprague-Dawley rats), as illustrated not least bythe marked difference in tolerance half-life outcome.

I. Overall Conclusions of the Case Studies

The eight selected case studies were chosen for theirspecific data patternswith respect to the onset, intensity,and duration of response, rather than representations oftherapeutic class. Whereas case studies 1–7 are repre-sentative single-case studies of the aforementioned, casestudy 8 is an example of applying an extendedmodel andmixed-effects modeling to meta-analysis of several sour-ces of data.

Time series are necessary for a correct assessmentof biomarker response data, particularly with timedelays between the plasma concentration of test com-pound and the biomarker response. Since different

TABLE 11Final estimates of system parameter half-lives from the meta-analysis

(biophase- vs. exposure-driven input)Data are the CV% and corresponding S.D. for the different parameters.

Parameter,t1/2 (h)

Lean Sprague-Dawley Rats Obese Zucker Rats

DRT Driven Exposure Driven DRT Study Exposure Driven

koutI 0.03 (16) 0.11 (14) 0.04 (45) 0.06 (17)ktolI 0.78 (30) 64. (28) 4.3 (6.0) 5.8 (48)koutRI 0.02 (16) 0.01 (17) 28. (54) 11. (27)kNI — — 5.0 (170) 29. (35)koutF 0.001 (1) 0.002 (140) 0.002 (150) 0.004 (120)ktolF 0.36 (1000) 0.57 (67) 1.2 (4.3) 1.1 (24)koutRF 0.08 (300) 0.71 (29) 77 (58) 41 (38)kNF 140 (330) 110 (65) 190 (24) 18 (14)

TABLE 10Information about study length, washout profile, dose schedule, route, and nutritional conditions

Number of saline control animals appears within parentheses.

Treatment Study Protocol Route Lean Rats Obese Rats

Acute On-off 1 h 1 h infusion i.v. 4 (3) 5 (3)On-off 5 h 5 h infusion s.c. 7 (2) 7 (5)On-off 12 h 12 h infusion s.c. 5 (2) 4 (2)Step-down 1 h 1 h infusion plus 3.5 h of

step-down infusioni.v. 5 (2) 5 (2)

Step-down 12 h 12 h infusion plus 3.5 h ofstep-down infusion

s.c. 5 (3) 4 (3)

Semichronic Continuous 120 h 120 h pretreatment infusionplus 5 h infusion

s.c. 6 (2) 8 (2)

Intermittent 120 h 120 h pretreatment intermittentinfusion (12 h infusion plus12 h washout) plus 5 h infusion

s.c. 6 (2) 8 (3)


model parameters contribute to varying extents atspecific phases (onset, intensity, and duration) of apharmacological response, close scrutiny of the experi-mental protocol becomes an integral part of design. Afull characterization of determinants behind the onset,intensity, and duration of response often requiresmultiple routes, rates, and extents of dosing regimens.DRT analyses typically use more parameters (e.g.,

biophase availability, biophase rate constant) thanplasma concentration-driven models, since the driver(biophase) of pharmacological response has to be pre-dicted as such. Plasma half-lives are also typicallymuchshorter than are pharmacodynamic responses (e.g.,shown in case studies 3 and 8). Finally, DRT modelsare generally compound, biomarker, and species de-pendent and are therefore difficult to generalize withinand across therapeutic classes. Table 12 lists charac-teristics, mechanisms, and features, along with somesuggestions for improvements to experimental design,applied to the case studies discussed in this article.

IV. Discussion

A. Background

This review focuses on DRT data analysis whenknowledge on test compound (drug) exposure is scarceor entirely lacking. We start by dissecting the differentphases of a pharmacodynamic response-time course(Fig. 2), and we propose what parameters may berelated. We then move on to exemplify structures ofbiophase functions from the literature (Fig. 3). Theserange from a simple bolus input coupled to monoexpo-nential decline and first-order and saturable input/out-put rates to parallel exogenous and endogenous inputfunctions. We provide a brief mathematical descriptionof biophase, drug mechanism, and commonly appliedturnover functions. We also discuss alternative ratemodels and put them into a biologic perspective. Thisreview contains eight case studies. These case studieswere selected to demonstrate how to tackle baseline/nobaseline, time delays, peak shifts, saturation, cellgrowth/kill data, functional adaptation, rebound, mul-tiple sources of drug provocations and biomarkers, andrapid exposure-response data (Table 2).Isaksson et al. (2009) and Andersson et al. (2016, 2017)

successfully performed meta-analyses on preclinical dataon NiAc (nonlinear absorption and elimination)–inducedfatty acid lowering, with accurate and precise param-eter estimates. The kinetic nonlinearities were wellseparated from the pharmacodynamic nonlinearities.Pharmacological data of two intertwined biomarkersand of multiple doses, rates, routes, and scheduleswere simultaneously regressed. This attests to theview that the success of a DRT analysis is a matter ofgood experimental design (Gabrielsson and Weiner,2010, 2016; Andersson et al., 2016) rather than the

number of observations, as was recently suggested(González-Sales et al., 2017).

The applicability of DRT analyses in both preclinicaland clinical situations is illustrated in Table 13, whichcompiles more than 60 examples found in a thoroughsearch of the literature. We also list some situations inwhich DRT analyses may be potentially useful but maybe challenging (Table 14).

To this end, references listed in Table 13 represent acompilation of reports published up to and including2017. References to abstracts and oral communicationshave intentionally been excluded from this review.These studies cover a large range of therapeutic andexperimental areas. Some of the preclinical studies usetwo or more modes of administration and are moreexperimental in nature (Schoenwald and Smolen, 1971;Smolen, 1971a,b, 1976a,b, 1978; Smolen et al., 1972;Smolen and Weigand, 1973; Isaksson et al., 2009;Andersson et al., 2017). Other studies are performedin critically ill patients (Salem et al., 2016), in neonates(Trefz et al., 2015), under disease progression(Musuamba et al., 2015), for postoperative pain man-agement (Abou Hammoud et al., 2009), in drug in-teraction studies (Gruwez et al., 2005, 2007), and inmetabolic systems (Fasanmade and Jusko, 1995;Hamberg et al., 2010; Ternant et al., 2014; Saffianet al., 2016), to mention just a few.

B. What Do Different Doses, Routes, and Rates ofInput Add?

When different doses (see case study 1) are givenin vivo, the process of onset of action, potential responsepeak shifts with increasing doses, saturation of re-sponse at higher doses, and functional adaptation ofduration of action may be observed in the pharmaco-logical response-time courses. When different routes ofadministration (e.g., intravenous and subcutaneous; seecase study 1) are applied, the biophase availability maybe estimated separately from the other kinetic anddynamic parameters. Nonlinearities in biophase kinet-ics may also be revealed (case study 8).

When different doses and routes are tested simulta-neously, information about saturation of either theabsorption or disposition of drug or both processesmay be discerned from saturable pharmacodynamicprocesses (Andersson et al., 2017). Functional adapta-tion may be observed and quantified when repeateddosing is done (case study 2; Andersson et al., 2016,2017). The two routes of administration in case study2 also revealed absorption-rate–limited elimination inthe pharmacodynamic data.

Neither the pharmacokinetics nor the pharmacody-namics need to be linear first order, time invariant, orinstantaneous or display a constant baseline. A meta-analysis of complex response-time data of fatty acids(FFAs) and insulin turnover in plasma after multipleprovocations of NiAc clearly demonstrated the full


TABLE

12Ove

rview

ofcase

studies,mecha

nismsinvo

lved

,de

sign

featur

esan

dsu

ggestedim

prov

emen

tsof

design

withresp

ectto

DRT

analyses

No.

Cas

eStudy

Com

men

tson

Mecha

nism

Fea

turesof

Data

DesignFea

tures

Sugg

estedIm

prov

emen

tsto

Design

1Tailflick

Ana

lgetic

drug

action

smay

invo

lvecentral(spina

l,su

pras

pinal)as

wellas

periph

eral

compo

nents

Bas

eline,

peak

shift,

saturation

,ab

sorp

tion

rate–limited

duration

ofresp

onse,timeseries

ofresp

onse,multipledo

sean

droutech

alleng

es

3an

d10

mgi.v

.bolus

esan

d10

,50,

and10

0mgs.c.;

10-mgdo

sesa

mefori.v

.an

ds.c.

dosing

Fne

edsi.v

.and

s.c.

data

Kamay

requ

irebe

tter

resolution

ofbo

thon

setan

doffset

ofresp

onse

Krequ

ires

better

resolution

ofthe

offset

ofresp

onse

k inrequ

ires

robu

stba

seline

data

k outde

term

ines

onsetan

doffset

ofresp

onse

SD

50de

term

ines

duration

and

offset

ofresp

onse

Smaxde

term

ines

intens

ityof

resp

onse

nHwillimpa

ctthestee

pnessof

the

rise

andde

clineof

resp

onse

2Locom

otor

activity

stim

ulation

Indirect-actingdo

pamine

receptor

stim

ulan

t(relea

ser);

psycho

stim

ulan

t

Noba

seline

,pe

aksh

ift,

zero-ord

errise

and

declineof

resp

onse,

saturation

ofbu

ildu

pan

dloss,high

-resolution

resp

onse-tim

eseries

3.12

and5.82

mg/kg

give

ni.p

.F

biop

hase

availability

need

si.v

.an

di.p

.da

ta

K’ne

edsi.v

.an

di.p

.da

tato

discriminatebe

tween

absorp

tion

anddisp

ositionrate-

limited

processes

k outne

edsreas

onab

leresolution

duringon

set

k Mne

edsreas

onab

leresolution

oflow

resp

onse

data

SD

50requ

ires

robu

stda

taon

the

offset

ofresp

onse

Smaxrequ

ires

data

atmax

imum

resp

onse

nHrequ

ires

robu

stda

taat

thetime

ofrapidrise

andde

cline

The

presen

tde

sign

may

sufficefor

assessmen

tof

theacute

locomotor

resp

onse.Tocaptur

ech

ronicpa

tterns

intheresp

onse,

extend

edtest

compo

und

expo

sure

duringseve

ralda

ysis

nee

ded,

whichin

turn

may

also

requ

iread

justmen

tof

themod

eliffunc

tion

alad

aptation

occu

rs(see

case

stud

y8)

(con

tinued

)


TABLE

12—Con

tinued

No.

Cas

eStudy

Com

men

tson

Mechan

ism

Fea

turesof

Data

DesignFea

tures

Sugg

estedIm

prov

emen

tsto

Design

3Fatty

acid

resp

onse

Antilipolytic

Bas

eline,

saturation

,rebo

und,

toleranc

e,multipleinterven

tion

s

Multiplei.v

.infusion

spluswas

hou

tKrequ

ires

robu

stda

taon

theon

set

andoffset

ofresp

onse

R0requ

ires

robu

stda

taat

baseline

k outrequ

ires

robu

stda

taat

onset

andoffset

ofresp

onse

k tolrequ

ires

robu

stda

tadu

ring

extend

edex

posu

reto

drug

ID50requ

ires

robu

stda

taon

the

offset

ofresp

onse

nHrequ

ires

robu

stda

taat

thetime

ofrapidrise

andde

cline

The

presen

tde

sign

may

sufficefor

assessmen

tof

theacute

antilipo

lyticresp

onse.To

captur

ech

ronicpa

tterns

inthe

resp

onse,ex

tend

edex

posu

reto

test

compo

unddu

ring

seve

ral

days

isne

eded

,which

also

requ

ires

adjustmen

tof

the

mod

el(see

case

study

8)4

Psych

osis

score(B

PRS)

Cen

tral

dopa

minereceptor

antago

nism

Baseline,

timeto

PD

steady

state,

categorizing

respon

ders,o

nset

ofaction

120–

600mgda

ilyp.o.

dosing

k inrequ

ires

robu

stda

taat

baseline

k outrequ

ires

robu

stda

taat

onset

I maxrequ

ires

data

atmax

imum

resp

onse

Exten

dedda

tabe

yondthe

observationa

ltimerang

emay

cast

ligh

ton

potentialfun

ctiona

lad

aptation

.Bas

elineda

taof

BPRSan

dplaceb

ocompa

risons

arealso

necessaryforcorrect

assessmen

tof

pure

drug-

indu

cedresp

onse

inde

pende

ntly

ofconc

entration-dr

iven

orDRT

appr

oach

5Bacterial

grow

than

dkill

Antim

icrobial

Bas

eline,

grow

thlimit,

saturation

,first-orde

rgrow

th/kill

Multiplei.v

.do

ses

k grequ

ires

robu

stve

hiclecontrol

data

k krequ

ires

killan

dregrow

thda

tafrom

twoor

moredo

ses

6Cortisolreleas

edu

ringan

dafterex

posu

reto

ACTH

Phy

siolog

icinterp

layin

the

cortisol

axis.Sug

gested

mecha

nism

smay

also

invo

lvech

ange

sin

cortisol

hydr

oxylation,

and/or

other

unkn

own

action

sof

ACTH

onsteroido

gene

sis

Bas

eline,

rebo

und,

toleranc

eturn

over,

endo

geno

usag

onist,

squarewav

eof

ACTH

interven

tion

Exp

erim

entalmod

elof

ACTH

expo

sure

driving

thereleas

erate

ofcortisol

k inrequ

ires

robu

stda

taat

baseline

(con

tinued

)


TABLE

12—Con

tinued

No.

Cas

eStudy

Com

men

tson

Mechan

ism

Fea

turesof

Data

DesignFea

tures

Sugg

estedIm

prov

emen

tsto

Design

k outrequ

ires

robu

stda

taat

onset

SD

50requ

ires

data

during

offset

ofresp

onse

Smaxrequ

ires

data

atmax

imum

intens

ity

nHrequ

ires

robu

stda

taat

thetime

ofrapidrise

andde

cline

Lon

g-term

adap

tation

requ

ires

extend

edex

posu

reto

test

compo

undov

ertime.

Oscillatory

beha

vior

inex

perimen

talda

tamay

need

amod

elex

tens

ion

similar

toAhlström

etal.(20

11)

7Mioticresp

onse

inthecat

eye

Prosten

oidF

receptor

interactionin

thesm

ooth

muscle

oftheiris

Bas

eline,

partialinhibition

,sa

turation

,rapid

equilibrium

mod

el

Thr

eedo

ses(0.1,1.0,

and

10mglatano

prost)

give

ntopically

Karequ

ires

robu

stda

taat

onset

andoffset

Krequ

ires

robu

stda

taat

onset

andoffset

t lagrequ

ires

robu

stda

taat

onset

ID50requ

ires

data

ontheoffset

I maxrequ

ires

data

atmax

imum

intens

ity

nHrequ

ires

robu

stda

taat

thetime

ofrapidrise

andde

cline

Lon

g-term

adap

tation

ofthe

pharmacolog

ical

resp

onse

requ

ires

extend

edex

posu

reto

test

compo

undov

ertime

8Meta-an

alysis

offattyacid

andinsu

linresp

onse

tomultiple

NiA

cpr

ovocations(N

LME

analysis)

Antilipo

lytic

Bas

eline,

saturation

,rebo

und,

toleranc

eturn

over,multiple

biom

arke

rs,multiple

drug

prov

ocations

,en

doge

nous

agon

ist,

inpu

t-stim

ulated

was

hout

profile

Multiplei.v

.an

ds.c.

doses,

rates,

andmod

esof

NiA

cpluswas

hou

t;multiple,

intertwined

,biom

arke

rresp

onses

For

athorou

ghde

scriptionmod

elan

dsu

ggestion

sab

out

expe

rimen

talde

sign

and

iden

tifiab

ilityof

KaM,Vmax,

K,

k in,k o

ut,SD

50,S

maxan

dnH,see

And

ersson

etal.(201

7).System

parametersafterDRT

analysis

arestillreas

onab

lycons

istent

withestimates

from

aconc

entration-dr

iven

analysis

NLME,n

onlinea

rmixed

effects.


TABLE

13Com

pilation

ofDRT

data

analyses

intheliterature

Exp

erim

entalArea

TestCom

pound

Rou

teBioph

aseParam

eter

Rea

sonforDRT

Referen

ce

AChE

inhibitionby

Sarin

Ade

nosineA1ag

onist

s.c.,i.m

.First-ord

erinpu

t/ou

tput

Noex

posu

reda

taBue

ters

etal.(200

3)

Anem

iaIn

ducedby

riba

virin

p.o.

First-ord

erou

tput

Noex

posu

reda

taTod

etal.(200

5)EPO

s.c.

Fas

tEPO

expo

sure

Ueh

ling

eret

al.(19

92),Portet

al.

(199

8)Antiarrh

ythmics

Amioda

rone

i.v.bo

lusplus

infusion

First-ord

erinpu

t/ou

tput

Noex

posu

reda

taSalem

etal.(201

6)

p.o.

Anticoag

ulants

Warfarin

p.o.

First-ord

erinpu

t/ou

tput

Spa

rseex

posu

reHam

berg

etal.(20

10,20

13),

Wrigh

tan

dDuffull(201

1)p.o.

First-ord

erinpu

t/ou

tput

Noex

posu

reda

taKim

etal.(20

15),Ooi

etal.(20

17)a

Antidep

ressan

tsClomipramine,

lithium

p.o.

First-ord

erinpu

t/ou

tput

Noex

posu

reda

taGru

wez

etal.(20

07)

Parox

etine,

pind

olol

p.o.

First-ord

erou

tput

Noex

posu

reda

taGru

wez

etal.(20

05)

TC-173

4s.c.

First-ord

erinpu

t/ou

tput

Utility

ofDRT

Gatto

etal.(200

4)Antilipo

lytic

NiA

ci.v

.infusion

First-ord

erou

tput

Utility

ofDRT

Isak

sson

etal.(200

9),Gab

rielsson

andWeine

r(201

0),And

ersson

etal.(201

6)i.v

.infusion

,p.o.

Micha

elis–Men

ten

inpu

t(p.o.)/first-orde

rou

tput

Utility

ofDRT

Antino

ciceptive

Morph

ine

i.v.infusion

First-ord

erou

tput

Noex

posu

reda

taAbo

uHam

mou

det

al.(200

9)a

Exp

erim

entaldr

ugi.v

.infusion

,s.c.

First-ord

erinpu

t/ou

tput,

biop

hase

availability

Utility

ofDRT

Gab

rielsson

etal.(200

0),

Gab

rielsson

andPeletier(201

4)

Arthr

itis

Fosda

grocorat,pr

ednison

ep.o.

First-ord

erou

tput

NoPDN

expo

sure

Sho

jiet

al.(20

17)a

CNSstim

ulan

tDex

amph

etam

ine

i.p.

Utility

ofDRT

Gab

rielsson

andWeine

r(201

6)Caffeine

p.o.

First-ord

erinpu

t/ou

tput

Noex

posu

reda

taRam

akrish

nanet

al.(201

3)COPD

andas

thma

Salmeterol

PUL

First-ord

erou

tput

Noex

posu

reda

taMus

uamba

etal.(20

15)

Salmeterol,PF-006

1035

5PUL

Equ

alinpu

t/ou

tput

rate

Noex

posu

reda

taNielsen

etal.(201

2)a

Exp

erim

entalcompo

und

PUL

First-ord

erinpu

t/ou

tput

Noex

posu

reda

taWuet

al.(20

11)a

LABA

bron

chod

ilators

PUL

Noex

posu

reda

taAgo

ram

etal.(200

8)Cortisolsecretion

Adr

enocorticotrop

ini.v

.infusion

Squ

arewav

eof

inpu

tSho

wutility

ofDRT

Gab

rielsson

andPeletier(201

4)Diabe

tes

Glucokinas

eactiva

tor

p.o.

First-ord

erinpu

t/ou

tput

Noex

posu

reda

taJa

uslinet

al.(20

12)

Diabe

ticmacular

edem

aTriam

cinolon

eaceton

ide

Intrav

itreal

Aud

renet

al.(200

4)

Glaucoma

PF-044

7527

0Top

ical

Lim

ited

expo

sure

Luu

etal.(200

9)Hyp

er-

parathyroidism

R-568

p.o.

First-ord

erinpu

t/ou

tput

Noex

posu

reda

taLalon

deet

al.(199

9)

Hyp

er-

phen

ylalan

inem

iaTetrahy

drob

iopterin

i.v.infusion

First-ord

erou

tput

Noex

posu

reda

taTrefz

etal.(201

5)a

Hyp

oten

sion

Sod

ium

nitrop

russide

i.v.infusion

First-ord

erou

tput

Fas

tSNP

expo

sure

Barrett

etal.(201

5)Mus

clerelaxa

ntTubo

curarine

i.m.

First-ord

erou

tput

Noex

posu

reda

taLev

y,19

64)

Vecuronium

i.v.bo

lus

First-ord

erou

tput

Noex

posu

reda

taBragg

etal.(199

4),Fishe

ran

dWrigh

t(199

7),Warwicket

al.

(199

8)Myd

riatics

Tridihe

xethyl

chloride

trop

icam

ide

p.o.,topical

Noex

posu

reda

taSch

oenw

aldan

dSmolen

(197

1),

Smolen

(197

1a,19

76b,

1978

),Smolen

andWeiga

nd(197

3)

(con

tinued

)


TABLE

13—

Con

tinued

Exp

erim

entalArea

TestCom

pound

Rou

teBioph

aseParam

eter

Rea

sonforDRT

Referen

ce

Myo

card

ialnecrosis

Indu

cedby

hand

ling

Mikae

lian

etal.(20

13)a

Obe

sity

PF-052

3102

3i.v

.injection

First-ord

erinpu

t/ou

tput

Noex

posu

reda

taTho

mps

onet

al.(201

6)a

Oncology

Cap

ecitab

ine

p.o.

Noex

posu

reda

taPau

leet

al.(20

12)a

Cap

ecitab

ine/do

cetaxe

lp.o.

(cap

ecitab

ine),

i.v.(docetax

el)

Noex

posu

reda

taFranc

eset

al.(201

1)

Carbo

platin,pe

gylated

lipo

somal

doxo

rubicin,

paclitax

el

i.v.infus

ion

First-ord

erou

tput

Noex

posu

reda

taWilba

uxet

al.(201

4)a

Carbo

platin,

cyclop

hosph

amide,

thiotepa

i.v.infus

ion

First-ord

erou

tput

Noex

posu

reda

taRam

on-Lop

ezet

al.(200

9)a

Chem

otherap

yhormon

otherap

yi.v

.infus

ion

First-ord

erou

tput

Noex

posu

reda

taWilba

uxet

al.(201

5)a

Cisplatin,carbop

latin,

etop

oside

i.v.infus

ion

First-ord

erou

tput

Noex

posu

reda

taBuil-Bru

naet

al.(201

4)a

Cya

A-E

7i.v

.infus

ion

First-ord

erou

tput

Noex

posu

reda

taParra-G

uillen

etal.(201

3)a

Gem

citabine

,carbop

latin

i.v.infus

ion

First-ord

erou

tput

Noex

posu

reda

taTha

met

al.(200

8)Irinotecan

,MBLI87

i.p.

First-ord

erou

tput

Noex

posu

reda

taSostellyet

al.(201

4)a

Paclitaxe

l,nab

-paclitaxe

l,ixab

epilon

ei.v

.infus

ion

First-ord

erou

tput

Noex

posu

reda

taMeh

rotraet

al.(201

7)

Tem

ozolom

ide

p.o.

First-ord

erou

tput

Noex

posu

reda

taMaz

zoccoet

al.(201

5),Bog

da� nsk

aet

al.(20

17)

Onc

olog

yan

dau

toim

mune

diseas

e

Mon

oclonal

antibo

dies

i.v.infus

ion,

s.c.

First-ord

erinpu

t/ou

tput

Lan

gean

dSch

midli(201

4,20

15)

Osteopo

rosis,

nutrient

supp

lemen

tation

Bisph

osph

onate,

denosumab

i.v.infus

ion,

p.o.

First-ord

erou

tput

Noex

posu

reda

tava

nSch

aick

etal.(201

5)

Iban

dron

ate

i.v.infus

ion,

p.o.

First-ord

erou

tput

Noex

posu

reda

taPillaiet

al.(20

04),Reg

inster

and

Gieschke

(200

6),Zaidi

etal.

(200

6)Calcium

p.o.

First-ord

erinpu

t/ou

tput

Noex

posu

reda

taAhn

etal.(201

4)a

Psych

omotor

Phen

cyclidine,

dextroam

phetam

ine

s.c.

First-ord

erinpu

t/ou

tput

Noex

posu

reda

taJa

cobs

etal.(20

10)a

The

rmoreg

ulation

8-OH-D

PAT

p.o.

First-ord

erinpu

t/ou

tput

Noex

posu

reda

taGab

rielsson

etal.(200

0)Thrombo

cytope

nia

Carbo

platin,radiation

i.v.infus

ion

Noex

posu

reda

taKrzyz

ansk

iet

al.(201

5)Rom

iplostim

s.c.

First-ord

erou

tput

Noex

posu

reda

taPerez-R

uixo

etal.(201

2)Virusinfection

HPAIH5N

1viru

sNoex

posu

reda

taKitajim

aet

al.(201

1)

8-OH-D

PAT,7

-(dipr

opylam

ino)-5,6,7,8-tetrahyd

rona

phthalen

-1-ol;AChE

,acetylcho

line

steras

e;CNS,cen

tral

nervou

ssystem

;COPD,chr

onic

obstru

ctivepu

lmon

arydiseas

e;EPO,e

rythropo

ietin;H

PAI,highly

pathog

enic

Asian

avianinfluen

za;L

ABA,lon

g-actingbe

taag

onists;M

BLI87,

acrido

nede

riva

tive

unde

rgoingpr

e-clinical

deve

lopm

ent;PDN,p

redn

ison

e;PF-044

7527

0,5-{3-[(2S)-2-{(3R

)-3-hy

drox

y-4-[3-(trifluorom

ethy

l)ph

enyl]butyl}-5-ox

opyrrolidin-

1-yl]propy

l}thioph

ene-2-carbox

ylate(PF-044

7527

0);P

F-052

3102

3,along-actingFGF21

analog

;PF-006

1035

5,be

ta-adr

enocep

torag

onist;PUL,p

ulm

onary;

R-568

,calcimim

etic

R-568

,atype

IIindirect

allosteric

CaS

Rmod

ulator;

TC-173

4,nicotinic

acetylch

olinereceptor

mod

ulator.

aDen

otes

publicationsreferringto

theK-PD

appr

oach

butap

plyingthebiop

has

eam

ountAboriginally

prop

osed

bySmolen

(197

1a).


TABLE

14Com

pilation

ofch

alleng

ingex

perimen

talsituations

forDRT

data

analyses

Challenge

MainProblem

Poten

tial

Solution

Rea

sonforCau

tiou

sApp

lication

ofDRT

Referen

ce

Endo

genou

sag

onists

Nosystem

icex

posu

reto

test

compo

und

Multipleinpu

troutes

andrates

may

reve

altherelative

biop

hase

availability

Difficu

ltto

discriminatebe

tween

endo

genou

san

dex

ogen

ous

contribu

tion

toPD

And

ersson

etal.(201

7)

Tim

e-va

rying

baseline

Impa

ctof

potential

endo

geno

usag

onist(s)

Dose-rang

efind

ingstud

ies

includ

ingthezero

dose

Noex

posu

reda

taon

endo

genou

sag

onist(s)

Lalon

deet

al.(199

9)

Com

bination

therap

ies

Nosystem

icex

posu

reof

the

individu

aldr

ugs

Systematic

dose-ran

gefind

ing

(inc

luding

zero

dose,differen

troutes,r

ates,a

ndmod

es)de

sign

islack

ing

Ifindividu

alpo

tenc

iesne

edto

beestablishe

d,thecontribu

tion

has

tobe

basedon

expo

sure

mea

sures

Gru

wez

etal.(20

05,20

07)

Multipletarget

sites

The

contribu

tion

ofea

chtarget

difficultto

assess

ifex

posu

remea

suresare

not

know

n

Multipleag

onistan

dan

tago

nist

dose

leve

lsmay

beap

plied

Ifmultipleag

onistan

dan

tago

nist

dose

leve

lsareno

tav

ailable

And

ersson

etal.(201

7)

Syn

ergistic

system

sNosystem

icex

posu

reof

the

individu

aldr

ugs

Systematic

dose-ran

gefind

ing

(inc

luding

zero

dose,differen

troutes,r

ates,a

ndmod

es)de

sign

islack

ing

Whe

nasystem

atic

dose-ran

gefind

ing(inc

luding

zero

dose)

design

islack

ing

Wilba

uxet

al.(20

15)

Safetystudies

Exp

osur

eto

test

compo

und

not

know

nan

dis

nee

ded

forsa

fety

assessmen

t

Systematic

dose-ran

gefind

ing

(inc

luding

zero

dose,differen

troutes,r

ates,a

ndmod

es)de

sign

islack

ing

Difficu

ltto

assess

safety

margin

witho

utex

posu

remea

suresor

biom

arke

r

Ahnet

al.(201

4)

Oligo

nucleotide

kine

tics

Extremedisp

osition

patterns

inplas

ma

Systematic

dose-ran

gefind

ing

(inc

luding

zero

dose,differen

troutes,r

ates,a

ndmod

es)de

sign

islack

ing.

Use

ofex

posu

re-

resp

onse

mod

el(e.g.,PBPK)as

onebu

ilding

block

Whe

nasystem

atic

dose-ran

gefind

ing(inc

luding

zero

dose)

design

islack

ing

—

Tim

e-de

pende

nt

chan

gesin

disp

ositionor

target

parameters

Exp

osure

totest

compo

und

not

know

nSystematic

dose-ran

gefind

ing

(inc

luding

zero

dose,differen

troutes,r

ates,a

ndmod

es)de

sign

islack

ing

Whe

nasystem

atic

dose-ran

gefind

ing(inc

luding

zero

dose

and

repe

ated

dose)de

sign

islack

ing

And

ersson

etal.(201

7)

Iontop

horetic

system

sExa

ctdr

ugdo

sean

dinpu

tratesareseldom

know

nDose-rang

efind

ingstud

ies

includ

ingthezero

dose,cu

rren

trang

ingstud

ies(see

Liu

etal.,

2016

,foraltern

ativede

sign

)

Whe

nasystem

atic

dose-ran

gefind

ing(inc

luding

zero

dose)

design

islack

ing

Liu

etal.(201

6)

PBPK,ph

ysiologicallyba

sedph

armacok

inetic;P

D,p

harmacod

ynam

ics.


capacity of the DRT approach (Isaksson et al., 2009;Andersson et al., 2017). Not only were the estimatedsystem parameters (kout, ktol, etc.) similar to thoseobtained from a full exposure-driven analysis, but theinteraction between FFAs and insulin was also correctlycaptured. Simultaneous sampling of multiple interact-ing biomarkers (e.g., FFAs and insulin) offers anopportunity to take the analysis to an even higher leveland deconvolute interaction patterns between the bio-markers, but still keep the assessment focus on bothdrug properties (e.g., ED50) and target properties (e.g.,kin and kout) (Andersson et al., 2017). The use ofmultiplesources of biomarker data obtained from diverse pat-terns of drug (NiAc) intervention allowed accurate andprecise parameter estimates. The mechanistic modelmay therefore have a higher translational potential.Some of the preclinical studies use two or more modes

of administration and are more experimental in nature(Schoenwald and Smolen, 1971; Smolen, 1971a,b,1976a,b, 1978; Smolen et al., 1972; Smolen and Wei-gand, 1973; Isaksson et al., 2009; Gabrielsson et al.,2015; Andersson et al., 2017). Smolen (1971a, 1976b)described a model for bioavailability and biokineticbehavior of a mydriatic drug. He extended the analysisto application of DRT data for bioequivalence testing.Others have applied a systems analysis approach(Veng-Pedersen and Modi, 1993). Fasanmade andJusko (1995) developed a kinetic/dynamic model usinga hypothetical reactive metabolite to explain formationof methemoglobin. These studies clearly demonstratethe potential of DRT analyses in a preclinical settingand their applications to drug discovery data. There areseveral clinical examples of how DRT modeling isapplied to analyze (sparse) human data (see Table 13).The statistical methodology of DRT modeling is ele-gantly illustrated with data from clinical studies oftherapeutic biologics in autoimmune disease (Langeand Schmidli, 2014, 2015), and Salem et al. (2016)applied a DRT approach to critically ill patients wheretraditional exposure-driven approaches are difficultand/or unethical. The drug mechanism function wasdriven by the biophase amount rather than the rate.Trefz et al. (2015) developed a pharmacodynamic DRTmodel to improve the description of individual sensitiv-ity to tetrahydrobiopterin responsiveness in neonateswith hyperphenylalaninemia. Abou Hammoud et al.(2009) demonstrated how a DRT model, which charac-terizes the time course of morphine-induced analgesiain the immediate postoperative period, could be used forpain management. Wilbaux et al. (2015) successfullyapplied a DRT approach for the assessment of treat-ment efficacy in metastatic castration-resistant pros-tate cancer, using the count of circulating tumor cells(CTCs) as a promising surrogate marker. The model isthe first to quantify the dynamics of prostate-specificantigen and CTC count during treatment in patientswith metastatic castration-resistant prostate cancer.

It has great potential as a general framework for showingthe combination effects of chemotherapy in other cancertypes to compare the properties of CTCs.

Gruwez et al. (2005, 2007) developed a DRT-interaction model between selective serotonin reuptakeinhibitors and pindolol for the assessment of clinicaltreatment of depression (Montgomery–Åsberg Depres-sion Rating Scale score). They concluded that the DRTmodel of Montgomery–Åsberg Depression Rating Scalescore kinetics was able to capture individual data of aclinical trial and to yield estimates of the typical valuesand the interindividual variability of the parameters. Aclinical trial simulation based on the model illustratedhow it might be used as a tool for clinical trial planningin the field of research on antidepressants and adjunc-tive treatments.

Ahn et al. (2014) used a semimechanistic approach tocharacterize the Ca2+-parathyroid hormone (PTH) sys-tem after intake of thermal spring water containingcalcium or calcium carbonate tablets. Without notice-able differences in plasma Ca2+ levels, the changes inPTH responses were used as a surrogate marker forcalcium absorption. With a more enhanced study de-sign, it is believed that their approach can be applied toevaluate calcium absorption, as determined by PTH-time responses.

A recent review on challenge tests of reaction inducedby Sephadex, interleukin 1b, collagen, lipopolysaccha-ride, and glucose and changes caused in eosinophil count,interleukin 6, paw-swelling, tumor necrosis factor a, andblood glucose, respectively, applied the DRT approach asa substitute to mimic time courses of the challengingagents (Sephadex, collagen, lipopolysaccharide, and glu-cose) when exposure to these challengers was lacking(Gabrielsson et al., 2015). Table 14 shows some de-manding experimental situations that identify problemsrelated to DRT analyses, with literature references topotential solutions.

C. Drug Delivery by Means of Iontophoresis

Drug administration through the skin has beeninvestigated for several decades, and many deliverytechniques have been explored to facilitate drug per-meation across this outer bodily barrier. Among these,iontophoresis is a novel promising drug delivery system,which enhances skin penetration and the release rate ofa number of drugs that are otherwise poorly permeablethrough the skin (for a review, see Iredahl et al., 2015).DRT analyses may play an important role in iontopho-retic systems due to their time- and concentration-wisedetachment from systemic exposure, although informa-tion that fully characterizes the rate and extent of druginput is still a key factor that needs to be sorted out priorto proposing a biophase structure. However, promisingresults were recently collected during histamine ionto-phoresis with laser Doppler monitoring (Liu et al.,2016). This approach allowed a fixed dose of histamine


to be delivered and an objective, continuous, anddynamic measurement of histamine epicutaneous re-sponse to be recorded in children and adults. Thereduced data set was then analyzed by means of aDRT model of the local histamine response.

D. Permutations of Smolen’s Model: The Kinetic-dynamic K-PD Rate of Change Model

DRT analyses were pioneered by Smolen and Levy inthe 1960s and 1970s with a series of classic paperspaving some of the theoretical ground (Levy, 1971;Schoenwald and Smolen, 1971; Smolen, 1971a,b;Smolen et al., 1972; Smolen and Weigand, 1973). Adifferent permutation, denoted kinetic-dynamic K-PD,was more recently suggested (Jacqmin et al., 2007). Thelatter bolus model suggests that the elimination ratefrom the biophase k × Ab drives the pharmacologicalresponse rather than the biophase amount (Ab orexposure) per se (eq. 38).8>>>>>>><

>>>>>>>:

dAb

dt¼ 2 k × Ab

Sðk × AbÞ ¼ 1þ Smax × ðk × AbÞSD50 þ ðk × AbÞ

dRdt

¼ kin × Sðk × AbÞ2 kout × R

ð38Þ

The SD50 is the rate of change of drug in the biophasethat results in half-maximal drug-induced response,which makes this parameter difficult to interpret andcompare across species. There are few examples in thepharmacological literature where the rate of change inbiophase amount rather than the biophase amount perse, is the “driver” of the pharmacological response. Thedivision of SD50 by k in eq. 38 (middle line) may thencause a parameter identifiability problem.In contrast to the biophase bolus model in K-PD,

there are numerous examples in the literature (Table 1)and case studies 1–3, 7, and 8 that demonstrate moreelaborate structures. Several authors refer to the K-PD(k × Ab) approach but have in fact applied the classicSmolen biophase kinetics (Ab) for driving the pharma-codynamic model (Table 13; see footnoted references).

E. General Conclusions

This review aims to raise cross–research and devel-opment discipline awareness about DRT data analysesto explore and improve interpretation of data. Althoughthe DRT methodology is not new, it is still compara-tively rare in experimental and clinical pharmacology.This review examines how the time course of a phar-macological response can be dissected to gain furtherinsight. It emphasizes the significance of time courses,illustrates the importance of multiple dose information,and addresses issues in the examination of time-response relationships. Furthermore, this review shows

how quantitative pharmacology adds significant valueto target turnover information and why ignoring thismay cause misleading results and conclusions. Inconclusion, access to robust and discrete baseline data(absolute numbers rather than differences or otherconversions) from multiple doses, routes, rates, andadministration modes will provide the best prerequisitefor fruitful DRT analysis.

Smolen et al. (1972) stated that

The use of pharmacological data for pharmacokineticsystems analysis would be obviously limited if re-sponses were restricted to only those effects that canbe directly observed and utilized without further, oronly minimal, treatment. When an assay for a drug inbiological media does not exist or is difficult to perform,it can be considered that the development of a pharma-cological method could instead provide the most expe-ditious means of accomplishing a pharmacokineticanalysis of the drug’s behavior.

He also suggested that “the use of pharmacologicaldata for bioavailability studies should be consideredirrespective of whether an assay method for the drugexists. The decision to use either or both types of datashould depend on the relative sensitivity precision,convenience, and economy” Smolen (1976b). It is ourhope that our account will further underscore hissentiments, and that the examples and suggestedsolutions in this article will inspire (and renew interestin) DRT analysis of pharmacodynamic responses.

Authorship Contributions

Participated in research design: Gabrielsson, Andersson, Jirstrand,Hjorth.Conducted experiments: Gabrielsson.Performed data analysis: Gabrielsson, Andersson.Wrote or contributed to the writing of the manuscript: Gabrielsson,

Andersson, Jirstrand, Hjorth.

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