Pattern Recognition in Pharmacokinetic Data Analysis
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Abstract
Pattern recognition is a key element in pharmacokinetic data analyses when first selecting a model to be regressed to data. We call this process going from data to insight and it is an important aspect of exploratory data analysis (EDA). But there are very few formal ways or strategies that scientists typically use when the experiment has been done and data collected. This report deals with identifying the properties of a kinetic model by dissecting the pattern that concentrationtime data reveal. Pattern recognition is a pivotal activity when modeling kinetic data, because a rigorous strategy is essential for dissecting the determinants behind concentrationtime courses. First, we extend a commonly used relationship for calculation of the number of potential model parameters by simultaneously utilizing all concentrationtime courses. Then, a set of points to consider are proposed that specifically addresses exploratory data analyses, number of phases in the concentrationtime course, baseline behavior, time delays, peak shifts with increasing doses, flipflop phenomena, saturation, and other potential nonlinearities that an experienced eye catches in the data. Finally, we set up a series of equations related to the patterns. In other words, we look at what causes the shapes that make up the concentrationtime course and propose a strategy to construct a model. By practicing pattern recognition, one can significantly improve the quality and timeliness of data analysis and model building. A consequence of this is a better understanding of the complete concentrationtime profile.
KEY WORDS
absorption area under the curve biexponential halflife induction intravenous and extravascular dosing lag time monoexponential multicompartment nonlinear elimination plasma concentrationtime courses targetmediated drug disposition transportersINTRODUCTION
Pattern recognition is a key element in pharmacokinetic data analyses when first selecting a model to be regressed to data. We call this process going from data to insight. But there are no formal best practices that scientists typically use. This report deals with identifying the properties of a kinetic model by dissecting the pattern that concentrationtime data reveal graphically. Pattern recognition is a pivotal activity when modeling pharmacokinetic data, because a rigorous strategy is essential for dissecting the determinants behind concentrationtime courses. In the pharmacology field, pattern recognition has also been proposed for interpreting results of drugdrug interactions (1).

In case studies 1 and 2, we explore intravenous iv bolus dosing of a one (two subjects with different clearances and similar volumes) and two compartments (two populations, clamped and normal with different clearances and effective halflives).

Then, we move on to extravascular dosing in case studies 3 and 4, where a onecompartment firstorder input/output systems with and without lag time, and iv and extravascular dosing for a twocompartment system, respectively.

In case study 5, we revisit extravascular (po) dosing although we observe rapid absorption, lag time, and multiexponential decline postpeak. This dataset does not contain iv data.

Case study 6 returns to iv bolus dosing and monoexponential decline in plasma but is extended also with urinary data so renal clearance or fraction excreted into the urine can be estimated simultaneously with total clearance.

Case studies 7 and 8 are also iv dosing covering mono and biexponential decline for two systems that exhibit saturable (nonlinear) clearance terms.

Two nonlinear systems are also presented after oral dosing in case studies 9 and 10. The former captures nonlinear elimination and the latter saturable absorption via transporters.

We extend the analyses by looking at iv (case study 11) and subcutaneous (sc) (case study 12) of two systems with endogenous levels of the test compound. This extends the model with one additional parameter, the turnover rate.

The last four case studies (case studies 13–16) are extensions looking at targetmediated drug disposition (case study 13), timedependent induction of clearance (case study 14), simultaneous fitting of nonlinear multicompartment kinetics of parent compound and metabolite after three different iv doses, and then finally fitting a first and zeroorder absorption model to an oral dosing dataset.
We encourage the analyst to regress several sources of data simultaneously if possible. In case study 4, a full twocompartment system is revealed in iv data but not in ev data. This dataset is contrasted with case study 5 where only ev data are available. However, the latter still displays a multicompartment behavior due to the rapid absorption. Both case studies 4 and 5 are commonly encountered situation which is the reason why they may be of interest to the reader. Several sources of data are further elaborated on in case studies 6, 7, 9, 10, 13, and 15. Further comparisons across two or more case studies are done at the end of relevant cases.
A set of points to consider are proposed that specifically addresses exploratory data analyses, number of phases in the concentrationtime course, convex or concave curvature, baseline behavior, time delay, lag time, peak shifts with increasing doses, flipflop phenomena, saturation, and other potential nonlinearities that the eye catches in the data. We look at what causes the shapes that make up the concentrationtime course. By practicing pattern recognition, one can significantly improve the quality of data analysis and model building. A consequence of this is a better understanding of the complete concentrationtime profile. We have therefore collected a set of patterns extracted from literature data and then modified them so the typical features emerge more clearly (2). We also propose alternative solutions to the patterns.
Equation 1 is applicable if sufficient and accurate data are obtained and can be extended. For example, if we have data after an intravenous bolus dose that decline in a biexponential fashion (i.e., with an α and a β phase), then it is possible to estimate 2 EX = 4 parameters (i.e., A, α, B, and β). If we also have measured drug excretion in urine, we could estimate renal clearance or fraction of dose excreted via the urine PE. If, in addition to the biexponential decline in plasma 2 EX, we have a nonlinear feature NL and urinary data PE, we might be able to estimate six parameters.
Equation 1 was suggested as a practical tool for the identification of a possible number of estimable parameters based on observable patterns of the drug moiety as such in plasma. We will also see how applicable the extended Eq. 1 is to the 16 datasets provided. Equation 1 should be used with careful attention to the quality of data, the design of the experiment, and when the number of data sources beyond parent compound in plasma and urine are available. Simultaneous fitting of all available sources of data is always recommended.
This report focuses on the practical identifiability of pharmacokinetic parameters based on visual inspection of experimental data. First, we extend a commonly used relationship for calculation of the number of potential model parameters by simultaneously utilizing all concentrationtime courses. Then, a set of points to consider is proposed that specifically addresses exploratory data analyses, number of phases in the concentrationtime course, baseline behavior, time delays, peak shifts with increasing doses, flipflop phenomena, saturation, and other potential nonlinearities that an experienced eye catches in the data. Finally, we set up a series of equations related to the patterns and themes. Theoretical a priori identifiability of pharmacokinetic parameters has been discussed by others (4) and will not be addressed here.
Case study 1
dC/dt is the rate of change of the plasma concentration, C is the plasma concentration, and K is the firstorder rate constant associated with the elimination process. A new parameter, clearance Cl, is then introduced. Clearance is defined as the volume of blood or plasma that is totally cleared of its content of drug per unit time (mL min^{−1} or L min^{−1}). The other parameter of primary interest is the volume of distribution V. This is the apparent space that test compound distributes into.
Should the two subjects have had the same clearance but different volumes, the AUCs would have been the same and the intercepts different. So, by inspecting the shapes (slopes and intercepts and areas), one can make conclusions about the relative clearances and volumes.
The key features of this pattern are monoexponential decline in plasma of two subjects with different clearances and same volumes. Applying Eq. 1, the number of parameters NP that can be estimated from the data is two for each subject (=2 EX such as Cl and V or K and V).
Case study 2
A biexponential decline is observed in plasma after an intravenous bolus dose to two different populations of rats (Fig. 4, case study 2 (2)). Both populations received the same amount of test compound (20 mg). Test compound is a large molecular weight chemical which is primarily cleared via the kidney and a small fraction is metabolized. One group of rats (denoted diseased animals) had the blood supply to and from the kidneys shut off by a clamp.
dC/dt and dC _{t} /dt are the rate of change of test compound in plasma and tissue compartments, C is the plasma concentration, C _{t} the tissue concentration, Cl plasma clearance, V _{c} central volume, V _{t} peripheral volume, and Cl_{ d } the intercompartmental distribution parameter. Cl_{ d } has the units of volume per time (mL min^{−1} or L min^{−1}) and is related to transport of test compound out into the tissues via the blood flow, transporters, and diffusion/convection forces. The total volume of distribution V _{ss} is the sum of V _{c} and V _{t}, which is the apparent space that test compound distributes into in a twocompartment system.
We typically observe and measure what is going on in the central compartment represented by plasma, but we need the additional peripheral compartment to make up for the two phases observed in the plasma concentrationtime course and the input rate from the gastrointestinal region. In this case study, there is still a distinct difference between the initial and terminal phases. In other situations, one may want to reduce or increase the number of exponentials. Model discrimination then has to be based on residual analysis, goodnessoffit criteria (objective function value, Akaike information criterion), parameter precision, and correlation (see Gabrielsson and Weiner, 2010, for a discussion (2)).
The key features of the studied patterns are biexponential decline in plasma of two groups of animals with different clearances. In spite of similar terminal halflives, their effective halflives differ almost tenfold. If we then apply Eq. 1, the number of estimable parameters is four (=2 EX = 2 × 2 = 4 namely Cl, Cl_{ d }, V _{c}, and V _{t}).
Case study 3
This case study deals with a typical pattern observed for an orally administered test compound that displays a delay in the onset of absorption. Experimental data are shown together with two modelpredicted time courses in Fig. 4 (case study 3 (2)). The most obvious signature in experimental data is a slight time delay of 10–20 min before plasma test compound concentrations start to rise, followed by a rapid rise and a peak concentration at about 60 min. Experimental data are then obtained up to 6 h after dosing. By plotting the data on a semilogarithmic scale, one observes a monoexponential postpeak decline which suggests that a onecompartment model with firstorder input (absorption) / output (elimination) may be a good start. Data reveal that when the terminal phase is compared to the disposition of test compound after intravenous dosing absorption ratelimited elimination prevails.
dC/dt is the rate of change of the plasma concentration, C is the plasma concentration, F is bioavailability, K _{a} absorption rate constant, V volume, and Cl is clearance (or reparameterized with the elimination rate constant K = Cl/V) associated with the elimination process. D _{po} and t _{lag} denote oral dose and lag time, respectively.
In this case study, there is a distinct difference between the lag time (model of choice) and no lag time models (systematic deviations throughout the modelpredicted concentrationtime course), which is already shown in the function plots. In other cases, this may not be so obvious and model discrimination then has to be based on residual analysis, goodnessoffit criteria (objective function value, Akaike information criterion), parameter precision, and correlation (see Gabrielsson and Weiner, 2010, for a discussion (2)).
Four parameters that are estimable from the data (K _{a}, t _{lag}, K, and V/F). For this case study, absorption ratelimited elimination is shown in the terminal portion of the oral data—also known as the flipflop pharmacokinetics.
Case study 4
This dataset shows a biexponential decline after intravenous dosing which is very much masked when test compound is given via the oral route (Fig. 4, case study 4 (2)). If the rate of absorption is relatively slow, oral data typically display a monoexponential decline postpeak. When iv data are added to the picture, a clearer picture of the twocompartment disposition emerges. Iv data are needed to correctly analyze the po data which shows a weak tendency of biexponential decline after C _{max}. We may also want to fit a lag time model for the oral absorption process.
Input_{po}, F, D _{po}, and K _{a} are the input rate, bioavailability, oral dose, and absorption rate constant, respectively. The dC/dt and dC _{t} /dt are the rate of change of test compound in plasma and tissue, C is the plasma concentration, C _{t} the peripheral concentration, Cl plasma clearance, V _{c} central volume, V _{t} peripheral volume, and Cl_{ d } the intercompartmental distribution parameter.
We typically observe and measure what is going on in the central compartment represented by plasma, but we need the additional peripheral compartment to make up for the two phases observed in the plasma concentrationtime course and the input rate from the gastrointestinal region.
The number of estimable parameters from data is then seven, namely Cl, Cl_{ d }, V _{c}, V _{t}, F, K _{a}, and t _{lag}.
Case study 5
Input_{po}, F, D _{po}, and K _{a} are the input rate, bioavailability, oral dose, and absorption rate constant, respectively. The dC/dt and dC _{t} /dt are the rate of change of compound in plasma and tissue, C is the plasma concentration, C _{t} the peripheral concentration, Cl plasma clearance, V _{c} central volume, V _{t} peripheral volume, and Cl_{ d } the intercompartmental distribution parameter.
We first fitted a no lag time model to the data which then failed to acceptably fit the upswing, peak, and initial postpeak phase. By adding a lag time, the systematic deviations were removed.
The number of estimable parameters from data is six, namely Cl/F, Cl_{ d }/F, V _{c}/F, V _{t}/F, K _{a}, and t _{lag}.
Case study 6
Data were collected from both plasma and urine after an intravenous bolus dose of test compound X (Fig. 6, case study 6 (2)). A monoexponential decline in plasma concentration coupled to cumulative amount in urine was modeled by a onecompartment drug model with a urine compartment as one of the elimination pathways. Plasma and urinary data were simultaneously fitted by a simple differential equation model (Eq. 12).
A model for cumulative amount of drug excreted into urine is selected in combination with the onecompartment plasma model. Equation 12 includes three parameters, of which Cl occurs in both the plasma and cumulative urine equations. Therefore, try whenever possible to utilize as many sources of data (such as plasma concentrations and urine amounts) simultaneously when fitting a model to the data to increase accuracy and precision of the parameter estimates. The model for cumulative amount in urine has a very robust model structure and generally allows accurate and precise estimation of f _{ e } or Cl_{ R } in our experience.
The number of estimable parameters from data is three, namely Cl, V, and Cl_{ R } or f _{ e }.
Case study 7
A convex decline is observed in two concentrationtime profiles after intravenous dosing of two doses to two individuals (Fig. 6, case study 7 (2)). This pattern shows a typical signature that cannot be modeled by adding more exponential terms. Since the slope of decline gets shallower in both subjects the higher the plasma concentration becomes, it suggests some kind of nonlinearity in possibly the elimination process. One typically would think of capacitylimited elimination (often called dose or concentrationdependent elimination), but if total concentrations are displayed, this could also be explained by saturable plasma protein binding. In this case study, we fit a model with capacitylimited (MichaelisMenten) elimination to the data. Note that the terminal portions of the concentrationtime courses have different slopes. Dosenormalized concentrations displayed the same initial concentration but deviated from each other during the remaining concentrationtime courses. We suggest that volume V and maximum rate of elimination V _{max} are the same for the two subjects but that their K _{m} values differ since total concentrationtime data are available for this highly plasmabound test compound. The underlying assumption is that K _{m} is more affected by plasma protein binding differences across subjects than V _{max}. We also tested using the same K _{m} but different V _{max} parameters for the two subjects but that failed to produce an acceptable fit (see Gabrielsson and Weiner, 2010, for a detailed description of the analysis).
C and Cl_{MM} are the plasma concentrations and MichaelisMenten type of capacitylimited clearance, respectively, and V _{max} and K _{m} are maximum rate of elimination and the MichaelisMenten constant. It is the MichaelisMenten expression in Eq. 14 that allows the model to capture the lack of superposition across doses and the timedependent halflife.
NL_{1} and NL_{2} are the observed different nonlinear placements in the two datasets that Eq. 14 was simultaneously fitted to. The number of estimable parameters from data is four, namely V _{max}, V, K _{m1}, and K _{m2}.
Case study 8
The kinetics of ethanol was characterized following a 30min constant rate intravenous infusion (Fig. 6, case study 8 (5)). A number of volunteers were infused intravenously with a dose of 0.4 g ethanol per kg body weight. Plasma samples were obtained for 7 h. Ethanol displayed capacitylimited clearance and a volume of distribution equal to total body water. This problem highlights some of the complexities in modeling multicompartment disposition with nonlinear capacitylimited elimination. For details about study design and a review of ethanol kinetics, see Norberg et al. (5,6).
The model has five parameters (V _{max}, K _{m}, Cl_{ d }, V _{c}, and V _{t}) and three constants (two doses, one duration of infusion). We fit the data simultaneously.
Case study 9
Case study 10
Three oral solutions of a test compound with increasing doses to human subjects displayed a nonlinear pattern at the peak concentrations (Fig. 7, case study 10 (2)). The initial rise of the plasma concentration is very rapid with a peak concentration occurring within 10 min after dosing at the first observation. Not only was a peak shift observed with increasing doses but also a flat portion at the highest dose lasting for about 100 min. Dosenormalized areas, obtained from noncompartmental analysis, superimposed, suggesting that the extent of absorption is complete but the rate is saturable. The compound utilizes a transporter system for endogenous compounds like amino acids, hormones, and other food ingredients. Data also displays a biexponential (concave) decline at concentrations below 100 μg L^{−1}. Figure 7 (case study 10) shows the highresolution data from a single individual. The data pattern is interesting as it displays a brief period with absorption ratelimited kinetics during the first 100 min, and then displays disposition ratelimited kinetics in the terminal phase.
However, there are only five parameters that are estimable from data (V _{max} and K _{m}, Cl_{ d }/F, V _{c}/F, V _{t}/F,). Since we apply a nonlinear transport to the absorption process, both ABS and NL relate to the same process.
Case study 11
Estradiol was given as a rapid intravenous injection to a postmenopausal woman. Estradiol concentrations in plasma were measured prior to dosing and during 32 h postdosing (Fig. 7, case study 11 (2)). A twocompartment model with endogenous turnover and clearance was fit to the data. Initial parameters were obtained by graphical methods.
dC/dt and dC _{t} /dt are the rate of change of test compound in plasma and tissue, C is the plasma concentration, C _{t} the tissue concentration, Cl plasma clearance, V _{c} central volume, V _{t} peripheral volume, and Cl_{ d } the intercompartmental distribution parameter. Cl_{ d } has the units of volume per time (mL min^{−1} or L min^{−1}) and is related to transport via blood flow, transporters, and diffusion/convection forces. Input_{sc} and R _{in} denote the exogenous bolus dose and endogenous secretion of estradiol, respectively. R _{in} is a model parameter that will be estimated together with Cl, Cl_{ d }, V _{c}, and V _{t} when fitting the model to the data. Here, we assume the endogenous production of estradiol is constant during the observational time period.
Case study 12
This case study of pattern recognition demonstrates the turnover concept including turnover rate and turnover time. A healthy volunteer received a 40 μg kg^{−1} dose D of growth hormone subcutaneously (sc). The plasma concentrations of growth hormone, which were measured before and during 72 h postdosing, are shown in Fig. 7 (case study 12 (2)) together with predicted concentrations. The data pattern includes a pre and postdose baseline concentration, a plasma concentration peak at about 2 h, and a rapid return back to baseline concentrations within 24 h.
We assume that the baseline concentration (initial condition) can be written as R _{in}/Cl (synthesis or turnover rate divided by Cl), and that F is equal to unity.
Case study 13
The TMDD model is schematically depicted in Fig. 8 (case study 13). The typical shapes of a plasma concentrationtime course that TMDD displays start with a rapid decline within the minute to hour range due to the secondorder reaction between ligand and soluble target. This phase is extended in both the concentration and the time range with diminishing doses. Remember that the rate process –k _{on} L R is dependent on both ligand and target concentrations and their relative sizes. The initial drop may easily be missed if the first plasma sample is 12–24 h postdose. Then, the curve displays a concave bend towards a slower decline. One often has firstorder linear (doseproportional) kinetics at higher exposure of ligand. The concentrationtime course displays biexponential decline at higher doses after intravenous dosing because the target, as a clearing route, is saturated.
The third typical phase is then a convex bend downward with a shorter apparent halflife as we approach lower concentrations. This phase is where TMDD starts to be of importance. The kinetics is now nonlinear. The appearance of the downward bend will occur at the same ligand concentrations independently of ligand dose. Throughout this phase, the target route of elimination is more or less saturable, but less saturable at low concentrations and therefore a more dominating clearing route in that concentration range.
Finally, the ligand enters a slower terminal phase, again after a concave bend, with a longer apparent halflife. This phase is very much governed by the elimination rate constant k _{ e }(RL) of complex RL and in some instances also by unspecific distribution (Cl_{ d }, V _{t}) of ligand.
C _{L}, input_{L}, Cl_{ L }, Cl_{ d }, k _{on}, R, k _{off}, C _{RL}, C _{T}, and V _{t} denote the ligand concentration, input of ligand, firstorder clearance of ligand, intercompartmental distribution of ligand, secondorder rate constant for the ligandtarget interaction, target level, firstorder dissociation rate constant of the ligandtarget complex, complex concentration, concentration of ligand in tissue due to nonspecific distribution, and the volume of distribution of nonspecific distribution of ligand. The k _{ e(RL)} is the firstorder rate constant of irreversible removal of the complex.
It should also be remembered that the concentrationtime courses similar to those following intermediate doses may in some instances also be observed for therapeutic proteins that do not undergo TMDD but exhibit the formation of clearing antidrug antibodies (ADA) due to an immunological reaction (8). This usually takes some time to develop and is sometimes seen after repeated dose administration of, for example, monoclonal antibodies.
This gives nine parameters based on the very simple relationship in Eq. 27. Still, additional concentrationtime data on target (=2 parameters) and complex (=3 parameters) need to be included to improve the precision of certain parameters. See Peletier and Gabrielsson (7) for a thorough discussion of this dataset.
Case study 14
Case study 14 has demonstrated the consequences of induction of the responsible metabolizing enzymes by another compound (heteroinduction by pentobarbital on nortriptyline metabolism). Induction or inhibition by the parent compound itself or a metabolite is also possible (e.g., carbamazepine (10)). This is manifested as a lower (induction, causing the halflife to decrease) or higher (inhibition, causing the halflife to increase) exposure to the test compound over time. Saturable tissue binding can also lead to a lack of predictive power by singledose data. It is therefore suggested that chronic indications require chronic dosing, and consequently pharmacokinetic assessment must be based on repeated dose information.
Case study 15
The concentration of drug A and metabolite M were measured in plasma at different times after intravenous bolus doses of 10, 50, and 300 μmol kg^{−1}. Figure 8 (case study 15 (2)) depicts the experimental concentration data for drug (solid lines) and metabolite (dashed lines).
The parent compound data show a biexponential decline with a concave curvature at about 30 min. The halflife of parent compound increases with increasing concentrations (doses). Notice that the metabolite peak concentration increases in a less than proportional manner and occurs later in time with increasing doses. The slope of the initial or intermediate portion of the plasma concentrationtime profiles increases with dose and the separation of the parent and metabolitetime courses increases with higher doses. The terminal portion obeys firstorder kinetics and should therefore be independent of dose (concentration). This pattern suggests a twocompartment system of differential equations for the parent compound (drug) with saturable elimination. The nonlinear elimination from parent becomes nonlinear formation input to a onecompartment metabolite (M) model. The threecompartment system is given by Eq. 30. In this model, an intravenously administered drug is fully converted to metabolite through its metabolic clearance and then excreted as the metabolite.
The Cl_{ME} parameter is the firstorder clearance parameter of the metabolite. Note that Cl_{M} is the metabolic clearance of the drug, which is the same as the formation clearance of the metabolite. Cl_{M} will be more and more saturated, the higher the doses are of the parent compound. This results in formationlimited elimination of the metabolite and is observed as a flatter concentrationtime course (longer apparent halflife of metabolite) at higher exposure to the parent compound.
Case study 16
A volunteer was given 20 mg orally of a highly polar drug (Fig. 8, case study 16 (2)). Data show an initial time delay followed by a late peak at about 4 h and a postpeak monoexponential decline. The objectives of this exercise are therefore to identify and fit the most suitable of two different types of absorption models to a dataset obtained after extravascular dosing with compound A. One is a firstorder model including a lag time, the other is a zeroorder input model.
In this case study, there is a distinct difference between the zeroorder input (model of choice) and firstorder input models (systematic deviations throughout the modelpredicted concentrationtime course), which is already shown in the function plots. In other words, the latter model is not an option and there is no need to extend the analysis to inspection of residuals or use other tools in the statistical battery (parameter precision, correlation, F test) (see Gabrielsson and Weiner (2) for a discussion).
The key pattern of this dataset is a somewhat delayed onset of absorption, a concentration maximum at about 4 h, and a monoexponential decline postpeak.
DISCUSSION
Pattern recognition is a pivotal aspect of exploratory data analysis when modeling pharmacokinetic and pharmacodynamic data. Therefore, a rigorous strategy is essential for dissecting the patterns that concentrationtime profiles reveal. As an alternative solution, one may utilize a set of points to consider that specifically addresses number of phases, convex or concave bending, time lags, peak shifts, baseline behavior, effective halflives, dosenormalized areas, concentration plateaus, and similar phenomena. Pattern recognition has also been proposed for interpreting results of drugdrug interactions. “A quicker and better understanding about the processes, which dominate a DDI, has been achieved using this approach by focusing on integration of all information available and mechanistic interpretation” (1).
The application of the extended Eq. 1 has been successful for the presented case studies but should in general be used cautiously and only from an exploratory point of view. One may decipher other characteristics of the data by for example simultaneously fitting several time courses.
Considerations and Methodologies for Comparison and Experimental Design
We advocate an iterative process for discrimination between rival models. This is done by starting with a simpler model (say biexponential or no lag time model), fit that model to the data, perform a thorough residual analysis (2), and look at other goodnessoffit criteria (objective function value, Akaike information criteria, etc.) together with parameter correlation and precision. The next step is then to systematically extend the model (adding an exponential term or a lag time if necessary), refit the updated model to the data, and inspect the residuals in combination with the statistical battery (goodnessoffit, parameter precision, parameter correlation). In some cases, an F test analysis may be a final check of the model of choice, although the residual analysis (again visual inspection of transformed data) is, in our experience, a powerful approach in model selection. The most parsimonious model is preferable in most modeling situations.
We advocate an iterative approach to practical experimental design. Start by running a pilot study with a single dose, a few animals, and logarithmic spacing of data in time. Then fit a model to the data to get an acceptable fit as possible without overdoing the analysis. Simulate the new design(s) with the model using the final parameter estimates from the pilot study. Propose alternative doses, alternative sampling time points, and/or a repeated dose design if necessary. Run the study and collect data according to revised design. Now fit all data from pilot and redesigned study simultaneously. If the model mimics all data, it is probably a relatively robust model. We commonly use this iterative approach (running a doserange finding limited animal/sample approach) prior to the more expensive repeated dose (e.g., 1, 3, or 12month safety) studies. There are several reallife case studies in Gabrielsson and Weiner where simultaneous fitting of data from two or more (incomplete) experiments have proven to be useful (2). One may also want to consider sparse sampling in combination with a mixedeffects modeling approach to save both animals and cost. The mixedeffects modeling approach has of course great potential but is beyond the focus of this report.
Some General Points to Consider with Respect to Visual Inspection of Data
 (1)the number of exponential phases in a semilogarithmic concentrationtime plot corresponds to the number of compartments in a linear mammillary compartment system after a rapid bolus injection or a short constant intravenous infusion.$$ \mathrm{Number}\ \mathrm{of}\ \mathrm{compartments}=\mathrm{Number}\ \mathrm{of}\ \mathrm{phases}\ \mathrm{in}\ \mathrm{plasma} $$(35)
 (2)Nonlinearities such as capacity, time, binding, and flowdependent phenomena (Eq. 36) are revealed by convex bends in the concentrationtime profile (see case studies 7–9, 13, and 15), which suggest that one or more nonlinear terms are needed in the model. Also, dose normalize the concentrationtime curves and check whether they superimpose. A set of nonlinear expressions are collated below.V _{max}(t), K _{d}, n, [P _{T} ], Q _{H}, and Cl_{int} denote the timedependent maximum metabolic capacity, affinity constant between drug and protein, number of binding sites on the protein, protein concentration, hepatic blood flow, and intrinsic clearance, respectively. Cl_{H}, Q _{H,B}, f _{u}, Cl_{u int,H}, C _{B}, and C _{p} are the hepatic blood clearance, hepatic blood flow, free fraction in plasma, unbound hepatic intrinsic clearance, total blood concentration, and total plasma concentration, respectively (11).$$ \mathrm{Nonlinearities}=\left\{\begin{array}{l}\mathrm{C}\mathrm{apacity}\kern0.5em \mathrm{C}\mathrm{l}=\frac{V_{\max }}{K_{\mathrm{m}}+C}\\ {}\mathrm{Time}\kern0.5em \mathrm{C}\mathrm{l}(t)=\frac{V_{\max }(t)}{K_{\mathrm{m}}+C}\\ {}\mathrm{Binding}\kern0.75em {f}_u=\frac{C_{\mathrm{u}}+{K}_d}{C_{\mathrm{u}}+{K}_d+n\cdot \left[{P}_T\right]}\\ {}\\ {}\mathrm{Flow}\kern0.75em \mathrm{C}\mathrm{l}=\frac{Q_{\mathrm{H}}\cdot {f}_u\cdot {\mathrm{Cl}}_{\mathrm{int}}}{Q_{\mathrm{H}}+{f}_u\cdot {\mathrm{Cl}}_{\mathrm{int}}}\\ {}\\ {}{\mathrm{Cl}}_{\mathrm{H}}=\frac{Q_{\mathrm{H},\mathrm{B}}\cdot {f}_u\cdot {{\mathrm{Cl}}_{\mathrm{u}}}_{\mathrm{int},\mathrm{H}}}{Q_{\mathrm{H},\mathrm{B}}+{f}_u\cdot {{\mathrm{Cl}}_{\mathrm{u}}}_{\mathrm{int},\mathrm{H}}/\left(\frac{C_{\mathrm{B}}}{C_{\mathrm{P}}}\right)}\end{array}\right. $$(36)
 (3)Extravascular data (oral data or data from alternative extravascular dosing) reveal time delays (t _{lag}), rate (K _{a}), and extent (F) of absorption, or even capacitylimited (V _{max}, K _{m}) input that impacts the onset of absorption, absorption rate, the AUC, and peak shifts in C _{max} /t _{max} when two or more doses are given, respectively. Useful expressions related to absorption profiles are shown in Eq. 37.A _{ g }, f _{ a }, and f _{H} denote the amount at the absorption site (e.g., in the gut), fraction absorbed into blood, and fraction that passes through the liver, respectively. All other parameters are explained above. One should remember, however, that data from only the oral route may confound the interpretation of slopes, clearance, and volume terms. A potential solution to this is to utilize iv and oral data simultaneously.$$ \mathrm{Absorption}=\left\{\begin{array}{l}\mathrm{Input}\ \mathrm{rate}\kern0.5em F\cdot {\mathrm{Dose}}_{\mathrm{po}}\cdot {e}^{{K}_{\mathrm{a}}\cdot \left(t{t}_{\mathrm{lag}}\right)}\hfill \\ {}\mathrm{Extent}\kern0.5em F={f}_{\mathrm{a}}\cdot {f}_{\mathrm{H}}\hfill \\ {}\mathrm{Capacity}\kern0.5em \mathrm{input}\ \mathrm{rate}=\frac{V_{\max }}{K_{\mathrm{m}}+{A}_{\mathrm{g}}}\hfill \end{array}\right. $$(37)
 (4)Baseline concentrations of endogenous compounds need to be considered by adding some production term (turnover rate) simultaneously with the clearance parameterWhen baseline values are observed for exogenous compounds one does not have to consider an endogenous turnover rate but rather setting the initial condition of the state variable(s) to the measured predose value.$$ \mathrm{Baseline}=\frac{\mathrm{Turnover}\ \mathrm{rate}}{\mathrm{Cl}} $$(38)
 (5)Measurements in other body tissues and fluids, for example, urinary data, may contribute to the estimation of either renal clearance Cl_{R} or fraction excreted via urine f _{ e }where dA _{u} /dt and C are the rate of excretion of drug into urine and the plasma concentration.$$ \frac{d{A}_{\mathrm{u}}}{dt}={\mathrm{Cl}}_{\mathrm{R}}\cdot C={f}_{\mathrm{e}}\cdot \mathrm{C}\mathrm{l}\cdot C $$(39)
The objective of this communication has been to focus on visual inspection of “shapes” of concentrationtime profiles in the exploratory analysis of pharmacokinetic data. We have tried to decompose the shapes and to systematically interpret what determines the rise, intensity, and decline of exposure. This approach may serve as a road map to pattern recognition of concentrationtime data.
Additional sources of data from two or more doses, urine, metabolite information, and repeated dose data should, whenever possible, be considered as part of a simultaneous fitting procedure.
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