MEDSCI 719

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Professor Emeritus Nick Holford

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Objectives

The objectives are to:

1. Define common models for the time course of drug effect.
2. Learn how to perform a simultaneous fit of concentration and effect data.
3. Use simulation to understand the properties of the turnover model for delayed drug effect.

Introduction

The time course of drug effect can be described by linking separate models for concentration and effect.

1. Immediate: Drug effects are determined by the concentration in a compartment of the pharmacokinetic model.
2. Delayed:
1. Effect Compartment: Drug effects are determined by the concentration in a hypothetical effect compartment whose input is from a compartment of the pharmacokinetic model.
2. Physiological mediator: Drug effects are determined by the concentration of a physiological mediator. The concentration of the mediator is influenced by the drug concentration in one of 4 basic ways:
1. Decreased synthesis of mediator
2. Increased synthesis of mediator
3. Decreased elimination of mediator
4. Increased elimination of mediator

'It had long been believed that there is no relationship between the drug concentration in plasma and time course of action for many drugs...' (1)

We prescribe and administer drugs to produce effects. Clinical preoccupation with merely what dose to give misses the point, and assumes an easy equivalence between dose and effect. What effect are we hoping to achieve, what concentration is this effect associated with, and what dose will give this concentration? This kind of thinking involves cognisance of the many variables known and unknown which can influence each of these steps.

Dosing results in drug concentration. So achieving an appropriate drug concentration is the first goal of drug administration. However plasma concentration monitoring is only readily available for a small number of drugs with low therapeutic index including digoxin, theophylline, and a handful of antibiotics, immunosuppressants, and anticonvulsants. In clinical practice adjusting drug dosing to concentration can be fraught with difficulty due to misconceptions about pharmacokinetics and lack of appreciation of factors causing individual variation.

Drug doses are more commonly adjusted according to clinical effect rather than concentration. This titration is more rapidly accomplished in settings of more intensive clinical monitoring, such as intensive care and anaesthesia. Titration to effect is most satisfying to the clinician for drugs with faster apparent onset and offset of effect, potentially resulting in instant gratification for the drug administrator, and hopefully more effective targeting of drug response for the patient.

Pharmacokinetics gives us drug concentration versus time, while pharmacodynamics gives us drug effect versus concentration. Concentration is the link between drug dosing and effect. Linking pharmacokinetics and pharmacodynamics gives us drug effect versus time. This is called the time course of effect.

Both the timing of onset and offset of drug effect may be important. When will the effect start to be observed, when will the effect peak, when will the effect be at steady state (where applicable), and when will the effect decrease and then cease to be observable? The Emax model is the most fundamental description of the relationship between drug concentration and effect. This model is named after the parameter Emax, which describes the maximum effect of a drug. Since this implies effect at infinite drug concentration, Emax can never be measured, but can only ever be estimated from the shape of the response curve, approaching its asymptote.

Drugs work by having action on physiological systems. The drug action produces a response in the physiological systems and associated control mechanisms. These changes lead to an observed drug effect. The unbound portion of the drug is responsible for its action, however plasma concentration measures total concentration (both bound and unbound).

Which effects are important? For ease of data collection faster onset effects that are easy to measure are most often reported scientifically and used for clinical titration of dose. However more meaningful effects are often delayed and sometimes cumulative. For example antihypertensives are used for cardiovascular disease risk reduction where the goal effect is not just reduction in blood pressure but reduction in myocardial ischaemia and stroke rates. The goal effects of many drugs relate not to easily measurable short term physiological change, but longterm reduction in morbidity and mortality. There are many examples in intensive care medicine where a focus on short term physiological change observed as drug effect does not equate to long term beneficial outcome. For example early studies on inotropic drugs in heart failure ( eg dobutamine) reported improved haemodynamic parameters, which may be clinically insignificant, while subsequent outcome studies demonstrated an increase or no effect on mortality.

The timing of drug effects may be classified as immediate, delayed, or cumulative. Very few drugs have immediate effects, heparin being a rare example. Most drugs have a delayed effect. This delay may be due to many different pharmacokinetic and pharmacodynamic factors eg absorption after administration more peripherally; distribution and transport to effect side eg (target organ, cell membrane, organelle), receptor binding interactions, protein binding interactions, effects on enzymes and other physiological mediators.

Three main causes of delay in time course of effect will be explored: absorption, effect compartments, and indirect or physiological substance mediated effects.

Absorption: Any drug that is not administered directly into a central compartment usually has to be absorbed. Typically this is described for the following routes of administration: oral, rectal, subcutaneous, intramuscular, but may also include the systemic and local effects of topical application such as transcutaneous, transmucosal, conjunctival. Inhalational drugs are typically described in terms of uptake rather than absorption, but the basic concept is similar.

Absorption is complex process: it may involve diffusion down concentration gradients, or osmotic gradient, or specific transport factors. An absorption constant may be estimated to explain delays in drug concentration and effect due to absorption.

Effect compartments: Theoretical effect compartments were introduced to help explain the time delay between plasma concentration and observed effect for some drugs. This delay may occur because the effect site is not the central compartment, and hence time is required for drug delivery to effect site, by perfusion, diffusion or transport. At steady state plasma concentrations, then there will be a constant rate of input into the effect compartment, so the time to steady state effect site concentration will be determined by the rate of equilibration half-life. Equilibration half-life is determined by volume of distribution (organ size, tissue binding), and clearance (blood flow, diffusion).

Physiological substance mediated effects: Drug effects may be defined as immediate or delayed. In reality almost no drugs have a truly immediate effect due to complex physiological control systems and interactions that exist at baseline and after drug administration. Drugs act typically at receptors, but the observed effect is only seen later. Delayed effects may be due to drug effect on a physiological mediator, often an enzyme system. A mediator is something which affects a transition between one stage and another. It can be thought of as a go between or intermediatory, occupying an intermediate position, forming a connecting link between one thing and another.

Physiological mediators may be defined as physiological substances that can be affected by a administered drug or physiological process to produce an effect on another physiological process. A mediator may be an enzyme, or other biologically active molecule, for example nitric oxide.

Drugs that have an effect via a physiological mediator can be modelled using delay due to the turnover of the mediator, hence these models are known as turnover models. Time course of effect models link pharmacokinetic and pharmacodynamic models to describe changes in effect over time. Differential equations can be used to describe delays in observed drug effect due to absorption, effect compartment disposition and drug action via physiological mediators. Changes in concentration over time are described by changes in rate in and changes in rate out. In the case of drugs that work via physiological mediators, the change in concentration of the mediator over time is used as a monitor of drug effect.

Introduction by Anita Sumpter (2008).

Reference

1. Shoenwald RD. Pharmacokinetics in Drug Discovery and Development. CRC Press, Boca Raton 2002. p. 34

Find the file Pharmacometrics Data\Time Course of Effect\pkpd.xls

1. Open pkpd.xls with Excel.
2. Look at the ka1+emaxc worksheet.
3. Identify
1. The model code
2. The model parameters
3. The independent variables
4. Export the simulated ka1emaxc data from pkpd.xls to a format that can be used by other programs
5. Create a new Excel workbook and add the following headings in the top row. It is important to put the # before ID so that the same file can be used for NONMEM.
   #ID TIME DV DVID
6. Fill the ID column with the value 1 down to row 23. 
7. Rows 2 to 12 will be populated with concentration observation records. Copy the Time values from pkpd.xls into the TIME column and copy the Conc values to the DV column in rows 2 to 12. Fill the DVID column with the value 1 for rows 2 to 11. The DVID value is used to identify the type of observation (1=conc, 2=effect).
8. Rows 13 to 23 will be populated with effect observation records. The ID remains as 1, because the data is from the same single subject. Copy the Time values from pkpd.xls into the TIME column and copy the Effect values to the DV column in rows 13 to 23. Fill the DVID column with the value 2 for these rows.
   
   Your worksheet should look similar to Table 1, but your DV values will be slightly different.
   
   

   	A	B	C	D
   1	#ID	TIME	DV	DVID
   2	1	0	-0.1	1
   3	1	0.25	1.7	1
   4	1	0.5	2.5	1
   5	1	0.75	3.5	1
   6	1	1	4.1	1
   7	1	1.5	5.1	1
   8	1	2	5.4	1
   9	1	3	5.1	1
   10	1	4	4.4	1
   11	1	6	2.6	1
   12	1	8	1.5	1
   13	1	0	-0.1	2
   14	1	0.25	33.7	2
   15	1	0.5	47.3	2
   16	1	0.75	54.5	2
   17	1	1	58.7	2
   18	1	1.5	62.5	2
   19	1	2	63.8	2
   20	1	3	62.2	2
   21	1	4	58.5	2
   22	1	6	46.9	2
   23	1	8	33.7	2

   Table 1. Data file for ka1emaxc.
9. Now save the data file in 'My Pharmacometrics Data\Time Course of Effect' using 'Save As' and choose CSV (Comma delimited, *.csv) format. Name the file ka1emaxc.csv.

Emax model

1. Open Berkeley Madonna using the shortcut in the Pharmacometrics Programs folder.
2. Add code to the Equations window defining the STARTTIME and STOPTIME, the output interval (DTOUT), the model parameters and the model equation (Figure 1).
3. Click on Run to run the model .
4. Save your model in your Time Course of Effect folder with the name Ka1EmaxC.mmd.
5. Click on the Graph window and use the Graph>Choose Variables option to show Conc and Effect.
6. View a table of times, concentrations and effects by clicking on the Table icon in the Run 1 graph window.

Figure 1. Code for Ka1EmaxC.mmd

Effect compartment model

1. Save the Ka1EmaxC Model in your Time Course of Effect folder with the new name:
   
   Ka1EmaxCe.mmd
2. Add an effect compartment for delayed concentrations (Ce) (Figure 2).
3. Click on Run to run the model.
4. Save your model. Click on the Graph window and use the Graph>Choose Variables option to show Conc, Ce and Efffect.
5. View a table of times, concentrations and effects by clicking on the Table icon in the Run 1 graph window.

1. Open the MONOLIX shortcut in the folder 'Pharmacometrics Programs'.
2. Click on the 'New Project' icon.
3. In MONOLIX, click 'Data' and use 'Browse' to locate your ka1emaxc.csv file and click open. The data information window will appear.
4. MONOLIX should identify that your input file has a header row and your data should appear under the headings: ID TIME OBSERVATION and OBSERVATION ID. Click 'Accept' and 'Next' if Monolix has assigned the correct headings.
   
5. Instead of using the model library, we will use MLXTRAN to write out the ka1emaxc model. Start by opening a text editor (e.g. Editplus) and enter the code shown in Figure 3. Note that the variable name 'effect' cannot be used in MLXTRAN so use 'eff'.

;One compartment first order input and elimination, immediate Emax effect

INPUT:
parameter={cl,v,tabs,emax,c50}

EQUATION:
dose=100 ; nominal dose
k=cl/v
ka=log(2)/tabs
conc=dose/v*ka/(ka-k)*(exp(-k*t)-exp(-ka*t))
eff=emax*conc/(c50+conc)

OUTPUT:
output= {conc,eff}

 Figure 3. Code for ka1emaxc_mlxt.txt



IMPORTANT: MLXTRAN is case sensitive. Take care to be consistent with upper and lower case letters in names.

1. Create a "Time Course of Drug Effect\Monolix" folder and save the model as ka1emaxc_mlxt.txt in the "Time Course of Effect\Monolix" folder. You can create the Monolix folder using Windows Explorer or from the Monolix Save project dialog box.
2. Return to the Monolix window, click 'Structural model' and then 'Browse'. Navigate to your "Time Course of Effect\Monolix" folder, select ka1emaxc_mlxt.txt and open it. You may need to wait a few seconds for the model to compile.
   
   

   If you get a compile error: Double check that your code is identical to that shown in the Figure and be sure to press ENTER after the last line of code - this is required to 'end' the last statement.

3. Under the 'Initial Estimates' tab, set the 'Residual error parameters' to 1 (SD of residual error) for both types of observation (concentration, effect).
4. Change the initial parameter estimates by editing their values in the 'Check Initial Estimates' tab. A plot of predictions based on the model and initial parameter estimates is displayed along with observed values. Once these are reasonable, click 'Set as initial values'. Do this for both models using the dropdown 'Output' tab on the top right of the window.
5. Click on 'Statistical Model and Tests'.
6. Set the 'Stand. dev. of the random effects' to 0 for each parameter by selecting 'None' under 'Random Effects'.

   Because these are data from a single individual, there is no between subject variability (random effects). The SD of random effects is therefore 0.

7. Set the error model to 'const' for both models.
8. IMPORTANT: Save the project as ka1_emaxc_project.mlxtran in your Time Course of Effect\Monolix folder.
9. Set the task options at the top of the 'Statistical model & Tasks' tab by ticking 'Standard Err', 'Likelihood' and 'Plots'.
10. Click on the grey bar next to the blue part of 'Plots' and click to enable 'Observed data', 'Individual fits' and 'SAEM'. Disable all other options. Close the 'Plots' options window.
11. Estimate the parameters by clicking on 'Run'. This will take a while depending on the complexity of the model. During the estimation process you can see how the parameter estimates are being changed and settle down towards the final value.
12. When the estimation finishes close the Monolix Scenario window. 
13. Click on 'Plots' and look at the individual fit. Use the 'Display' options to show 'Predicted Median' and 'Prediction Interval' for the 'Individual predictive check'. You can alter the names of plot axes using the ‘Axes’ tab. When you are ready, you can export all of your plots together into your project folder automatically using the Export option in the main menu.
14. View the parameter estimates by clicking 'Results'. A text file containing these results is saved in a "pop_parameters.txt" in the project folder.

1. Open the NONMEM shortcut in the folder 'Pharmacometrics Programs'.
2. Change directory to the Time Course of Drug Effect folder by typing this command in the NONMEM window then press <ENTER>.
   cd Time*
   
   

   All commands shown in the green boxes should end by pressing <Enter>.

   
   
3. Make a NONMEM directory by typing this command in the NONMEM window.
   md NONMEM
   
   

   You only have to make the directory once.

   
   
4. Change to the NONMEM directory from the Time Course of Effect directory.
   cd NONMEM
5. Use EditPlus from Windows Explorer to create the following code in a file named bo1.ctl.
6. Enter the code for ka1emaxc.ctl shown in Figure 2.
   
   
   $PROB one compartment first-order input and elimination plus emax effect
   $INPUT ID TIME DV DVID
   $DATA ..\..\ka1emaxc.csv
   $ESTIM METHOD=CONDITIONAL
   $COV
   $THETA
   (0,3,) ; POP_CL
   (0,10,) ; POP_V
   (0,1,) ; POP_TABS
   (0,100,) ; POP_EMAX
   (0,3,) ; POP_c50
   $OMEGA
   0 FIX ; ETA_CL
   0 FIX ; ETA_V
   0 FIX ; ETA_TABS
   0 FIX ; ETA_EMAX
   0 FIX ; ETA_C50
   $SIGMA 1 ; ERR_CONC
   $SIGMA 25 ; ERR_EFFECT
   $PRED
   DOSE=100
   CL=THETA(1)*EXP(ETA(1))
   V=THETA(2)*EXP(ETA(2))
   TABS=THETA(3)*EXP(ETA(3))
   EMAX=THETA(4)*EXP(ETA(4))
   C50=THETA(5)*EXP(ETA(5))
   K=CL/V
   KA=LOG(2)/TABS
   CONC=DOSE/V*KA/(KA-K)*(EXP(-K*TIME)-EXP(-KA*TIME))
   EFFECT=EMAX*CONC/(C50+CONC)
   IF (DVID.EQ.1) THEN
   Y=CONC + ERR(1)
   ELSE
   Y=EFFECT + ERR(2)
   ENDIF
   $TABLE ID TIME CL V TABS EMAX C50 DVID Y
   NOPRINT ONEHEADER FILE=ka1emaxc.fit
   

   Figure 2. NM-TRAN code for ka1emaxc.ctl (NONMEM)

   The $OMEGA parameters are FIXed to 0. This is because there is only one individual being modelled. These random effects parameters are used to describe between subject variability when there is more than one individual.

   
   
7. Save the file. Check using Windows Explorer that you can find the file 'ka1emaxc.ctl' in the User Defined Models\NONMEM folder.
8. The data file name must match in the $DATA record of the ka1emaxc.ctl file.
   

   In order for NONMEM to find the data file the $DATA record has to include a path relative to the folder where NONMEM is executed. This is why the '..\..\' is put before the name of the data file which is located in the Time Course of Effect folder.

   
   
9. Execute NONMEM with this command in the NONMEM window:
   
   nmgo ka1emaxc
   
   

   When you get errors from NONMEM with the nmgo command then please read the error message carefully and try to understand what it is telling you. The usual errors that occur with these example problems will give you some clues to what you might need to change in your ctl file.

   Commands can be recalled by using the Up arrow on your keyboard. You can easily repeat the command without more typing or edit it to save the amount of typing you do.

10. Use Excel to open the ka1emaxc.fit file in the ka1emaxc.reg results folder.   
11. Select column A then click on the 'Data' menu item then click on 'Text to Columns'. Click on 'Finish'. This will separate the values into separate columns.
12. Select all cells in the worksheet (e.g. press ctrl-A) then right click on a cell and click on 'Format cells'. Click on the 'General' option. This will make the numeric values easier to read.
13. Delete row 1 which contains the value 'TABLE 1'.
14. Click on cell A1 then click on the 'Data' menu item then click on 'Data Filter'. This will let you choose rows with specific values of DVID by clickiing on the arrow in the corner of the column with the heading 'DVID'.
15. Use the Data Filter to select rows with DVID=1 (concentrations). Create a graph of TIME  versus Y and DV. 
16. Use the Data Filter to select rows with DVID=2 (effects). Create a graph of TIME  versus Y and DV. 
17. Save the Excel file with the graphs as ka1emaxc.xlsx in the Time Course of Effect\NONMEM folder.
18. Use EditPlus to look at the saved results in the '.smr' file in the ka1emaxc.reg folder. The results are saved in a folder with the same name as the ctl file but with the extension '.reg' e.g. use the following command or open the file using the Windows Explorer.
    
    
    notepad ka1emaxc.reg\ka1emaxc.smr
    
    

Physiological mediator model

Turnover models are used to describe delayed responses due to the turnover of a physiological mediator. The physiological mediator (M) without drug is formed at a rate proportional to its initial concentration (M0) and its elimination rate constant (kout). There are 4 basic ways that drugs can affect a turnover model. These give rise to characteristic time courses of response.

1. Use Berkeley Madonna to open Pharmacometrics Data\Time Course of Effect\turnover.mmd
2. Look at the model equations
3. Run the model and make a graph of Time vs Conc (left axis) and Time vs M1, M2, M3, M4 (right axis)
4. Use the Parameters Batch Runs option to vary Dose from 10 to 1000 in 5 steps
5. Explain the pattern of times of peak effect shown for each of the response curves (M1, M2, M3, M4)

Parameter Estimation

1. Write up the results of parameter estimation using MONOLIX and NONMEM.
2. Describe the differences between the parameters used for simulation and the parameter estimates obtained from MONOLIX and NONMEM. 
3. Discuss how you might change the experimental design in order to get better agreement between the simulation parameters and the parameter estimates.