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

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.
7. METHOD RK4
   STARTTIME = 0
   STOPTIME=10
   DT = 0.02
   DTOUT=1
   Dose=100
   CL=3
   V=10
   Tabs=1
   KA=logn(2)/Tabs
   Emax=100
   EC50=3
   E0=0
   
   Conc=Dose*Ka/V/(Ka-CL/V)*(EXP(-CL/V*Time)-EXP(-Ka*Time))
   Effect=E0+Emax*Conc/(EC50+Conc)
   
   Figure 1. Code for Ka1EmaxC.mmd
   
   Effect compartment model
8. Save the Ka1EmaxC Model in your Time Course of Effect folder with the new name:
   
   Ka1EmaxCe.mmd
9. Add an effect compartment for delayed concentrations (Ce) (Figure 2).
10. Click on Run to run the model.
11. Save your model. Click on the Graph window and use the Graph>Choose Variables option to show Conc, Ce and Efffect.
12. 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 folder "Pharmacometrics Programs".
2. Click on the New Project icon (top left of MONOLIX GUI).
3. To export the simulated ka1emaxc data from pkpd.xls to a format that MONOLIX can read, 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
4. The next 11 rows will be populated with concentration observation records. Fill the ID column with the value 1. Copy the Time values from pkpd.xls into the TIME column and copy the Conc values to the DV column. Fill the DVID column with the value 1 for these rows.
5. 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. 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
   3	1	0.25	1.505	1
   4	1	0.5	2.754	1
   5	1	0.75	3.081	1
   6	1	1	3.847	1
   7	1	1.5	4.303	1
   8	1	2	5.299	1
   9	1	3	4.878	1
   10	1	4	4.522	1
   11	1	6	3.043	1
   12	1	8	1.490	1
   13	1	0	0	2
   14	1	0.25	32.607	2
   15	1	0.5	44.963	2
   16	1	0.75	53.086	2
   17	1	1	61.409	2
   18	1	1.5	63.671	2
   19	1	2	62.322	2
   20	1	3	65.999	2
   21	1	4	54.659	2
   22	1	6	49.954	2
   23	1	8	35.624	2

   Table 1. MONOLIX input file for ka1emaxc.
6. Now save the data file using 'Save As…' and choose CSV (Comma delimited, *.csv) format. Name the file ka1emaxc.csv.
7. In MONOLIX, click 'The data', locate your ka1emaxc.csv file and click open. The data information window will appear.
8. MONOLIX should identify that your input file has a header row and your data should appear under the headings: ID DOSE TIME Y. Place a check in the 'Use header' box so that MONOLIX can identify the data by your header row. Click accept.
9. 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.
   $PROBLEM One compartment first order input and elimination, immediate Emax effect
   
   $PSI
   cl v tabs ; PK parameters
   emax ec50 ; PD parameters
   
   $EQUATION
   dose=100 ; nominal dose
   k=cl/v
   ka=log(2)/tabs
   conc=dose/v*ka/(ka-k)*(exp(-k*T)-exp(-ka*T))
   effect=emax*conc/(ec50+conc)
   
   $OUTPUT
   OUTPUT1 = conc
   OUTPUT2 = effect
   
    Figure 3. Code for ka1emaxc_mlxt.txt
   
   
   IMPORTANT: MLXTRAN is case sensitive. Please use all lower case letters except for the $RECORD names, the time variable T and the OUTPUT1 and OUTPUT2 variables.
10. Save the model as ka1emaxc_mlxt.txt in your Time Course of Effect folder.
11. Click 'The structural model' and the model library will appear. Click 'Other list' and navigate to your Time Course of Effect folder and click 'OK'. Select ka1emaxc_mlxt.txt and click 'Compile', then 'Accept'.
    
    
    If you get a compile error. Double check that your code is identical to that shown in Figure 3 and be sure to press ENTER after the last line of code - this is required to 'end' the last statement.
12. Change the initial parameter estimates under Fixed effects.
    
13. Set the 'Stand. dev. of the random effects' to 0 for each parameter and fix them by right clicking on each box and selecting fix.
14. Set the residual error model to 'exp'.
15. Set the 'Residual error parameters' to 0.1 (first parameter in each row). These are the initial estimates for a exponential residual error parameter for concentration and effect.
16. Save the project as ka1emaxc.mat in your User Defined Models folder.
17. Run the model by clicking he initial estimates for a proportional residual error parameter for concentration and effect.
18. Save the project as ka1emaxc.mat in your User Defined Models folder.
19. Run the model by clicking 'Estimate the Population Parameters' at the top of the MONOLIX window.
20. When the estimation finishes click on 'Settings Results'. Choose 'Individual Fits' then 'VPC'. Look at the graphs to see how well the model and the final parameters fit the data.
    
    When you have more than one output you will need to select the 'y_1' or y_2' output variable for each graph.
21. View the parameter estimates by clicking 'Last Results'. The parameter estimates are also displayed in the command prompt that is behind the main Monolix window.
    
    

1. Open the NONMEM shortcut in folder "Pharmacometrics Programs".
2. Change directory to the User Defined Models folder by typing this command in the NONMEM window then press <ENTER>.
   
   
   cd User*
   
   
   
   All commands shown in the green boxes should end by pressing <Enter>.
   
   
3. Make a NONMEM directory then change to the NONMEM folder by typing this command in the NONMEM window.
   
   
   md NONMEM
   
   
   
   You only have to make the directory once. You dont have to give the md NONMEM command when you are working on the K01 and emaxc problems.
   
   
4. Change to the NONMEM directory if you are in the User Defined Models directory..
   
   
   cd NONMEM
   
   
   
5. Use the Windows Notepad to create the following code in a file named bo1.ctl e.g. by using the command:
   
   
   notepad bo1.ctl
   
   
6. Enter the code for bo1.ctl shown in Figure 2.
   
   
   $PROB one compartment bolus
   $INPUT ID TIME AMT DV
   $DATA bo1.csv
   IGNORE #
   IGNORE (AMT.GT.0)
   $ESTIM METHOD=CONDITIONAL
   $COV
   $THETA (0,3,) ; POP_CL
   $THETA (0,10,) ; POP_V
   $OMEGA 0 FIX ; ETA_CL
   $OMEGA 0 FIX ; ETA_V
   $SIGMA 0.01 ; ERR_CV
   $PRED
   DOSE=100
   CL=THETA(1)*EXP(ETA(1))
   V=THETA(2)*EXP(ETA(2))
   F= DOSE/V*EXP(-CL/V*TIME)
   Y=F + F*ERR(1)
   $TABLE ID TIME Y
   NOPRINT ONEHEADER FILE=bo1.fit
   

   Figure 2. NM-TRAN code for bo1.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 “bo1.ctl” in the User Defined Models\NONMEM folder.
8. Copy your bo1.csv file from your User Defined Models\Monolix folder to your NONMEM folder. The data file name must match in the $DATA record of the bo1.ctl file.
9. Open each .csv file and type # before the the ID column values of rows 1,2 and 3 (header and dose record and first observation at TIME=0).
   
   
   NONMEM ignores rows that start with # (defined by IGNORE # in the control stream [.ctl file]). You could just delete these rows, but if you open the file again at a later date, you may not remember what your data represent.
   
   NONMEM can also ignore rows based on the values in the column. In this example the AMT row is dropped. This is because we are defining the dose directly in this $PRED model. More commonly the PREDPP library is used and this requires the dose to be defined in the AMT column.
   
   
10. Execute NONMEM with this command in the NONMEM window:
    
    
    nmgo bo1
    
    
    
    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.
    
    
11. Use the Windows notepad or EditPlus to open this file and look at the saved results e.g. with the following command:
    
    
    notepad bo1.std\bo1.smr
    
    
12. Repeat the above steps for the K01 and emaxc models. For Monolix you will need to find a suitable model in the Monolix library of models. For NONMEM you should change the bo1.ctl file and save it with a suitable name such as K01.ctl or emaxc.ctl..