# Presentation: 301562

## Principles of Clinical Pharmacology

- Noncompartmental versus Compartmental Approaches to Pharmacokinetic Data Analysis
- David Foster, Professor Emeritus
- Department of Bioengineering
- University of Washington

## Questions asked

- What does the body do to the drug?
- Pharmacokinetics
- What does the drug do to the body?
- Pharmacodynamics
- What is the effect of the drug on the body?
- Disease progression and management
- What is the variability in the population?
- Population pharmacokinetics

## What is needed?

- A means by which to communicate the answers to the previous questions among individuals with diverse backgrounds
- The answer: **pharmacokinetic parameters**

## Pharmacokinetic parameters

- Definition of pharmacokinetic parameters
- Formulas for the pharmacokinetic parameters
- Methods to estimate the parameters from the formulas using data

## Pharmacokinetic parameters

- Descriptive or observational
- Quantitative (requiring a formula and a means to estimate using the formula)

## Quantitative parameters

- Formula reflective the definition
- Data
- Estimation methods

## Models for estimation

- Noncompartmental
- Compartmental

## Goals of this lecture

- Description of the quantitative parameters
- Underlying assumptions of noncompartmental and compartmental models
- Parameter estimation methods
- What to expect from the results

## Goals of this lecture

- Not to conclude that one method is better than another
- What are the assumptions, and how can these affect the conclusions
- Make an intelligent choice of methods depending upon what information is required from the data

## A drug in the body:constantly undergoing change

- Absorption
- Transport in the circulation
- Transport across membranes
- Biochemical transformation
- Elimination

## A drug in the body:constantly undergoing change

- How much?
- What’s happening?

- How much?

- How much?

- How much?

- How much?

- What’s happening?

## Kinetics

- The temporal and spatial distribution of a substance in a system.

## Pharmacokinetics

- The temporal and spatial distribution of a drug (or drugs) in a system.

## Definition of kinetics: consequences

- _Temporal_:  *When* in the system
- If               are spatial coordinates and
- 	is the measurement of a substance at a specific     , then the rate of change of the measurements depends upon     and t:

## A drug in the body:constantly undergoing change

- How much?
- What’s happening?

- How much?

- How much?

- How much?

- How much?

- What’s happening?

## A drug in the body:constantly undergoing change

- How much?
- What’s happening?

- How much?

- How much?

- How much?

- How much?

- What’s happening?

## Using partial derivatives

- Requires a knowledge of physical chemistry, irreversible thermodynamics and circulatory dynamics.
- Difficult to solve.
- Difficult to design an experiment to estimate parameter values.
- While desirable, normally not practical.
- Question:  What can one do?

## Resolving the problem

- Reducing the system to a finite number of components
- Lumping processes together based upon time, location or a combination of the two
- Space is not taken directly into account

## Lumped parameter models

- Models which make the system discrete through a lumping process thus eliminating the need to deal with partial differential equations.
- Classes of such models:
- _Noncompartmental models_ (algebraic equations)
- _Compartmental models_ (linear or nonlinear differential equations)

## The system

- Accessible pools:  These are pools that are available to the experimentalist for test input and/or measurement.
- Nonaccessible pools:  These are pools comprising the rest of the system which are not available for test input and/or measurement.

## An accessible pool

- SYSTEM

- AP

## Characteristics of the accessible pool

- Kinetically homogeneous
- Instantaneous and well-mixed

## Kinetic homogeneity

## Instantaneous and well-mixed

**S1**

**S2**

**S1**

**S2**

**B**

**A**

## Instantaneous and well-mixed

## The single accessible pool

- SYSTEM

- AP

- E.g.  Direct input into plasma with plasma samples.

## The two accessible pools

- SYSTEM

- AP

- AP

- E.g. Oral dosing or plasma and urine samples.

## The pharmacokinetic parameters

- The pharmacokinetic parameters estimated using kinetic data characterize both the kinetics in the accessible pool, and the kinetics in the whole system.

## Accessible pool parameters

- Volume of distribution
- Clearance rate
- Elimination rate constant
- Mean residence time

## System parameters

- Equivalent volume of distribution
- System mean residence time
- Bioavailability
- Absorption rate constant

## Models to estimate the pharmacokinetic parameters

- The difference between noncompartmental and compartmental models is how the nonaccessible portion of the system is described.

## The noncompartmental model

## Single accessible pool model

- SYSTEM

- AP

## Two accessible pool model

- SYSTEM

- AP

- AP

**1**

**2**

## Recirculation-exchange arrow

- Recirculation/
- Exchange

## Recirculation-exchange arrow

- AP

- Recirculation/
- Exchange

## Single accessible pool model

- Parameters (bolus and infusion)
- Estimating the parameters from data

## Single accessible pool model parameters

- _Bolus_

- _Infusion_

*d* – dose; *u* – infusion rate; *C(t)* – concentration; *AUC* – area under
- curve (1st moment); *AUMC* – mean area under curve (2nd moment);
-   – steady state concentration.

## What is needed?

- Estimates for C(0),           and/or     .
- Estimates for AUC and AUMC.
-    All require extrapolations beyond the time frame of the experiment.  Thus this method is not model independent as often claimed.

## The integrals

## Estimating AUC and AUMC using sums of exponentials

- _Bolus_

- _Infusion_

## Bolus Injection

- And in addition:

## Infusion

- And in addition:

## Estimating AUC and AUMC using other methods

- Trapezoidal
- Log-trapezoidal
- Combinations
- Other
- Extrapolating

## The Integrals

- The other methods provide formulas for the integrals between
- t1 and tn leaving it up to the researcher to extrapolate to time
- zero and time infinity.

## Trapezoidal rule

## Log-trapezoidal rule

## Extrapolating from tn to infinity

- Terminal decay is a monoexponential often called z.
- Half-life of terminal decay calculated:
- 	tz/1/2 = ln(2)/ z

## Extrapolating from tn to infinity

- _From last datum:_

- _From last calculated value:_

## Estimating the Integrals

- To estimate the integrals, one sums up the individual
- components.

## Advantages of using sums of exponentials

- Extrapolation done as part of the data fitting
- Statistical information of all parameters calculated
- Natural connection with the solution of linear, constant coefficient compartmental models
- Software easily available

## The Compartmental Model

## Single Accessible Pool

- SYSTEM

- AP

## Single Accessible Pool

**AP**

## A model of the system

**PLASMA**

- INPUT

**A**

**C**

**B**

**D**

- PLASMA
- CONCENTRATION
- SAMPLES

**TIME**

**Key Concept**: Predicting inaccessible features of
-         the system based upon measurements in the
-         accessible pool, while estimating specific parameters
-         of interest.

- Inaccessible Pools

- Accessible Pool

## A model of the system

- Accessible Room

- Inaccessible Rooms

## Compartmental model

- Compartment
- instantaneously well-mixed
- kinetically homogeneous
- finite number of compartments
- specifically connected
- specific input and output

## Kinetics and the compartmental model

- _Time and Space_

- _Time_

## Slide 59

## Notation

- Q

- i

- Fij are transport in units mass/time.

## The Fij

- Describe movement among, into or out of a compartment
- A composite of metabolic activity
- transport
- biochemical transformation
- both
- Similar time frame

## The Fij

- (ref:  see Jacquez and Simon)

## The kij

- The kij are called fractional transfer functions.

- If a

- is constant, the kij is called

- a fractional transfer or rate constant.

## Compartmental models and systems of ordinary differential equations

- Well mixed:  permits writing Qi(t) for the ith compartment.
- Kinetic homogeneity: permits connecting compartments via the kij.

## The ith compartment

- Rate of
- change of
- Qi

- Fractional
- loss of
- Qi

- Fractional
- input from
- Qj

- Input from
- “outside”

## Linear, constant coefficient compartmental models

- All transfer rates kij are constant.
- Assumes “steady state” conditions.

## The ith compartment

## The compartmental matrix

## Compartmental model

- A postulation of how one believes a system functions.
- The need to perform the same experiment on the model as one did in the laboratory.

## Example using SAAM II

## Example using SAAM II

## Example using SAAM II

## Experiments

- Need to recreate the laboratory experiment on the model.
- Need to specify input and measurements
- Key:  UNITS

## Model of the System?

## A Model of the System

**PLASMA**

- INPUT

**A**

**C**

**B**

**D**

- PLASMA
- CONCENTRATION
- SAMPLES

**TIME**

**Key Concept**: Predicting inaccessible features of
-         the system based upon measurements in the
-         accessible pool, while estimating specific parameters
-         of interest.

- Inaccessible Pools

- Accessible Pool

## Parameter Estimation

**MODEL FIT**

**PARAMETERS**

## Parameter estimates

- Model parameters: kij and volumes
- Pharmacokinetic parameters:  volumes, clearance, residence times, etc.
- Reparameterization -  changing the paramters from kij to the PK parameters.

## Recovering the PK parameters from the compartmental model

- Parameters based upon the model primary parameters
- Parameters based upon the compartmental matrix

## Parameters based upon the model primary parameters

- Functions of model primary parameters
- Clearance = volume * k(0,1)

## Parameters based upon the compartmental matrix

- Theta, the negative of the inverse of the compartmental
- matrix, is called the _**mean residence time matrix**_.

## Parameters based upon the compartmental matrix

- The average time the drug entering compartment j
- for the first time spends in compartment i before
- leaving the system.

- The probability that a drug particle in
- compartment j will eventually pass through
- compartment i before leaving the system.

## Compartmental models:advantages

- Can handle non-linearities
- Provide hypotheses about system structure
- Can aid in experimental design
- Can be used to estimate dosing regimens for Phase 1 trials

## Noncompartmental versus Compartmental Approaches to PK Analysis: A Example

- Bolus injection of 100 mg of a drug into plasma.  Serial plasma samples taken for 60 hours.
- Analysis using:
- WinNonlin (“trapezoidal” integration)
- Sums of exponentials
- Linear compartmental model

## Slide 84

## Slide 85

## Results

- Sum of Exponentials

- Compartmental Model

- Volume

- 10.2 (9%)

- 10.2 (3%)

- Clearance

- 1.02

- 1.02 (2%)

- 1.02 (1%)

- MRT

- 19.5

- 20.1 (2%)

- 20.1 (1%)

- 

- z

- 0.0504

- .0458 (3%)

- .0458 (1%)

- AUC

- 97.8

- 97.9 (2%)

- 97.9 (1%)

- AUMC

- 1908

- 1964 (3%)

- 1964 (1%)

## Take Home Message

- To estimate the traditional pharmacokinetic parameters, either model is probably okay.
- Noncompartmental models cannot help in prediction
- Best strategy is probably a blend of compartmental to understand “system” and noncompartmental for FDA filings.

## Some References

- JJ DiStefano III.  Noncompartmental vs compartmental analysis: some bases for choice.  Am J. Physiol. 1982;243:R1-R6
- DG Covell et. al.  Mean Residence Time.  Math. Biosci. 1984;72:213-2444
- Jacquez, JA and SP Simon.  Qualitative theory of compartmental analysis.  SIAM Review 1993;35:43-79
- Jacquez, JA.  _Compartmental Analysis in Biology and Medicine_.  BioMedware 1996.  Ann Arbor, MI.
- Cobelli, C, D Foster and G Toffolo. _Tracer Kinetics in Biomedical Research_. Kluwer Academic/Plenum Publishers. 2000, New York.

## SAAM II

- A general purpose kinetic analysis software tool
- Developed under the aegis of a Resource Facility grant from NIH/NCRR
- Available from the SAAM Institute:
- 		http://www.saam.com

## Moments

- Moments play a key role in estimating pharmacokinetic parameters via noncompart-mental models.
- Modern use:  Yamaoka, K et al.  Statistical moments in pharmacokinetics.  J. Pharma. Biopharm. 1978;6:547
- Initial use:  Developed in late 1930’s

## Moments