# Modeling

Explain the concept of pharmacokinetic modeling of single and multiple compartment models.

**Pharmacokinetics** describes what the body does to a drug. Pharmacokinetic models are mathematical concepts used to predict plasma concentrations of drugs at different time points.

## Basic Pharmacokinetic Terms

Key concepts in pharmacokinetics include:

**Volume of distribution, V**_{D}

The volume of distribution is defined as the**theoretical volume into which an amount of drug would be distribute to produce the observed plasma concentration**.- Units are ml.kg
^{-1} - It is a way to describe what proportion of a drug is confined to plasma, and what proportion distributes to other tissues
- It does not correspond to any particular volume, however a V
_{D}of:- Less than 40ml.kg
^{-1}indicates a drug is confined to plasma - Up to 200ml.kg
^{-1}indicates a drug is confined to the ECF - Up to 600ml.kg
^{-1}indicates a drug is dissolved into the TBW - Greater than 1L.kg
^{-1}indicates a drug is highly protein bound or lipophilic

Agents which cross the blood brain barrier typically have a V_{D}of 1-2L.kg^{-1}.

- Less than 40ml.kg
- Subtypes of the volume of distribution are used to describe drug distribution at different times or with different models
- These include:
- V
_{1}

Volume of central compartment. - V
_{D}ss

Volume of distribution at steady state. - V
_{D}pe

Volume of distribution at peak effect.

- V
- Which volume to use depends on the pharmacological question
- e.g. Intubating dose for opioid should use a volume between V
_{1}(very small) and V_{D}ss (very large) - V_{D}pe is ideal as it will allow a target concentration to be selected for the time at which intubation will occur relative to drug administration

- e.g. Intubating dose for opioid should use a volume between V

- These include:

- Units are ml.kg

**Half-life**(t_{1/2})

The time it takes for a process to be 50% complete. With respect to drug clearance, it is the time it takes for concentration (typically in plasma) to fall by 50%.- A process is considered to be complete after 4-5 half-lives

Concentration will decrease by 50% after each half-life, so after 5 half-lives concentration will be 3.125% of its starting value.- This also applies to wash in - it will take ~4-5 elimination half-lives of a drug for a constant-rate infusion to reach its final concentration

- Half-life is mathematically related to many other key pharmacokinetic terms:

, where:- is the time constant
- is the rate constant for elimination
- is the volume of distribution
- is the clearance

- Various types of half-life are described:
- t
_{1/2}α describes the rapidity of the distribution phase following drug administration - t
_{1/2}β describes the rapidity of the*elimination*phase occurring after drug distribution equilibrium

This only evaluates clearance from plasma, and so is a composite of both excretion from the body (e.g. renal and hepatic clearance) and ongoing distribution to peripheral tissues.- The elimination half-life is generally not useful to predict drug offset, as this is affected by many factors

However, it does set an**upper limit**on how long it will take plasma concentration to fall by 50%.

- The elimination half-life is generally not useful to predict drug offset, as this is affected by many factors

- t

- A process is considered to be complete after 4-5 half-lives

**Time-constant**()

The time taken for a process to complete if it continued at its initial rate of change. Time constants are related to half-life, but are better suited when modeling change in exponential processes.- Time constants are discussed in more detail under respiratory time constants
- Elimination will be virtually complete after three time constants
- A time constant is the
**inverse of the rate constant for elimination**, i.e. - Illustration of the relationship between half-life and time constant:

**Clearance**

The clearance is volume of plasma completely cleared of a drug per unit time.- In a one compartment model, this can be expressed as: in ml.min
^{-1}.- As the
**time constant**is the inverse of**k**, clearance can also be expressed as:

- As the
- Since and are constants, clearance is also a constant
- Total clearance is a sum of the clearance of each individual clearance organ

- In a one compartment model, this can be expressed as: in ml.min

**Rate of elimination**

Amount of drug removed by the body per unit time.- Rate of elimination is the product of the clearance and the current concentration: , in mg.min
^{-1} - This is not the rate constant for elimination

- Rate of elimination is the product of the clearance and the current concentration: , in mg.min

## Compartmental Modeling

The simplest model imagines the body a single, well-stirred compartment.

In a one compartment model, the concentration of a drug () at time is given by the equation:

Where:

- is the concentration at time 0

As drug can only be eliminated from the compartment, this is also the peak concentration. **k**is the**rate constant for elimination**

This is the fraction of the Vd from which the drug is removed per unit time. The rate constant**determines**the**slope**of the curve.- A high rate constant for elimination results in a steep curve and therefore a short time constant

### Steady state

At steady state, **input is equal to output**. Therefore concentration at steady state is:

- Proportional to the concentration of the infusion and infusion rate
- Inversely proportional to the clearance:
- Concentration of drug can therefore be determined by the amount infused and the clearance

- Note steady state requires peripheral compartments to be saturated, and so will only occur after an infusion of many hours

### Multiple Compartment Models

- Models with multiple compartments have a better fit with experimental data than single compartment models
- Three-compartment models are typically used, as additional compartments typically offer no extra fidelity but are mathematically more complex
- A three-compartment model
*can*be conceptualised as a plasma (or central) compartment, a well-perfused compartment, and a poorly-perfused compartment

This doesn't mean that they*should*be thought of in this way - they are a mathematical technique used to calculate plasma concentration at a given time.

Plasma concentration in multi-compartment models is:

- Predicted through the net effect of several negative exponential equations x This is covered under two-compartment models below.
- Dependent on the effects of:
**Distribution**

Distribution describes the movement of drug from the central compartment (V_{1}) to the peripheral compartment(s).- Rapid fall in plasma concentration of a drug after administration is generally due to distribution

Distribution is an important method for drug offset in short-acting drugs.

- Rapid fall in plasma concentration of a drug after administration is generally due to distribution
**Redistribution**

Redistribution refers to the movement of drug from the peripheral compartment(s) back into plasma.- Drugs which have a large V
_{D}in a peripheral compartment tend to**distribute quickly**along this concentration gradient, and**redistribute slowly**back into plasma - Drugs which tend to
**distribute slowly**tend to**redistribute quickly**once administration has ceased

- Drugs which have a large V
**Excretion**

Excretion is the removal of drug from the body.

#### Clearance in Two-Compartment Models

Removal of drug in two-compartment models is via:

- Distribution from the central to the peripheral compartment
- Elimination from the central compartment
- This produces a
**bi-exponential**fall in plasma concentration

Consists of two phases:- Phase α

Distribution phase: A rapid decline in plasma concentration due to distribution to peripheral tissues. - Phase β

Elimination phase: Slow decline in plasma concentration due to:- Elimination from the body
- Redistribution into plasma

- Phase α

This curve is given by the equation , where:

- is the concentration of drug in plasma
- is the y-intercept of the distribution exponent

Used to calculate distribution half-life. - is the y-intercept of the elimination exponent

Used to calculate elimination half-life. - is the rate constant for distribution

The value of is dependent on the ratio of rate constants for distribution and redistribution (i.e. ).- If distribution greatly exceeds redistribution, the gradient of will be very steep and plasma concentration will fall rapidly after administration

- is the rate constant for elimination

- Note that the distribution and elimination curves appear straight because the y-axis is log-transformed
- If plasma concentration was plotted on the y-axis, then each of these curves would be a negative exponential (wash-out curve)

#### Effect Site

Pharmacokinetic models typically display the plasma concentration.

- Clinically however, we are interested in drug concentrations at the site of action (e.g. the brain)
- Concentration at the effect site (also known as biophase) is given by
**Ce**- This cannot be measured, and so is a calculated value

- Effect site concentration be different from plasma concentration (
**Cp**) prior to reaching steady state

The delay between plasma and effect site concentrations is an example of hysteresis.

- Concentration at the effect site (also known as biophase) is given by
- The effect site can be modeled as an additional compartment in three-compartment models

The effect site is modeled as a**compartment of negligible volume**contained**within V**, but does have rate constants_{1}- Effect site volume changes as V
_{1}changes - The k
_{e1}is the rate constant for drug diffusion from plasma into the effect site - The
**k**is the rate constant for elimination of drug_{e0}**from the effect site**

This is a theoretical elimination pathway - drug is not usually metabolised at the effect site.- The t
_{1/2}ke0 describes the effect-site equilibration time

It describes how rapidly the effect site reaches equilibrium with plasma.- A large ke0 (rapid drug flow) gives a short t
_{1/2}ke0 - After one t
_{1/2}ke0, 50% of the final effect site concentration will be reached provided plasma concentration remains constant

- A large ke0 (rapid drug flow) gives a short t
**A shorter t**_{1/2}ke0 indicates that that the effect site concentration will reach equilibrium with plasma more rapidly, and therefore a more rapid clinical effect following administration is seen- Note that:
- The t
_{1/2}ke0 is not the time to peak effect- Neither is k
_{e0}

- Neither is k
- For an infusion run at constant plasma concentration the peak effect will be seen at 3-5x the t
_{1/2}ke0 - The time to peak effect is a function of both plasma kinetics and the t
_{1/2}ke0- e.g. adenosine has such a short elimination t
_{1/2 the effect site concentration will reach its peak rapidly regardless of the ke0}

- e.g. adenosine has such a short elimination t

- The t

- The t

- Effect site volume changes as V

## Non-Compartmental Models

Compartment models are not appropriate for describing the behaviours of all drugs. Non-compartmental models are used when drug:

- Clearance is organ-independent
- Elimination does not occur solely from the central compartment

These models use **AUC**, which is calculated by measuring the plasma concentration of a drug at different time intervals, and **plotting the area under the curve (AUC)**. This can be used to:

- Determine
**clearance** - Determine
**Bioavailability**

Difference between the AUC of the same dose of drug administered IV and via another route.

## Footnotes

The formula for half-life can be derived from the equation for a wash-in exponential as follows:

- Wash in exponential is given by:
- can then be substituted and the equation solved for as follows:

## References

- Peck TE, Hill SA. Pharmacology for Anaesthesia and Intensive Care. 4th Ed. Cambridge University Press. 2014.
- Plasma Volume. Mosby's Medical Dictionary, 8th edition. 2009.
- Stanski RD, Shafer SL. The Biophase Concept and Intravenous Anesthesia.
- Petkov V. Essential Pharmacology For The ANZCA Primary Examination. Vesselin Petkov. 2012.