Introduction into Pharmacokinetics

The logarithmic base is

10 (log 10 = 1; log 100 or 102 = 2, 1000 or 103 = 3, 0.001 or 10-3 = −3). Example: pH = 7 means [H3O+] = 10−7 mol/L or [H3O+] = 10−pH

ln x =

= 2.303 log x

Plot Cp versus time on *semi-log* graph

A *linear relationship → first-order* process.

Compartment models of PK

A simplified, but useful model of PK is compartmentally-based models. A diagram showing input, distribution into specific compartments (e.g. blood, interstitial fluid, tissue binding) and output of drug. In a compartment it is assumed that the drug is uniformly distributed. It also describes any variables that control these processes (e.g. k rate constants; volume of distributions or parameters).

pharmacokinetics encompasses 3 components:

Absorption Distribution Elimination (metabolism and excretion)

Analytical methods of drug measurements: Analytical methods of drug measurements:

Analytical methods of drug measurements: Chromatography (e.g. HPLC) Mass spectrometry Test kits (e.g. fluorescence immune assays)

Clinical Pharmacokinetics:

Application of pharmacokinetic methods to drug therapy in patient care. Age, gender, genetic predisposition, and ethnic differences affect the outcome of drug therapy. Population pharmacokinetics/pharmacogenomics/pharmacogenetics Even several diseases influence drug kinetics (e.g. renal failure, hepatic diseases).

Integration of the differential equation yields:

C = C0 e -k1t

Catenary Model

Catenary Model The compartments are connected in a sequence. Since this is not a representative model of normal physiology, this model is rarely used.

Experimental pharmacokinetics

* Development of biological sampling techniques. * Analytical methods to measure drug concentrations and their metabolites. * Procedures that facilitate data collection and manipulation.

theoretical pharmacokinetics

* Development of pharmacokinetic models that predict drug disposition. * Mathematical and statistical models to design optimal dosage regimens.

Plot Cp versus time on *normal* graph

*Not* a linear relationship → probably first-order process. Try plotting values on semi-log paper/graph

The rate of first-order reaction is

*dependent* upon the concentration of the drug: Radionuclide decay is an example of a first-order reaction.

Basic PK and PK models

Complex nature between drug disposition and distribution Mathematical models are used to predict drug concentrations in the body using specific parameters. Independent (time) and dependent (concentration) variables. Example: the drug/metabolite concentration in the liver of an oral tablet that was administered depends on the time after administration and the specific elimination rate constant k of the drug.

Clinical Pharmacology

Deals with the pharmacognosy (drugs obtained from nature), pharmacokinetics (PK), pharmacodynamics (PD) (relationship between drug concentration and action = receptor binding/signaling effects), pharmacotherapeutics, and toxicology (drug safety).

Calculation of t½ for a zero-order process

Hence, the *t½ is not constant*, it *depends on the C* and is *inversely* related to k0. C = C0 - k0t If C = 0.5C0 then: 0.5C0 = C0 - kot½ 0.5C0 = kot½ t½ = 0.5C0/k0

one-compartment

In a one-compartment model, drug is both added to and eliminated from a central compartment. For example, during an IV injection, the drug enters directly into the plasma compartment. Elimination occurs via the kidneys and liver which are well-perfused organs.

two-compartment

In a two-compartment model, drug can move between the plasma compartment to and from the tissue compartment.

Integral Calculus

Integral calculus is the reverse of differentiation and is considered the summation of f(x)∙dx. Example: integration of the function y = mx yields ∫mx.dx. For the expression C = −2t + 10, integration gives: ∫−2t∙dt. It basically calculates the sum of the individual areas (over a dt or Dt period) under the graph (or area under the curve = AUC).

Measurement of drug concentrations invasive:

Invasive: Blood (red blood cells, platelets, white blood cells, proteins) Serum (supernatant after clotting) Plasma (supernatant before clotting) Drugs are mostly bound to plasma proteins (bound/unbound drug)

The trendline shows the equation that describes the curve cp=30e^-0.35t shows 3 important things

It is a one exponential equation meaning a one-compartment model The value "− 0.35" is the slope and the elimination rate constant k is 0.35 h−1 The y-axis intercept is the value "30" and is the plasma concentration (Cp) at t = 0 or the initial Cp0. (Cp = 30e−0.35x0 = 30e0 = 30)

Flow model of PK s19

Lung: highly vascularized, rapid equilibration with blood Adipose: lipophilic, low blood supply, slow accumulation

Mammillary Model

Most common model and represents a system of compartments connected to a central component.

In a zero-order process, t½ is...

NOT constant, but depends on the concentration!

Analytical methods of drug measurements: Non-invasive:

Non-invasive: Urine Feces Saliva Expired air Breast milk

Physiological PK model (flow model)

Physiological PK model (flow model) Uses actual blood flow data for each tissue (e.g. portal vein blood flow measurement). Drug concentrations in the various tissues are predicted according to blood velocities and organ tissue size. Data form animal studies (mice or rats) can be extrapolated to human studies.

PK models are used to:

Predict drug levels from any dosage regimen Calculate the optimum dosage tailored to each patient Estimate accumulation Correlate drug concentrations with pharmacologic (nontoxic) and toxic effects. Examine bioequivalence (rate, extent) Describe how changes in disease affect absorption, distribution, or elimination. Explain drug interactions

Flow model of PK s18

SET: slowly equilibrating tissue RET: rapid equilibrating tissue Q: blood perfusion rate ke: first-order rate constant of urine excretion km: rate constant for hepatic metabolism

Measurement of drug concentrations Sampling of biologic specimens:

Sampling of biologic specimens: * spinal fluid * synovial fluid * tissue biopsy or any biologic tissue obtained during surgery

Zero Order Reactions

The *rate* of a zero-order process is one that proceeds over time (t) *independent from the concentration* (C) of the drug: k0 unit e.g. in (mg/mL)/h dC/dt = − k0 dC = − k0dt C = C0 - k0t Example: the k0 of a drug is 5.5 ng/ml/min, the initial C of the drug was 200 ng/mL. Calculate the time when the C of the drug is 100 ng/mL? Answer: C = C0 - k0t → 100 = 200 −5.5t → 100-200 = −5.5t → t = − 100/ − 5.5 = 18.2 min In PK, the time to require for a drug to be cleared for 50% is called the half-life time (t½). Substitute C with 0.5C0 in the above equation and you get the equation to calculate t½.

natural logarithm

The natural logarithm has the symbol "ln" and its base is e (or 2.71828).

What is Pharmacokinetics?

The science of the kinetics of a drug in an organism: Pharmacokinetics = absorption + distribution + elimination

Differential calculus involves finding the

_________ calculus involves finding the *rate* at which a variable quantity is changing. For example, the concentration (C) of a drug changes linearly as a function of time.

practice problem 2 The PK model depicted below represents a drug eliminated by renal excretion, biliary excretion, and drug metabolism. The metabolite distribution is described by a one-compartment open model. Questions: a) How many parameters are needed to describe the model if the drug is injected via IV? b) What compartments can be sampled? c) What would be the overall k from department 1? d) Write an expression describing the rate of change of drug concentration in compartment 1 (dC1/dt)

a- b- c- k = ke + kb + km d-Answer: dC1/dt = k21C2 - (k12+km+ke+kb)C1

The integral of y = mx is the

area under the curve (AUC).

The differential or rate equation is:

dC/dt = − k1C

In a first-order process, t1/2

is constant

This is a linear relationship

lnC = lnC0 - k1t

This is an exponential curve

log C = log C0 - k1t ∙ loge (loge = 0.434 =1/2.3)

Logarithm Rules log mn = log m/n = log (1/m) = log 1 = lne = log 10 = log 10n = log mn =

log mn = log m + log n log m/n = log m − log n log (1/m) = − log m log 1 = 0 lne = 1 log 10 = 1 log 10n = n log mn = nlog m

The pH scale is a

logarithmic scale: pH = − log [H3O+]

Exponential Rules m0 = m1 = m−1 = mp/mq = mp∙mq = (mp)q = mp/np = mp∙np =

m0 = 1 m1 = m m−1 = 1/m mp/mq = m p-q (example: 102/103 = 10−1) mp∙mq = mp+q (example: 102 ∙ 103 = 105) (mp)q = mpq (example: (102)3 = 106) mp/np = (m/n)p (example: 82/42 = 22) mp∙np = (m∙n)p (example: 82 ∙ 42 = 322)

Order

refers to the way in which the C influences the rate of the process

Plasma level-time relationships after oral administration of a drug

s10 graphs

Draw a diagram describing a thee-compartment model with first-order absorption and drug elimination from compartment 1.

s16

Overall scheme demonstrating the dynamic interplay between the drug and its pharmacologic effect

s4

In PK two orders are of importance

the zero- and first-order.

AUC can be calculated using the

trapezoidal rule:

t½ in a first-order reaction:

t½ is *constant* in a first-order reaction: t½ = 0.693/k1

Rate

velocity or speed of the reaction (elimination or absorption ), expressed as change in concentration of drug over a change in time: dC/dt