# Basic Pharmacokinetics and Toxicokinetics

Crispin H. Pierce, Ph.D.

(206) 685-9537 -- crispo@u.washington.edu

*Absorption*

### Absorption assumes primary importance in
oral, inhalation, and dermal exposures. The two kinetic parameters of concern
are the rate of absorption and the extent of absorption (or bioavailability).

###

### · Rate of Absorption

###

### -- The rate of absorption determines the time of onset
and the degree of acute toxicity. This is largely because time to peak (Tpeak)
and maximum concentration (Cmax) after each exposure depend on the rate
of absorption.

### Slowing of absorption (AÆB)

### - prolonged Tp

### - lower Cmax

###

### -- In instances when the absorption rate is slower than
elimination rate, the rate of washout of toxicant becomes rate-limited by
absorption rather than by elimination (i.e., a depot effect).

### · Systemic Availability

### The actual extent of exposure as defined by the amount
of toxicant reaching the systemic circulation is determined by (1) entry
barrier permeability, and (2) the extent of "first-pass" metabolism.
The fraction of dose reaching the system circulation in intact form, or
systemic availability (F), is estimated from either the AUCs, AUCroute

### F = ------------

### AUCiv

### or from the amount of intact toxicant excreted in urine
or exhaled via the lungs (Aex).

### Aex-route

### F = ---------

### Aex-iv

###

### Modelling absorption

### A. Intravenous dosing

### IVrate = IVdose / Tinf

### and

### input into venous blood.

###

### B. Percutaneous dosing

### Percrate = (Percdose*exp(-KA*percTime))*KA, perc

### and

### input into venous blood

###

### C. Oral dosing

### Oralrate = (Oraldose*exp(-KA*oralTime))*KA, oral

### and

### input into liver

###

### D. Inhalation dosing

### Qp * (Cinh- Calv) = Qc * (Cart - Cven)

### kblood/air = Pblood/air = Cart / Calv

### then

### Cart = (QpPb/aCinh + CvenPb/aQc)/(QcPb/a + Qp)

### Inhalationrate = CartQc

*Volume of Distribution*

### The Volume of Distribution is the apparent
volume into which a drug or toxicant distributes, and provides a proportionality
constant between blood (or plasma) concentration and the amount in the body:

### Volume of Distribution = Amount / Concentration

### The volume of distribution can be readily calculated after
an intravenous bolus dose of a substance that exhibits "one-compartment
model" characteristics:

###

### Volume of Distribution = Dose / Initial Concentration

### However, because of the uncertainty in the estimate of
Co, volume can be more accurately estimated by V = Dose / (k·AUC),
where AUC is the area under the concentration-time curve.

###

### The volume of distribution does not necessarily correspond
to any physiologic volume, and is influenced by binding to plasma and tissue
constituents. Volume can range from about 3 liters (as is seen with Tolbutamide,
which is distributed in blood only, to about 50,000 liters (as is seen with
Quinacrine, which distributes and binds to many tissues).

###

### -- The volume of distribution relates blood conc. to the
total body burden of a toxicant, i.e., AB = V·Cb·

### -- Physiologic Meaning? A measure of extravascular distribution.
Two determinants of distribution into a tissue region: - Tissue or organ
volume Vti - Distribution or Partition ratio Pi = Cti/Cb Æ a constant
@ pseudo-distribution equilibrium or steady state. Accordingly,

### ith Tissue Load = Vti·Cti = Vti·(Pi·Cb)

### n

### Total Tissue Load = S Vti·Pi·Cb (n = # of
tissues)

### i=1

### Total Body Load = Amount in blood + Amount in tissues

### n n

### AB = Vb·Cb + S Vti·Pi·Cb = (Vb + S
Vti·Pi)·Cb

### i=1 i=1

### By definition, n Vb, Vti, Pi are

### V = AB/Cb = Vb + S Vti·Pi constant

### i=1

### blood apparent tissue volume volume

### Since Pi can assume a value ~0Æ·, V varies
from a minimum of Vb to many times the body size. Because the volume of
distribution reflects the degree of xenobiotic dispersal and binding to
all tissues, the following relationship is observed:

###

### Vinitial < Vsteady-state < Vterminal phase

*Clearance*

### Clearance is a measure of the body's ability
to completely clear a drug or toxicant from blood or plasma. Clearance is
the rate of elimination by all routes relative to the concentration in a
systemic biologic tissue, and is measured in units of flow, or volume per
unit time.

### CL (units of volume/time) = Rate of elimination (units
of mass/time) / Concentration (units of mass/volume)

###

### Clearance is normally measured by collecting blood concentration-time
data following a known dose, and using the following equation:

###

### CL (units of volume/time) = F * Dose (units of mass) /
AUC (units of time-mass/volume)

###

### where F is the bioavailability (fraction of dose entering
systemic circulation), and AUC is the area under the blood concentration-time
curve.

###

### Clearance also plays a role in determining the steady-state
concentration of a drug or toxicant:

###

### Csteady-state = Rate of administration/ CL

###

### -- Area Under the Blood Concentration Time Curve (AUC):
an internal or systemic exposure index.

###

###

### & since C0 = Dose/V, then Dose

### AUC = ------

### k·V

### The product k·V is equal to clearance.

### Dose Dose

### \ AUC = ------ or CL = ------

### CL AUC

### i.e., clearance governs the extent of systemic exposure
as represented by AUC for a given dose of toxicant.

### -- Physiologic basis of clearance: relationship to eliminating
organ function.

### - Blood clearance can be resolved into components representing
the various metabolic and excretory pathways of elimination, e.g., CL =
CLm + CLex

### or further resolved into organ clearances, e.g., CL = (CLh
+ CLg + ...) + (CLk + CLp + ...)

### liver gut kidney lung

### mucosa

### - Individual organ clearance can in turn be related to
organ blood flow (Qi) and extraction efficiency (Ei). For instance, Hepatic
Clearance (CLh) = Qh·Eh, note that Eh varies from 0Æ 1 (i.e.,
0 to 100% extraction) \ the upper limit of CLh ~ Qh when Eh ~ 1

### Extraction ratio can in turn be related to transit time
through organ, transport parameters, metabolic enzyme activities (Michaelis
constants: Km, Vmax), intracellular binding, etc.

*Half-Life*

### The Half-life is a measure of how rapidly
a steady-state concentration will be achieved during constant exposure,
and conversely how rapidly the concentration will fall after cessation of
exposure.

### Half-life is related to the elimination rate constant k
by the formula:

### t1/2 = ln 2 / k

### The elimination rate constant, like the clearance, is a
fractional rate of decline: k = Rate of elimination / Amount

###

### Since CL = Rate of elimination/Concentration, the elimination
rate constant can be estimated: k = CL / Volume of Distribution

### Half-life can then be found by t1/2 = ln 2/ k = ln 2 *
V / CL

###

### · Elimination Half-life (t0.5, t1/2) is a characteristic
of First-order kinetics

### For a one-compartment model:

### Rate Equation: dAB ---- = - k · AB

### dt

### AB = Amount of toxicant in body k = 1st order elimination
rate constant

### -- Since AB declines with time, elimination rate also decreases!
However, the fractional rate is a constant, i.e.,

### - 1 dAB -DAB/AB

### -- · --- ª ------- = k Units: time-1

### AB dt Dt hr-1, min-1

### -- Upon integration, AB = A0·e-k·t A0 = Body
load @ t=0

### -- But blood conc. rather than body load is measured.

### AB a Cb and AB = V · Cb V = Apparent Volume of Distribution

### \ Cb = C0·e-k·t or Ln Cb = Ln C0 - k·t
where C0=Blood conc. @ t=0

###

### This explains the monoexponential decline.

### Note that when Cb = 1/2C0, t = 0.693/k = t1/2,

###

### -- It always takes 1 t1/2 to reach 50% of any starting
conc. (i.e., t1/2 independent of C0)

###

### -- Takes about 3-4 t1/2s to effect 90% of elimination or
to achieve 90% of the steady-state value under constant exposure.

###

### -- For compounds with multicompartmental kinetics, there
will be a t1/2 estimate for each of the exponential phases. The terminal
t1/2 is often quoted as the "Elimination t1/2," whereas the t1/2s
of the earlier phases are referred to as "Distribution t1/2s."

### For example, in a two compartment model described by Cb
= A·e-a·t + B·e-b·t,

### t1/2,a = 0.693/a and t1/2,b = 0.693/b

*Physiologic Model*

*· Body is viewed as grouped tissue
compartments which are interconnected by blood flows.*

*· Modeling is mechanistic, and
compartments are defined by physiologic volumes and partition coefficients.*

###

*· Model parameters are fit within
physiologic bounds to observed data. Use of minimum number of compartments
that adequately describe the data.*

### · Flow-limited delivery of xenobiotics to tissue
groups.

### where: At = Amount in tissue, Qt = Blood flow to tissue,
Ca = Arterial blood concentration, Kp = Tissue/blood partition coefficient,
Vt = Tissue volume

###

###

### · Michaelis-Menten metabolism

### where:

### At = Amount in tissue

### Qt = Blood flow to tissue

### Ca = Arterial blood concentration

### Kp = Tissue/blood partition coefficient

### Vt = Tissue volume

### Vmax = Maximum metabolic rate

### Km = Michaelis constant (toxicant concentration at half-maximum
metabolic rate)

### · Uses of the physiologic model

### -- Use across a wide range of doses - low to saturating.

### -- Interspecies scaling, in particular with regards to
risk.

### -- Determination of target tissue dose.

### -- Simulating complex risk assessment conditions.

### -- Searching for relevant interindividual differences.

###

###

### · Drug/toxicant-specific considerations

### -- Molecular weight, conformation, charge >>>
Flow or diffusion limited delivery and filtration by kidney.

### -- Vapor pressure over blood >>> Elimination by
exhalation

### -- Molecular weight, conformation, charge (lipophilicity)
>>> Tissue distribution and accumulation

###

### · Human physiologic considerations

### -- Genetic polymorphism >>> Metabolic rates

### -- Sex and age differences >>> Blood flows and
metabolic rates

### -- Weight >>> Blood flows and tissue volumes

### -- Body fat percentage >>> Adipose tissue volume

### -- Working conditions >>> Alveolar ventilation
rate, cardiac output, blood flows

### -- Route of administration >>> First-pass effects

### References

###

### Clearly written textbook on pharmacokinetic principles
and applications:

### Malcolm Rowland and Thomas N. Tozer, Clincial Pharmacokinetics:
Concepts and Applications, third edition, Lea & Febiger, Philadelphia.

###

### Good overall reference:

### Clewell, H.J., III and M.E. Anderson, Risk assessment extrapolations
and physiological modeling. Tox and Ind Health, 1985. 1(4): p. 111-131.

###

### Many models presented, including discussion of interspecies
scale-up and partitioning within a compartment.

### Bischoff, K.B., Physiologically based pharmacokinetic modeling,
in Pharmacokinetics in risk assessment. 1987, National Academy Press: Washington,
D.C.

###

### Linkage of toxicokinetic and toxicodynamic models:

### Conolly, R.B., R.H. Reitz, and M.E. Anderson, Mutation
accumulation: a biologically based mathematical model of chronic cytotoxicant
exposure, in Pharmacokinetics in risk assessment. 1987, National Academy
Press: Washington, D.C.

###

### Model application to cancer risk assessment:

### Krewski, D., D.J. Murdoch, and J.R. Withey, The application
of pharmacokinetic data in carcinogenic risk assessment, in Pharmacokinetics
in risk assessment. 1987, National Academy Press: Washington, D.C.