Chap 6'

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Chap 6. Modeling the Body as Compartments,
Sources and Streams
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III. Examples of Pharmacokinetic Models
Example 6.1 A Simple Drug Distribution Model
Assumptions of Pharmacokinetic modeling

• Transfer from one compartment to another is
  irreversible.
• The rate of transfer of drug from a compartment
  is proportional to the amount of drug in that
  compartment (1st order kinetics)
• Thus, absorption, excretion and
  metavolization are 1st order processes, with ka,
  ke, and km.
• The rate of release of drug from the dosage
  form is constant (while it lasts)
• The drug is not metavolized until after
  absorption.

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III. Examples of Pharmacokinetic Models
Example 6.1 A Simple Drug Distribution Model
M
D drug G GI tract U Urine B Body except urine
and GI tract
km
Rate
G
B
D
ke
R
ka
U
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III. Examples of Pharmacokinetic Models
Example 6.1 A Simple Drug Distribution Model
M
D drug G GI tract U Urine B Body except urine
and GI tract
km
Rate
G
B
D
ke
R
ka
U
B.C. At t 0, D Do and G0
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III. Examples of Pharmacokinetic Models
Example 6.1 A Simple Drug Distribution Model
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Fig. 6.4 Concentration of drug in the body (B) as
a function of time for two types of
drug dosage forms
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VIII. Concentration-Time Behavior in Simple
Compartmental Systems
A. The OneCompartment Open Model
Rest of Body
Urine, Feces, Metabolization
GI Tract

• The conc. of active drug in the body is affected
  by the rates of absorption and elimination steps.
• Should consider drug in GI tract (Drug swallowed
  in liq. form)
• Elimination metabolization, urine and feces
• Thus, 1st order irreversible kinetics can be
  assumed.

k1
k2
A amount of drug in the GI tract B amount of
drug in the rest of body C drug eliminated by
urine, feces and metabolization
A ? B ? C
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k1
k2
A ? B ? C
Conditions and Resulting Equations for the Time
Dependency of Each Components
General case
Rapid final step
Rapid initial step
If k1 k2,
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B. Data Analysis Using the OneCompartment Open
Model Rapid Initial Step (k1 gt k2) Case
Drug concentration in the body B
As A?0 (after considerable time),
or
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Fig 6.18, 6.19, 6.20, 6.21
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D. The Special Case of Intravenous Injection

• Body as a single perfectly mixed tank
• Drug is slowly eliminated and metabolized, and
  thus an uniform concentration of drug in the body
  in spite of a short time after the injection

k2
B
C
Drug concentration in the body B
exponential decay
t
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E. The Special Case of Constant Intravenous
Injection
For the continuous intravenous injection to the
body,
k2
R
R constant drug release rate
B
A
C
or
Integrating factor
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F. The Two-Compartment Open Model
Body Compartment Blood Tissue
k2
k1
P
C
A
k12
k21
T
Two-Compartment Model
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F. The Two-Compartment Open Model
A simple model for drug uptake, distribution and
elimination
Drug metabolism in tissues etc.
Metabolites in blood
Drug in tissues and other fluids of distribution
Urinary excretion of intact drug
Drug in depot
Drug in blood
Urinary excretion of metabolites
Drug metabolism
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F. The Two-Compartment Open Model
General and realistic model for drug uptake,
distribution and metabolism and excretion
Drug fixed to tissues
Metabolites in the other fluid
Drug in saliva
Drug in other fluids
Metabolites fixed to tissues
Drug(sol) in stomach fluids
Solid drug in stomach
Solid drug in dosage form
Drug in extracellular fluid
Metabolites in extracellular fluid
Metabolites in urine
Metabolites in bile
Drug(sol) in intestinal fluids
Solid drug in intestine
Drug in bile
Metabolites in intestinal tract
Metabolites in feces
Drug in urine
Drug in feces
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G. The Case of Intravenous Injection
k2
P
C
k12
k21
or
T
Bi-exponential
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IX. BLOOD/TISSUE MODELS (LOCAL REGION MODELS)

• Lumped modeling of one-compartment
• Thus, no fluid streams through the compartment
• Local region of the blood flow
• No convective solute transport

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A. Two-Compartment Local Model Assuming
Equilibrium
blood
tissue
Mass balance for a solute(drug) across the
compartment
Since blood volume is much less than the tissue
volume,
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A. Two-Compartment Local Model Assuming
Equilibrium (cont)
blood
tissue
Equilibrium between blood and tissue
and
If the drug is suddenly introduced at a constant
rate into the blood,
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A. Two-Compartment Local Model Assuming
Equilibrium (cont)
All mass transport processes (chemical-kinetic
model)
k2
k1
X
D
E
or
For a step change, at t0, D0 to DDo
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A. Two-Compartment Local Model Assuming
Equilibrium (cont)
k2
k1
blood
X
D
E
tissue
Thus,
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X. USE OF INDICATORS TO DETERMINE REGIONAL
BLOOD VOLUMES AND BLOOD FLOW RATES
Indicators (usually dyes)
blood
ave. accumulation rate 0
M (mass of dye introduced)
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X. USE OF INDICATORS TO DETERMINE REGIONAL
BLOOD VOLUMES AND BLOOD FLOW RATES
Mean residence time of a tracer in the region
and
Thus, the volume of the blood is
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A single compartment model for dye infusion. CBi
- incoming dye concentration CCBi - inc outgoing
dye concentration Q - blood flow rate VB -
volume of blood in the organ.
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2. Indicator Dilution Dye
Technique
Addition of 1 cc of dye solution with a known
concentration of 2000 dye particles per cc into
an unknown volume of solution. If the final dye
concentration becomes 1 particle per cc then
2000 particles x(1cc)1 particle x unknown vol 1
cc 1cc Unknown vol 2000 x 1cc/particle
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Procedure

• 1) A dye of known volume and concentration is
  injected into the blood via the
• right atrium or pulmonary artery.
• 2) The concentration of the dye after
  equilibration measured at a downstream
• site usually the femoral of radial artery
• 3) The greater the final concentration, the
  smaller the flow (volume)
• 4) A densitometer spectrophotometrically
  determines the moment to moment
• dye concentration producing a resulting curve
• Cardiac output 60 x the amount of dye injected
  divided by the area under
• the curve (average dye concentration x time)

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Sources of error

• As the blood is in motion, the addition of the
  dye must be rapid and the
• measurement of the dye concentration downstream
  site continuous until all of
• the added dye flows past the sampling site.
• Inaccurate measurement
• Prolonged injection time
• Inadequate mixing of dye blood
• Intracardiac shunts, low cardiac output,
  valvular insufficiency
• Rapid recirculation may affect calculation

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Concentration-time curve for dye infusion
X - time of injection at the injection site Y -
time of first arrival of the dye at the sampling
site A - rising part of the dye concentration
curve B - exponential decay part of the curve
C - beginning of recirculation of the dye D -
part of the curve if no recirculation had
occurred.
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Concentration-time curve data. the rectangular
areas represent the calculated average
concentration of dye in the arterial blood for
the duration in the respective curves. (Redrawn
from Guyton 1986, with permission from the
author and W. B. Saunders Co.).
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Cardiac output measurement by indicator dilution

• Fick Technique

• Requires the simultaneous collection of arterial
  mixed venous (pulmonary artery) blood samples
  while a sample of expired gas is taken
• The difference in oxygen content of the arterial
  venous blood is the AV O2 difference. Oxygen
  consumption is calculated from the oxygen content
  of the inspired minus the expired gas and the
  respiration rate
• Accurate in low cardiac output states, valvular
  regurgitation and shunts

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Fick principle for the measurement of cardiac
output
C. O. - cardiac output A - arterial V - venous
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Indicator dilution thermal
Technique

• An indicator dilution technique which uses
  temperature change as the indicator

• A chilled solution is added to the blood
  (usually via right atrium), and the resultant
  drop in temperature is recorded at a downstream
  site (usually in pulmonary artery)
• Cardiac output is determined from 4 variables

• volume of injectate (often iced water to
  increase signal to noise ratio)
• the area under the curve
• temperature of the blood (use patients
  temperature)
• the correction factor for the injectate warming

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Indicator dilution thermal
Sources of error

• Not accurate in presence of shunts, or pulmonary
  or tricuspid regurgitation
• May be inaccurate in low cardiac output states
• Specific heat and gravity of blood changes with
  Hct (a factor in calculation of
• cardiac output)
• Patients respiration
• Poor injection technique