Transcript Two Compartment Model

Principles of
Pharmacokinetics
Pharmacokinetics of IV
Administration, 1-Compartment
Karunya Kandimalla, Ph.D.
Associate Professor, Pharmaceutics
[email protected]
Pharmacokinetics & Pharmacodynamics
ADME
R
R
R
Target
organ
R
R
R
2
Kinetics From the Blood or Plasma Data
Pharmacokinetics of a drug in
plasma or blood
Absorption (Input)
Disposition
Distribution
Excretion
Elimination
Metabolism
3
Objectives
• Be able to:
• To understand the properties of linear models
• To understand assumptions associated with
•
•
first order kinetics and one compartment
models
To define and calculate various one
compartment model parameters (kel, t½, Vd,
AUC and clearance)
To estimate the values of kel, t½, Vd, AUC and
clearance from plasma or blood concentrations
of a drug following intravenous administration.
4
Recommended Readings
• Chapter 3, p. 47-62
•
•
•
•
IV route of administration
Elimination rate constant
Apparent volume of distribution
Clearance
5
Intravascular Administration
• IV administration (bolus or infusion):
• Drugs are injected directly into central
•
compartment (plasma, highly perfused
organs, extracellular water)
• No passage across membranes
Population or individual elimination rate
constants (kel) and volumes of distribution
(Vd) enable us to calculate doses or
infusion rates that produce target (desired)
concentrations
6
Disposition Analysis (Dose Linearity)
70
Plasma Concentration (g/ml)
60
Dose 1 (100 mg)
Dose 2 (200 mg)
50
40
30
20
10
0
0.0
2.5
5.0
7.5
10.0 12.5 15.0 17.5 20.0
Time (hr)
Plasma Concentration / Dose (1  105)
35
30
Dose 1 (100 mg)
Dose 2 (200mg)
25
20
15
10
5
0
0.0
2.5
5.0
7.5
10.0 12.5 15.0 17.5 20.0
Time (hr)
7
Disposition Analysis (Time Variance)
Plasma concentration ( g/ml)
40
35
1st Adminitration
2nd Administration
30
3rd Administration
25
20
15
10
5
0
0
5
10
15
20
Time (hr)
8
Linear Disposition
• The disposition of a drug molecule is not
•
affected by the presence of the other
drug molecules
Demonstrated by:
a)
Dose linearity
Saturable hepatic metabolism may result in
deviations from the dose linearity
b)
Time invariance
Influence of the drug on its own metabolism and
excretion may cause time variance
9
Disposition Modeling
• A fit adequately describes the
•
•
experimental data
A model not only describes the
experimental data but also makes
extrapolations possible from the
experimental data
A fit that passes the tests of linearity will
be qualified as a model
10
One Compartment Model (IV Bolus)
• Schematically, one compartment model
can be represented as:
kel
Drug in
Body
Drug
Eliminated
Xp = Vd • C
Where Xp is the amount of drug in the body, Vd is the
volume in which the drug distributes and kel is the first
order elimination rate constant
11
One Compartment Data (Linear Plot)
Plasma concentration ( g/ml)
35
30
25
20
15
10
5
0
0
5
10
15
20
Time (hr)
12
One Compartment Data (Semi-log Plot)
Plasma concentration ( g/ml)
100
10
1
0.1
0
5
10
15
20
Time (hr)
13
Two Compartment Model (IV Bolus)
Blood,
kidneys,
liver
K 12
Drug in
Central
Compartment
Drug in
Peripheral
Compartment
K 21
kel
Fat, muscle
Drug
Eliminated
• For both 1- and 2-compartment models, elimination
takes place from central compartment
14
Two Compartment Data (Linear Plot)
Plasma concentration ( g/ml)
18
16
14
12
10
8
6
4
2
0
0
5
10
15
Time (hr)
15
Two Compartment Data (Semi-log Plot)
Plasma concentration ( g/ml)
100
10
1
0.1
0
5
10
15
Time (hr)
16
One Compartment Model-Assumptions
• 1-Compartment—Intravascular drug is in
rapid equilibrium with extravascular drug
• Intravascular drug [C] proportional to
extravascular [C]
• Rapid Mixing—Drug mixes rapidly in blood
•
and plasma
First Order Elimination Kinetics:
• Rate of change of [C]  Remaining [C]
17
Derivation-One Compartment Model
Bolus IV
Central
Compartment (C)
Kel
dC
  KelC
dt
dC
  Keldt
C
Integratin g the above equation between
C0 at time t  0 and C at time t
C  C  e  Kelt
0
Linearizin g the above equation
ln C  ln C0  Kelt
log C  log C0 
Kelt
2.303
18
IV Bolus Injection: Graphical Representation
Assuming 1st Order Kinetics
C0 = Dose/Vd C0
= Dose/Vd
• C0 = Initial [C]
Slope =
-K/2.303
Slope
= -Kel/2.303
• C0 is calculated by
Concentration versus time, semilog paper
back-extrapolating
the terminal
elimination phase to
time = 0
19
Elimination Rate Constant (Kel)
• Kel is the first order rate constant
describing drug elimination
(metabolism + excretion) from the body
dC
  KelC
dt
• Kel is the proportionality constant
relating the rate of change of drug
concentration and the concentration
• The units of Kel are time-1, for example
hr-1, min-1 or day-1
20
Half-Life (t1/2)
• Time taken for the plasma concentration to
•
reduce to half its original concentration
Drug with low half-life is quickly eliminated
from the body
C0
 C0  e  Kel t1/ 2
2
1
 e  Kel t1/ 2
2
1
ln   Kel  t1 / 2
2
ln 2 0.693
t1 / 2 

Ke /
Ke /
t/t1/2
% drug
remaining
1
50
2
3
4
5
25
12.5
6.25
3.125
21
Change in Drug Concentration as a
Function of Half-Life
Percent of Drug Remaining
120
% Remaining
100
80
60
40
20
0
t=0
1
2
3
4
5
6
7
Number of Half-Lives
22
Apparent Volume of Distribution (Vd)
• Vd is not a physiological volume
• Vd is not lower than blood or plasma volume
but for some drugs it can be much larger
than body volume
• Drug with large Vd is extensively distributed
to tissues
• Vd is expressed in liters and is calculated as:
V
Dose
C0
• Distribution equilibrium between drug in
tissues to that in plasma should be achieved
to calculate Vd
23
Volume of Distribution—The Concept
Plasma [C] Tissue [C]
“Apparent” Vd
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NB: For lipid-soluble drugs, Vd changes with body size
and age (decreased lean body mass, increased fat)
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Area Under the Curve (AUC)
• AUC is not a parameter; changes with Dose
• Toxicology: AUC is used as a measure of drug
exposure
• Pharmacokinetics: AUC is used as a measure of
bioavailability and bioequivalence
•Bioavailability: criterion of clinical effectiveness
•Bioequivalence: relative efficacy of different drug
products (e.g. generic vs. brand name products)
• AUC has units of concentration  time (mg.hr/L)
AUC 
Dose
Dose

Clearance Vd Kel
25
Calculation of AUC using trapezoidal rule
C
1
C
2
Concentration
Area t  12 (t2  t1 )(C1  C2 )
t2
1
t
1
t
2
Time
Area 0 
tn
1
2
(C1  C2 )(t 2  t1 )  12 (C2  C3 )(t3  t 2 )  ...
 12 (Cn 1  Cn )(t n  t n 1 )
26
Clearance (Cl)
• The most important disposition parameter that
describes how quickly drugs are eliminated,
metabolized and distributed in the body
• Clearance is not the elimination rate
• Has the units of flow rate (volume / time)
Dose
Cl 
AUC
• Clearance can be related to renal or hepatic
function
• Large clearance will result in low AUC
27
Clearance -The Concept
Cinitial
ORGAN
Cfinal
elimination
If Cfinal < Cinitial, then it is a clearing organ
28
Practical Example
• IV bolus
administration
• Dose = 500 mg
• Drug has a linear
disposition
Time
(hr)
1
2
3
4
6
10
12
Plasma
Conc.
(mg/L)
9.46
7.15
5.56
4.74
3.01
1.26
0.83
ln
(Plasma
Conc.)
2.25
1.97
1.71
1.56
1.10
0.23
-0.19
29
Linear Plot
Plasma concentration (mg/L)
10
9
8
7
6
5
4
3
2
1
0
0
2
4
6
8
10
12
Time (hr)
30
Natural logarithm Plot
Ln (Plasma concentration)
2.5
Kel
2
ln (C0)
y = -0.218x + 2.4155
R2 = 0.9988
1.5
1
0.5
0
0
2
4
6
8
10
12
-0.5
Time (hr)
31
Half-Life and Volume of Distribution
t1/2 = 0.693 / Kel = 3.172 hrs
Vd = Dose / C0 = 500 / 11.12 = 44.66
ln (C0) = 2.4155
C0 = Inv ln (2.4155) = 11.195 mg/L
32
Clearance
Cl = D/AUC
Cl = VdKel
Cl = 44.66  0.218 = 9.73 L/hr
33
Home Work
Determine AUC and
Calculate clearance from AUC
34
Principles of
Pharmacokinetics
Pharmacokinetics of IV
Administration, 2-Compartment
Karunya Kandimalla, Ph.D
[email protected]
Objectives
• Be able to:
• Describe assumptions associated with multi•
•
•
•
compartment models
Describe processes that take place during
distribution and terminal elimination
Define and calculate , β, t½, Vi, VdSS, Cl and
AUC
Understand influence of Volume of distribution
on loading doses and toxicity
Design appropriate experiments to determine
proper modeling of drug disposition
36
Recommended Readings
• Chapter 4, p. 73-92, 95-97
•
•
•
•
•
•
•
•
Multicompartment model assumptions (73-4)
Two-compartment open model (75-9)
Method of residuals (79-81)
Digoxin simulation (81-84)
Apparent volume of distribution (84-90)
Drug in tissue compartment (90-91)
Clearance and elimination constant (92)
Determination of compartment models (95-7)
37
Physiological Perspective
One
compartment
Quick
Two
compartments
k12
Quick
Slow
38
Notes on Two-Compartment Modeling
• Ideal model should mimic
distribution and disposition
• Full set of rate processes
seldom taken into account
• Tissue [C] often unknown
• Because tissue [C]
correlates with plasma [C],
response often (but not
always) correlates with
plasma [C]
• Invasive nature of tissue
sampling limits
sophistication
Blood or Plasma Pharmacokinetics
(2 compartment model)
Absorption (Input)
Disposition
Distribution
Elimination
Excretion
Metabolism
39
Multicompartment Modeling
k12
Two compartment
model
Vi
k21
Vt
k10
Three compartment
model
k31
Vt3
k13
k12
Vi
k21
Vt2
k10
Vi = Volume of central compartment
Vt 2 or 3 = Volume of peripheral compartments
40
Assumptions (Two-Compartment Model)
k12
Vi
k21
Vt
k10
• Drug in peripheral compartment (bone, fat, muscle etc.)
equilibrates with drug in central compartment
•Plasma, highly perfused organs, extracellular water
• [C] in a given compartment is uniform
• Two-compartment drugs distribute into various tissues at
different, first order rates
• Elimination follows a single 1st order rate process only after
distribution equilibrium is reached
41
Two-Compartment Model
(Mathematical Perspective)
• Ct is a bi-exponential decaying function that
depends on 2 hybrid constants (A and B),
which can be determined graphically, and the
distribution () and elimination (β) rate
constants
Ct = A • e -t + B • e –βt
Because  >> than β, this term
goes to zero at greater t values
A function of k10,
k12 and k21
42
2-Compartment Data (Linear Plot)
Concentration-Time Course of Caffeine IV Bolus
20.00
Cp (mg/dL)
16.00
12.00
8.00
4.00
0.00
0.00
2.00
4.00
6.00
8.00
10.00
Time (hr)
Clinical Pharmacology and Therapeutics. 1993;53:6-14
43
2-Compartment Data (Semi-log Plot)
Concentration-Time Course of Caffeine IV Bolus
Cp (mg/dL)
100.00
10.00
1.00
0.00
2.00
4.00
6.00
8.00
10.00
Time (hr)
Clinical Pharmacology and Therapeutics. 1993;53:6-14
44
2-Compartment Data (Semi-Log Plot)
Concentration-Time Course of Caffeine IV Bolus
100.00
Distribution or
Alpha Phase
Cp (mg/dL)
A
Elimination or Beta
Phase
Slope = β/2.303
10.00
B
Slope =
/2.303
1.00
0.00
2.00
4.00
6.00
8.00
10.00
Time (hr)
Note the bi-exponential decline in drug concentration
45
Calculation of Micro-constants
• k21 =  • B + β • A
A+B
• k10 =  • β
k21
k12
Vi
k21
Vt
k10
• k12 =  + β – k10 – k21
Note: Micro-constants cannot be calculated by direct means
46
Two-Compartment Elimination
Rate Constants
• k10 represents elimination from central
compartment only
• Larger than β
• Not dependent on drug transfer into tissue
compartment
• β represents elimination when distribution
equilibrium attained
• Influenced by drug transfer into deep tissues
• Clinically more useful than k10
47
Initial Concentration (Time = 0)
Question 1:
Based on the information gathered
thus far, what is the drug
concentration at time Zero?
• Answer: The initial concentration at time = 0
is equal to the sum of the intercepts A and B
48
Half-Life
• Compounds demonstrating two compartment
kinetics will have t1/2 estimates for each
exponential phases
• Distribution half-life
• t ½ Dist = ln2/
• Elimination half-life
• t ½ Elim = ln2/β
• Terminal Half life is the elimination half life
for most of drugs
49
What is the Elimination Half-Life
(Aspirin Vs. Gentamicin)?
• Aspirin
• Distribution phase accounts for 31% of the dose
• Elimination phase accounts for 69% of the dose
• Terminal half life is the elimination half-life for aspirin
• Gentamicin
• Distribution phase accounts for 98% of the dose
• Elimination phase accounts for 2% of the dose
• Distribution half life is the appropriate half-life for
gentamicin
Clinical Pharmacokinetics Concepts and Applications, Third edition, Lippincott
50
Volume of Distribution (Vd)
• One compartment model Vd is constant:
Dose
Vd 
C0
• Two compartment model Vd changes with
time and reaches a plateau at the
distribution equilibrium
51
Two Compartment Model (Vd vs. Time)
Volume distribution
35
30
Vt
25
20
Vdss
15
10
5
Vi
0
0
5
10
15
20
25
Time (hr)
52
Determination of Vi, Vdss and Vd
From Hybrid Constants
• Vi = Dose
• Vt = Vi k12
k21
A+B
• VdSS = Vi [1 + k12]
k21
• VdSS = Dose
β • AUC 0  ∞
• Note that VdSS is a
function of transfer rate
constants
• The more extensively a
drug distributes (i.e., the
higher k12) the larger the
volume of distribution
53
Vdss- The Concept
• Vt is mostly influenced by the elimination rate and
•
doesn’t reflect distribution
Vdss is mostly influenced by distribution
• Volume term of the steady state when a drug is infused at
a constant rate
Lies between Vi and Vt
•
• Generally, difference between Vdss and Vt is small
• Aspirin
• Vdss = 10.4 L, Vt = 10.5 L
• Gentamicin
• Vdss = 345 L, Vt = 56 L
• Substantially eliminated before distribution equilibrium
achieved
54
Loading Doses
• Loading doses are designed to achieve
therapeutic concentrations faster
 A: 45 mg/h constant
IV infusion
DL = Cp target • Vd
F
 B: Plasma [C]
 C: Drug remaining
from 530 mg IV
loading dose
55
Two-Compartment Distribution,
Loading Doses & Site of Action
• Some 2-compartment drugs exert their
therapeutic and toxic effects on target
organs located in the central compartment
• Lidocaine, quinidine, procainamide
Question 2:
How should loading doses
for these drugs be handled?
56
Loading Doses for TwoCompartment Drugs Acting in Vi
Answer:
Slow administration to allow for drug
distribution into Vt
OR…
Small bolus doses such that
Cp does not exceed
predetermined concentrations
DL = VC • CSS
57
Two-Compartment Distribution,
Loading Doses & Site of Action
• Some 2-compartment drugs exert their
therapeutic and toxic effects on target
organs located in Vt
• Digoxin has a myocardium distribution half-life
of 35 min and requires 8 to 12 h to completely
distribute
Question 3:
How should loading doses
for these drugs be handled?
58
Loading Doses for TwoCompartment Drugs Acting in Vt
Answer:
Quick administration is fine since
the initially observed high Cps are
not dangerous.
These concentrations, however,
cannot be used to predict
therapeutic effects.
DL = VSS • CSS
59
Loading Doses: The Case of
Lidocaine
• A: Loading dose +
Seizures
Hypotension
Infusion using Vi (volume
of central compartment)
• B: Loading dose +
Infusion using VSS
• Doted line: Constant
infusion with no loading
dose
• Dashed line: Loading
dose using Vi, no infusion
60
Tip
• If Vdss is unknown, use a value that falls
between Vt and Vi
61
Estimation of AUC
From hybrid constants:
• AUC0∞ = A + B

Area
β
t2 t3
= ½ (t3 – t2)(C2 + C3)
AUC by Trapezoidal Method
62
Clearance –Two Compartment Model
Q.CA
ORGAN
Q.Cv
elimination
• CA = arterial blood concentration; Cv= Venous blood concentration; Q =
blood flow
• Clearance = Q(Ca-Cv)
Ca
63
Clearance (Two Compartment Model)
Question 4:
Clearance (1- compartment Model):
Vd • Kel
Clearance (2- compartment model):
?
• Answer: Clearance is model independent. However we
need to use different rate constants depending on the
choice of volume term
• Example: Cltotal = k10 • Vi
64
Model-Independent Calculation of
Clearance
• Cl = Dose
AUC 0  ∞
• No modeling consideration needed, but
requires accurate measurement of AUC
• Early & frequent sampling essential
• Units = Volume/time
• Theoretical volume of blood or plasma
completely cleared of drug per unit time
65
One vs. Two Compartment Dilemma
• Distribution phase may be missed entirely
if blood is sampled too late or at wide
intervals after drug administration
66
Use of One Compartment Modeling
for Two-Compartment Drugs
• If no concentration-time data points lie above
back-extrapolated terminal line (semilog paper),
assume one-compartment kinetics
• One-compartment modeling can be used in place of
two-compartment modeling provided:
• Duration of distribution is small compared with
elimination half-life
• Elimination is minimal during distribution
• Referred to as “non-significant” 2-compartment kinetics
• Pharmacokinetic parameters must be computed after
distribution is over
67
Tips For Solving the
Problem Set
68
• Plot Cp against time on semilog paper
• Extrapolate terminal phase to t = 0
• Intercept = B
• Slope = b/2.303
• Read at least 3 extrapolated [C]s during
•
distribution
Calculate residual [C]s
• Measured – extrapolated
• Plot residuals against time (semilog paper)
• Intercept of “feathered” line = A
• Slope = /2.303
69