Transcript Drugs - University of Alberta

Drugs: Determination of the
Appropriate Dose
Laura Rojas and Rita Wong
Biological Background
• Drug: is any chemical substance that, when
absorbed into the body of a living organism,
alters normal bodily function[1].
• Lethal dose: The amount of drug that would
induce toxicity.
• Therapeutic range: The amount of drug that
would produce a desired effect on cells.
Scientific Motivation
• What is the minimum effective dose?
• What is the maximum safe dose?
• Distinguish between prescription drugs and
non-prescription drugs.
• How should the drug be administered?
• How does the drug move from the small
intestine to the bloodstream?

The Basic Model
Assumptions
•Instantaneous absorption of drug after
injection.
•Natural decay of the drug
Cn 1  Cn  k1Cn  bf (t)
dC
 k2C  bf (t)
dt
•C [mass/volume] = Drug concentration
•K1 = decay rate constant, proportion that is lost each time step
•b [C/t] = administered dose
•f(t) = different modes of administration
•k2[1/t] = decay rate
The Significance of k and Half-Life
K is the decay rate constant, but how could we relate it to
the time steps?
1.2
1
0.8
0.6
0.4
0.2
0
0
1
1 k  m
2
5
m
10
15
t
 1/ 2
m=number of time steps per half life
Cn1  Cn  kCn  b
The Discrete-Time Model
• Fixed point=b/k for f(t)=1 (an instantaneous injection
every time step)
• Let g(x)=x-kx+b
g’(x)=1-k
• Since 0<k<1, g’(b/k)<1 and therefore there is always
a stable steady state at C=b/k
2.5
For: k=0.3 and b=0.7,
b 0.7

 2.33
k 0.3
For: k=0.3 and b=0.5,
b 0.5

 1.67
k 0.3
Concentration
2
1.5
1
0.5
0
0
5
10
15
20
Time steps
25
30
35
The Discrete-Time Model
Cn1  Cn  kCn  b
• There is always a stable steady state at C=b/k
3
2.33
2.5
k=0.3 and b=0.7,
2
Cn+1
k=0.3 and b=0.5,
1.5
1.67
1
0.5
0
0
0.5
1
1.5
Cn
2
2.5
3
The Continuous Model
dC
 k 2C  bf (t )
dt
•b [C/t]=administration of drug
•f(t) is the function the determine how the dose
would be administrated.
•k [1/t] is similar to the decay rate constant in
the discrete model.
•k=ln2/half-life
The Continuous Model
1.
dC
 kC  bf (t)
dt
f(t)=H(t-a)-H(t-b) (H(t)=Heaviside function) for a
constant injection for a timeperiod of length b-a
2. f(t)=δ(t-a) for an instantaneous injection of
magnitude b at time a
dY
dY
t
t
The Continuous Model

dC
 kC  bf (t)
dt
This is an example of
dextromethorphan, a cough
suppressant.
The total uptake per day is
673.15mg.
The lethal dose is 2100mg.
Doctors recommend to take
the drug up to seven days
(148h).
Here f(t) is an instantaneous injection every 4 hours.
The red line represents an injection every four hours continuously for 5
days, and the blue line represents an injection every four hours taking into
account that during the night you don't get any shot.
The Continuous Model
Concentration (mg)

dC
 kC  bf (t)
dt
This is an example of
Tylenol, usually taken for
cold, flu and headaches.
Tylenol, has a half life of 4 h.
The lethal dose of Tylenol is
7.5g.
The therapeutic range is at
10-30µg/mL of blood which is
50mg for an average man.
Here f(t) is an instantaneous injection every 4 hours.
The red line represents an injection every four hours continuously for 3
days, and the blue line represents an injection every four hours taking into
account that during the night you don't get any shot.
Compartmentalized Model
Drug
administration
Stomach
Blood
Blood stream
Transport
removal
decay
b(t)
k1
k2
dS (t )
 b(t )  k1S (t )  transport
dt
dB(t )
 transport k 2 B (t )
dt
•Transport: refers to diffusion from the small intestine to the
blood due to a gradient in the concentration
•Transport= p( S (t )  B(t )) where p is the permeability in the
membrane of the blood cell
•b(t)=Drug administration is a combination of Heaviside functions
•k1=removal from the small intestine
•k2=decay rate inside the bloodstream
Linear stability analysis
dS (t )
  k1S (t )  p ( S (t )  B (t ))
dt
dB(t )
 p ( S (t )  B (t ))  k 2 B (t )
dt
The steady state of the system is (0,0)
J ( 0, 0 )
p 
  k1  p

 
 k2  p 
 p
Re( )  k1  p  k2  p  0
 (0,0)
Is always stable
Compartmentalized Model
Plasma concentration-time profiles of acetaminophen after oral administration
at a dose of 7,7 mg/kg in fasted cynomolgus monkeys.
Figure: Takahashi et al. 2007
Compartmentalized Model
k1=0.1 [1/h]
p=0.02 [1/h]
k2=0.8 [1/h]
Dose=27mg every 6 hours
Blue line: Concentration in the stomach
Green line: Concentration in the bloodstream
Compartmentalized Model
Concentration (mg)
k1=0.173 [1/h]
p=0.02 [1/h]
k2=0.1 [1/h]
Dose=650mg every 6 hours
Blue line: compartmentalized model
Red line: single model
Green line: bloodstream
Compartmentalized Model
k1=0.173 [1/h]
p=0.22 [1/h]
k2=0.1 [1/h]
Dose=650mg every 6 hours
Drug administration
Further work
Transport
removal
uptake
decay
dS(t )
 b(t )  k1S (t )  p( S (t )  B(t ))
dt
dB(t )
m B(t )
 p( S (t )  B(t ))  k 2 B(t ) 
dt
a  B(t )
dC(t )
m B(t )

 k3C (t )
dt
a  B(t )
Acknowledgements
• Gerda De Vries
• Petro Babak