© D. Wong 2002 © D. Wong 2003 1 CS610 / CS710 Database Systems I Daisy Wong.
Drugs: Determination of the Appropriate Dose Laura Rojas and Rita Wong.
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Transcript of 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.
• 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?
Scientific Motivation
The Basic Model
Cn1 Cn k1Cn bf (t)dC
dt k2C bf (t)
•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
Assumptions•Instantaneous absorption of drug after injection.•Natural decay of the drug
The Significance of k and Half-Life
mk
2
11
m=number of time steps per half life
K is the decay rate constant, but how could we relate it to the time steps?
2/1t
m
• 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
For: k=0.3 and b=0.7, 33.23.0
7.0
k
b
For: k=0.3 and b=0.5, 67.13.0
5.0
k
b
bkCCC nnn 1
0
0.5
1
1.5
2
2.5
0 5 10 15 20 25 30 35
Time steps
Co
nc
en
tra
tio
n
The Discrete-Time Model
The Discrete-Time Model• There is always a stable steady state at C=b/k
k=0.3 and b=0.7,
0
0.5
1
1.5
2
2.5
3
0 0.5 1 1.5 2 2.5 3
Cn
Cn+
1
2.33
1.67
bkCCC nnn 1
k=0.3 and b=0.5,
The Continuous Model
)(2 tbfCkdt
dC
•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 Model1. f(t)=H(t-a)-H(t-b) (H(t)=Heaviside function) for a
constant injection for a time period of length b-a2. f(t)=δ(t-a) for an instantaneous injection of
magnitude b at time a
dC
dt kC bf (t)
dY
t
dY
t
The Continuous Model
dC
dt kC bf (t)
The total uptake per day is 673.15mg.
The lethal dose is 2100mg.
Doctors recommend to take the drug up to seven days (148h).
This is an example of dextromethorphan, a cough suppressant.
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
dC
dt kC bf (t)
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.
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.
Con
cent
ratio
n (m
g)
k2
)()(
)()()(
2
1
tBktransportdt
tdB
transporttSktbdt
tdS
b(t)
k1
•Transport: refers to diffusion from the small intestine to the blood due to a gradient in the concentration •Transport= 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
))()(( tBtSp
Linear stability analysis
)())()(()(
))()(()()(
2
1
tBktBtSpdt
tdB
tBtSptSkdt
tdS
pkp
ppkJ
2
1
)0,0(The steady state of the system is (0,0)
)0,0(
0)Re( 21
pkpk
Is always stable
Figure: Takahashi et al. 2007
Plasma concentration-time profiles of acetaminophen after oral administration at a dose of 7,7 mg/kg in fasted cynomolgus monkeys.
Compartmentalized Model
Compartmentalized Model
Blue line: Concentration in the stomachGreen line: Concentration in the bloodstream
k1=0.1 [1/h]p=0.02 [1/h]k2=0.8 [1/h]Dose=27mg every 6 hours
Blue line: compartmentalized modelRed line: single modelGreen line: bloodstream
k1=0.173 [1/h]p=0.02 [1/h]k2=0.1 [1/h]Dose=650mg every 6 hours
Compartmentalized ModelC
once
ntra
tion
(mg)
Further workDrug administration
removal decay
Transport uptake
)()(
)()(
)(
)()())()((
)(
))()(()()()(
3
2
1
tCktBa
tmB
dt
tdC
tBa
tmBtBktBtSp
dt
tdB
tBtSptSktbdt
tdS