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)

Compartmentalized Model

Drug administration

removal decay

Transport

Blood streamBloodStomach

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)

Compartmentalized Modelk1=0.173 [1/h]p=0.22 [1/h]k2=0.1 [1/h]Dose=650mg every 6 hours

Further workDrug administration

removal decay

Transport uptake

)()(

)()(

)(

)()())()((

)(

))()(()()()(

3

2

1

tCktBa

tmB

dt

tdC

tBa

tmBtBktBtSp

dt

tdB

tBtSptSktbdt

tdS

Acknowledgements

• Gerda De Vries• Petro Babak