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 C [mass/volume] = Drug concentration K 1 = decay rate constant, proportion that is lost each time step b [C/t] = administered dose f(t) = different modes of administration k 2 [1/t] = decay rate Assumptions Instantaneous absorption of drug after injection. Natural decay of the drug

The Significance of k and Half-Life m=number of time steps per half life K is the decay rate constant, but how could we relate it to the time steps?

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, For: k=0.3 and b=0.5, The Discrete-Time Model

There is always a stable steady state at C=b/k k=0.3 and b=0.7, k=0.3 and b=0.5,

The Continuous Model 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.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 t t

The Continuous Model The total uptake per day is mg. 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 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. Concentration (mg)

Compartmentalized Model Drug administration removaldecay Transport Blood stream Blood Stomach

k2k2 b(t) k1k1 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 k 1 =removal from the small intestine k 2 =decay rate inside the bloodstream

Linear stability analysis The steady state of the system is (0,0) Is always stable

Figure: Takahashi et al Plasma concentration-time profiles of acetaminophen after oral administration at a dose of 7,7 mg/kg in fasted cynomolgus monkeys. Compartmentalized Model

Blue line: Concentration in the stomach Green line: Concentration in the bloodstream k 1 =0.1 [1/h] p=0.02 [1/h] k 2 =0.8 [1/h] Dose=27mg every 6 hours

Blue line: compartmentalized model Red line: single model Green line: bloodstream k 1 =0.173 [1/h] p=0.02 [1/h] k 2 =0.1 [1/h] Dose=650mg every 6 hours Compartmentalized Model Concentration (mg)

Compartmentalized Model k 1 =0.173 [1/h] p=0.22 [1/h] k 2 =0.1 [1/h] Dose=650mg every 6 hours

Further work Drug administration removal decay Transport uptake

Acknowledgements Gerda De Vries Petro Babak