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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 k 2C 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 Cn1 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 Cn1 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 timeperiod 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 k 2 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 ) mB(t ) p( S (t ) B(t )) k 2 B(t ) dt a B(t ) dC (t ) mB(t ) k3C (t ) dt a B(t ) Acknowledgements • Gerda De Vries • Petro Babak