Cold Pills & Compartmental Modeling

Ronnie SchumannMA 354- Math ModelingDr. Jyoti ChampanerkarMonday, Dec. 5, 2005

Background

Idea: Treat body as a set of homogenous compartments through which medication must pass: GI Tract, Bloodstream, Organs, Excretory System.

Medication moves from compartment A to compartment B at a rate proportional to amount of drug in compartment A.

Important to know: WHERE? WHEN? HOW MUCH?

Sensitivity to One Dose vs. Continuous Doses

One Dose: Models

x[t]: drug amount in GI at time t y[t]: drug amount in bloodstream at time t k1 & k2: clearance coefficients of GI & bloodstream

Assume medication dissolves instantly upon entering body

dx(t)

dt= -k1x(t), x(0) = A

dy(t)

dt

= k1x(t) - k2y(t), y(0) = 0

One Dose: Models

Provided k1, k2 > 0 & k1 ≠ k2

As t ∞, x[t] 0, y[t] 0: drug is eliminated from system

k1 & k2 depend on type of drug, specific compartments, age & health of patient

x(t) = Ae-k1t

y(t) = k1A

k1 - k2

(e-k2t – e-k1t)

Antihistamine in Bloodstream (y[t]) vs. Time (t)

Max: y[23.78 hr] = .57 units

k2= .0231 hr-1

k1= .06931 hr-1

A = 1

Period: 24 hr

Antihistamine in Bloodstream (y[t]) vs. Time (t)

Max: y[17.96 hr] = .66 units

k2= .0231 hr-1

k1= .11 hr-1

A = 1

Period: 24 hr

Antihistamine in Bloodstream (y[t]) vs. Time (t)

Max: y[9.26 hr] = .807 units

k2= .0231 hr-1

k1= .3 hr-1

A = 1

Period: 24 hr

Antihistamine in Bloodstream (y[t]) vs. Time (t)

Max: y[5.08 hr] = .889 units

k2= .0231 hr-1

k1= .6931 hr-1

A = 1

Period: 24 hr

Antihistamine in Bloodstream (y[t]) vs. Time (t)

Max: y[3.86 hr] = .915 units

k2= .0231 hr-1

k1= 1.0 hr-1

A = 1

Period: 24 hr

Antihistamine in Bloodstream (y[t]) vs. Time (t)

Max: y[2.83 hr] = .937 units

k2= .0231 hr-1

k1= 1.5 hr-1

A = 1

Period: 24 hr

One Dose: Variance in k1

Larger k1 values higher & quicker peaks in y[t]

TRAFFIC JAM!!!

Restrictions

Effective range of a drug Fast-acting Long-lasting

Effective range: 0.2 – 0.8 units1. y[t] ≤ 0.8 units, for all t2. 0.2 units ≤ y[2 hr] ≤ 0.8 units3. 0.2 units ≤ y[24 hr] ≤ 0.8 units Pronounce k1 values desirable

or undesirable

Graphical Examination: Box Plots

Antihistamine in Bloodstream (y[t]) vs. Time (t)

y[2 hr] = 0.126 units0.126 units; y[24 hr] = 0.577 units0.577 units UndesirableUndesirable

k2= .0231 hr-1

k1= .06931 hr-1

A = 1

Period: 24 hr

Antihistamine in Bloodstream (y[t]) vs. Time (t)

y[2 hr] = 0.193 units0.193 units; y[24 hr] = 0.637 units0.637 units UndesirableUndesirable

k2= .0231 hr-1

k1= .11 hr-1

A = 1

Period: 24 hr

Antihistamine in Bloodstream (y[t]) vs. Time (t)

y[2 hr] = 0.44 units0.44 units; y[24 hr] = 0.622 units0.622 units Undesirable Undesirable Maximum > 0.8 Maximum > 0.8

k2= .0231 hr-1

k1= .3 hr-1

A =1

Period: 24 hr

Antihistamine in Bloodstream (y[t]) vs. Time (t)

y[2 hr] = 0.729 units0.729 units; y[24 hr] = 0.594 units0.594 units Undesirable Undesirable Maximum > 0.8 Maximum > 0.8

k2= .0231 hr-1

k1= .6931 hr-1

A = 1

Period: 24 hr

Antihistamine in Bloodstream (y[t]) vs. Time (t)

y[2 hr] = 0.839 units0.839 units; y[24 hr] = 0.588 units0.588 units UndesirableUndesirable

k2= .0231 hr-1

k1= 1.0 hr-1

A = 1

Period: 24 hr

Antihistamine in Bloodstream (y[t]) vs. Time (t)

y[2 hr] = 0.919 units0.919 units; y[24 hr] = 0.583 units0.583 units UndesirableUndesirable

k2= .0231 hr-1

k1= 1.5 hr-1

A = 1

Period: 24 hr

Range of Effective k1 for k2 = .0231

Upper Bound: k1 = .282

Lower Bound: k1 = .115

Want to maximize range of k1 for which drug is safe & effective.

Small k1 range: Reformulation

Different drug Different concentration Different path

Limit Users

Continuous Doses: Model

x[t]: drug amount in GI at time t y[t]: drug amount in bloodstream at time t k1 & k2: clearance coefficients of GI & bloodstream

R measured in (units / hr) Assume medication dissolves instantly upon entering

body

dx(t)

dt= R - k1x(t), x(0) = 0

dy(t)

dt

= k1x(t) - k2y(t), y(0) = 0

Continuous Doses: Model

Provided k1, k2 > 0 & k1 ≠ k2

As t ∞, x[t] (R/k1), y[t] (R/k2): equilibrium levels

k1 & k2 depend on type of drug, specific compartments, age & health of patient

k1 & k2 used for decongestant & antihistamine scenarios

x(t) = (1-e-k1t)

y(t) =R

k2

(e-k2t – e-k1t)]

R

k1

1

k1 – k2

[1+

Clearance Coefficients & Equilibrium Levels for Young & Healthy

Decongestant Antihistamine

K1

(GI)

1.386 hr-1

Eq: 0.722 units

0.6931 hr-1

Eq: 1.443 units

K2

(blood)

0.1386 hr-1

Eq: 7.215 units

0.0231 hr-1

Eq: 43.29 units

Clearance Coefficients & Equilibrium Levels for Old & Infirm

Decongestant Antihistamine

K1

(GI)

0.462 hr-1

Eq: 2.165 units

0.231 hr-1

Eq: 4.328 units

K2

(blood)

0.0462 hr-1

Eq: 21.65 units

0.0077 hr-1

Eq: 129.87 units

Young’uns vs. Elders: GI/decongestant Equilibrium: .7215 units 30 hours

Young’uns vs. Elders: Blood/decongestant Equilibrium: 7.215 units 30 hours

Young’uns vs. Elders: GI/decongestant Equilibrium: 2.165 units 30 hours

Young’uns vs. Elders: Blood/decongestant Equilibrium: 21.645 units 72 hours (3 days)

Young’uns vs. Elders: GI/antihistamine Equilibrium: 1.443 units 30 hours

Young’uns vs. Elders: Blood/antihistamine Equilibrium: 43.29 units 120 hours (5 days)

Young’uns vs. Elders: GI/antihistamine Equilibrium: 4.328 units 30 hours

Young’uns vs. Elders: Blood/antihistamine Equilibrium: 129.87 units 336 hours (14 days)

Restrictions: Elderly Blood/antihistamine Effective range of a drug Fast-acting Long-lasting

Effective range: 25-50 units

1. y[t] ≤ 50 units, for all t

2. 25 units ≤ y[24 hr] ≤ 50 units

3. 25 units ≤ y[120 hr] ≤ 50 units

Elders: Blood/antihistamine

Function increases too rapidly between 24 hours & 120 hours

k1= . 231 hr-1

k2= .0077 hr-1

R = 1 unit/hr

y[24] = 18.21 units

y[120] = 76.544 units

Elders: Blood/antihistamine

Function increases too rapidly between 24 hours & 120 hours

k1= . 231 hr-1

k2= .0077 hr-1

R = 1.4 unit/hr

Elders: Blood/antihistamine

Function increases too rapidly between 24 hours & 120 hours

k1= . 231 hr-1

k2= .0077 hr-1

R = .65 unit/hr

Elders: Blood/antihistamine

No constant continuous dosage amount satisfies both conditions

Solution: Discontinuous dosage

Vary treatment times Vary treatment amounts

Applications & Drawbacks

Cannot track movement AND absorption of drugs at one time—can devise set of related systems to describe nearly all attributes of drug flow.

Versatile technique for absorption modeling.

Complex compartmental models useful in epidemiology when several possible paths are present.

Bibliography

Borrelli, Robert L. & Courtney S. Coleman. Differential Equations: A Modeling Perspective. Preliminary ed. New York: John Wiley & Sons, Inc., 1996.

Foster, David. Principles of Clinical Pharmacology. University of Washington. Undated. http://www.cc.nih.gov/researchers/training/principles/ppt/foster_slides_2002-2003.ppt#1.

Giordano, Frank, M. Weir, & W. Fox. A First Course in Mathematical Modeling. 3rd ed. Pacific Grove, CA: Brooks/Cole-Thomson Learning, 2003.

White, Emma. Epidemic Models for Drug Use. NUI Maynooth. Undated. http://www.nuim.ie/rosie/documents/EpidemicModelsforDrugUseIHRCConferenceMarch2005-Emma.pdf

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QUESTIONS?