Cold Pills & Compartmental Modeling Ronnie Schumann MA 354- Math Modeling Dr. Jyoti Champanerkar Monday, 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 k 1 & k 2 : clearance coefficients of GI & bloodstream Assume medication dissolves instantly upon entering body dx(t) dt = -k 1 x(t), x(0) = A dy(t) dt = k 1 x(t) - k 2 y(t), y(0) = 0

One Dose: Models Provided k 1, k 2 > 0 & k 1 ≠ k2 As t  ∞, x[t]  0, y[t]  0: drug is eliminated from system k 1 & k 2 depend on type of drug, specific compartments, age & health of patient x(t) = Ae -k 1 t y(t) = k 1 A k 1 - k 2 (e -k 2 t – e -k 1 t )

Antihistamine in Bloodstream (y[t]) vs. Time (t) Max: y[23.78 hr] =.57 units k 2 =.0231 hr -1 k 1 = hr -1 A = 1 Period: 24 hr

Antihistamine in Bloodstream (y[t]) vs. Time (t) Max: y[17.96 hr] =.66 units k 2 =.0231 hr -1 k 1 =.11 hr -1 A = 1 Period: 24 hr

Antihistamine in Bloodstream (y[t]) vs. Time (t) Max: y[9.26 hr] =.807 units k 2 =.0231 hr -1 k 1 =.3 hr -1 A = 1 Period: 24 hr

Antihistamine in Bloodstream (y[t]) vs. Time (t) Max: y[5.08 hr] =.889 units k 2 =.0231 hr -1 k 1 =.6931 hr -1 A = 1 Period: 24 hr

Antihistamine in Bloodstream (y[t]) vs. Time (t) Max: y[3.86 hr] =.915 units k 2 =.0231 hr -1 k 1 = 1.0 hr -1 A = 1 Period: 24 hr

Antihistamine in Bloodstream (y[t]) vs. Time (t) Max: y[2.83 hr] =.937 units k 2 =.0231 hr -1 k 1 = 1.5 hr -1 A = 1 Period: 24 hr

One Dose: Variance in k 1 Larger k 1 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 units 1. y[t] ≤ 0.8 units, for all t units ≤ y[2 hr] ≤ 0.8 units units ≤ y[24 hr] ≤ 0.8 units Pronounce k 1 values desirable or undesirable

Graphical Examination: Box Plots

Antihistamine in Bloodstream (y[t]) vs. Time (t) units0.577 units y[2 hr] = units; y[24 hr] = units Undesirable Undesirable k 2 =.0231 hr -1 k 1 = hr -1 A = 1 Period: 24 hr

Antihistamine in Bloodstream (y[t]) vs. Time (t) units0.637 units y[2 hr] = units; y[24 hr] = units Undesirable Undesirable k 2 =.0231 hr -1 k 1 =.11 hr -1 A = 1 Period: 24 hr

Antihistamine in Bloodstream (y[t]) vs. Time (t) 0.44 units0.622 units y[2 hr] = 0.44 units; y[24 hr] = units Undesirable  Maximum > 0.8 Undesirable  Maximum > 0.8 k 2 =.0231 hr -1 k 1 =.3 hr -1 A =1 Period: 24 hr

Antihistamine in Bloodstream (y[t]) vs. Time (t) units0.594 units y[2 hr] = units; y[24 hr] = units Undesirable  Maximum > 0.8 Undesirable  Maximum > 0.8 k 2 =.0231 hr -1 k 1 =.6931 hr -1 A = 1 Period: 24 hr

Antihistamine in Bloodstream (y[t]) vs. Time (t) units0.588 units y[2 hr] = units; y[24 hr] = units Undesirable Undesirable k 2 =.0231 hr -1 k 1 = 1.0 hr -1 A = 1 Period: 24 hr

Antihistamine in Bloodstream (y[t]) vs. Time (t) units0.583 units y[2 hr] = units; y[24 hr] = units Undesirable Undesirable k 2 =.0231 hr -1 k 1 = 1.5 hr -1 A = 1 Period: 24 hr

Range of Effective k 1 for k 2 =.0231 Upper Bound: k 1 =.282 Lower Bound: k 1 =.115 Want to maximize range of k 1 for which drug is safe & effective. Small k 1 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 k 1 & k 2 : clearance coefficients of GI & bloodstream R measured in (units / hr) Assume medication dissolves instantly upon entering body dx(t) dt = R - k 1 x(t), x(0) = 0 dy(t) dt = k 1 x(t) - k 2 y(t), y(0) = 0

Continuous Doses: Model Provided k 1, k 2 > 0 & k 1 ≠ k2 As t  ∞, x[t]  (R/k 1 ), y[t]  (R/k 2 ): equilibrium levels k 1 & k 2 depend on type of drug, specific compartments, age & health of patient k 1 & k 2 used for decongestant & antihistamine scenarios x(t) = (1-e -k 1 t ) y(t) = Rk2Rk2 (e -k 2 t – e -k 1 t )] R k 1 1 k 1 – k 2 [1+

Clearance Coefficients & Equilibrium Levels for Young & Healthy DecongestantAntihistamine K 1 (GI) hr -1 Eq: units hr -1 Eq: units K 2 (blood) hr -1 Eq: units hr -1 Eq: units

Clearance Coefficients & Equilibrium Levels for Old & Infirm DecongestantAntihistamine K 1 (GI) hr -1 Eq: units hr -1 Eq: units K 2 (blood) hr -1 Eq: units hr -1 Eq: units

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

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

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

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

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

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

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

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

Restrictions: Elderly Blood/antihistamine Effective range of a drug Fast-acting Long-lasting Effective range: units 1. y[t] ≤ 50 units, for all t units ≤ y[24 hr] ≤ 50 units units ≤ y[120 hr] ≤ 50 units

Elders: Blood/antihistamine Function increases too rapidly between 24 hours & 120 hours k 1 =. 231 hr -1 k 2 =.0077 hr -1 R = 1 unit/hr y[24] = units y[120] = units

Elders: Blood/antihistamine Function increases too rapidly between 24 hours & 120 hours k 1 =. 231 hr -1 k 2 =.0077 hr -1 R = 1.4 unit/hr

Elders: Blood/antihistamine Function increases too rapidly between 24 hours & 120 hours k 1 =. 231 hr -1 k 2 =.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., Foster, David. Principles of Clinical Pharmacology. University of Washington. Undated. slides_ ppt#1. slides_ ppt#1 Giordano, Frank, M. Weir, & W. Fox. A First Course in Mathematical Modeling. 3 rd ed. Pacific Grove, CA: Brooks/Cole- Thomson Learning, White, Emma. Epidemic Models for Drug Use. NUI Maynooth. Undated. IHRCConferenceMarch2005-Emma.pdf. IHRCConferenceMarch2005-Emma.pdf QUESTIONS ?