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Cold Pills & Compartmental Modeling Ronnie Schumann MA 354- Math Modeling Dr. Jyoti Champanerkar Monday, Dec. 5, 2005
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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
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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 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](http://images.slideplayer.com./25/8118902/slides/slide_3.jpg "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")
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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 )
![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 )](http://images.slideplayer.com./25/8118902/slides/slide_4.jpg "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 )")
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Antihistamine in Bloodstream (y[t]) vs. Time (t) Max: y[23.78 hr] =.57 units k 2 =.0231 hr -1 k 1 =.06931 hr -1 A = 1 Period: 24 hr
![Antihistamine in Bloodstream (y[t]) vs.](http://images.slideplayer.com./25/8118902/slides/slide_5.jpg "Time (t) Max: y[23.78 hr] =.57 units k 2 =.0231 hr -1 k 1 = hr -1 A = 1 Period: 24 hr.")
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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.](http://images.slideplayer.com./25/8118902/slides/slide_6.jpg "Time (t) Max: y[17.96 hr] =.66 units k 2 =.0231 hr -1 k 1 =.11 hr -1 A = 1 Period: 24 hr.")
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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.](http://images.slideplayer.com./25/8118902/slides/slide_7.jpg "Time (t) Max: y[9.26 hr] =.807 units k 2 =.0231 hr -1 k 1 =.3 hr -1 A = 1 Period: 24 hr.")
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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.](http://images.slideplayer.com./25/8118902/slides/slide_8.jpg "Time (t) Max: y[5.08 hr] =.889 units k 2 =.0231 hr -1 k 1 =.6931 hr -1 A = 1 Period: 24 hr.")
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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.](http://images.slideplayer.com./25/8118902/slides/slide_9.jpg "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.")
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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
![Antihistamine in Bloodstream (y[t]) vs.](http://images.slideplayer.com./25/8118902/slides/slide_10.jpg "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.")
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One Dose: Variance in k 1 Larger k 1 values higher & quicker peaks in y[t] TRAFFIC JAM!!!
![One Dose: Variance in k 1 Larger k 1 values higher & quicker peaks in y[t] TRAFFIC JAM!!!](http://images.slideplayer.com./25/8118902/slides/slide_11.jpg "One Dose: Variance in k 1 Larger k 1 values higher & quicker peaks in y[t] TRAFFIC JAM!!!")
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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 2. 0.2 units ≤ y[2 hr] ≤ 0.8 units 3. 0.2 units ≤ y[24 hr] ≤ 0.8 units Pronounce k 1 values desirable or undesirable
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Graphical Examination: Box Plots
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Antihistamine in Bloodstream (y[t]) vs. Time (t) 0.126 units0.577 units y[2 hr] = 0.126 units; y[24 hr] = 0.577 units Undesirable Undesirable k 2 =.0231 hr -1 k 1 =.06931 hr -1 A = 1 Period: 24 hr
![Antihistamine in Bloodstream (y[t]) vs.](http://images.slideplayer.com./25/8118902/slides/slide_14.jpg "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.")
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Antihistamine in Bloodstream (y[t]) vs. Time (t) 0.193 units0.637 units y[2 hr] = 0.193 units; y[24 hr] = 0.637 units Undesirable Undesirable k 2 =.0231 hr -1 k 1 =.11 hr -1 A = 1 Period: 24 hr
![Antihistamine in Bloodstream (y[t]) vs.](http://images.slideplayer.com./25/8118902/slides/slide_15.jpg "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.")
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Antihistamine in Bloodstream (y[t]) vs. Time (t) 0.44 units0.622 units y[2 hr] = 0.44 units; y[24 hr] = 0.622 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.](http://images.slideplayer.com./25/8118902/slides/slide_16.jpg "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.")
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Antihistamine in Bloodstream (y[t]) vs. Time (t) 0.729 units0.594 units y[2 hr] = 0.729 units; y[24 hr] = 0.594 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.](http://images.slideplayer.com./25/8118902/slides/slide_17.jpg "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.")
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Antihistamine in Bloodstream (y[t]) vs. Time (t) 0.839 units0.588 units y[2 hr] = 0.839 units; y[24 hr] = 0.588 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.](http://images.slideplayer.com./25/8118902/slides/slide_18.jpg "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.")
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Antihistamine in Bloodstream (y[t]) vs. Time (t) 0.919 units0.583 units y[2 hr] = 0.919 units; y[24 hr] = 0.583 units Undesirable Undesirable k 2 =.0231 hr -1 k 1 = 1.5 hr -1 A = 1 Period: 24 hr
![Antihistamine in Bloodstream (y[t]) vs.](http://images.slideplayer.com./25/8118902/slides/slide_19.jpg "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.")
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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
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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 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](http://images.slideplayer.com./25/8118902/slides/slide_21.jpg "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")
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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+
![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+](http://images.slideplayer.com./25/8118902/slides/slide_22.jpg "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+")
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Clearance Coefficients & Equilibrium Levels for Young & Healthy DecongestantAntihistamine K 1 (GI) 1.386 hr -1 Eq: 0.722 units 0.6931 hr -1 Eq: 1.443 units K 2 (blood) 0.1386 hr -1 Eq: 7.215 units 0.0231 hr -1 Eq: 43.29 units
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Clearance Coefficients & Equilibrium Levels for Old & Infirm DecongestantAntihistamine K 1 (GI) 0.462 hr -1 Eq: 2.165 units 0.231 hr -1 Eq: 4.328 units K 2 (blood) 0.0462 hr -1 Eq: 21.65 units 0.0077 hr -1 Eq: 129.87 units
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Young’uns vs. Elders: GI/decongestant Equilibrium:.7215 units 30 hours
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Young’uns vs. Elders: Blood/decongestant Equilibrium: 7.215 units 30 hours
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Young’uns vs. Elders: GI/decongestant Equilibrium: 2.165 units 30 hours
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Young’uns vs. Elders: Blood/decongestant Equilibrium: 21.645 units 72 hours (3 days)
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Young’uns vs. Elders: GI/antihistamine Equilibrium: 1.443 units 30 hours
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Young’uns vs. Elders: Blood/antihistamine Equilibrium: 43.29 units 120 hours (5 days)
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Young’uns vs. Elders: GI/antihistamine Equilibrium: 4.328 units 30 hours
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Young’uns vs. Elders: Blood/antihistamine Equilibrium: 129.87 units 336 hours (14 days)
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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
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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] = 18.21 units y[120] = 76.544 units
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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
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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
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Elders: Blood/antihistamine No constant continuous dosage amount satisfies both conditions Solution: Discontinuous dosage Vary treatment times Vary treatment amounts
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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.
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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. 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. 3 rd 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/EpidemicModelsforDrugUse IHRCConferenceMarch2005-Emma.pdf. http://www.nuim.ie/rosie/documents/EpidemicModelsforDrugUse IHRCConferenceMarch2005-Emma.pdf QUESTIONS ?
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