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Computational Biology, Part 18 Compartmental AnalysisCompartmental SystemsSlide 3Problems in Compartmental AnalysisDefinition of CompartmentFirst-order Compartment ModelsHandling first-order compartment modelsExample: Lead AccumulationLead Accumulation ModelSlide 10Slide 11Slide 12PharmacokineticsSlide 14Slide 15Computational Biology, Part 18Compartmental AnalysisComputational Biology, Part 18Compartmental AnalysisRobert F. MurphyRobert F. MurphyCopyright Copyright 1996, 1999-2007. 1996, 1999-2007.All rights reserved.All rights reserved.Compartmental SystemsCompartmental SystemsCompartmental systemCompartmental systemmade up of a finite number of macroscopic made up of a finite number of macroscopic subsystems, called subsystems, called compartmentscompartments, each of , each of which is which is homogeneoushomogeneous and and well-mixedwell-mixedinteractions between compartments consist of interactions between compartments consist of exchanging materialexchanging materialCompartmental SystemsCompartmental SystemsAll interactions between compartments are All interactions between compartments are transfers of material in which some type of transfers of material in which some type of mass conservation condition holdsmass conservation condition holdsInputs from/outputs to the environment are Inputs from/outputs to the environment are permittedpermittedIf they occur, systems is If they occur, systems is openopen (otherwise (otherwise closedclosed))Problems in Compartmental AnalysisProblems in Compartmental AnalysisDevelopment of Development of plausible models plausible models for for particular biological systemsparticular biological systemsDevelopment of Development of analytic theory analytic theory for each for each class of compartmental systemsclass of compartmental systemsEstimation of model parameters Estimation of model parameters and and determination of “best” model - so-called determination of “best” model - so-called ““inverse probleminverse problem””Definition of CompartmentDefinition of Compartment““A A compartment compartment is an is an amount of a amount of a material material that acts kinetically like a that acts kinetically like a distinct, distinct, homogeneous, well-mixed amount homogeneous, well-mixed amount of the of the material.” (Jacquez)material.” (Jacquez)Not a physical volume or spaceNot a physical volume or spaceFirst-order Compartment ModelsFirst-order Compartment ModelsA common, important category of A common, important category of compartment models is that set of models in compartment models is that set of models in which the rates of all transfers between which the rates of all transfers between compartments are given by first-order rate compartments are given by first-order rate constantsconstantsHandling first-order compartment modelsHandling first-order compartment modelsDon’t need to solve (e.g. dsolve) the model Don’t need to solve (e.g. dsolve) the model from the differentials, since the general form from the differentials, since the general form of the solution is knownof the solution is knownJust need to enter the rate constants for the Just need to enter the rate constants for the allowed transfers into the matrix allowed transfers into the matrix AA, the , the environmental transfers into vector environmental transfers into vector ff, the , the initial concentrations into vector initial concentrations into vector XX00 and and evaluateevaluateX =eAtX0+A−1f[ ]−A−1fExample: Lead AccumulationExample: Lead AccumulationYeargers, section 7.10 (pp. 220-224)Yeargers, section 7.10 (pp. 220-224)Three compartments: blood, soft tissues, Three compartments: blood, soft tissues, boneboneOpen system (input from environment only Open system (input from environment only into blood)into blood)First-order compartment modelFirst-order compartment modelLead Accumulation ModelLead Accumulation ModelCompartment 1 = blood, Compartment 2 = soft Compartment 1 = blood, Compartment 2 = soft tissue, Compartment 3 = skeletal system, tissue, Compartment 3 = skeletal system, Compartment 0 = environmentCompartment 0 = environmentxxii for for ii=1..3 is amount of lead in compart. =1..3 is amount of lead in compart. iiaaijij for for ii=0..3,=0..3,jj=1..3 is rate of transfer to =1..3 is rate of transfer to compartment compartment ii from compartment from compartment jjIILL(t)(t) is the rate of intake into blood from is the rate of intake into blood from environmentenvironmentLead Accumulation ModelLead Accumulation Model€ dx1dt= −(a01+ a21+ a31)x1+ a12x2+ a13x3+ Il(t)dx2dt= a21x1− (a02+ a12)x2dx3dt= a31x1− a12x3Lead Accumulation ModelLead Accumulation Model€ dx1dt= −(a01+ a21+ a31)x1+ a12x2+ a13x3+ Il(t)dx2dt= a21x1− (a02+ a12)x2dx3dt= a31x1− a12x3€ X'= AX + fwhere X =x1x2x3 ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ and f =IL(t)00 ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ andA =−(a01+ a21+ a31)a12a13a21−(a02+ a12) 0a310 −a13 ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥Lead Accumulation ModelLead Accumulation Model(Maple sheet 1)(Maple sheet 1)PharmacokineticsPharmacokineticsYeargers, section 7.11 (pp. 226-229)Yeargers, section 7.11 (pp. 226-229)xx = amount of drug in GI tract = amount of drug in GI tractyy = amount of drug in blood = amount of drug in bloodD(t)D(t) is dosing function is dosing functiondrug taken every six hours and dissolves within one drug taken every six hours and dissolves within one half-hourhalf-houraa = half-life of drug in GI tract = half-life of drug in GI tractbb = half-life of drug in blood = half-life of drug in bloodPharmacokineticsPharmacokinetics€ dxdt= −ax + Ddydt= ax − byPharmacokineticsPharmacokinetics(Maple sheet 2)(Maple sheet

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