Andersen, Kim Emil

Brooks, Stephen P.

Højbjerre, Malene

    Aalborg UniversitetBayesian model discrimination for glucose-insulin homeostasisAndersen, Kim Emil; Brooks, Stephen P.; Højbjerre, MalenePublication date:2007Document VersionPublisher's PDF, also known as Version of recordLink to publication from Aalborg UniversityCitation for published version (APA):Andersen, K. E., Brooks, S. P., & Højbjerre, M. (2007). Bayesian model discrimination for glucose-insulinhomeostasis. Poster session presented at COBAL 2, San José del Cabo, Mexico.General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            ? Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            ? You may not further distribute the material or use it for any profit-making activity or commercial gain            ? You may freely distribute the URL identifying the publication in the public portal ?Take down policyIf you believe that this document breaches copyright please contact us at vbn@aub.aau.dk providing details, and we will remove access tothe work immediately and investigate your claim.Downloaded from vbn.aau.dk on: April 30, 2017Bayesian Model Discriminationfor Glucose-Insulin HomeostasisCobal 2February 2005Kim E. Andersen∗, Stephen P. Brooks∗∗ and Malene Højbjerre∗∗Department of Mathematical Sciences, Aalborg University, Denmark∗∗Statistical Laboratory, University of Cambridge, UKCobal 2, February 2005 – p. 1/17DataIVGTT: Intra-Venous Glucose Tolerance TestSampling of glucose and insulin concentrations in plasma following an intravenousglucose injection.Healthy Type II diabetic0 50 100 150 20068101214Time (minutes)Plasma glucose concentration (mmol/l)0 50 100 150 20010152025Time (minutes)Plasma glucose concentration (mmol/l)0 50 100 150 200100300500700Time (minutes)Plasma insulin concentration (pmol/l)0 50 100 150 200405060708090Time (minutes)Plasma insulin concentration (pmol/l)Data on the human body’s response to increased blood sugar levels.Cobal 2, February 2005 – p. 2/17Models – The Minimal ModelThe Minimal Model: Bergman et al (1979) and Toffolo et al (1980)γt p3 p2EliminationnPlasma InsulinI(t)RemoteInsulin ActionX(t)LiverPlasma GlucoseG(t)PeripheryIntravenous glucose doseG1: ˙G(t) = −p1(G(t) − Gb) − X(t)G(t) G(0) = G0˙X(t) = −p2X(t) + p3(I(t) − Ib) X(0) = 0I2: ˙I(t) = −n(I(t) − Ib) + γJ+(G(t) − h)t I(0) = I0Cobal 2, February 2005 – p. 3/17Models – The Minimal ModelParameters of Interest in the Minimal Model:insulin sensitivity: SI = p3/p2glucose effectiveness: SG = p1pancreatic responsiveness: ϕ1 = (I0 − Ib)/[n(G0 − Gb)]ϕ2 = γ× 104Current Approach:Iterative nonlinear least squares technique (MINMOD PROGRAMME)ä not a unified system (the insulin is treated as known).ä SI estimated close to zero with negative confidence intervals.ä ϕ1 and ϕ2 not estimated (the insulin is treated as known).ä the positive truncation J is physiologically questionable.ä the multiplicative effect of time t is difficult to justify biologically.ä no account of individual variability or process error.Alternative models and/or other approaches are called for.Cobal 2, February 2005 – p. 4/17Models – Variations on the Minimal ModelVariations on the Minimal Model:Three additional variants of the insulin component:I2: ˙I(t) = −n(I(t) − Ib) + γJ+(G(t) − h) I(0) = I0I3: ˙I(t) = −n(I(t) − Ib) + γ(G(t) − h)t I(0) = I0I4: ˙I(t) = −n(I(t) − Ib) + γ(G(t) − h) I(0) = I0Cobal 2, February 2005 – p. 5/17Models – The de Gaetano & Arino ModelThe GA Model: De Gaetano & Arino (2000)-1 0 1 2 3 4 5 6 7 8 9 10 118910111213b6/b5Eliminationb2Plasma InsulinI(t)LiverPlasma GlucoseG(t)PeripheryIntravenous glucose doseG2: ˙G(t) = −b1G(t) − b4I(t)G(t) + b7 G(0) = Gb + b0I5: ˙I(t) = −b2I(t) +b6b5∫tt−b5G(s) ds I(0) = Ib + b3b0where G(t) ≡ Gb for t ∈ [−b5, 0). Cobal 2, February 2005 – p. 6/17Models – The de Gaetano & Arino ModelParameters of interest:insulin sensitivity: SI = b4glucose effectiveness: SG = b1pancreatic responsiveness: ϕ1 = b3/b2ϕ2 not estimated under the GA modelReformulating the modelsSame approach as Andersen & Højbjerre (2005):ä The glucose and insulin concentrations are log-transformed (same scale -common variance).ä The DE’s are discretised.ä Impose random errors both on the system and the measurements.ä Extend to population modelling.ÙA stochastic state space modelling framework that allows for likelihoodconstructions.Cobal 2, February 2005 – p. 7/17You may flip this page if you dareCobal 2, February 2005 – p. 8/17Reformulating the ModelsLog-Transformation:g(t) = log G(t), x(t) = log X(t) and i(t) = log I(t)g˙(t) = ˙G(t)/G(t), x˙(t) = ˙X(t)/X(t) and ˙i(t) = ˙I(t)/I(t)Reparameterizing by SI, SG, ϕ1, ϕ2 , b0 and b3.Log-Transformed Minimal Model:G1: g˙(t) = −SG(1 − Gb e−g(t)) − ex(t) g(0) = log(Gb + b0)x˙(t) = −p2(1 − SI(ei(t) − Ib)e−x(t)) x(0) → −∞I1: ˙i(t) = −b3ϕ1(1 − e−i(t)Ib) + 10−4e−i(t)ϕ2J+(eg(t) − h)t i(0) = log(Ib + b3b0)Discretised Log-transformed Minimal ModelΛ = {t1, t2, . . . , t|Λ|} and new notation: g(tk) = gstk , x(tk) = xstk and i(tk) = istkgstk=gstk−1−(tk−tk−1)(SG(1 − Gb e−gstk−1 ) + exstk−1)+ gsxstk=xstk−1−(tk−tk−1)p2(1 − SI(eistk−1 − Ib)e−xstk−1)+ xsistk=istk−1−(tk−tk−1)( b3ϕ1(1−e−istk−1 Ib)−10−4e−istk−1 ϕ2J+(egstk−1 −h)tk−1)+ iswhere gs , xs and is follows N (0, ν−1(tk − tk−1)).Cobal 2, February 2005 – p. 8/17Reformulating the ModelsSystem processes:gstk | gstk−1, xstk−1 , ν ∼ N (fgtk−1, ν−1(tk − tk−1))xstk | xstk−1, istk−1 , ν ∼ N (fxtk−1, ν−1(tk − tk−1)), tk ∈ Λistk | istk−1, gstk−1 , ν ∼ N (fitk−1, ν−1(tk − tk−1))wherefgtk−1=gtk−1 −(tk−tk−1)(SG(1 − Gb e−gtk−1 ) + extk−1)fxtk−1 =xtk−1 −(tk−tk−1)p2(1 − SI(eitk−1 − Ib)e−xtk−1)fitk−1 = itk−1 −(tk−tk−1)( b3ϕ1(1 − e−itk−1 Ib) − 10−4e−itk−1 ϕ2J+(egtk−1 −h)tk−1)Observed processes:gotk | gstk, ν−1go ∼ N (gstk, νgo)iotk | istk, ν−1io ∼ N (istk, νio), tk ∈ T ⊆ ΛThe other models can be reformulated similarly.Cobal 2, February 2005 – p. 9/17Population ModellingSystem processes for individual j in model m:Φsjm ={{gsjmtk , xsjmtk, isjmtk}tk∈Λ for m = 1, . . . , 5, (G1 × I1, . . . , I5){gsjmtk , isjmtk}tk∈Λ for m = 6, . . . , 10, (G2 × I1, . . . , I5)Observed processes for individual j:Φoj = {gojtk, iojtk }tk∈TDistributional assumptions for individual j in model m:pm(Φsjm | θjm)∝{ν|Λ|j exp(−Vm(Φsjm, θjm)) for m = 1, . . . , 5ν3|Λ|/2j exp(−Vm(Φsjm, θjm)) for m = 6, . . . , 10pm(Φoj | θjm, Φsjm) ∝ (νgoj νioj)|T |/2 exp(−W(Φoj , Φsjm, θjm))whereθjm={(SGj, SIj, ϕ1j, b3j, b0j, Gbj, Ibj, ϕ2j, p2j, hj, νj, νgoj , νioj) for m = {1, . . . , 9} \ 5(SGj, SIj, ϕ1j, b3j, b0j, Gbj, Ibj, b5j, νj, νgoj , νioj) for m = 5 and 10Cobal 2, February 2005 – p. 10/17Population Modelling – cont’dandVm(Φsjm,θjm)=12νj∑tk∈Λ(gsjmtk−fgjmtk)2+(xsjmtk−fxjmtk)2+(isjmtk−fijmtk)2 m=1, . . . , 512νj∑tk∈Λ(gsjmtk −fgjmtk)2+(isjmtk−fijmtk)2 m=6, . . . , 10W(Φoj , Φsjm, θjm)=12∑tk∈Tνgoj(gojtk − gsjmtk)2 + νioj(iojtk − isjmtk)2Likelihood for population of L individuals in model m:L(Ψm, θm, Φsm | Φo) =L∏j=1pm(Φoj | θjm, Φsjm)pm(Φsjm | θjm)pm(θjm | Ψm)whereΦsm = (Φs1m, Φs2m, . . . , ΦsLm)Φo = (Φo1 , Φo2 , . . . , ΦoL)θm = (θ1m, θ2m, . . . , θLm)and pm(θjm | Ψm) is the product of log-normal (for the system parameters)and gamma distributions (for the precisions).Cobal 2, February 2005 – p. 11/17Bayesian AnalysisPosterior Distribution: MCMC Methods (Metropolis-Hastings)pi(Ψ, θ, Φs | Φo) ∝L∏j=1p(Φoj | θj, Φsj )p(Φsj | θj)p(θj | Ψ)p(Ψ)where p(Ψ) prior for the population parameters - a product of normal (for themeans) and gamma (for the precisions) distributions.Model Uncertainty: Reversible Jump MCMCpi(Ψm,θm,Φsm,m|Φo)∝L∏j=1pm(Φoj |θjm, Φsjm)pm(Φsjm|θjm)pm(θjm|Ψm)p(Ψm|m)p(m)where p(m) prior for model index - uniform.Improving Mixing: Simulated Tempering (RJ)MCMCpiτ(Ψm, θm, Φsm, m | Φo) ∝L∏j=1(pm,τ(Φoj | θjm, Φsjm)pm,τ(Φsjm | θjm))s(τ)× pm(θjm | Ψm)p(Ψm | m)p(τ | m)p(m)where p(τ | m) prior for temperature τ (coarseness level of discritisation) ands(τ) = 2−(τ−1)n for n > 0. Note, τ = 1 provides the posterior.Cobal 2, February 2005 – p. 12/17ResultsPosterior Model Probability:Posterior model probabilityPopulation Coarseness 1 2 3 4 5 6 7 8 9 101 0.00 0.21 0.76 0.03 0.00 0.00 0.00 0.00 0.00 0.00Healthy 2 0.10 0.15 0.23 0.20 0.08 0.02 0.07 0.08 0.05 0.014 0.11 0.13 0.12 0.10 0.10 0.16 0.07 0.05 0.09 0.061 0.00 0.00 0.07 0.00 0.01 0.00 0.00 0.75 0.00 0.17Diabetic 2 0.05 0.09 0.15 0.07 0.08 0.04 0.13 0.20 0.08 0.114 0.13 0.05 0.11 0.10 0.07 0.08 0.14 0.13 0.12 0.09Cobal 2, February 2005 – p. 13/17ResultsSimulated values of m and τ:Healthy population Type II diabetic populationCoarsenessτ = 1124Resolution level124Resolution levelτ = 2234Model345678910Modelτ = 3246810Model246810Model0 1 2 3 4 5246810Iterations in millionsModel0 1 2 3 4 5246810Iterations in millionsModelCobal 2, February 2005 – p. 14/17ResultsPosteriorPopulationGlucoseandInsulinConcentrations:HealthyTypeIIdiabetic0501001502004 5 6 7 8 9 1 0Time (minutes)P l a s m a  g l u c o s e  c o n c e n t r a t i o n  ( m m o l / l )0501001502001 0 1 5 2 0 2 5 3 0Time (minutes)P l a s m a  g l u c o s e  c o n c e n t r a t i o n  ( m m o l / l )0501001502000 5 0 1 5 0 2 5 0Time (minutes)P l a s m a  i n s u l i n  c o n c e n t r a t i o n  ( p m o l / l )0501001502004 0 6 0 8 0 1 0 0 1 2 0 1 4 0Time (minutes)P l a s m a  i n s u l i n  c o n c e n t r a t i o n  ( p m o l / l )Cobal2,February2005–p. 15/17DiscussionResults:ä Discriminating among 10 glucose and insulin models.ÙHealthy population: Minimal model with no positive relection (original model)Diabetic population: Original insulin minmal model and new glucose model.ä Providing model-averaged inference on parameters of interest.ä Unified systems of both glucose and insulin.ä No SI = 0 problems for diabetic population.ä Possible to estimate ϕ1 and ϕ2.ä Random errors on system and measurements.ä Population modelling.For more details see Andersen, Brooks and Højbjerre(2004) - download fromhttp://www.math.aau.dk/research/reports/R-2004-15.pdfCobal 2, February 2005 – p. 16/17ReferencesBergman, R. N., Y. Z. Ider, C. R. Bowden and C. Cobelli (1979), Quantitativeestimation of insulin sensitivity. American Journal of Physiology 236, E667 – E677Toffolo, G., D. R.N. Bergman, C. R. Finegood, Bowden and C. Cobelli (1980),Quantitative estimation of beta cell sensitivity to glucose in the intact organism: aminimal model of insulin kinetics in the dog. Diabetes 29, 979–990de Gaetano, A. and O. Arino (2000), Mathematical modelling of the intravenousglucose tolerance test. Journal of Mathematical Biology 40, 136 – 168Andersen, K. E. and M. Højbjerre (2005), A Population based Bayesian Approach tothe Minimal Model of Glucose and Insulin Homeostasis. To appear in Statistics inMedicine.Andersen, K. E., Brooks, S. P. and M. Højbjerre (2004), Bayesian ModelDiscrimination for Glucose-Insulin Homeostasis. Technical Report R-2004-15,Department of Mathematical Sciences, Aalborg University.Cobal 2, February 2005 – p. 17/17

Bayesian model discrimination for glucose-insulin homeostasis

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Bayesian model discrimination for glucose-insulin homeostasis

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