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Claude Beigel, PhD. Exposure Assessment Senior Scientist Research Triangle Park, USA Practical session metabolites Part I: curve fitting

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1 Using Compartment Models for Metabolite Curve Fitting Parent + metabolite(s) data sets can be fitted with compartment models based on the same principles shown for parent substance Model parameters are defined A compartment is added for each metabolite Flows are added between parent and metabolite(s), and metabolite(s) and sink Each flow is defined with differential equation corresponding to appropriate kinetic model, using defined parameters Model is fitted to parent and metabolite measured data If metabolite was applied to test system, data set treated as for parent substance Metabolite decline treated as parent substance, with time 0 starting as time of maximum, and initial amount (estimated) as maximum amount of metabolite

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2 Two Approaches to Defining Flows Individual Rate Constants and Formation Fractions Overall degradation rate of a substance is defined by differential equation corresponding to selected model (SFO, FOMC, DFOP) Basic simplifying assumption: degradation to different compartments (metabolite(s) and sink) follows same kinetic model Overall rate is split between metabolite(s) formed and sink Substance SFO, two options: –Use individual first-order rate constant for each flow with sum = overall degradation rate constant (because first-order rates are additive) –Multiply overall rate constant by formation fraction for each metabolite, and 1- ffM i for sink Substance biphasic –Multiply overall rate equation by formation fraction for each metabolite, and 1- ffM i for sink

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3 Metabolite Curve-fitting Summary of required steps to follow (1) Always build simplest model representative of pathway Follow metabolic pathway Initially include all flows to sink, reduce when applicable Data handling Set metabolite time-0 to 0 and eventually correct parent time-0 Deal with metabolite

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4 Ask yourself: what type of endpoints are needed? Trigger DT50/90 best-fit kinetics PEC soil endpoints (formation + degradation rate parameters, formation fraction) best-fit kinetics Modeling endpoints (formation + degradation rate parameters, formation fraction) restricted kinetic models Use stepwise approach for complex cases Determine parent kinetics first Add metabolites stepwise Free all parameters in final fit Metabolite Curve-fitting Summary of required steps to follow (2)

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5 Hands-on Example 1 Exercise 1 Same substance 1 as fitted yesterday in parent session Proposed pathway shows substance degrading to primary metabolite 1 Measured data for metabolite 1 given in Excel spreadsheet 2.2_metabolites examples input.xls Derive trigger and modeling endpoints for metabolite 1 Trigger endpoints: metabolite DT50/90 Modeling endpoints: parent degradation rate, metabolite formation fraction and metabolite degradation rate

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6 Building the Compartment Model Step-by-step Results from yesterdays exercise showed that SFO model was appropriate for both trigger and modeling endpoints for parent We will add metabolite 1 using a model formulation with formation fraction We will follow the stepwise approach to fitting 1.Fix parent parameters and fit metabolite parameters 2.Use fitted parameters as initial values, and fit parent and metabolite parameters together

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7 Building the Compartment Model Step-by-step Start from parent – sink model with appropriate kinetic model for endpoints of interest (here SFO) –Open 2.2_Example1_parent.mod ModelMaker file provided

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8 Building the Compartment Model Step-by-step Define SFO parameters for primary metabolite(s) –In this example, formation fraction ffM1 and first-order rate constant kM1 –Select initial value of 0.5 for ffM1 and constrain between 0 and 1 –Select initial value of 0.01 for kM1 (unconstrained)

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9 Building the Compartment Model Step-by-step Add metabolite compartment(s) –Here create one compartment for Metabolite 1 (no space in symbol/name) –Leave metabolite initial value set to 0.0

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10 Building the Compartment Model Step-by-step Add flows from parent to metabolite compartment(s) and metabolite(s) to sink –Here create flow parent to Metabolite 1 and Metabolite 1 to Sink –Red arrows mean that flows are not defined yet

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11 Building the Compartment Model Step-by-step Define flow from parent to metabolite with appropriate differential equation for kinetic model (multiplied by formation fraction) –Here define fP_M1 with SFO equation = ffM1*kP*Parent

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12 Building the Compartment Model Step-by-step Define flow from metabolite to sink with differential equation for SFO model –Here define fM1_S with SFO equation = kM1*Metabolite1

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13 Building the Compartment Model Step-by-step Modify flow from parent to sink to account for formation of metabolite(s) (multiply by 1- ffMi) –Here modify fP_S to equation = (1-ffM1)*kP*Parent –Compartment model is now fully defined

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14 Building the Compartment Model Step-by-step Create variables for calculating metabolite DT50/90 values –In main page, click on variable icon, create DT50_M1 = LN(2)/kM1 and DT90_M1 = LN(10)/kM1

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15 Building the Compartment Model Step-by-step Add Metabolite1 compartment and DT50/90 variables to Table –In table page, right-click and go to selection, add the components to selection by double-clicking in component list or use >> and << buttons to select and unselect components

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16 Building the Compartment Model Step-by-step Add metabolite data to model data –Type or paste metabolite data in Not Used column, if necessary, insert column, highlight column and define as Metabolite1 –Always check that data correspond to correct times, ModelMaker tends to disregard empty cells and move data up or left

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17 Building the Compartment Model Step-by-step Add metabolite to graph –In graph page, right-click and go to selection window, add Metabolite 1 from components by double-clicking on component or use >> button –Modify series appearance by right-clicking and go to series window, you can remove error bar and change line and symbol

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18 Building the Compartment Model Step-by-step Run model (model – integrate)

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19 Building the Compartment Model Step-by-step Optimize metabolite parameters –In parameters page, select metabolite parameters by clicking on optimize, leave parent parameters unchecked at this point –Fit to data by clicking on Model - Optimize

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20 Building the Compartment Model Step-by-step Repeat optimization changing initial parameter values to check that results do not change Your results should be the following (minimal variation if different initial values used): Update parameters (in parameter results page, select parameters, right-click outside of selection, and update)

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21 Building the Compartment Model Step-by-step Run model with optimized parameters (model – integrate)

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22 Building the Compartment Model Step-by-step Final step: optimize parent and metabolite parameters together –In parameters page, select all parameters by clicking on optimize, keep initial values to previously optimized values –Fit to data by clicking on Model – Optimize –Update all parameters, run model and save –Write-down final optimization results, and calculated DT50/90 values

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23 Additional Notes on Example 1 The Modelmaker file for the equivalent model formulated with individual rate constants is provided in your training material (2.2_Example1_individualrates.mod file). You can check that you obtain similar results with the two model formulations (minimal variation due to initial value of parameters). The stepwise approach is recommended for complex cases, and would not be necessary for a well-behaved data set such as this. You can try a simultaneous fit approach by changing the initial parameter values to reasonable estimates such as Pini = 100, kP = 0.1, ffM1 = 0.5 and kM1 = 0.01 and fit all parameters together. You should obtain similar results as in the stepwise final fit (minimal variation due to initial value of parameters).

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24 Hands-on Example 2 Exercise 2 Same substance 2 as fitted yesterday in parent session Proposed pathway shows substance degrading to one metabolite Measured data for metabolite of substance 2 given in Excel spreadsheet 2.2_metabolites examples input.xls Derive trigger and modeling endpoints for metabolite Trigger endpoints: metabolite DT50/90 Modeling endpoints: parent degradation rate, metabolite formation fraction and metabolite degradation rate

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25 Hands-on Example 2 General Guidance Parent substance Results from yesterdays exercise on parent showed that parent degradation is biphasic –FOMC model of choice for parent trigger endpoints –DFOP model may be used for modeling endpoints Add metabolite using a model formulation with formation fraction Follow the stepwise approach to fitting 1.Fix parent parameters and fit metabolite parameters 2.Use fitted parameters as initial values, and fit parent and metabolite parameters together

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26 Hands-on Example 2 Guidance for Deriving Trigger Endpoints Start from parent FOMC fit Use 2.2_Example2_parentFOMC.mod ModelMaker file provided Add metabolite parameters and compartment (same as for example 1) Split parent flow with metabolite formation fraction: –kP_M1 = ffM1*alphaP/betaP*Parent/(t/betaP+1) –kP_S = (1-ffM1)*alphaP/betaP*Parent/(t/betaP+1) Further steps same as for example 1

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27 Hands-on Example 2 Guidance for Deriving Modeling Endpoints Start from parent DFOP fit Use 2.2_Example2_parentDFOP.mod ModelMaker file provided Add metabolite parameters and compartment (same as for example 1) Split parent flow with metabolite formation fraction: –kP_M1 = ffM1*(k1*g*exp(-k1*t)+k2*(1-g)*exp(-k2*t))/(g*exp(-k1*t)+(1-g)*exp(-k2*t))*Parent –kP_S = (1-ffM1)*(k1*g*exp(-k1*t)+k2*(1-g)*exp(-k2*t))/(g*exp(-k1*t)+(1-g)*exp(-k2*t))*Parent (tip: use copy/paste, ctrl-c/ctrl-v) Further steps same as for example 1

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28 For Those Who Have Time to Go Further Exercise 1 (continued) Add second metabolite (metabolite 2) formed from metabolite 1 and derive trigger and modeling endpoints Measured data for metabolite 2 of substance 1 given in Excel spreadsheet examplesinput.xls Exercise 2 (continued) Fit metabolite decline data (from maximum onward) with SFO model to derive decline rate constant and DT50 value

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