Claude Beigel, PhD. Exposure Assessment Senior Scientist Research Triangle Park, USA

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

2. 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

3. Two Approaches to Defining FlowsIndividual 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-ffMi for sink • Substance biphasic • Multiply overall rate equation by formation fraction for each metabolite, and 1-ffMi for sink

4. Metabolite Curve-fittingSummary 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 <LOD/LOQ data as recommended • Set first data point <LOD/LOQ before first detect and after last detect to half of LOD or half (LOQ+LOD)

5. Metabolite Curve-fittingSummary of required steps to follow (2) • 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

6. 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

7. Building the Compartment Model Step-by-step • Results from yesterday’s 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 • Fix parent parameters and fit metabolite parameters • Use fitted parameters as initial values, and fit parent and metabolite parameters together

8. 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

9. 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)

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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

19. Building the Compartment Model Step-by-step • Run model (model – integrate)

20. 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

21. 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)

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

23. 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

24. 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).

25. 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

26. Hands-on Example 2 General Guidance • Parent substance • Results from yesterday’s 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 • Fix parent parameters and fit metabolite parameters • Use fitted parameters as initial values, and fit parent and metabolite parameters together

27. Hands-on Example 2Guidance 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

28. Hands-on Example 2Guidance 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

29. 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 endpointsMeasured 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 modelto derive decline rate constant and DT50 value