Parameter estimation

1. Benjamin Strauch Parameter estimation Application in influenza treatment

2. Influenza • Infectious disease caused by RNA viruses • Vaccination available, but antiviral drugs desired • Severe epidemics occur in seasonal patterns

3. Oseltamivir • Antiviral drug, developed by Gilead Sciences • Commonly marketed by Roche as Tamiflu • Rose to prominence during the 2009 flu pandemic (swine flu) • Effectiveness controversial • Use parameter estimation to assess effectivity on different strains

4. Parameter estimation in pharmacology • Determine how virus loads decrease after drug treatment • Compare responses of different virus strains Treatment begins

5. Parameter estimation – the model • Virus loads seem to exhibit an exponential decay • The constant is called clearance, in our pharmacological context. • Estimate parameters and in accordance with the data.

6. Parameter estimation – problem setting • Compare two strains of H1N1 and one Influenza B strain • How does the virus clearance differ? H1N1 B sensitive (42 patients) resistant (17 patients) sensitive (32 patients)

7. Study data

8. Parameter estimation in pharmacology • Determine how virus loads decrease after drug treatment • Compare responses of different virus strains

9. Parameter estimation • Use the data to fit the model function • Take normalization constants into account

10. Parameter estimation in MATLAB • We use the lsqcurvefit routine • An gradient-based trust-region approach lsqcurvefit(model, parameters, times, virus loads)

11. Errors • The data could have a mixture of additive and proportional errors

12. Errors • Dealing with proportional errors in our model

13. Errors • Possibility of dealing with proportional errors in our model: • Estimate parameters using a linearized model

14. Errors • Two weighting factors used to account for errors • Mean value, divided by the number of datapoints • Median value , divided by the the number of datapoints

15. Methods • Two main approaches will be considered: • Pool data of multiple individuals together and estimate parameters • Pool both for all strains together and for each set of strains • Estimate parameters for each individual • Can compare average individual parameters with pooled parameters. • (For each estimate, 2-5 data points will be available) • Try both linearized and normal model function for least squares

16. Methods • Pool data of multiple individuals together and estimate parameters • At each time point t use mean of virus loads M(t) at that point as the data • Captures parameters typical for the whole population or

17. Methods • Estimate parameters for each individual • Use data V(i,t), for individual i at time point t directly • Gives unique statistics even for heterogenous populations • Can compare mean of indvidual estimates and estimate of the pooled data

18. Results • Function fits pooled data moderately well • Non-linearized model function seems to fit worse in the semilog-plot

19. Results • Estimates on individuals fit very well for the non-linearized model

20. Pharmacological implications • Initial question: How did the strains differ in response to treatment

21. Pharmacological implications • Using the fit, we predict the time to reach non-detectable virus load • Non-detectable: Less than 10 copies/reaction in RT-PCR Set to 10

22. Pharmalogical implications • Resistant strain requires significantly longer treatment

23. Pooled vs. Individual estimates • Parameters differ strongly, but difference in eradication times remains

24. Conclusion • The least-squares fit was able to identify differing treatment responses • Resistant H1N1 strains take significantly longer to treat • Considering errors and normalization is essential • Proportional errors might benefit from a transformation of the data • Weighted least-squares can also account for the error distribution

25. Outlook • Improving the data: • In extreme cases, only 2 data points for each patient available in this study • No untreated control available to assess baseline effectiveness of treatment • Try different estimation methods: • Gauß-Newton instead of trust-region • Stochastic methods: EM-Algorithm, etc.

26. Thank you for your attention.