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Parameter estimation Application in influenza treatment 1 Benjamin Strauch

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

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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 3

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

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Parameter estimation – the model 5

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Parameter estimation – problem setting Compare two strains of H1N1 and one Influenza B strain How does the virus clearance differ? H1N1B sensitive (42 patients) resistant (17 patients) sensitive (32 patients) 6

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Study data PatientIDDays after treatment initiationVirus load [1/ml]Virus Strain 10663354A/H1N1 (2009) sensitive 15756A/H1N1 (2009) sensitive... 304605312A/H1N1 (2009) resistant 35346042A/H1N1 (2009) resistant... 21025900B sensitive 213857B sensitive 2151256B sensitive... 7

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

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Parameter estimation 9

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

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

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Errors Dealing with proportional errors in our model 12

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

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Errors 14

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Methods Two main approaches will be considered: 1.Pool data of multiple individuals together and estimate parameters Pool both for all strains together and for each set of strains 2.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 15

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Methods 1.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 16

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Methods 2.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 17

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

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

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

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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 21

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Pharmalogical implications Resistant strain requires significantly longer treatment 22

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Pooled vs. Individual estimates Median individual estimatesPooled estimate AllH1N1 res.H1N1 sens.B sens.AllH1N1 res.H1N1 sens.B sens. Time to eradication Parameters differ strongly, but difference in eradication times remains 23

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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 24

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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. 25

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Thank you for your attention. 26

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