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Parameter estimationApplication in influenza treatment

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

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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

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

Treatment begins

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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.

Δ𝑉=−𝑘𝑉 ,𝑉 (0 )=𝑉 0

𝑉=𝑉 0𝑒−𝑡 ⋅ 𝑘 ,𝑉 (0 )=𝑉 0

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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)

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Study dataPatientID Days after treatment initiation Virus load [1/ml] Virus Strain

1 0 663354 A/H1N1 (2009) sensitive

1 5 756 A/H1N1 (2009) sensitive

... ... ... ...

3 0 4605312 A/H1N1 (2009) resistant

3 5 346042 A/H1N1 (2009) resistant

... ... ... ...

21 0 25900 B sensitive

21 3 857 B sensitive

21 5 1256 B sensitive

... ... ... ...

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

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

min∑𝑖=1

𝑛 (𝑉 (𝑖 , 𝑡 )−𝑉 (𝑖 , 𝑡 )𝜖 𝑖

)2

𝑉 (𝑖 , 𝑡)=𝑉 (𝑖 )0𝑒− 𝑡 ⋅ 𝐶𝐿( 𝑖 ) ,𝑉 (𝑖 )0=𝑉 0

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Parameter estimation in MATLAB• We use the lsqcurvefit routine• An gradient-based trust-region approach

lsqcurvefit(model, parameters, times, virus loads)

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

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Errors• Dealing with proportional errors in our model

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

log𝑉 (𝑖 , 𝑡 )= log𝑉 (𝑖 )0−𝑡 ⋅𝐶𝐿 (𝑖 ) ,𝑉 (𝑖 )0= log𝑉 0

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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

𝜖 𝑖=𝜇𝑖

𝑛𝑖

𝜖 𝑖=𝑚𝑖

𝑛𝑖

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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

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Methods1. 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

𝑉 0 ,𝐶𝐿=argmin∑𝑡

𝑀 (𝑡 )−𝑀 (𝑡 )′

𝜖𝑡

𝜖𝑡=𝑀𝑡

𝑛𝑡𝜖𝑡=

𝑚𝑡

𝑛𝑡or

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Methods2. 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

𝑉 0 (𝑖 ) ,𝐶𝐿 (𝑖 )=argmin∑𝑡

𝑉 (𝑖 ,𝑡 )−𝑉 (𝑖 , 𝑡 )𝑉 (𝑖 , 𝑡 )

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

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

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

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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

𝑉 (𝑖 , 𝑡 )=𝑉 0 ⋅𝑒− 𝑡𝐶 𝐿𝑣 (𝑖 )

Set to 10

𝑡=ln𝑉 0

10⋅ 1𝐶𝐿𝑣 (𝑖 )

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Pharmalogical implications• Resistant strain requires significantly longer treatment

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Pooled vs. Individual estimates

Median individual estimates Pooled estimate

All H1N1 res. H1N1 sens. B sens. All H1N1 res. H1N1 sens. B sens.

Time to eradication

• Parameters differ strongly, but difference in eradication times remains

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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

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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.

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Thank you for your attention.