Download Parameter estimation - Systems Pharmacology And Disease Control

Benjamin Strauch
Parameter estimation
Application in influenza treatment
1
Influenza
• Infectious disease caused by RNA viruses
• Vaccination available, but antiviral drugs desired
• Severe epidemics occur in seasonal patterns
2
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
Parameter estimation in pharmacology
• Determine how virus loads decrease after drug treatment
• Compare responses of different virus strains
Treatment begins
4
Parameter estimation – the model
• Virus loads seem to exhibit an exponential decay
Δ𝑉 = −𝑘 𝑉,
𝑉 0 = 𝑉0
𝑉 = 𝑉0 𝑒 −𝑡⋅𝑘 ,
𝑉 0 = 𝑉0
• The constant 𝑘 is called clearance, in our pharmacological context.
Estimate parameters 𝑉0 and 𝑘 in accordance with the data.
5
Parameter estimation – problem setting
• Compare two strains of H1N1 and one Influenza B strain
• How does the virus clearance differ?
H1N1
B
sensitive
resistant
sensitive
(42 patients)
(17 patients)
(32 patients)
6
Study data
PatientID
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
...
...
...
...
7
Parameter estimation in pharmacology
• Determine how virus loads decrease after drug treatment
• Compare responses of different virus strains
8
Parameter estimation
• Use the data to fit the model function
• Take normalization constants 𝜖 into account
𝑛
min
𝑖=1
𝑉 𝑖, 𝑡 − 𝑉(𝑖, 𝑡)
𝜖𝑖
𝑉(𝑖, 𝑡) = 𝑉 𝑖 0 𝑒 −𝑡⋅𝐶𝐿 𝑖 ,
2
𝑉 𝑖
0
= 𝑉0
9
Parameter estimation in MATLAB
• We use the lsqcurvefit routine
• An gradient-based trust-region approach
lsqcurvefit(model, parameters, times, virus loads)
10
Errors
• The data could have a mixture of additive and proportional errors
11
Errors
• Dealing with proportional errors in our model
12
Errors
• Possibility of dealing with proportional errors in our model:
• Estimate parameters using a linearized model
log 𝑉(𝑖, 𝑡) = log 𝑉 𝑖
0
− 𝑡 ⋅ 𝐶𝐿 𝑖 ,
𝑉 𝑖
0
= log 𝑉0
13
Errors
• Two weighting factors used to account for errors
• Mean value, divided by the number of datapoints
𝜇𝑖
𝜖𝑖 =
𝑛𝑖
• Median value 𝑚𝑖 , divided by the the number of datapoints
𝑚𝑖
𝜖𝑖 =
𝑛𝑖
14
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
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
𝑉0 , 𝐶𝐿 = arg min
𝑡
𝑀𝑡
𝜖𝑡 =
𝑛𝑡
or
𝑀 𝑡 −𝑀 𝑡
𝜖𝑡
′
𝑚𝑡
𝜖𝑡 =
𝑛𝑡
16
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
𝑉0 𝑖 , 𝐶𝐿 𝑖 = arg min
𝑡
𝑉(𝑖, 𝑡) − 𝑉(𝑖, 𝑡)
𝑉(𝑖, 𝑡)
• Can compare mean of indvidual estimates and estimate of the pooled data
17
Results
• Function fits pooled data moderately well
• Non-linearized model function seems to fit worse in the semilog-plot
18
Results
• Estimates on individuals fit very well for the non-linearized model
19
Pharmacological implications
• Initial question: How did the strains differ in response to treatment
20
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
𝑉0
1−𝑡 𝐶𝐿
𝑣
𝑉
𝑡 = 𝑉⋅ 0 ⋅ 𝑒
𝑡 =𝑖, ln
10 𝐶𝐿𝑣 𝑖
𝑖
Set to 10
21
Pharmalogical implications
• Resistant strain requires significantly longer treatment
22
Pooled vs. Individual estimates
• Parameters differ strongly, but difference in eradication times remains
Median individual estimates
All
H1N1 res.
Pooled estimate
H1N1 sens.
B sens.
All
H1N1 res.
H1N1 sens.
B sens.
𝑉0
3 ⋅ 104
2.6 ⋅ 106
1.4 ⋅ 105
6132
2 ⋅ 106
4.7 ⋅ 106
1.6 ⋅ 106
1.2 ⋅ 106
𝐶𝐿
1.12
0.75
1.4
0.8
2.5
1
2.35
2.48
Time to
eradication
7.14
16.16
7.64
7.65
4.82
13.02
5.1
4.72
23
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
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
Thank you for your attention.
26