Statistical Analysis of the Whole Body Specific Absorption Rate using Human Body Characteristics El Habachi A, Conil E, Hadjem A, Gati A, Wong M-F and Wiart J, Whist lab & Orange Labs Vazquez E and Fleury G, Supélec 05/11/2009,
Contents section 1 Context & problematic section 2 WBSAR surrogate model section 3 Sequential Bayesian experiment planning section 4 Conclusion
Contents section 1 Context & problematic section 2 WBSAR surrogate model section 3 Sequential Bayesian experiment planning section 4 Conclusion
Context Normative context Basic Restrictions (BR): SAR over 10 grams and over the whole body Difficult to check in situ Reference Levels (RL) : max. allowed EMF and power density Guaranty the compliance to BR Established several years ago Dosimetric context Anatomical phantoms Large variability of the exposure in the existing set of phantoms How to characterize the exposure for an entire population?
Problematic : How to characterize the exposure for an entire population? 18 phantoms X Monte Carlo method WBSAR surrogate model Focus on the probability of failure. WBSAR density for the population Region of interest : Upper 95 % bound WBSAR Exposure configuration : frontal plane wave polarized vertically at the frequency 2100 MHz and incident power of 1W/m² How to build such surrogate model?
Contents section 1 Context & problematic section 2 WBSAR surrogate model section 3 Sequential Bayesian experiment planning section 4 Conclusion
Surrogate model building Observation on phantom's families WBSAR (W/Kg) 14 x 10-3 12 10 8 6 4 VH Familiy JM Family JF Family Norman Family Korean Family Zubal Family 2 0.03 0.04 0.05 0.06 0.07 0.08 BMI (-1) WBSAR calculated using the FDTD method Percentage 90 80 70 60 50 40 30 20 10 0 5 years Percentage of main tissues for Zubal family 8 years 12 years 1 2 3 4 Tissues Skin Fat Muscles Bones Adult y = βx + ε, estimation error WBSAR inverse of the Body Mass Index (BMI =weight/height²) unknown parameter depending on internal morphology β estimated by Least square Method 30% of error on the WBSAR
Description of the surrogate model Y = β.x Input data X = inverse of BMI statistical available data in literature β = internal morphology no statistical data for populations Unknown statistical distribution : Estimation of the distribution by different types of statistical laws ( parametric ones and Gaussian mixture) Output data Threshold of WBSAR at 95% What is our knowledge on β?
Knowledge on β Physical knowledge β depends on internal morphology β is bounded (human morphology is bounded) : β low and β upp (lower and upper bound) β is positive (WBSAR is positive) Additional constraints Mean value : most of phantoms match to the mean of population they come from β i= 1 < β >= 18 β low = 0 (WBSAR >0) β upp = 0.3: chosen by assuming a symmetry around the mean 18 i 0.15 Quantile at 95% - Mean 30 25 20 15 10 Anthropometric Database The independence between β and X (to be relaxed) 5 0 0 5 10 15 20 25 30 Mean - Qauntile at 5%
Parametric laws : Gamma, Beta, Normal, Weibull and Log-normal Most of morphological external factors belongs to the parametric laws family. Application to the French population aged of 20 years : BMI~N(22.29, 2.9²) Density Densité 15 10 Gamma Weibull Log-normale Beta Normal Param. laws Gamma Beta Normal Weibull Log-normal Threshold of the WBSAR at 95 % (mw/kg) 9.2 9.3 11 10.5 9.1 5 Std/mean 9% 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 alpha Beta Weak variability of the WBSAR at 95% whatever the parametric law
Relaxation of the independency between β and X 35 BMI 30 25 20 15 10 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 Beta negative correlation treshold of the WBSAR at 95 % Relaxation of the independency 0.0135 0.013 0.0125 0.012 0.0115 0.011 0.0105 0.01 Beta belongs to [0, 0.3] Beta~Beta Beta~Gamma Beta~Log-Normal Beta~Weibull Beta~Normal 0.0095-1 -0.8-0.6-0.4-0.2 0 Correlation correlation WBSAR at 95%
Gaussian Mixture Density probability function for β that maximizes the threshold of the WBSAR. n pβ ( β) = pig ( β) m i, σ i i= 1 n pi = 1 i= 1 0 pi 1 From knowledge to Constraints : β positive and belongs to [β low, β upp ] : m n + 3σ n = β upp and m 1-3σ 1 = β low 99.74% of the probability density belongs to [β low, β upp ] Additional constraints : n Known mean : p i = 1 i 0.15 σ i does not appear in the estimated CDF : choice of σ i = σ m i Smoothness : m i m i-1 = σ i (Rayleigh Criterion) Unimodality : PDF with single mode (p i p i+1 p j et p j p j+1 p n )
Number of Gaussian in mixture Taylor development CDF WBSAR 0.014 estimated threshold by approximation of the CDF compared to the empirical threshold obtained using Monte Carlo PDF obtained for 20 Gaussian French population 6 Threshold of the WBSAR at 95 %(W/Kg) 0.013 0.012 0.011 0.01 0.009 Empirical threshold Estimated threshold Example for the 1 st mode 0.008 0 10 20 30 40 50 60 70 Number of Gaussians Density 5 4 3 2 1 0-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Beta Threshold of the WBSAR at 95 % (mw/kg) Gamma 9.2 Beta 9.3 Norma l 11 Weib ull 10.5 Lognormal 9.1 Gaussian mixture 13.2 1st mode median mode last mode
Bell-shaped Gaussian mixture To obtain bell-shaped PDF, a constraint on the variance is introduced : n 2 2 var( β) = p (m 2 < β > m ) + σ + < β > i= 1 i i i 2 Bound The initial variance depends on the mode and the number of Gaussians: For a mixture of 20 Gaussians the variance belongs to [4.5.10-3,5.3.10-3 ] Density 30 Bound = 2.3.10-4 25 20 15 10 5 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Beta Gamma Beta Normal Weibull Log-normal Gaussian mixture Threshold of the WBSAR at 95 % (mw/kg) 9.2 9.3 11 10.5 9.1 8.9
Contents section 1 Context & problematic section 2 WBSAR surrogate model section 3 Sequential Bayesian experiment planning section 4 Conclusion
Context How to separate the internal and external factors impacting the WBSAR? use of homogeneous phantoms Homogeneous phantoms Equivalent liquid (IEC) Morphing technique to deform the external shapes of the existing phantoms Monte Carlo is still expensive to characterize the threshold of the WBSAR at 95 % for a given population. Sequential Bayesian Experiment Planning : allow finding the morphology corresponding to the region of 95 %. WBSAR density for the population Region of interest : Upper 95 % bound WBSAR
How to extend the database of phantoms? Morphing technique to deform homogeneous phantoms Deforming factors used: height, inside leg height, front shoulder breadth, chest and waist Laws of the deforming factors: parametric laws Studied population Anthropometric database of French population Sample of 3800 adults Log-normal and normal laws to estimate the density of the deforming factors Examples of morphed phantoms Density 0.04 0.035 0.03 0.025 0.02 0.015 0.01 0.005 Stature Log-normale Normale Gamma 0 140 150 160 170 180 190 200 Stature Height
Parametric model Using existing simulations, a suitable model is written as follow chest waist WBSAR = θ1height + θ2 + θ3 + θ4 front shoulder breadth front shoulder breadth Initial observations D-optimal design to choose observations that allow obtaining a small confident region for the parameters 6 initial observations F n = (x i, y i ) i= 1,...,n Xi = factors of the parametric model (height, waist ) Yi = WBSAR calculated with FDTD + ε Height F.S.B chest waist WBSAR (mw/kg) 1,57 1,81 1,918 2,02 0,346 0,417 0,389 0,524 1,144 1,246 1,147 1,114 1,056 0,899 1,197 0,896 3,87 4,7 4,48 5,68 Choose observations to refine the region of interest (WBSAR at 95%) 1,98 0,435 0,968 1,024 4,75 1,38 0,372 0,782 0,663 7,64
Sequential Bayesian Experiment Planning Prior Law : Non-informative initial observations F n = (x i, yi ) i= 1,...,n Bayes Formula P( Θ \ F n ) = P( Θ).P(F n \ Θ) where Θ = [ θ 1, θ 2, θ 3, θ 4 ] Posterior law Likelihood Criteria to choose the next candidate: diminution of the variance of the threshold of the WBSAR at 95 % Θ 1 PDF WBSAR 1 PDF WBSAR at 95 % Θ 2 Θ 3 PDF WBSAR 2 PDF WBSAR 3 σ Initial observations x
Results Variance of the quantile of the WBSAR at 95 % 3.5 3 2.5 2 1.5 1 0.5 4 x 10-7 0 0 5 10 15 20 25 30 iteration number Mean of the quantile of the WBSAR (mw/kg) 7.6 7.4 7.2 7 6.8 6.6 6.4 6.2 0 5 10 15 20 25 30 Iteration number One phantom having a WBSAR of 0,007 W/kg Stability of the mean of the quantile at 0.007 W/Kg Height =147 cm Chest =76.5 cm Waist = 65 cm Frontal shoulder breadth= 34 cm Weight = 49 Kg WBSAR = 0,007W/kg
Contents section 1 Context & problematic section 2 WBSAR surrogate model section 3 Sequential Bayesian experiment planning section 4 Conclusion
Conclusion WBSAR surrogate model for heterogeneous phantoms External morphology characterized by the BMI Internal morphology described by different types of statistical laws weak variability of the WBSAR at 95% in term of the type of the laws WBSAR at 95% maximized using Gaussian mixture (0,013W/kg) WBSAR parametric model for homogeneous phantoms Morphing technique to obtain additional phantoms Baysian inference to choose new observations after 25 iterations, stability of the variance of the WBSAR at 95% 0,007 W/kg: mean value obtained for the WBSAR at 95% Future work : to assess the influence of the dielectric properties in the parametric model
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