Modeling of the Absorption of the Electromagnetic Wave Energy in the Human Head Induced by Cell Phone

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1 Journal of Appled Mathematcs and Physcs, 14,, Publshed Onlne November 14 n ScRes. Modelng of the Absorpton of the Electromagnetc Wave Energy n the uman ead Induced by Cell Phone Daman Gok, Marcn Senek, Potr Gurgul, Mace Paszyńsk Department of Computer Scence, AG Unversty of Scence and Technology, Kraków, Poland Emal: mace.paszynsk@agh.edu.pl Receved October 14 Abstract In ths paper, we present the Proecton Based Interpolaton (PBI) technque for constructon of contnuous approxmaton of MRI scan data of the human head. We utlze the result of the PBI algorthm to perform three-dmensonal (3D) Fnte Element Method (FEM) smulatons of the heatng of the human head nduced by cell phone. In partcular, we utlze the Pennes equaton to descrbe the boheat transfer wth the rght hand sde representng the heat generated by cell phone. We utlze our own non-statonary tme dependent mult-thread parallel drect solver for the soluton of ths computatonal problem. From our numercal results t follows that 15 mnutes (1 seconds) exposure to the cell phone radaton mples up to degrees Celsus ncrease of the temperature of the bran n the range close to the cell phone. Keywords Fnte Element Method, Proecton Based Interpolaton, Pennes Equaton, eatng of the uman ead 1. Introducton In ths paper we present desgn and mplementaton of three dmensonal fnte element method h adaptve code for effcent and accurate soluton of the challengng engneerng problem of heatng of the human head nduced by the propagaton of electromagnetc waves generated by cell phone. We use the Pennes boheat transfer equaton for the modelng of the heatng of the human head. The heat source n the equaton s based on the numercal soluton for the Maxwell equatons modelng the propagaton of the electromagnetc waves over the human head generated by cell phone. Ths proect wll allow for the frst tme fnd relable answer to the queston of the nfluence of the cellular phones on the bran tssue, compare Fgure 1. Ths s the proect wth collaboraton wth the group of Professor Leszek Demkowcz from ICES (Insttute for Computatonal and Engneerng Scence, The Unversty of Texas n Austn), n the frame of the Polsh Natonal Scence Center grant ARMONIA. The collaboratng group of prof. Leszek Demkowcz provdes the soluton of the Maxwell equatons, namely the heat source for the rght hand sde of the Pennes equaton. The geometry of the human head s obtaned by usng my own mplementaton of the proecton based nterpolaton algorthm ow to cte ths paper: Gok, D., Senek, M., Gurgul, P. and Paszyńsk, M. (14) Modelng of the Absorpton of the Electromagnetc Wave Energy n the uman ead Induced by Cell Phone. Journal of Appled Mathematcs and Physcs,,

D. Gok et al. Fgure 1. Left panel: Thermographc mage of the head wth no expose to harmful cell phone radaton, Rght panel: Thermographc mage of the head after a 15-mnute phone call.

2 D. Gok et al. Fgure 1. Left panel: Thermographc mage of the head wth no expose to harmful cell phone radaton, Rght panel: Thermographc mage of the head after a 15-mnute phone call. Yellow and red areas ndcate thermal (heatng) effects that can cause negatve health problems (mages obtaned from Prof. Leszek Demkowcz, The Unversty of Texas at Austn). [1] generatng the three dmensonal h refned mesh based on the MRI scan of the human head of prof. Mace Paszyńsk. Ths s the challengng problem due to the fact that of the complexty of the human head geometry and the numercal equatons nvolved, whch results n huge sze of the system of lnear equatons.. Smulaton of the eatng of the uman ead The smulaton of the absorpton of the electromagnetc wave energy over the human head, n partcular human bran, nduced by cell phone usage, and possble adverse health effects of EM waves are of the great mportance. The smulatons avalable so far concerned the smplfed geometrcal model of the human head []. In ths paper we present a three dmensonal fnte element method smulatons of the heatng of the human head nduced by electromagnetc waves generated by cell phone for the case of the real human head. In partcular we perform the followng steps n our smulaton. Frst, we perform a MRI scan of the head of one of the authors of ths paper (M. Paszyńsk), compare Fgure 1. We create a three dmensonal btmap from the 9 cross-sectons of the head wth resoluton pxels obtaned from MRI scan data. The range of the values at dfferent pxels vares from [,55]. These values correspond to dfferent materal data. Next, we utlze a proecton based nterpolaton technque [3] [4] to generate a contnuous approxmaton of the materal data, compare Fgure. We utlze three dmensonal h adaptaton algorthm [5] wth hexahedral fnte elements, wth the error estmator based on the L norm dfference between the local element approxmaton and the MRI scan data. avng the approxmaton a globally C and locally C1 functon, we may dstngush dfferent materal data. In partcular we assume ar (approxmaton < 1), bran (1 <= approxmaton < 4) and skull (approxmaton >= 4). avng the PBI approxmaton, we can formulate the boheat Pennes equaton, takng nto account thermal conductvty k as well as metabolsm q m, perfuson Wc b b, and heat source from the cellphone q sar : u ρc = ( k u) + Wc ( ) n b b ua u + qm+ qsar Ω (1) t The unknown (, ) u xt s the temperature dstrbuton n space and tme. The values of the parameters are summarzed n Fgure 1. They are selected accordng to the knd of the tssue, as suggested by the results of the PBI approxmaton. 3. Proecton Based Interpolaton The locaton of materal data nsde the human head model has been obtaned by usng the proecton based nterpolaton algorthm that conssts the followng sub-steps, related to hexahedral fnte element vertces, edges, faces and nterors. For each fnte element, we are lookng for a coeffcents n a partcular order. The computatonal mesh can be generated by usng a lnear computatonal cost proecton based nterpolaton routne, frst proposed n [1] [5]. We start wth vertces, snce ther coeffcents are the most straghtforward to compute. There s only one functon per each of eght vertexes wth a support on t and the nterpolatng functon s requred to be equal to the nterpolant, whch yelds: ( ) ( v ) U v a = v 1,, 8 ϕ =, () 18

Frst, on each one of the 1 edges of tetrahedral element: dmv e 8 U uv, mn 1,,1 ule = = 1 l= 1 (3) where dmve sgnfes the number of edge shape functons n space V wth supports on edge e.

3 D. Gok et al. Fgure. Sngle slce of the MRI scan of the human head and ts PBI representaton. On nodes other than vertces, the nput functon cannot be represented exactly, so nstead we are tryng to mnmze the representaton error. Frst, on each one of the 1 edges of tetrahedral element: dmv e 8 U uv, mn 1,,1 ule = = 1 l= 1 (3) where dmve sgnfes the number of edge shape functons n space V wth supports on edge e. Such a problem can be reduced to a lnear system and solved wth a lnear solver, but f we assume the adaptaton order p = on each node, for each edge there exsts only one shape functon wth a support on t. Not only s ths restrcton ustfed performance-wse (one local equaton nstead of a system), but t also suffces n most cases, accordng to our experments. Thus equaton (3) reduces to: 1 ( e ) 1 ( e ) where U vanshes on the element s vertces. After rewrtng the norm: we have whch leads to U 8 U uv u, e = 1 mn = 1,,1, (4) e k = 1 (, e ) 3 U u dx mn = 1,,1, (5) 3 duoe, dx mn = 1,,1 and uoe, = a eϕ (6) e k = 1 dx d k xk duoe, du oe, dx dx+ dx mn = 1,,1 dx dx dx dx (7) k 1 k e = k k e = k 3 du 3 oe, d d U uoe, dx dx mn = 1,,1 dx dx dx k e = k k (8) snce the other term s constant and can be omt n mnmzaton., l u s a lnear form defned as: Let be buv ( ) s a blnear, symmetrc form and ( ) 181

4 D. Gok et al. 3 d 3 u du buv x l v x dx dx dx oe, oe, (, ) = d, ( ) = d (9) k k e = k It s proven that mnmzng 1 buv (, ) l( u) s reducble to solvng buv (, ) l( u) By applyng ths lemma we obtan whch leads to du dv du dx dx mn 1,,1 dx dx = dx dx = 3 3 oe, oe, k k e = k k = for all test functons v. (1) 3 d 3 ϕe dϕ d e ϕ e ae = dx d x 1,,1 = dx dx dx dx (11) k k e = k k The next step conssts n an optmzaton on sx faces of hexahedral element: where U vanshes on vertces and edges. Ths leads to: U 8 1 U uv u, e u, f = 1 = 1 mn = 1,, 6, (1) 1 ( f ) 3 d 3 ϕf dϕ d f ϕ f af = dx d x 1,...,6 = dx dx dx dx f k= 1 k 1 k k f = k k Fnally, an analogcal optmzaton n the nteror of the fnte element: (where U vanshes everywhere except from the nteror) yelds: (13) U U uv u, e u, f u, I = 1 = 1 = 1 mn, (14) f k= 1 k 1 k k f = k k 1 ( I) 3 3 dϕi dϕi dϕi ai = dx dx dx dx dx dx (15) It s worth notng that usng ths method the global matrx s not constructed at all. Thanks to the p = restrcton, we have a sngle equaton over each vertex, edge, face and nteror. Ths algorthm requres a computatonal cost lnear wth respect to the mesh sze, because t nvolves constant number of operatons for each vertex, edge, face and nteror and the number of respectve nodes s proportonal to n the number of fnte elements. 4. Potental for Mult-Thread Implementaton The proecton-based nterpolaton s a fully local operaton. The PBI nvolves four global loops, and all the computatons are performed over element vertces, edges, faces and nterors. These operatons may be fully parallelzable usng Open MP archtecture. for vertex n (vertces of elements of the mesh) compute values of coeffcent a v gven by () for PBI for vertex store a v at vertex for edge n (edges of elements of the mesh) compute values of coeffcent a e gven by (11) for PBI for edge store a at edge e 18

I gven by (15) for PBI for vertex store a at nteror I 5.

5 D. Gok et al. for face n (faces of elements of the mesh) compute values of coeffcent a f gven by (13) for PBI for face store a at face f for nteror n (nterors of elements of the mesh) compute values of coeffcent a I gven by (15) for PBI for vertex store a at nteror I 5. Numercal results In ths secton we present the numercal results for the heatng of the human head nduced by cell phone wth the materal data selected accordng to the knd of the tssue, as suggested by the results of the PBI approxmaton, usng values from Table 1. The PBI algorthm has been executed on the 3D MRI scan data, compare Fgure. For the numercal soluton of (1) we utlze the fnte element method wth Crank-Ncolson scheme, proved to be uncondtonally stable. In our work we solve the Pennes equatons over the real human head model, but the heat source from the cell-phone s based on the soluton obtaned by [1] (pcture 6.14, the dpol located at cm dstance from the human head). From our numercal results t follows that 15 mnutes (1 seconds) exposure to the cell phone radaton mples up to Celsus ncrease of the temperature of the bran n the range close to the cell phone, compare Fgure 3. Table 1. Materal data used n the smulaton. Materal Parameters Ar Bran Skull ρ C k q m Wc 4 33 b b Fgure 3. Left panel: The soluton to Pennes equaton after 1 seconds wthout a cell phone exposure. Unform temperature Rght panel: The soluton to the Pennes equaton at two crosssectons close to the ear (where the cell-phone s located), wth and wthout the cell phone usage, after 1 seconds (16 mnutes of a cell phone exposure). Maxmum temperature (red areas) reaches 38.4 Celsus. 183

6 D. Gok et al. 6. Concluson In ths paper we presented a three dmensonal smulatons on the heatng of the human head tssues nduced by the cell phone usage. We showed that 16 mnutes of the exposure to the cell phone radaton results n a local ncrease of temperature up to 38.4 Celsus. Our solver can be appled for such the smulatons for arbtrary nput data, namely the results of the MRI scan of a human head. The heat source nduced by cell phone was obtaned from the work of [] for the dpole antenna. In our future work we plan to develop software solvng the Maxwell equatons over the PBI data, n order to be able to experment wth dfferent cell phone antennas. Acknowledgements The work presented n ths paper has been supported by Polsh Natonal Scence Center grants No. DEC- 11/3/N/ST6/1397 (Marcn Senek, PBI algorthm; Potr Gurgul, analyss of the concurrency), 1/ 7/B/ST6/19 (Daman Gok, non-statonary solver) and 1/6/M/ST1/363 (Mace Paszyńsk, human head modelng) as well as the Dean s grant no (Potr Gurgul, mult-threadng). References [1] Demkowcz, L. (13) Proecton Based Interpolaton. The Unversty of Texas, Austn. [] Km, K. (13) Fnte Element Modelng of the Radaton and Induced ear Transfer n the uman Bodz. Ph.D. Dssertaton, The Unversty of Texas, Austn [3] Senek, M., Paszyńsk, M., Made, Ł. and Gok, D. (14) Adaptve Proecton Based Interpolaton as a Pre-Processng Tool n the Fnte Element Method Workflow for Elastcty Smulatons of the Dual Phase Mcrostructures. Steel Research Internatonal, 85, [4] Senek, M. and Paszyńsk, M. (14) Subtree Reuse n Mult-Frontal Solvers for Regular Grds n Step-and-Flash Imprnt Nanolthography Modelng. Advanced Engneerng Materals, 16, [5] Demkowcz, L., Kurtz, J., Pardo, D., Paszyńsk, M., Rachowcz, W. and Zdunek, A. (7) Computng wth hp-adaptve Fnte Elements, Vol. II. Three Dmensonal Ellptc and Maxwell Problems wth Applcatons. Chapmann & all, CRC Press. 184

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