An Efficient Deterministic-Stochastic Model of the Human Body Exposed to ELF Electric Field

<div>Induced electric field in the human body with three RVs—length, radius, and conductivity <svg height="11.5564pt" id="M129" style="vertical-align:-2.26807pt" version="1.1" viewbox="-0.0498162 -9.28833 40.89 11.5564" width="40.89pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z" id="g113-41"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="346" vert-adv-y="346"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,4.498,0)"><path d="M541 160L512 170C485 116 462 88 439 67C411 41 376 35 318 35C272 35 238 37 224 48C207 61 205 85 212 121L290 533C305 611 308 615 391 622L397 650H139L133 622C217 615 221 612 206 533L126 118C111 41 103 34 23 26L17 0H474C489 31 528 124 541 160Z" id="g113-77"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="553" 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transform="matrix(.013,0,0,-0.013,23.667,0)"><path d="M95 130C70 130 46 113 46 88C46 72 54 64 59 64C93 55 121 33 121 -3C121 -41 93 -68 44 -88L55 -117C117 -98 186 -56 186 22C186 91 131 130 95 130Z" id="g113-45"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="228" vert-adv-y="228"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,28.81,0)"><path d="M548 455L523 469C505 448 490 439 461 439C416 439 374 447 316 447C144 447 23 310 23 161C23 47 89 -12 179 -12C312 -12 429 106 429 244C429 303 419 344 368 386L371 388C404 383 440 378 474 378C507 378 530 411 548 455ZM350 274C350 191 308 25 198 25C144 25 108 75 108 157C108 264 170 395 276 395C296 395 320 384 333 360C347 334 350 316 350 274Z" id="g113-240"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="565" vert-adv-y="565"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,36.158,0)"><path d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z" id="g113-42"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="346" vert-adv-y="346"></glyph.data></g></svg>: <svg height="13.6781pt" id="M130" style="vertical-align:-2.150599pt" version="1.1" viewbox="-0.0498162 -11.5275 29.6808 13.6781" width="29.6808pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M360 -163L167 270L360 703H298L106 273V267L298 -163H360Z" id="g113-61"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="413" vert-adv-y="413"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,5.369,0)"><path d="M593 650H142L136 622C219 616 224 611 209 531L130 120C114 39 108 34 23 28L17 0H501C518 32 557 130 568 158L540 169C519 132 495 93 469 70C439 43 409 35 332 35C272 35 241 37 225 50C211 61 211 89 222 149L253 320H350C441 320 443 312 441 239H471L510 434H481C455 366 448 358 360 358H260L303 572C311 613 312 615 348 615H431C509 615 522 603 532 581C544 555 546 531 545 491L574 493C583 555 592 631 593 650Z" id="g113-70"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="616" vert-adv-y="616"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,13.377,-5.741)"><path d="M138 535C169 535 191 560 191 589C191 620 168 644 139 644S85 620 85 589C85 560 110 535 138 535ZM257 0V30C192 35 185 43 185 109V453C142 435 94 422 40 414V386C101 378 104 372 104 311V109C104 41 97 35 33 30V0H257Z" id="g58-103"></path><glyph.data ascent="1007" descent="-369" horiz-adv-x="275" vert-adv-y="275"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,15.88,-5.741)"><path d="M532 0V30C473 38 467 38 467 107V298C467 395 416 451 336 451C307 451 279 439 251 420C226 404 205 389 185 374V453C142 434 94 422 43 413V386C98 378 104 373 104 309V107C104 41 100 38 31 30V0H251V30C191 35 185 40 185 107V337C216 364 255 389 295 389C360 389 386 347 386 276V110C386 44 380 35 321 30V0H532Z" id="g58-108"></path><glyph.data ascent="1007" descent="-369" horiz-adv-x="557" vert-adv-y="557"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,20.766,-5.741)"><path d="M303 37L293 66C280 59 256 49 231 49S173 63 173 143V396H281C294 404 298 428 287 439H173V575L158 577L92 509V439H46L17 408L22 396H92V107C92 27 128 -12 192 -12C203 -12 218 -9 237 2L303 37Z" id="g58-114"></path><glyph.data ascent="1007" descent="-369" horiz-adv-x="312" vert-adv-y="312"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,24.142,0)"><path d="M307 267V273L115 703H53L246 270L53 -163H115L307 267Z" id="g113-63"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="413" vert-adv-y="413"></glyph.data></g></svg> is stochastic expected value, <svg height="9.32509pt" id="M131" style="vertical-align:-3.1815pt" version="1.1" viewbox="-0.0498162 -6.14359 13.0737 9.32509" width="13.0737pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M548 455L523 469C505 448 490 439 461 439C416 439 374 447 316 447C144 447 23 310 23 161C23 47 89 -12 179 -12C312 -12 429 106 429 244C429 303 419 344 368 386L371 388C404 383 440 378 474 378C507 378 530 411 548 455ZM350 274C350 191 308 25 198 25C144 25 108 75 108 157C108 264 170 395 276 395C296 395 320 384 333 360C347 334 350 316 350 274Z" id="g113-240"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="565" vert-adv-y="565"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,6.76,3.132)"><path d="M615 650H141L132 618C223 612 227 607 212 526L136 123C122 42 112 38 27 32L18 0H522C539 33 580 135 592 166L558 179C536 139 512 100 486 75C454 48 420 41 345 41C287 41 252 42 237 55C221 68 220 100 231 158L261 316H357C455 316 455 309 455 235H489L529 437H494C468 366 460 360 369 360H269L305 565C315 609 316 611 357 611H450C525 611 535 599 548 578C559 556 561 526 560 482L595 486L615 650Z" id="g50-70"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="639" vert-adv-y="639"></glyph.data></g></svg> is standard deviation, and <svg height="11.5773pt" id="M132" style="vertical-align:-0.04979992pt" version="1.1" viewbox="-0.0498162 -11.5275 18.9025 11.5773" width="18.9025pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M593 650H142L136 622C219 616 224 611 209 531L130 120C114 39 108 34 23 28L17 0H501C518 32 557 130 568 158L540 169C519 132 495 93 469 70C439 43 409 35 332 35C272 35 241 37 225 50C211 61 211 89 222 149L253 320H350C441 320 443 312 441 239H471L510 434H481C455 366 448 358 360 358H260L303 572C311 613 312 615 348 615H431C509 615 522 603 532 581C544 555 546 531 545 491L574 493C583 555 592 631 593 650Z" id="g113-70"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="616" vert-adv-y="616"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,8.008,-5.741)"><path d="M138 535C169 535 191 560 191 589C191 620 168 644 139 644S85 620 85 589C85 560 110 535 138 535ZM257 0V30C192 35 185 43 185 109V453C142 435 94 422 40 414V386C101 378 104 372 104 311V109C104 41 97 35 33 30V0H257Z" id="g58-103"></path><glyph.data ascent="1007" descent="-369" horiz-adv-x="275" vert-adv-y="275"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,10.511,-5.741)"><path d="M532 0V30C473 38 467 38 467 107V298C467 395 416 451 336 451C307 451 279 439 251 420C226 404 205 389 185 374V453C142 434 94 422 43 413V386C98 378 104 373 104 309V107C104 41 100 38 31 30V0H251V30C191 35 185 40 185 107V337C216 364 255 389 295 389C360 389 386 347 386 276V110C386 44 380 35 321 30V0H532Z" id="g58-108"></path><glyph.data ascent="1007" descent="-369" horiz-adv-x="557" vert-adv-y="557"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,15.397,-5.741)"><path d="M303 37L293 66C280 59 256 49 231 49S173 63 173 143V396H281C294 404 298 428 287 439H173V575L158 577L92 509V439H46L17 408L22 396H92V107C92 27 128 -12 192 -12C203 -12 218 -9 237 2L303 37Z" id="g58-114"></path><glyph.data ascent="1007" descent="-369" horiz-adv-x="312" vert-adv-y="312"></glyph.data></g></svg> is the value obtained from numerical deterministic model. (a) Confidence interval of 95.45% is given as <svg height="11.5303pt" id="M133" style="vertical-align:-3.18148pt" version="1.1" viewbox="-0.0498162 -8.34882 26.9967 11.5303" width="26.9967pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M535 290V340H323V533H265V340H52V290H265V84H323V290H535ZM535 -22V28H52V-22H535Z" id="g117-37"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="587" vert-adv-y="587"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,7.631,0)"><path d="M412 140C382 77 369 73 315 73H129L270 222C362 320 402 379 402 466C402 571 322 635 234 635C177 635 130 609 99 576L42 495L64 475C90 514 133 568 201 568C274 568 318 519 318 435C318 349 255 267 193 193C144 135 87 78 32 23V0H405C417 45 427 89 440 131L412 140Z" id="g113-51"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="480" vert-adv-y="480"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,13.871,0)"><path d="M548 455L523 469C505 448 490 439 461 439C416 439 374 447 316 447C144 447 23 310 23 161C23 47 89 -12 179 -12C312 -12 429 106 429 244C429 303 419 344 368 386L371 388C404 383 440 378 474 378C507 378 530 411 548 455ZM350 274C350 191 308 25 198 25C144 25 108 75 108 157C108 264 170 395 276 395C296 395 320 384 333 360C347 334 350 316 350 274Z" id="g113-240"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="565" vert-adv-y="565"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,20.631,3.132)"><path d="M615 650H141L132 618C223 612 227 607 212 526L136 123C122 42 112 38 27 32L18 0H522C539 33 580 135 592 166L558 179C536 139 512 100 486 75C454 48 420 41 345 41C287 41 252 42 237 55C221 68 220 100 231 158L261 316H357C455 316 455 309 455 235H489L529 437H494C468 366 460 360 369 360H269L305 565C315 609 316 611 357 611H450C525 611 535 599 548 578C559 556 561 526 560 482L595 486L615 650Z" id="g50-70"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="639" vert-adv-y="639"></glyph.data></g></svg> margins around mean. (b) Confidence interval of 99.7% is given as <svg height="11.5303pt" id="M134" style="vertical-align:-3.18148pt" version="1.1" viewbox="-0.0498162 -8.34882 26.9967 11.5303" width="26.9967pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M535 290V340H323V533H265V340H52V290H265V84H323V290H535ZM535 -22V28H52V-22H535Z" id="g117-37"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="587" vert-adv-y="587"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,7.631,0)"><path d="M285 378C315 398 338 416 353 432C373 451 384 474 384 503C384 579 325 635 236 635H235C182 635 136 610 108 579L65 516L85 496C110 533 150 575 205 575C258 575 300 543 300 481C300 407 232 369 141 339L147 310C163 315 188 321 211 321C268 321 338 284 338 192C338 94 288 40 217 40C160 40 119 68 93 91C85 98 77 97 69 91C60 84 47 71 46 58C44 46 48 35 62 22C75 10 116 -12 162 -12C234 -12 424 62 424 224C424 297 373 359 285 376V378Z" id="g113-52"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="480" vert-adv-y="480"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,13.871,0)"><path d="M548 455L523 469C505 448 490 439 461 439C416 439 374 447 316 447C144 447 23 310 23 161C23 47 89 -12 179 -12C312 -12 429 106 429 244C429 303 419 344 368 386L371 388C404 383 440 378 474 378C507 378 530 411 548 455ZM350 274C350 191 308 25 198 25C144 25 108 75 108 157C108 264 170 395 276 395C296 395 320 384 333 360C347 334 350 316 350 274Z" id="g113-240"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="565" vert-adv-y="565"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,20.631,3.132)"><path d="M615 650H141L132 618C223 612 227 607 212 526L136 123C122 42 112 38 27 32L18 0H522C539 33 580 135 592 166L558 179C536 139 512 100 486 75C454 48 420 41 345 41C287 41 252 42 237 55C221 68 220 100 231 158L261 316H357C455 316 455 309 455 235H489L529 437H494C468 366 460 360 369 360H269L305 565C315 609 316 611 357 611H450C525 611 535 599 548 578C559 556 561 526 560 482L595 486L615 650Z" id="g50-70"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="639" vert-adv-y="639"></glyph.data></g></svg> margins around mean.</div>

International Journal of Antennas and Propagation

fig6

Figure 6

Figure 6: An Efficient Deterministic-Stochastic Model of the Human Body Exposed to ELF Electric Field 