A Multiphase Flow in the Antroduodenal Portion of the Gastrointestinal Tract: A Mathematical Model

<div>Identifying subareas by functional and geometric features, <svg height="13.9678pt" id="M3" style="vertical-align:-4.67947pt" version="1.1" viewbox="-0.0498162 -9.28833 27.91 13.9678" width="27.91pt" 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="M471 379C471 433 467 511 427 578C379 658 305 712 208 712C154 712 98 699 70 684C70 668 95 634 111 634C141 651 180 665 234 665C306 665 350 627 372 575C391 530 393 480 393 442V415C371 436 344 448 313 448C280 448 227 437 176 411C100 372 23 279 23 148C23 67 64 -12 180 -12C293 -12 360 46 407 129C450 205 471 310 471 379ZM382 335C382 142 289 50 238 35C226 31 213 28 200 28C139 28 110 91 110 164C110 310 191 382 223 395C245 404 259 409 280 409C305 409 332 404 359 388C379 376 382 362 382 335Z" id="g113-222"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="494" vert-adv-y="494"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,6.424,0)"><path d="M719 146L691 147C666 77 657 72 612 72H528C572 125 688 224 688 364C688 546 558 665 384 665C212 665 77 541 77 361C77 221 188 124 239 72H150C110 72 97 80 73 147L46 146L63 0H307L314 18C284 71 178 203 178 360C178 529 256 631 383 631C509 631 588 523 588 363C588 197 485 67 453 18L458 0H701L719 146Z" id="g113-153"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="757" vert-adv-y="757"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,16.265,3.132)"><path d="M309 -142C211 -61 151 99 151 271S210 601 309 683L288 710C141 611 75 451 75 272V271C75 90 141 -72 288 -169L309 -142Z" id="g50-41"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="358" vert-adv-y="358"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,19.523,3.132)"><path d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 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360 46 407 129C450 205 471 310 471 379ZM382 335C382 142 289 50 238 35C226 31 213 28 200 28C139 28 110 91 110 164C110 310 191 382 223 395C245 404 259 409 280 409C305 409 332 404 359 388C379 376 382 362 382 335Z" id="g113-222"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="494" vert-adv-y="494"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,6.424,0)"><path d="M719 146L691 147C666 77 657 72 612 72H528C572 125 688 224 688 364C688 546 558 665 384 665C212 665 77 541 77 361C77 221 188 124 239 72H150C110 72 97 80 73 147L46 146L63 0H307L314 18C284 71 178 203 178 360C178 529 256 631 383 631C509 631 588 523 588 363C588 197 485 67 453 18L458 0H701L719 146Z" id="g113-153"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="757" vert-adv-y="757"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,16.265,3.132)"><path d="M309 -142C211 -61 151 99 151 271S210 601 309 683L288 710C141 611 75 451 75 272V271C75 90 141 -72 288 -169L309 -142Z" 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<svg height="13.9678pt" id="M5" style="vertical-align:-4.67947pt" version="1.1" viewbox="-0.0498162 -9.28833 27.91 13.9678" width="27.91pt" 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="M471 379C471 433 467 511 427 578C379 658 305 712 208 712C154 712 98 699 70 684C70 668 95 634 111 634C141 651 180 665 234 665C306 665 350 627 372 575C391 530 393 480 393 442V415C371 436 344 448 313 448C280 448 227 437 176 411C100 372 23 279 23 148C23 67 64 -12 180 -12C293 -12 360 46 407 129C450 205 471 310 471 379ZM382 335C382 142 289 50 238 35C226 31 213 28 200 28C139 28 110 91 110 164C110 310 191 382 223 395C245 404 259 409 280 409C305 409 332 404 359 388C379 376 382 362 382 335Z" id="g113-222"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="494" vert-adv-y="494"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,6.424,0)"><path d="M719 146L691 147C666 77 657 72 612 72H528C572 125 688 224 688 364C688 546 558 665 384 665C212 665 77 541 77 361C77 221 188 124 239 72H150C110 72 97 80 73 147L46 146L63 0H307L314 18C284 71 178 203 178 360C178 529 256 631 383 631C509 631 588 523 588 363C588 197 485 67 453 18L458 0H701L719 146Z" id="g113-153"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="757" vert-adv-y="757"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,16.265,3.132)"><path d="M309 -142C211 -61 151 99 151 271S210 601 309 683L288 710C141 611 75 451 75 272V271C75 90 141 -72 288 -169L309 -142Z" id="g50-41"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="358" vert-adv-y="358"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,19.523,3.132)"><path d="M462 177V227H365V632H320C217 496 116 350 21 208V177H284V109C284 43 280 38 190 31V0H451V31C369 38 365 43 365 108V177H462ZM284 227H88C155 336 218 430 282 519H284V227Z" id="g50-53"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="487" vert-adv-y="487"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,23.955,3.132)"><path d="M283 271C283 451 220 610 70 710L48 683C146 603 207 443 207 271S149 -59 48 -142L70 -169C220 -67 283 90 283 271Z" id="g50-42"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="358" vert-adv-y="358"></glyph.data></g></svg>, and <svg height="13.9678pt" id="M6" style="vertical-align:-4.67947pt" version="1.1" viewbox="-0.0498162 -9.28833 27.91 13.9678" width="27.91pt" 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="M471 379C471 433 467 511 427 578C379 658 305 712 208 712C154 712 98 699 70 684C70 668 95 634 111 634C141 651 180 665 234 665C306 665 350 627 372 575C391 530 393 480 393 442V415C371 436 344 448 313 448C280 448 227 437 176 411C100 372 23 279 23 148C23 67 64 -12 180 -12C293 -12 360 46 407 129C450 205 471 310 471 379ZM382 335C382 142 289 50 238 35C226 31 213 28 200 28C139 28 110 91 110 164C110 310 191 382 223 395C245 404 259 409 280 409C305 409 332 404 359 388C379 376 382 362 382 335Z" id="g113-222"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="494" vert-adv-y="494"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,6.424,0)"><path d="M719 146L691 147C666 77 657 72 612 72H528C572 125 688 224 688 364C688 546 558 665 384 665C212 665 77 541 77 361C77 221 188 124 239 72H150C110 72 97 80 73 147L46 146L63 0H307L314 18C284 71 178 203 178 360C178 529 256 631 383 631C509 631 588 523 588 363C588 197 485 67 453 18L458 0H701L719 146Z" id="g113-153"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="757" vert-adv-y="757"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,16.265,3.132)"><path d="M309 -142C211 -61 151 99 151 271S210 601 309 683L288 710C141 611 75 451 75 272V271C75 90 141 -72 288 -169L309 -142Z" id="g50-41"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="358" vert-adv-y="358"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,19.523,3.132)"><path d="M158 548H390L417 615L410 623H122L83 318C105 326 143 337 185 337C296 337 350 275 350 188C350 116 308 42 225 42C164 42 122 74 100 93C90 101 82 99 72 92C60 82 51 68 50 59C48 46 52 38 66 24C82 9 125 -12 172 -12C225 -11 292 15 346 59C408 108 437 166 437 226C437 309 371 397 242 397C214 397 170 382 133 369L158 548Z" id="g50-54"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="487" vert-adv-y="487"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,23.955,3.132)"><path d="M283 271C283 451 220 610 70 710L48 683C146 603 207 443 207 271S149 -59 48 -142L70 -169C220 -67 283 90 283 271Z" id="g50-42"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="358" vert-adv-y="358"></glyph.data></g></svg> are the alkaline secretion area, and <svg height="13.9678pt" id="M7" style="vertical-align:-4.67947pt" version="1.1" viewbox="-0.0498162 -9.28833 27.91 13.9678" width="27.91pt" 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="M471 379C471 433 467 511 427 578C379 658 305 712 208 712C154 712 98 699 70 684C70 668 95 634 111 634C141 651 180 665 234 665C306 665 350 627 372 575C391 530 393 480 393 442V415C371 436 344 448 313 448C280 448 227 437 176 411C100 372 23 279 23 148C23 67 64 -12 180 -12C293 -12 360 46 407 129C450 205 471 310 471 379ZM382 335C382 142 289 50 238 35C226 31 213 28 200 28C139 28 110 91 110 164C110 310 191 382 223 395C245 404 259 409 280 409C305 409 332 404 359 388C379 376 382 362 382 335Z" id="g113-222"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="494" vert-adv-y="494"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,6.424,0)"><path d="M719 146L691 147C666 77 657 72 612 72H528C572 125 688 224 688 364C688 546 558 665 384 665C212 665 77 541 77 361C77 221 188 124 239 72H150C110 72 97 80 73 147L46 146L63 0H307L314 18C284 71 178 203 178 360C178 529 256 631 383 631C509 631 588 523 588 363C588 197 485 67 453 18L458 0H701L719 146Z" id="g113-153"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="757" vert-adv-y="757"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,16.265,3.132)"><path d="M309 -142C211 -61 151 99 151 271S210 601 309 683L288 710C141 611 75 451 75 272V271C75 90 141 -72 288 -169L309 -142Z" id="g50-41"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="358" vert-adv-y="358"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,19.523,3.132)"><path d="M389 0V32C297 38 291 46 291 118V635C234 613 175 595 109 583V556L161 554C203 552 207 547 207 497V118C207 46 201 38 110 32V0H389Z" id="g50-50"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="487" vert-adv-y="487"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,23.955,3.132)"><path d="M283 271C283 451 220 610 70 710L48 683C146 603 207 443 207 271S149 -59 48 -142L70 -169C220 -67 283 90 283 271Z" id="g50-42"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="358" vert-adv-y="358"></glyph.data></g></svg>, <svg height="13.9678pt" id="M8" style="vertical-align:-4.67947pt" version="1.1" viewbox="-0.0498162 -9.28833 27.91 13.9678" width="27.91pt" 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="M471 379C471 433 467 511 427 578C379 658 305 712 208 712C154 712 98 699 70 684C70 668 95 634 111 634C141 651 180 665 234 665C306 665 350 627 372 575C391 530 393 480 393 442V415C371 436 344 448 313 448C280 448 227 437 176 411C100 372 23 279 23 148C23 67 64 -12 180 -12C293 -12 360 46 407 129C450 205 471 310 471 379ZM382 335C382 142 289 50 238 35C226 31 213 28 200 28C139 28 110 91 110 164C110 310 191 382 223 395C245 404 259 409 280 409C305 409 332 404 359 388C379 376 382 362 382 335Z" id="g113-222"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="494" vert-adv-y="494"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,6.424,0)"><path d="M719 146L691 147C666 77 657 72 612 72H528C572 125 688 224 688 364C688 546 558 665 384 665C212 665 77 541 77 361C77 221 188 124 239 72H150C110 72 97 80 73 147L46 146L63 0H307L314 18C284 71 178 203 178 360C178 529 256 631 383 631C509 631 588 523 588 363C588 197 485 67 453 18L458 0H701L719 146Z" id="g113-153"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="757" vert-adv-y="757"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,16.265,3.132)"><path d="M309 -142C211 -61 151 99 151 271S210 601 309 683L288 710C141 611 75 451 75 272V271C75 90 141 -72 288 -169L309 -142Z" id="g50-41"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="358" vert-adv-y="358"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,19.523,3.132)"><path d="M141 347C171 480 264 541 327 570C364 587 403 599 436 605L429 641C389 634 339 622 300 609C191 570 37 458 37 239C37 86 126 -12 245 -12C367 -12 454 89 454 210C454 314 380 397 272 397C252 397 230 390 208 380L141 347ZM231 338C323 338 366 257 366 174C366 108 340 27 263 27C180 27 129 123 129 244C129 268 130 292 135 310C158 323 193 338 231 338Z" id="g50-55"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="487" vert-adv-y="487"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,23.955,3.132)"><path d="M283 271C283 451 220 610 70 710L48 683C146 603 207 443 207 271S149 -59 48 -142L70 -169C220 -67 283 90 283 271Z" id="g50-42"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="358" vert-adv-y="358"></glyph.data></g></svg> are the inlet/outlet section.</div>

Computational and Mathematical Methods in Medicine

fig2

Figure 2

Figure 2: A Multiphase Flow in the Antroduodenal Portion of the Gastrointestinal Tract: A Mathematical Model 