Difference between revisions of "Team:GreatBay SCIE/Drug Loading"

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−
<p>As we can consider our nanoparticle/liposome as a perfect sphere ,the best ratio of aptamer and the particle follows the following equation :</p>
 
−
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y="-686"></use></g></g><use xlink:href="#E1-MJMAIN-29" x="5116" y="0"></use><use xlink:href="#E1-MJMAIN-3A" x="5783" y="0"></use><use xlink:href="#E1-MJMAIN-31" x="6339" y="0"></use><use xlink:href="#E1-MJMAIN-3D" x="7116" y="0"></use><use xlink:href="#E1-MJMATHI-6E" x="8172" y="0"></use><use xlink:href="#E1-MJMAIN-3A" x="9050" y="0"></use><use xlink:href="#E1-MJMAIN-31" x="9606" y="0"></use></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-1">(\frac{4\pi r^2}{\frac{\sqrt{3}}{4}a^2}  \times \frac{1}{2}):1 = n:1</script></div></div>
 
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<p>The samllest distance between two aptamer on a plane can be identified by the affective length of the HER2 protein ,then we can find the distance of two adjacent aptamer on the sphere by following equation :</p>
 
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y="0"></use><use transform="scale(0.707)" xlink:href="#E9-MJMAIN-32" x="637" y="408"></use></g></g></g><use xlink:href="#E9-MJMAIN-3D" x="6369" y="0"></use><g transform="translate(7425,0)"><use xlink:href="#E9-MJMAIN-63"></use><use xlink:href="#E9-MJMAIN-6F" x="444" y="0"></use><use xlink:href="#E9-MJMAIN-73" x="944" y="0"></use></g><use xlink:href="#E9-MJMATHI-3B8" x="8929" y="0"></use></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-8">\frac{r^2+r^2-4d^2}{2r^2}=\cos\theta</script></div></div>
 
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<p>r represent the radius of the particle ,a represent the distance of two adjacent aptamers conjugate with particle at their surface ,n represent the numbers of aptamers .</p>
 
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−
<p>We can single control the concentration of APT-NH2 to achieve the binding of [APT-LIP]</p>
 
−
<p>&nbsp;</p>
 
  
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<div id='write'  class=''><h1 id='maximum-drug-loading-amount'><span>Maximum Drug Loading Amount</span></h1><p><span>Because the doxorubicin is loaded onto the liposomes via diffusion, it must follow the pattern of the Fick&#39;s First Law, which is :</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n3" cid="n3" mdtype="math_block"><div class="md-rawblock-container md-math-container" tabindex="-1"><div class="MathJax_SVG_Display" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-1-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="13.16ex" height="2.565ex" viewBox="0 -788 5666 1104.4" role="img" focusable="false" style="vertical-align: -0.735ex; max-width: 100%;"><defs><path stroke-width="0" id="E4-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path><path stroke-width="0" id="E4-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="0" id="E4-MJMAIN-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path stroke-width="0" id="E4-MJMATHI-44" d="M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 1L236 0H125H72Q48 0 41 2T33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM703 469Q703 507 692 537T666 584T629 613T590 629T555 636Q553 636 541 636T512 636T479 637H436Q392 637 386 627Q384 623 313 339T242 52Q242 48 253 48T330 47Q335 47 349 47T373 46Q499 46 581 128Q617 164 640 212T683 339T703 469Z"></path><path stroke-width="0" id="E4-MJMAIN-25BD" d="M59 480Q59 485 61 489T66 495T72 498L75 500H814Q828 493 828 480V474L644 132Q458 -210 455 -212Q451 -215 444 -215T433 -212Q429 -210 342 -49T164 282T64 466Q59 478 59 480ZM775 460H113Q113 459 278 153T444 -153T610 153T775 460Z"></path><path stroke-width="0" id="E4-MJMATHI-43" d="M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 240Q610 253 612 255T628 257Q648 257 648 248Q648 243 647 239Q618 132 523 55T319 -22Q206 -22 128 53T50 252Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E4-MJMATHI-4A" x="0" y="0"></use><use xlink:href="#E4-MJMAIN-3D" x="910" y="0"></use><use xlink:href="#E4-MJMAIN-2212" x="1966" y="0"></use><use xlink:href="#E4-MJMATHI-44" x="2744" y="0"></use><use xlink:href="#E4-MJMAIN-25BD" x="3794" y="0"></use><use xlink:href="#E4-MJMATHI-43" x="4906" y="0"></use></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-1">J=-D▽C</script></div></div><p><span>In this formula, J represents the diffusion flux, D represents the diffusion coefficient, ▽ represents the gradient operator and C represents the concentration. We can work out the value of J and ▽C by carrying out experiments, which can help us work out the value of D. Which is:</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n5" cid="n5" mdtype="math_block"><div class="md-rawblock-container md-math-container" tabindex="-1"><div class="MathJax_SVG_Display" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-2-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="13.16ex" height="2.565ex" viewBox="0 -788 5666 1104.4" role="img" focusable="false" style="vertical-align: -0.735ex; max-width: 100%;"><defs><path stroke-width="0" id="E5-MJMATHI-44" d="M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 1L236 0H125H72Q48 0 41 2T33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM703 469Q703 507 692 537T666 584T629 613T590 629T555 636Q553 636 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y="0"></use><use xlink:href="#E5-MJMAIN-25BD" x="2161" y="0"></use><use xlink:href="#E5-MJMATHI-43" x="3050" y="0"></use><use xlink:href="#E5-MJMAIN-2212" x="4032" y="0"></use><use xlink:href="#E5-MJMATHI-4A" x="5033" y="0"></use></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-2">D=▽C-J</script></div></div><p><span>And the D value can be used in the Fick&#39;s Second Law, and because the diffusion is taken place on liposome which can be considered as a perfect sphere, the formula can be inferred as is:</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n7" cid="n7" mdtype="math_block"><div class="md-rawblock-container md-math-container" tabindex="-1"><div class="MathJax_SVG_Display" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-3-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="24.221ex" height="5.851ex" viewBox="0 -1678.8 10428.6 2519.2" role="img" focusable="false" style="vertical-align: -1.952ex; max-width: 100%;"><defs><path stroke-width="0" id="E6-MJMAIN-2202" d="M202 508Q179 508 169 520T158 547Q158 557 164 577T185 624T230 675T301 710L333 715H345Q378 715 384 714Q447 703 489 661T549 568T566 457Q566 362 519 240T402 53Q321 -22 223 -22Q123 -22 73 56Q42 102 42 148V159Q42 276 129 370T322 465Q383 465 414 434T455 367L458 378Q478 461 478 515Q478 603 437 639T344 676Q266 676 223 612Q264 606 264 572Q264 547 246 528T202 508ZM430 306Q430 372 401 400T333 428Q270 428 222 382Q197 354 183 323T150 221Q132 149 132 116Q132 21 232 21Q244 21 250 22Q327 35 374 112Q389 137 409 196T430 306Z"></path><path stroke-width="0" id="E6-MJMATHI-43" d="M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 240Q610 253 612 255T628 257Q648 257 648 248Q648 243 647 239Q618 132 523 55T319 -22Q206 -22 128 53T50 252Z"></path><path stroke-width="0" id="E6-MJMATHI-74" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z"></path><path stroke-width="0" id="E6-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 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transform="translate(8388,0)"><g transform="translate(120,0)"><rect stroke="none" width="1411" height="60" x="0" y="220"></rect><g transform="translate(60,676)"><use xlink:href="#E6-MJMAIN-2202" x="0" y="0"></use><use xlink:href="#E6-MJMATHI-43" x="531" y="0"></use></g><g transform="translate(214,-687)"><use xlink:href="#E6-MJMAIN-2202" x="0" y="0"></use><use xlink:href="#E6-MJMATHI-72" x="531" y="0"></use></g></g></g><use xlink:href="#E6-MJMAIN-29" x="10039" y="0"></use></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-3">\frac{∂C}{∂t}=D(\frac{∂^2C}{∂r^2}+\frac{2}{r}\frac{∂C}{∂r})</script></div></div><p><span>And through this formula, we can work out the relationship between the concentration of the doxorubicin, change in time and the change in position. Through which, we can indicate the maximum drug loading amount by the maximum concentration of doxorubicin in the liposome.</span></p><p>&nbsp;</p></div></div>
 
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Maximum Drug Loading Amount(Markdown)

Maximum Drug Loading Amount

Because the doxorubicin is loaded onto the liposomes via diffusion, it must follow the pattern of the Fick's First Law, which is :

In this formula, J represents the diffusion flux, D represents the diffusion coefficient, ▽ represents the gradient operator and C represents the concentration. We can work out the value of J and ▽C by carrying out experiments, which can help us work out the value of D. Which is:

And the D value can be used in the Fick's Second Law, and because the diffusion is taken place on liposome which can be considered as a perfect sphere, the formula can be inferred as is:

And through this formula, we can work out the relationship between the concentration of the doxorubicin, change in time and the change in position. Through which, we can indicate the maximum drug loading amount by the maximum concentration of doxorubicin in the liposome.