Mathematical Analysis for the Effects of Medicine Supplies to a Solid Tumor
<p>Diagram for medicine supplies: (i) <math display="inline"><semantics> <mrow> <msub> <mi>I</mi> <mn>1</mn> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>I</mi> <mn>2</mn> </msub> </mrow> </semantics></math> are medicines injected into heart and liver, respectively, (ii) <math display="inline"><semantics> <mrow> <msub> <mi>k</mi> <mn>1</mn> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>k</mi> <mn>2</mn> </msub> </mrow> </semantics></math> are flow rates from heart to vessels and from liver to heart, respectively.</p> "> Figure 2
<p>A cylindrical cross section domain: (<b>a</b>) Discretization for approximate solutions using a finite volume method, (<b>b</b>) Notations of finite control volumes.</p> "> Figure 3
<p>Validation of the results with the previously published articles [<a href="#B27-symmetry-13-01988" class="html-bibr">27</a>] based on the fraction killed cells along normalized time.</p> "> Figure 4
<p>TAF concentration <math display="inline"><semantics> <mi>c</mi> </semantics></math> along parameters the time <math display="inline"><semantics> <mi>τ</mi> </semantics></math> and the radius <math display="inline"><semantics> <mi>r</mi> </semantics></math> when the time values <math display="inline"><semantics> <mrow> <msub> <mi>t</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>t</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math> and the ratio value <math display="inline"><semantics> <mrow> <mfrac> <mrow> <msub> <mi>k</mi> <mn>1</mn> </msub> </mrow> <mrow> <msub> <mi>k</mi> <mn>2</mn> </msub> </mrow> </mfrac> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>.</p> "> Figure 5
<p>Concentration of medicine <math display="inline"><semantics> <mi>M</mi> </semantics></math> in blood vessels along the time <math display="inline"><semantics> <mi>τ</mi> </semantics></math> when the time values <math display="inline"><semantics> <mrow> <msub> <mi>t</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>t</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>.</p> "> Figure 6
<p>TAF concentration <math display="inline"><semantics> <mi>c</mi> </semantics></math> for various ratio <math display="inline"><semantics> <mrow> <mfrac> <mrow> <msub> <mi>k</mi> <mn>1</mn> </msub> </mrow> <mrow> <msub> <mi>k</mi> <mn>2</mn> </msub> </mrow> </mfrac> </mrow> </semantics></math> along the time <math display="inline"><semantics> <mi>τ</mi> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>r</mi> <mo>=</mo> <mi>R</mi> <mo>/</mo> <mn>2</mn> </mrow> </semantics></math> when the time values <math display="inline"><semantics> <mrow> <msub> <mi>t</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>t</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>.</p> "> Figure 7
<p>Concentration of TAF along the radius <math display="inline"><semantics> <mi>r</mi> </semantics></math> when the time values <math display="inline"><semantics> <mrow> <msub> <mi>t</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>t</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math>.</p> "> Figure 8
<p>Variations of tumor cells density <math display="inline"><semantics> <mo>∅</mo> </semantics></math> along the time <math display="inline"><semantics> <mi>τ</mi> </semantics></math> for various values <math display="inline"><semantics> <mi>r</mi> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>θ</mi> <mo>=</mo> <msup> <mn>0</mn> <mo>°</mo> </msup> </mrow> </semantics></math>.</p> "> Figure 9
<p>Variations of tumor cells density <math display="inline"><semantics> <mo>∅</mo> </semantics></math> along the radius <math display="inline"><semantics> <mi>r</mi> </semantics></math> for various values <math display="inline"><semantics> <mi>τ</mi> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>θ</mi> <mo>=</mo> <msup> <mn>0</mn> <mo>°</mo> </msup> </mrow> </semantics></math>.</p> ">
Abstract
:1. Introduction
2. Mathematical Modelling
2.1. Tumor Cell Mobility
2.2. Discretization for Finite Volume Method
2.3. Validation of Numerical Results
3. Results and Discussions
4. Conclusions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
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Go, J. Mathematical Analysis for the Effects of Medicine Supplies to a Solid Tumor. Symmetry 2021, 13, 1988. https://doi.org/10.3390/sym13111988
Go J. Mathematical Analysis for the Effects of Medicine Supplies to a Solid Tumor. Symmetry. 2021; 13(11):1988. https://doi.org/10.3390/sym13111988
Chicago/Turabian StyleGo, Jaegwi. 2021. "Mathematical Analysis for the Effects of Medicine Supplies to a Solid Tumor" Symmetry 13, no. 11: 1988. https://doi.org/10.3390/sym13111988