Numerical Analysis of an Osseointegration Model
<p>Asymptotic linear convergence.</p> "> Figure 2
<p>Graph of <span class="html-italic">p</span> defined as a sigmoid (red) and the piecewise form (dashed black), for <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mi>m</mi> </msub> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>.</p> "> Figure 3
<p>Concentration of platelets along the spatial domain at the final time.</p> "> Figure 4
<p>Concentration of growth factors <math display="inline"><semantics> <msub> <mi>s</mi> <mn>1</mn> </msub> </semantics></math> (<b>left</b>) and <math display="inline"><semantics> <msub> <mi>s</mi> <mn>2</mn> </msub> </semantics></math> (<b>right</b>) at the final time.</p> "> Figure 5
<p>Concentration of platelets along the spatial domain at the final time for the case of <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mi>m</mi> </msub> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math> (blue) and <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mi>m</mi> </msub> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math> (black).</p> "> Figure 6
<p>Concentration of osteogenic cells (<b>left</b>) and osteoblasts (<b>right</b>) at the final time for the case of <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mi>m</mi> </msub> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math> (blue) and <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mi>m</mi> </msub> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math> (black).</p> "> Figure 7
<p>Computational domain and mesh for the two-dimensional problem. Due to the symmetry of the problem, only half of the domain was simulated. Interior borders were added to ensure that at the regions of discontinuity for the derivatives of <span class="html-italic">p</span>, only edges were placed. Dimensions in mm.</p> "> Figure 8
<p>Concentration of platelets at the final time for the case of <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mi>m</mi> </msub> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math> (<b>left</b>) and <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mi>m</mi> </msub> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math> (<b>right</b>).</p> "> Figure 9
<p>Concentration of osteogenic cells at the final time for the case of <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mi>m</mi> </msub> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math> (<b>left</b>) and <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mi>m</mi> </msub> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math> (<b>right</b>).</p> "> Figure 10
<p>Concentration of osteoblasts at the final time for the case of <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mi>m</mi> </msub> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math> (<b>left</b>) and <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mi>m</mi> </msub> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math> (<b>right</b>).</p> ">
Abstract
:1. Introduction
2. Biological Problem and Its Variational Formulation
3. Fully Discrete Approximations and an a Priori Error Analysis
4. Numerical Results
4.1. Numerical Scheme
4.2. Numerical Convergence
4.3. One-Dimensional Examples
4.4. Two-Dimensional Example
5. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
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h↓ | |||||||||
---|---|---|---|---|---|---|---|---|---|
8.440 | 4.712 | 2.845 | 1.925 | 1.477 | 1.276 | 1.202 | 1.167 | 1.151 | |
7.919 | 4.199 | 2.322 | 1.387 | 0.927 | 0.719 | 0.644 | 0.610 | 0.593 | |
7.768 | 4.052 | 2.175 | 1.236 | 0.771 | 0.553 | 0.474 | 0.437 | 0.420 | |
7.707 | 3.992 | 2.114 | 1.175 | 0.708 | 0.482 | 0.400 | 0.362 | 0.343 | |
7.668 | 3.953 | 2.075 | 1.135 | 0.667 | 0.438 | 0.348 | 0.309 | 0.290 | |
7.634 | 3.919 | 2.041 | 1.100 | 0.631 | 0.400 | 0.301 | 0.262 | 0.243 | |
7.602 | 3.887 | 2.008 | 1.067 | 0.597 | 0.364 | 0.256 | 0.216 | 0.197 | |
7.570 | 3.855 | 1.976 | 1.034 | 0.563 | 0.329 | 0.214 | 0.170 | 0.151 | |
7.539 | 3.823 | 1.944 | 1.001 | 0.530 | 0.295 | 0.178 | 0.125 | 0.105 |
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Baldonedo, J.; Fernández, J.R.; Segade, A. Numerical Analysis of an Osseointegration Model. Mathematics 2020, 8, 87. https://doi.org/10.3390/math8010087
Baldonedo J, Fernández JR, Segade A. Numerical Analysis of an Osseointegration Model. Mathematics. 2020; 8(1):87. https://doi.org/10.3390/math8010087
Chicago/Turabian StyleBaldonedo, Jacobo, José R. Fernández, and Abraham Segade. 2020. "Numerical Analysis of an Osseointegration Model" Mathematics 8, no. 1: 87. https://doi.org/10.3390/math8010087