# Physically Consistent Scar Tissue Dynamics from Scattered Set of Data: A Novel Computational Approach to Avoid the Onset of the Runge Phenomenon

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## Abstract

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## 1. Introduction

## 2. Methods

## 3. Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Appendix A

## References

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**Figure 1.**Vandermonde-based interpolation (black squares = mean experimental values, solid lines = theoretical predictions): (

**a**) The huge difference among the element 9,9 and the other ones resulted in an ill-conditioned matrix. (

**b**) The interpolating polynomial function was affected by the Runge phenomenon, resulting in nonphysical predictions for inter-interval values (negative values). (

**c**) The normalization of times over the maximum experimental time lead to a better conditioned matrix. (

**d**) Moreover, in the case of time normalization, the performances of the interpolating polynomials were unchanged.

**Figure 2.**Interpolation through standard Lagrange polynomials: (

**a**) Standard Lagrange polynomials (solid lines) were able to fit all the experimental data points (black squares). In this case, the function was the same as the function in Figure 1a, which was obtained through the Vandermonde-based interpolation. As a consequence, the numerical problems affecting the prediction of the thickness over time were the same. (

**b**) Lagrange polynomials fitting all experimental points together with an extrapolated supplementary point. This function was still affected by the Runge phenomenon, but the magnitude of differences between predictions and experimental values was lower (i.e., about $1\times {10}^{7}$). (

**c**) Lagrange polynomials fitting all experimental points together with two extrapolated supplementary points. Numerical errors still affected the interpolating function leading to nonphysical predictions, even if the magnitude of errors was lowered to about $1\times {10}^{6}$. (

**d**) Lagrange polynomials fitting all experimental points together with three extrapolated supplementary points. The Runge phenomenon still affected the interpolating polynomial function, which again resulted in nonphysical predictions. The magnitude of errors was further lowered to about $1\times {10}^{5}$.

**Figure 3.**Values of the ${H}_{m,i}$ numerical coefficients in Equation (10). For each candidate function all the values of each ${H}_{m}$ vector are reported in lines. These values are the numerical coefficients needed to provide the linear combination of $JH{S}_{i}(t)$ functions to express each ${w}_{m}(t)$ function.

**Figure 4.**Reproduction of experimental data through the novel ${w}_{n}(t)$ functions (black squares = mean values, error bars = ± standard deviation, solid lines = theoretical predictions, dotted lines = $90\%$ confidence prediction bounds). (

**a**) The function ${w}_{0}(t)$ (m = 0) was unable to reproduce experimental values since it assumed a constant value. (

**b**) The function ${w}_{1}(t)$ (m = 1) was able to reproduce experiments with ${R}^{2}=0.8661$. (

**c**) The function ${w}_{2}(t)$ (m = 2) was able to reproduce experiments with ${R}^{2}=0.9195$. (

**d**) The function ${w}_{3}(t)$ (m = 3) reproduced experiments with ${R}^{2}=0.9197$. (

**e**) The function ${w}_{4}(t)$ (m = 4) reproduced experiments with ${R}^{2}=0.9221$. (

**f**) The function ${w}_{5}(t)$ (m = 5) reproduced experimental data with ${R}^{2}=0.9190$.

**Figure 5.**(

**a**) The natural logarithm of the ${\mathsf{\Lambda}}_{f}$ index was used to account for the intensity of the Runge phenomenon since it is able to graphically show whether approximating functions were affected by the Runge phenomenon. The first approximation obtained through the Vandermonde approach resulted in a positive value of $ln({\mathsf{\Lambda}}_{{g}_{9}(t)})$. Similarly, for ${g}_{9}(t),\cdots ,{g}_{12}(t)$, this logarithm was positive since the Runge phenomenon affected predictions. On the contrary, for ${w}_{1}(t),\cdots ,{w}_{5}(t)$, this logarithm resulted in negative values and the Runge phenomenon did not affect numerical predictions of inter-interval values. For the function ${w}_{0}(t)$, the index $ln({\mathsf{\Lambda}}_{{w}_{0}(t)})$ increased without bound towards $-\infty $ (not shown within the plot). (

**b**) Overall suitability: Considering both the ability of each candidate function to reproduce experimental values and the intensity of the Runge phenomenon affecting numerical predictions, the overall suitability shows that only the ${w}_{n}(t)$ family was able to reproduce experiments in a suitable way. Among the ${w}_{n}(t)$ functions, the ${w}_{4}(t)$ was also the most suitable to reproduce experimental data for inter-interval times.

**Table 1.**Elements of the $\mathbf{a}$ vector, fully characterizing the candidate function resulting from the classic Vandermonde approach.

Elements of the a Vector | Value |
---|---|

${a}_{0}$ | 0 |

${a}_{1}$ | −0.276 |

${a}_{2}$ | +0.015 |

${a}_{3}$ | $-1.640\times {10}^{-4}$ |

${a}_{4}$ | $+6.339\times {10}^{-7}$ |

${a}_{5}$ | $-1.019\times {10}^{-9}$ |

${a}_{6}$ | $+7.121\times {10}^{-13}$ |

${a}_{7}$ | $-2.057\times {10}^{-16}$ |

${a}_{8}$ | $+1.947\times {10}^{-20}$ |

**Table 2.**The elements of the $\mathbf{x}$ vector, fully characterizing the interpolating Lagrange polynomials. The numeric values of the $\mathbf{x}$ vector elements for the functions ${g}_{10}(t),{g}_{11}(t),{g}_{12}(t)$ are shown in columns 1, 2, 3, respectively.

Elements | Total Time Points ${\mathit{n}}_{\mathit{tot}}$ = 9 + 1 | Total Time Points ${\mathit{n}}_{\mathit{tot}}$ = 9 + 2 | Total Time Points ${\mathit{n}}_{\mathit{tot}}$ = 9 + 3 |
---|---|---|---|

${x}_{0}$ | 0 | 0 | 0 |

${x}_{1}$ | $-0.278$ | $-0.280$ | $-0.282$ |

${x}_{2}$ | $+0.015$ | $+0.015$ | $+0.015$ |

${x}_{3}$ | $-1.687\times {10}^{-04}$ | $-1.687\times {10}^{-04}$ | $-1.764\times {10}^{-04}$ |

${x}_{4}$ | $+6.776\times {10}^{-07}$ | $+7.162\times {10}^{-07}$ | $+7.532\times {10}^{-07}$ |

${x}_{5}$ | $-1.182\times {10}^{-09}$ | $-1.331\times {10}^{-09}$ | $-1.478\times {10}^{-09}$ |

${x}_{6}$ | $+9.692\times {10}^{-13}$ | $+1.225\times {10}^{-12}$ | $+1.494\times {10}^{-12}$ |

${x}_{7}$ | $-3.840\times {10}^{-16}$ | $-5.922\times {10}^{-16}$ | $-8.382\times {10}^{-16}$ |

${x}_{8}$ | $+7.081\times {10}^{-20}$ | $+1.530\times {10}^{-19}$ | $+2.714\times {10}^{-19}$ |

${x}_{9}$ | $-4.850\times {10}^{-24}$ | $-1.998\times {10}^{-23}$ | $-5.051\times {10}^{-23}$ |

${x}_{10}$ | $+1.035\times {10}^{-24}$ | $+5.016\times {10}^{-27}$ | |

${x}_{11}$ | $-2.060\times {10}^{-31}$ |

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**MDPI and ACS Style**

Sergi, P.N.; De la Oliva, N.; del Valle, J.; Navarro, X.; Micera, S.
Physically Consistent Scar Tissue Dynamics from Scattered Set of Data: A Novel Computational Approach to Avoid the Onset of the Runge Phenomenon. *Appl. Sci.* **2021**, *11*, 8568.
https://doi.org/10.3390/app11188568

**AMA Style**

Sergi PN, De la Oliva N, del Valle J, Navarro X, Micera S.
Physically Consistent Scar Tissue Dynamics from Scattered Set of Data: A Novel Computational Approach to Avoid the Onset of the Runge Phenomenon. *Applied Sciences*. 2021; 11(18):8568.
https://doi.org/10.3390/app11188568

**Chicago/Turabian Style**

Sergi, Pier Nicola, Natalia De la Oliva, Jaume del Valle, Xavier Navarro, and Silvestro Micera.
2021. "Physically Consistent Scar Tissue Dynamics from Scattered Set of Data: A Novel Computational Approach to Avoid the Onset of the Runge Phenomenon" *Applied Sciences* 11, no. 18: 8568.
https://doi.org/10.3390/app11188568