# State and Parameter Estimation of a Mathematical Carcinoma Model under Chemotherapeutic Treatment

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Experimental Data

#### 2.2. The Applied Tumor Growth Model

^{3}] and ${x}_{2}$ [mm

^{3}], respectively. The third variable ${x}_{3}$ [mg/kg] describes the drug concentration in the host. The dynamics is described by the equations given in [6,7] at time t as

#### 2.3. Extended Kalman Filter

#### 2.3.1. Measurement Noise Characteristics

#### 2.3.2. Kalman Filter Tuning

#### 2.4. Moving Horizon Estimation

## 3. Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Ethical Statement

## Abbreviations

EKF | Extended Kalman filter |

MHE | moving horizon estimation |

RMSE | root-mean-square error |

## References

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**Figure 1.**Scheme of the experiment from [14].

Parameter | PLD1 | PLD2 | PLD3 | PLD4 | PLD5 | PLD6 | PLD8 | PLD9 | Nominal | SD |
---|---|---|---|---|---|---|---|---|---|---|

a [1/day] | 0.333 | 0.307 | 0.307 | 0.310 | 0.289 | 0.299 | 0.308 | 0.311 | 0.306 | 0.0186 |

b [1/day] | 0.116 | 0.169 | 0.198 | 0.180 | 0.163 | 0.184 | 0.174 | 0.167 | 0.166 | 0.0302 |

c [1/day] | 0.235 | 0.297 | 0.304 | 0.272 | 0.312 | 0.365 | 0.187 | 0.161 | 0.257 | 0.0820 |

n [1/day] | 0.115 | 0.148 | 0.153 | 0.173 | 0.134 | 0.161 | 0.133 | 0.145 | 0.144 | 0.0235 |

${b}_{k}$$\left(\right)$ | 6.15 | 6.05 | 6.02 | 6.10 | 6.19 | 6.16 | 6.17 | 6.11 | 6.12 | 0.404 |

${K}_{B}$ [mg/kg] | 0.367 | 0.361 | 0.342 | 0.230 | 0.362 | 0.374 | 0.515 | 0.400 | 0.36 | 0.1242 |

$E{D}_{50}$ [${10}^{-5}$ mg/kg] | 8.89 | 9.03 | 10.4 | 13.3 | 8.64 | 7.91 | 7.79 | 8.94 | 9.71 | 1.48 |

w [1/day] | 0.346 | 0.344 | 0.331 | 0.341 | 0.341 | 0.339 | 0.336 | 0.342 | 0.34 | 0.0253 |

Penalization | Arguments | ||
---|---|---|---|

State | Disturbance | Parameter | |

Measurement difference | $\left|\right|\mathbf{z}-{h\left(\mathbf{x}\right)\left|\right|}_{{\mathbf{W}}_{R}}^{2}$ | $\left|\right|\mathbf{z}-{h\left(\mathbf{x}\left(\mathbf{w}\right)\right)\left|\right|}_{{\mathbf{W}}_{R}}^{2}$ | $\left|\right|\mathbf{z}-{h\left(\mathbf{x}\left(\mathbf{p}\right)\right)\left|\right|}_{{\mathbf{W}}_{R}}^{2}$ |

Modification | $\left|\right|\mathbf{x}-{\mathbf{x}}_{ol}{\left|\right|}_{{\mathbf{W}}_{Q}}^{2}$ | ${\left|\right|\mathbf{w}\left|\right|}_{{\mathbf{W}}_{Q}}^{2}$ | ${\left|\right|\mathrm{\Delta}\mathbf{p}\left|\right|}_{{\mathbf{W}}_{Q}}^{2}$ |

Arrival cost | $\left|\right|\widehat{\mathbf{x}}-\mathbf{x}{\left|\right|}_{{\mathbf{W}}_{P}}^{2}$ | $\left|\right|\widehat{\mathbf{x}}-\mathbf{x}{\left|\right|}_{{\mathbf{W}}_{P}}^{2}$ | $\left|\right|\widehat{\mathbf{x}}-\mathbf{x}{\left|\right|}_{{\mathbf{W}}_{P}}^{2}$ |

EKF | MHE 7 Days | MHE 14 Days | MHE 21 Days | MHE 28 Days | MHE 35 Days | |
---|---|---|---|---|---|---|

PLD3 | 121.6 | 114.6 | 115.5 | 121.5 | 121.4 | 169.5 |

PLD4 | 72.12 | 32.82 | 48.96 | 59.33 | 60.33 | 103.9 |

PLD5 | 74.53 | 28.68 | 40.48 | 52.16 | 61.89 | 88.76 |

PLD6 | 97.93 | 43.46 | 50.15 | 65.97 | 88.49 | 146.1 |

PLD9 | 59.55 | 49.05 | 55.67 | 61.77 | 72.52 | 86.39 |

MEAN | 85.14 | 53.73 | 62.16 | 72.16 | 80.94 | 118.9 |

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**MDPI and ACS Style**

Siket, M.; Eigner, G.; Drexler, D.A.; Rudas, I.; Kovács, L.
State and Parameter Estimation of a Mathematical Carcinoma Model under Chemotherapeutic Treatment. *Appl. Sci.* **2020**, *10*, 9046.
https://doi.org/10.3390/app10249046

**AMA Style**

Siket M, Eigner G, Drexler DA, Rudas I, Kovács L.
State and Parameter Estimation of a Mathematical Carcinoma Model under Chemotherapeutic Treatment. *Applied Sciences*. 2020; 10(24):9046.
https://doi.org/10.3390/app10249046

**Chicago/Turabian Style**

Siket, Máté, György Eigner, Dániel András Drexler, Imre Rudas, and Levente Kovács.
2020. "State and Parameter Estimation of a Mathematical Carcinoma Model under Chemotherapeutic Treatment" *Applied Sciences* 10, no. 24: 9046.
https://doi.org/10.3390/app10249046