# American Institute of Mathematical Sciences

January  2018, 23(1): 145-160. doi: 10.3934/dcdsb.2018009

## Sensitivity of combined chemo-and antiangiogenic therapy results in different models describing cancer growth

 Systems Engineering Group, Silesian University of Technology, Akademicka 16, Gliwice, 44-100, Poland

* Corresponding author: Jaroslaw Smieja

Received  November 2016 Revised  June 2017 Published  January 2018

Fund Project: This work was supported by the NCN grant DEC-2013/11/B/ST7/01713.

This paper is concerned with analysis of two anticancer therapy models focused on sensitivity of therapy outcome with respect to model structure and parameters. Realistic periodic therapies are considered, combining cytotoxic and antiangiogenic agents, defined on a fixed time horizon. Tumor size at the end of therapy and average tumor size calculated over therapy horizon are chosen to represent therapy outcome. Sensitivity analysis has been performed numerically, concentrating on model parameters and structure at one hand, and on treatment protocol parameters, on the other. The results show that sensitivity of the therapy outcome highly depends on the model structure, helping to discern a good model. Moreover, it is possible to use this analysis to find a good protocol in case of heterogeneous tumors.

Citation: Marzena Dolbniak, Malgorzata Kardynska, Jaroslaw Smieja. Sensitivity of combined chemo-and antiangiogenic therapy results in different models describing cancer growth. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 145-160. doi: 10.3934/dcdsb.2018009
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##### References:
Parameter sensitivity ranking calculated with first-order sensitivity functions for the model (2). Sensitivity indices for parameters were normalized to the maximum value
Parameter sensitivity rankings calculated with first-order sensitivity functions for different treatment protocols and the model (2): chemotherapy administered every (a) 2 days, (b) 4 days, (c) 10 days; antiangiogenic agents administered every (a) 10 days, (b) 13 days, (c) 20 days. Sensitivity indices for parameters were normalized to the maximum value to facilitate comparison of the rankings
The heatmap representing sensitivity of the treatment outcome, defined as the number of cancer cells at the end of treatment (right) and two sample solutions that illustrate its meaning (left)
Evaluation of various treatment protocols for the models (a, b) (2) and (c, d) (3); taking into account: low drugs toxicity (a, c) and high drugs toxicity (b, d). Axes are labeled by the number of days between consecutive doses of respective drugs
An example of the model (2) response for treatment protocol selected on the basis of J
Changes in therapy outcome caused by different administration times of the same total doses for the models (a, b) (2) and (c, d) (3). Left and right columns represent indices given by the tumor volume at the end of treatment and mean tumor volume during the treatment, respectively
Changes in therapy outcome for model (2) due to changes in parameter $\beta$: (a, b) increased by 50%; (c, d) decreased by 50%. Left and right columns represent indices given by the tumor volume at the end of treatment and mean tumor volume during the treatment, respectively
Changes in therapy efficiency caused by variability in parameters for model (2). Maximum change is presented. (a, b) $\beta$, (c, d) $\gamma$, (e, f) $\lambda$. Left and right columns represent indices given by the tumor volume at the end of treatment and mean tumor volume during the treatment, respectively
Changes in therapy efficiency caused by variability in parameters for model (2). Mean change is presented. (a, b) $\beta$, (c, d) $\gamma$, (e, f) $\lambda$. Left and right columns represent indices given by the tumor volume at the end of treatment and mean tumor volume during the treatment, respectively
Changes in therapy efficiency caused by variability in parameters for model (3). Max change is presented. (a, b) $\beta$, (c, d) $\gamma$, (e, f) $\lambda$. Left and right columns represent indices given by the tumor volume at the end of treatment and mean tumor volume during the treatment, respectively
Changes in therapy efficiency caused by variability in parameters for model (3). Mean change is presented. (a, b) $\beta$, (c, d) $\gamma$, (e, f) $\lambda$. Left and right columns represent indices given by the tumor volume at the end of treatment and mean tumor volume during the treatment, respectively
Nominal values of parameters
 Par. Values Description Model (2) Model (3) $N_0$ $10^6$ $10^6$ Initial no. of cancer cells $[mm^3]$ $K_0$ $N_0 \cdot 10$ $N_0 \cdot 10$ Initial no. of endothelial cells $[mm^3]$ $\beta$ $1.92 \cdot 10^{-1}$ $1.92 \cdot 10^{-1}$ Tumor growth parameter $[day^{-1}]$ $\gamma$ $1.755 \cdot 10^{1}$ $1.755 \cdot 10^{1}$ Endothelial stimulation parameter $[day^{-1}]$ $\lambda$ $8.73 \cdot 10^{-5}$ $8.73 \cdot 10^{-5}$ Endothelial inhibition parameter $[day^{-1} mm^{-\frac{2}{3}}]$ $\mu$ 0 0 Natural mortality of endothelial cells $[day^{-1}]$ $\psi$ $1.17 \cdot 10^{-2}$ - Cytostatic killing parameter (for cancer cells) $[mg^{-1} m^2]$ $\psi *$ - $1.17 \cdot 10^{-2}$ Maximum value of cytostatic killing parameter (for cancer cells) -$[mg^{-1} m^2]$ $\eta$ $1.17 \cdot 10^{-4}$ $1.17 \cdot 10^{-4}$ Cytostatic killing parameter (for endothelial cells) $[mg^{-1} m^2]$ $\xi$ $1.75 \cdot 10^{-2}$ $5.25 \cdot 10^{-2}$ Anti-angiogenic killing parameter $[mg^{-1} kg]$ $\sigma$ - $3.5\cdot 10^{-1}$ Parametr used in the efficacy curve of the drug $\rho_{opt}$ - 2 $\frac{K}{N}$ ratio, for which the most effective functionality of vasculature is observed
 Par. Values Description Model (2) Model (3) $N_0$ $10^6$ $10^6$ Initial no. of cancer cells $[mm^3]$ $K_0$ $N_0 \cdot 10$ $N_0 \cdot 10$ Initial no. of endothelial cells $[mm^3]$ $\beta$ $1.92 \cdot 10^{-1}$ $1.92 \cdot 10^{-1}$ Tumor growth parameter $[day^{-1}]$ $\gamma$ $1.755 \cdot 10^{1}$ $1.755 \cdot 10^{1}$ Endothelial stimulation parameter $[day^{-1}]$ $\lambda$ $8.73 \cdot 10^{-5}$ $8.73 \cdot 10^{-5}$ Endothelial inhibition parameter $[day^{-1} mm^{-\frac{2}{3}}]$ $\mu$ 0 0 Natural mortality of endothelial cells $[day^{-1}]$ $\psi$ $1.17 \cdot 10^{-2}$ - Cytostatic killing parameter (for cancer cells) $[mg^{-1} m^2]$ $\psi *$ - $1.17 \cdot 10^{-2}$ Maximum value of cytostatic killing parameter (for cancer cells) -$[mg^{-1} m^2]$ $\eta$ $1.17 \cdot 10^{-4}$ $1.17 \cdot 10^{-4}$ Cytostatic killing parameter (for endothelial cells) $[mg^{-1} m^2]$ $\xi$ $1.75 \cdot 10^{-2}$ $5.25 \cdot 10^{-2}$ Anti-angiogenic killing parameter $[mg^{-1} kg]$ $\sigma$ - $3.5\cdot 10^{-1}$ Parametr used in the efficacy curve of the drug $\rho_{opt}$ - 2 $\frac{K}{N}$ ratio, for which the most effective functionality of vasculature is observed
Pharmacokinetics parameters
 Parameter $k_u$ $k_v$ $T_{u1/2}$ $T_{v1/2}$ Model (2) $0.0117$ $0.0175$ 12[h] 3.5[days] Model (3) $0.0117$ $0.0525$ 12[h] 3.5[days] $u$ -chemotherapy $v$ -antiangiogenic therapy
 Parameter $k_u$ $k_v$ $T_{u1/2}$ $T_{v1/2}$ Model (2) $0.0117$ $0.0175$ 12[h] 3.5[days] Model (3) $0.0117$ $0.0525$ 12[h] 3.5[days] $u$ -chemotherapy $v$ -antiangiogenic therapy
Performance index parameters for models (2) and (3)
 Case (ⅰ) (ⅱ) Parameter $r_1$ $r_2$ $r_3$ $r_1$ $r_2$ $r_3$ Value $1$ $0.1 \cdot 10^4$ $0.05 \cdot 10^4$ $1$ $0.5 \cdot 10^5$ $0.25 \cdot 10^5$
 Case (ⅰ) (ⅱ) Parameter $r_1$ $r_2$ $r_3$ $r_1$ $r_2$ $r_3$ Value $1$ $0.1 \cdot 10^4$ $0.05 \cdot 10^4$ $1$ $0.5 \cdot 10^5$ $0.25 \cdot 10^5$
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