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Sensitivity and uncertainty analysis of a simplified Kirschner-Panetta model for immunotherapy of tumor-immune interaction
- Onyango Lawrence Omondi^{1, 2}Email author,
- Chuncheng Wang^{1} and
- Xiaoping Xue^{1}
https://doi.org/10.1186/s13662-015-0550-3
© Omondi et al. 2015
- Received: 11 February 2015
- Accepted: 18 June 2015
- Published: 11 July 2015
Abstract
In this study, we have simplified the Kirschner-Panetta model on the interaction of tumor cells and effector cells by considering linear growth term as opposed to logistic growth term used by Kirschner and Panetta. We have done a comprehensive mathematical analysis and established the existence of positive equilibrium. In addition, a fixed point bifurcation is investigated using the rate of spread of tumor as a varying parameter, suggesting that backward bifurcation can occur under reasonable choice of parameters. However, bistable dynamics is unlikely to happen in this case, which implies that the strategies that could reduce the rate of spread of tumor are the most influential to cancer treatment. Through mathematical deduction and numerical simulation, an elaborate uncertainty and sensitivity analysis of the rate of spread of tumor \(R_{s}\) is performed. The distribution of \(R_{s}\) is derived, and the sensitivity of the magnitude of \(R_{s}\) to the uncertainty in estimating values of input parameters is assessed. The results indicate that the external source of effector cells and its death rate are influential in the rate of spread of tumor.
Keywords
- tumor-immune system
- backward bifurcation
- uncertainty and sensitivity analysis
1 Introduction
Cancer is a term used to describe a disease in which abnormal cells divide without control and are able to invade other tissues. Cancer cells can spread to other parts of the body through blood and lymph systems. The main categories of cancer include carcinoma, sarcoma, leukemia, lymphoma, and myeloma [1–3]. Cancer is known as the leading cause of deaths in the world, and research has shown that new cases of cancer are on the rise. For instance, in 2008, there were estimated 12,667,500 new cases of cancer worldwide, with Eastern Asia having most of the cases (3,720,000). It has also been observed that one eighth of deaths in the world is due to cancer. In fact, it causes more deaths than AIDS, tuberculosis, and malaria combined [4–7].
2 Motivation
Kirschner and Panetta (KP) developed a cancer immunotherapy of tumor - immune systems model with logistic growth rate, and supported some of their findings through mathematical analysis and numerical simulations [11]. However, their formulations disregarded the rate of spread of tumor and the existence of a positive endemic equilibrium. Also, they did not identify the most influential parameters in their model. However, there is a need to address these two issues especially if the model is to be applied in a clinical setup. Rihan et al. tried to analyze the KP model (see [12]). They made certain assumptions, which reduced the KP model from a three-dimensional system of nonlinear ODEs to a two-dimensional one. However, explanation for the disappearance of the third variable is not provided. In addition, we observed that the type of non-dimensionalization pursued, which led to the elimination of the denominator variables and even transformation of some of the variables to parameters, fundamentally alters the KP model [12]. Moore studied the KP model and raised a concern that it inadequately represents the real situations [13].
The authors in [11] analyzed the KP model for tumors with logistic growth rate. However, testicular carcinomas, pediatric tumors, and some mesenchymal tumors are examples of rapidly proliferating cell populations, for which the tumor volume doubling time (TVDT) can be counted in days, and they exhibit a linear growth rate [14]. In this work, therefore, we have considered a later case by setting constant growth rate term, \(r_{2}(T)\), in the KP model. We have mathematically analyzed the KP model more comprehensively and proved that, indeed, a positive endemic equilibrium exists. Conditions for the occurrence of bifurcation are set out, and the type of anticipated bifurcation is specified. In addition, we have performed numerical simulations to confirm and demonstrate the validity of various assumptions and conclusions made during the analysis. The paper is organized as follows. In Section 1, we have provided some background information on cancer infection; in Section 2, we have given a brief discussion on the previous works about the KP model and provided some explanations regarding the necessity to analyze the model with a linear growth rate; in Section 3, we have presented the KP model and mathematically analyzed it; Section 4 presents the numerical simulations and discussions of the results to ground the assumptions and conclusions. The work is concluded in Section 5.
3 The model
Theorem 1
Proof
It is easy to see that \(\lambda_{1}<0\), therefore, the existence of boundary equilibrium and its stability is implied directly from (4) and (6), and the existence of interior equilibrium can be shown by Descartes’s rule of signs [21, 22]. In fact, if \(R_{s}>1\), then \(A_{0}<0\). Note that we assume \(A_{3}>0\); therefore, there are four possible sign changes of the coefficients of (9), that is, \((+,+,+,-)\), \((+,+,-,-)\), \((+,-,-,-)\), and \((+,-,+,-)\), and the last possibility can be ruled out by (10). By Descartes’ theorem, the number of positive roots of (9) is either equal to the sign changes between consecutive coefficients, or is less than them by an even number, which completes the proof. □
Corollary 2
Assume that \(\mu_{2}(s_{2}+\mu_{3})-p_{1}s_{2}>0\) and \(g_{3}=1\). When \(B_{2}>0\), system (3) has a unique positive equilibrium if \(R_{s}>1\), and has two positive equilibria if \(R_{s}<1\), \(B_{1}<0\), and \(\Delta:=B_{1}^{2}-4B_{0}B_{2}>0\). When \(B_{2}<0\), system (3) has a unique positive equilibrium if \(R_{s}<1\) and has two positive equilibria if \(R_{s}>1\), \(B_{1}>0\), and \(\Delta>0\).
However, we observe from formulation (7) of the rate of spread of tumor \(R_{s}\) that the dynamics of model (3) without the treatments \(s_{1}\) and \(s_{2}\) leads to a blow-up since \(R_{s}=+\infty\). This implies that the treatments are very critical in controlling the spread of malignant cancer.
4 Numerical simulations
Parameter values used for model in ( 3 )
Parameter | c | \(\boldsymbol{\mu _{2}}\) | \(\boldsymbol{p_{1}}\) | \(\boldsymbol{r_{2}}\) | a | \(\boldsymbol{p_{2}}\) | \(\boldsymbol{g_{3}}\) | \(\boldsymbol{\mu _{3}}\) | \(\boldsymbol{s_{2}}\) |
---|---|---|---|---|---|---|---|---|---|
Value | 0.025 | 0.16 | 0.69 | 1 | 4.5 | 0.2 | 1 | 7 | 0.05 |
Descriptive statistics from the uncertainty analysis
Minimum | Maximum | Mean | Variance | |
---|---|---|---|---|
\(R_{s}\) | 0.2408 | 3.9802 | 0.8913 | 0.2150 |
Partial rank correlation coefficients (PRCCs) for the basic reproduction number and each input parameter variable
Parameter | c | \(\boldsymbol {s_{1}}\) | \(\boldsymbol{\mu_{2}}\) | \(\boldsymbol{s_{2}}\) | \(\boldsymbol{\mu_{3}}\) | \(\boldsymbol{p_{1}}\) | \(\boldsymbol{p_{2}}\) |
---|---|---|---|---|---|---|---|
PRCCs | 0.048 | −0.982 | 0.872 | −0.127 | 0.091 | −0.038 | −0.072 |
p-value | 0.127 | 0 | 0 | 5.5 × 10^{−5} | 0.004 | 0.229 | 0.022 |
5 Conclusion
In this study, a mathematical analysis of the tumor-immune interaction model proposed in [11] has been simplified and analyzed. Proof of the existence of positive equilibrium has been established. Additionally, the rate of spread of tumor \(R_{s}\) was derived, and explanations regarding the behavior of \(R_{s}\) were provided. The fixed point bifurcation was investigated using \(R_{s}\) as free parameters. The results reveal that backward bifurcation could occur for a reasonable set of parameter values. However, the numerical results indicate that the equilibria at the bifurcated branch are all unstable, meaning that the bistable dynamics will never happen for the model. Unlike most epidemic models in the literature, the numerical simulations in this work show that when \(R_{s}>1\) the solution tends to infinity as time increases. Therefore, the control strategies (for instance, at least a certain amount of external effector \(s_{1}\)) that will reduce \(R_{s}\) to within 1 are of great importance for the possible tumor cell elimination.
Due to the difficulty of measure, or collection, of empirical data on some parameters in the model, we also conducted uncertainty analysis and sensitivity analysis based on Latin hypercube sampling in estimating \(R_{s}\). The distribution for \(R_{s}\) gives a wide range of estimates due to uncertainty in estimating values of seven selected parameters. Even though the upper bound of \(R_{s}\) could run up to 4, the 95% confidence interval of the distribution of \(R_{s}\) is \([0.2512, 2.9341]\). Hence, tumor cells are likely to be eliminated under proper treatment strategies, based on the interpretation of the model, because this interval contains the number 1. The results of sensitivity analysis imply that an increase in external effector would have lead to less severe growth of tumor size, because a high concentrate effector cells would have resulted in fewer number of tumor cells. A decrease in the per capita natural death rate of effector cells would also have led to fewer number of tumor cells. These results are obvious from the expression of (7), and our sensitivity analysis extends these conclusions quantitatively.
Declarations
Acknowledgements
The authors of this work are grateful to the journal editors and the anonymous reviewers for their comments and recommendations, which have greatly improved our manuscript and made it more suitable for readers of the journal. Additionally, we would like to appreciate B. Maiseli and E. Ogada for their contributions in editing the manuscript.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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