A stochastic Gompertz model highlighting internal and external therapy function for tumour growth

Moummou, El Kettani; Gutiérrez-Sánchez, Ramón; Melchor, M. C.; Ramos-Ábalos, Eva María

doi:10.1016/j.amc.2014.08.008

Cited by 10 publications

(14 citation statements)

References 20 publications

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“…Consider a Gompertz stochastic model without any control input. It is described as

\begin{matrix} right leftthickmathspace.5em \\ normald X (t) = & [α X (t) - β X (t) log X (t)] normald t \\ + σ X (t) normald W (t) \end{matrix}

Like [20],

α = 0.1

β = 0.3

, and

σ = 0.1

are assigned. Moreover,

X ()(, 0) = 1

is defined for tumour‐cell population.…”

Section: Simulation Resultsmentioning

confidence: 99%

“…In this work, the probabilistic characteristics of the model have been obtained by using its related It

\hat{o}

differential equation followed by the estimation of model parameters through the maximum likelihood method. Like [20, 25, 28], this paper has used the Gompertz model to describe tumour growth

d X ()(, t) = (][, α X ()(, t) - β X ()(, t) log X ()(, t)) d t + σ X ()(, t) d W ()(, t)

In (1),

(}{, X ()(, t), t \geq t_{0} \geq 0)

denotes the tumour‐cell population at time t , and

W ()(, t)

represents the standard Wiener process. The parameters

α, β

, and

σ

are positive constants representing birth rate, death rate, and stochastic oscillation width, respectively.…”

Section: Modelling the Systemmentioning

confidence: 99%

“…The parameters

α, β

, and

σ

are positive constants representing birth rate, death rate, and stochastic oscillation width, respectively. Like [20, 25, 28], the time‐dependent function

u ()(, t)

is used to insert the therapy effect into the model. Thus, (2) is obtained

d X ()(, t) = (][, ()(, α - u ()(, t)) X ()(, t) - β X ()(, t) log X ()(, t)) d t + σ X ()(, t) d W ()(, t)

In (2),

u ()(, t)

models the effect of therapy.…”

Section: Modelling the Systemmentioning

confidence: 99%

See 2 more Smart Citations

Adaptive non‐linear control for cancer therapy through a Fokker–Planck observer

2018

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In recent years, many efforts have been made to present optimal strategies for cancer therapy through the mathematical modelling of tumour-cell population dynamics and optimal control theory. In many cases, therapy effect is included in the drift term of the stochastic Gompertz model. By fitting the model with empirical data, the parameters of therapy function are estimated. The reported research works have not presented any algorithm to determine the optimal parameters of therapy function. In this study, a logarithmic therapy function is entered in the drift term of the Gompertz model. Using the proposed control algorithm, the therapy function parameters are predicted and adaptively adjusted. To control the growth of tumour-cell population, its moments must be manipulated. This study employs the probability density function (PDF) control approach because of its ability to control all the process moments. A Fokker-Planck-based non-linear stochastic observer will be used to determine the PDF of the process. A cost function based on the difference between a predefined desired PDF and PDF of tumour-cell population is defined. Using the proposed algorithm, the therapy function parameters are adjusted in such a manner that the cost function is minimised. The existence of an optimal therapy function is also proved. The numerical results are finally given to demonstrate the effectiveness of the proposed method.

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“…Consider a Gompertz stochastic model without any control input. It is described as

\begin{matrix} right leftthickmathspace.5em \\ normald X (t) = & [α X (t) - β X (t) log X (t)] normald t \\ + σ X (t) normald W (t) \end{matrix}

Like [20],

α = 0.1

β = 0.3

, and

σ = 0.1

are assigned. Moreover,

X ()(, 0) = 1

is defined for tumour‐cell population.…”

Section: Simulation Resultsmentioning

confidence: 99%

“…In this work, the probabilistic characteristics of the model have been obtained by using its related It

\hat{o}

differential equation followed by the estimation of model parameters through the maximum likelihood method. Like [20, 25, 28], this paper has used the Gompertz model to describe tumour growth

d X ()(, t) = (][, α X ()(, t) - β X ()(, t) log X ()(, t)) d t + σ X ()(, t) d W ()(, t)

In (1),

(}{, X ()(, t), t \geq t_{0} \geq 0)

denotes the tumour‐cell population at time t , and

W ()(, t)

represents the standard Wiener process. The parameters

α, β

, and

σ

are positive constants representing birth rate, death rate, and stochastic oscillation width, respectively.…”

Section: Modelling the Systemmentioning

confidence: 99%

“…The parameters

α, β

, and

σ

are positive constants representing birth rate, death rate, and stochastic oscillation width, respectively. Like [20, 25, 28], the time‐dependent function

u ()(, t)

is used to insert the therapy effect into the model. Thus, (2) is obtained

d X ()(, t) = (][, ()(, α - u ()(, t)) X ()(, t) - β X ()(, t) log X ()(, t)) d t + σ X ()(, t) d W ()(, t)

In (2),

u ()(, t)

models the effect of therapy.…”

Section: Modelling the Systemmentioning

confidence: 99%

See 1 more Smart Citation

Adaptive non‐linear control for cancer therapy through a Fokker–Planck observer

2018

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show abstract

“…Subjected to environmental randomness, many authors have studied the Gompertz population models with Brown white noises, see e.g. [1][2][3][4]. Zou, Li and Wang [5] have investigated the optimal harvesting policy for the Gompertz model driven by Levy noises.…”

Section: Introductionmentioning

confidence: 99%

A remark on one non-autonomous stochastic Gompertz model with delay

Zhao

2015

Applied Mathematics and Computation

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“…Specifically, the studies have focused on the following topics: (i) analysis of probabilistic characteristics, as the transition probability densities, the first-passage time density through a particular time dependent threshold (cf. Albano et al [18,19], Ghost and Prajneshu [20]); (ii) discovery of appropriate procedures to estimate the unknown parameters and the time dependent functions characterizing the infinitesimal moments of the diffusion process (see, Gutiérrez et al [21], Moummou et al [22,23], Albano et al [24,25], Román-Román et al [26]); (iii) study of stochastic Gompertz process with jumps that occur at random time and reset the process to a fixed value; after each jump, the process restarts with modified grows parameters (see Spina et al [27], Giorno et al [28]). The Gompertz growth model plays a special role also in nonlinear models of interacting populations (cf., for instance, Goel et al [29]) and in a non-autonomous predator-prey system (cf.…”

Section: Introductionmentioning

confidence: 99%

Restricted Gompertz-Type Diffusion Processes with Periodic Regulation Functions

Giorno

Nobile

2019

Mathematics

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We consider two different time-inhomogeneous diffusion processes useful to model the evolution of a population in a random environment. The first is a Gompertz-type diffusion process with time-dependent growth intensity, carrying capacity and noise intensity, whose conditional median coincides with the deterministic solution. The second is a shifted-restricted Gompertz-type diffusion process with a reflecting condition in zero state and with time-dependent regulation functions. For both processes, we analyze the transient and the asymptotic behavior of the transition probability density functions and their conditional moments. Particular attention is dedicated to the first-passage time, by deriving some closed form for its density through special boundaries. Finally, special cases of periodic regulation functions are discussed.

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A stochastic Gompertz model highlighting internal and external therapy function for tumour growth

Cited by 10 publications

References 20 publications

Adaptive non‐linear control for cancer therapy through a Fokker–Planck observer

Adaptive non‐linear control for cancer therapy through a Fokker–Planck observer

A remark on one non-autonomous stochastic Gompertz model with delay

Restricted Gompertz-Type Diffusion Processes with Periodic Regulation Functions

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