A study on modeling of rat tumors with the discrete-time Gompertz model

Levent Özbek
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Abstract

Cancer formation is one of the pathologies whose frequency has increased in the recent years. In the literature, the compartment models, which are non-linear, are used for such problems. In nonlinear compartment models, nonlinear state space models and the extended Kalman filter (EKF) are used to estimate the parameter and the state vector. This paper presents a discrete-time Gompertz model (DTGM) for the transfer of optical contrast agent, namely indocyanine green (ICG), in the presence of tumors between the plasma and extracellular extravascular space (EES) compartments. The DTGM, which is proposed for ICG and the estimation of ICG densities used in the vascular invasion of tumor cells of the compartments and in the measurement of migration from the intravascular area to the tissues, is obtained from the experimental data of the study. The ICG values are estimated online (recursive) using the DTGM and the adaptive Kalman filter (AKF) based on the experimental data. By employing the data, the results show that the DTGM in conjunction with the AKF provides a good analysis tool for modeling the ICG in terms of mean square error (MSE), mean absolute percentage error (MAPE), and . When the results obtained from the compartment model used in the reference [9] are compared with the results obtained with the DTGM, the DTGM gives better results in terms of MSE, MAPE and $R^2$ criteria. The DTGM and the AKF compartment model require less numerical processing when compared to the EKF, which indicates that DTGM is a less complicated model. In the literature, EKF is used for such problems.
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用离散时间Gompertz模型建立大鼠肿瘤模型的研究
癌症形成是近年来发病率增加的病理之一。在文献中,非线性的隔室模型被用于此类问题。在非线性隔室模型中,使用非线性状态空间模型和扩展卡尔曼滤波器(EKF)来估计参数和状态向量。本文提出了一种离散时间Gompertz模型(DTGM),用于在血浆和细胞外血管外间隙(EES)隔间之间存在肿瘤的情况下转移光学造影剂,即吲哚菁绿(ICG)。DTGM是为ICG和ICG密度估计而提出的,用于隔室肿瘤细胞的血管侵袭和测量从血管内区域到组织的迁移,它是从研究的实验数据中获得的。基于实验数据,使用DTGM和自适应卡尔曼滤波器(AKF)在线(递归)估计ICG值。通过使用这些数据,结果表明,DTGM与AKF相结合,为ICG的均方误差(MSE)、平均绝对百分比误差(MAPE)和建模提供了一个很好的分析工具。当将参考文献[9]中使用的隔间模型获得的结果与DTGM获得的结果进行比较时,DTGM在MSE、MAPE和$R^2$标准方面给出了更好的结果。与EKF相比,DTGM和AKF隔间模型需要较少的数值处理,这表明DTGM是一个不那么复杂的模型。在文献中,EKF被用于此类问题。
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