TY  - JOUR
T1  - A study on  modeling of rat tumours with the discrete-time Gompertz model
AU  - Özbek, Levent
PY  - 2022
DA  - December
Y2  - 2022
DO  - 10.31801/cfsuasmas.914887
JF  - Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics
JO  - Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat.
PB  - Ankara University
WT  - DergiPark
SN  - 1303-5991
SP  - 1169
EP  - 1179
VL  - 71
IS  - 4
LA  - en
AB  - 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.
KW  - Indocyanine green
KW  - tumor cell
KW  - discrete time Gompertz models
KW  - adaptive Kalman filter
KW  - estimation
CR  - Tofts, P.S., Modeling tracer kinetics in dynamic Gd-DTPA MR imaging, J. Magn. Reson. Imag., 7 (1997), 91-101. doi: 10.1002/jmri.1880070113
CR  - Su, M.Y., Jao, J.C., Nalcioglu, O., Measurement of vascular volume fraction and blood tissue permeability constants with a pharacokinetic model: studies in rat muscle tumors with dynamic Gd-DTPA enhanced MRI, Magn. Reson. Med., 32 (1994), 714-724. doi: 10.1002/mrm.1910320606
CR  - Ntziachristos, V., Yodh, A.G., Schnall, M., Chance, B., Concurrent MRI and diffuse optical tomography of breast after indocyanine green enhancement, Proc. Natl. Acad. Sci. USA, 97 (2000), 2767-2772. doi/10.1073/pnas.040570597
CR  - Botsman, K., Tickle, K., Smith, J.D., A Bayesian formulation of the Kalman filter applied to the estimation of individual pharmacokinetic parameters, Comput. Biomed. Res., 30 (1997), 83-93. doi/10.1006/cbmr.1997.1440
CR  - Özbek, L., Efe, M., An adaptive extended Kalman filter with application to compartment models, Communications In Statistics-Simulation and Computation, 33(1) (2004), 145-158. doi/10.1081/SAC-120028438
CR  - Alacam, B., Yazici, B., Chance, B., Extended Kalman filtering for the modeling and analysis of ICG pharmacokinetics in cancerous tumors using NIR optical methods, IEEE Transactions on Biomedical Engineering, 53(10) (2006), 1861-1871. doi:10.1109/TBME.2006.8817
CR  - Alacam, B., Yazici, B., Intes, X., Nioka, S., Chance, B., Pharmacokinetic-rate images of indocyanine green for breast tumors using near-infrared optical methods, Phys. Med. Biol., 53 (2008), 837-859. doi: 10.1088/0031-9155/53/4/002
CR  - Alacam, B., Yazici, B., Direct reconstruction of pharmacokinetic-rate images of optical fluorophores from NIR measurements, IEEE Transactions on Medical Imaging, 28(9) (2009), 1337-1353. doi: 10.1109/TMI.2009.2015294
CR  - Ozbek, L., Efe, M., Babacan, E.K., Yazihan, N., Online estimation of capillary permeability and contrast agent concentration in rat tumors, Hacettepe Journal of Mathematics and Statistics, 39(2) (2010), 283-293.
CR  - Gottam, O., Naik, N., Gambhirc, S., Parameterized level-set based pharmacokinetic fluorescence optical tomography using the regularized Gauss-Newton filter, Journal of Biomedical Optics, 24(3) (2019), 1-17. doi/10.1117/1.JBO.24.3.031010
CR  - Gottam, O., Naik, N., Gambhirc, S., Pandey, P.K., RBF level-set based fully-nonlinear fluorescence photoacoustic pharmacokinetic tomography, Inverse Problems in Science and Engineering, doi/10.1080/17415977.2021.1982934
CR  - Gompertz, B., On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies, Philosophical Transactions of the Royal Society of London, 115 (1825), 513-583.
CR  - Bertalanffy, L., Problems of organic growth, Nature, 163 (1949), 156-158.
CR  - Richards, F.A., Flexible growth function for empirical use, Journal of Experimental Botany, 10 (1959), 280-300.
CR  - Zwietering, M.H., Jongenburger, I., Rombouts, F.M., Van’t Riet, K., Modeling of the bacterial growth curve, Appl Environ Microbiol, 56(6) (1990), 1875-1881. doi: 10.1128/aem.56.6.1875-1881.1990
CR  - Gerlee, P., The model muddle: in search of tumor growth laws, Cancer Research, 73(8) (2013), 2407-2411. doi/10.1158/0008-5472.CAN-12-4355
CR  - Tjorve, K.M.C., Tjorve, E., The use of Gompertz models in growth analyses, and new Gompertz-model approach: An addition to the Unified-Richards family, PLoS One, 12(6) (2017), e0178691. doi/10.1371/journal.pone.0178691
CR  - Dennis, B., Ponciano, J.M., Subhash, R., Traper, L.M.L., Staples, D.F., Estimating density dependence, process noise and observation erros, Ecological Monographs, 76(3) (2006), 323-341. doi/10.1890/0012-9615.
CR  - Reddingius, J., Gambling for existence: A discussion of some theoretical problems in animal population ecology, Acta Biotheoretica, 20 (1971), 1-208.
CR  - Pollard, E., Lakhani, K.H., Rothery, P., The detection of density-dependence from a series of annual censuses, Ecology, 68 (1987), 2046-2055. doi: 10.2307/1939895
CR  - Dennis, B., Taper, M.L., Density dependence in time series observations of natural populations: estimation and testing, Ecological Monographs, 64 (1994), 205-224. doi/10.2307/2937041
CR  - Rotella, J.J., Ratti, J.T., Reese, K.P., Taper, M.L., Dennis, B., Long-term population analysis of Gray Partridge in eastern Washington, Journal of Wildlife Management, 60 (1996), 817-825. doi/10.2307/3802382
CR  - Cuccia, D.J., Bevilacqua, F., Durkin, A.J., Merritt, S., Tromberg, B.J., Gulsen, G., Yu, H., Wang, J., Nalcioglu, O., In vivo quantification of optical contrast agent dynamics in rat tumors by use of diffuse optical spectroscopy with magnetic resonance imaging coregistration, Appl. Opt., 42 (2003), 2940-2950. doi/10.1364/AO.42.002940
CR  - Jazwinski, A.H., Stochastic Processes and Filtering Theory, Academic Press, 1970.
CR  - Anderson, B.D.O., Moore, J.B., Optimal Filtering, Prentice Hall, 1979.
CR  - Chui, C.K., Chen, G., Kalman Filtering with Real-time Applications, Springer Verlag, 1991.
CR  - Ljung, L., Söderström T., Theory and Practice of Recursive Identification, The MIT Press, 1993.
CR  - Chen, G., Approximate Kalman Filtering, World Scientific, 1993.
CR  - Grewal, S.M., Andrews, A.P., Kalman Filtering: Theory and Practice, Prentice Hall, 1993.
CR  - Özbek, L., Kalman Filtresi, Akademisyen Kitabevi, 2017.
CR  - Kalman, R.E., A new approach to linear filtering and prediction problems, Journal of Basic Engineering, 82 (1960), 35-45. http://dx.doi.org/10.1115/1.3662552
CR  - Özbek, L., Aliev, F.A., Comments on adaptive Fading Kalman filter with an application, Automatica, 34(12) (1998), 1663-1664.
CR  - Efe, M., Özbek, L., Fading Kalman filter for manoeuvring target tracking, Journal of the Turkish Statistical Assocation, 2(3) (1999), 193-206.
UR  - https://doi.org/10.31801/cfsuasmas.914887
L1  - https://dergipark.org.tr/en/download/article-file/1702200
ER  -