Research Article
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Year 2022, Volume: 71 Issue: 4, 1169 - 1179, 30.12.2022
https://doi.org/10.31801/cfsuasmas.914887

Abstract

References

  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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.
  • 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
  • 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
  • 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.
  • Bertalanffy, L., Problems of organic growth, Nature, 163 (1949), 156-158.
  • Richards, F.A., Flexible growth function for empirical use, Journal of Experimental Botany, 10 (1959), 280-300.
  • 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
  • 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
  • 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
  • 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.
  • Reddingius, J., Gambling for existence: A discussion of some theoretical problems in animal population ecology, Acta Biotheoretica, 20 (1971), 1-208.
  • 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
  • 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
  • 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
  • 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
  • Jazwinski, A.H., Stochastic Processes and Filtering Theory, Academic Press, 1970.
  • Anderson, B.D.O., Moore, J.B., Optimal Filtering, Prentice Hall, 1979.
  • Chui, C.K., Chen, G., Kalman Filtering with Real-time Applications, Springer Verlag, 1991.
  • Ljung, L., Söderström T., Theory and Practice of Recursive Identification, The MIT Press, 1993.
  • Chen, G., Approximate Kalman Filtering, World Scientific, 1993.
  • Grewal, S.M., Andrews, A.P., Kalman Filtering: Theory and Practice, Prentice Hall, 1993.
  • Özbek, L., Kalman Filtresi, Akademisyen Kitabevi, 2017.
  • 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
  • Özbek, L., Aliev, F.A., Comments on adaptive Fading Kalman filter with an application, Automatica, 34(12) (1998), 1663-1664.
  • Efe, M., Özbek, L., Fading Kalman filter for manoeuvring target tracking, Journal of the Turkish Statistical Assocation, 2(3) (1999), 193-206.

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

Year 2022, Volume: 71 Issue: 4, 1169 - 1179, 30.12.2022
https://doi.org/10.31801/cfsuasmas.914887

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.

References

  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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.
  • 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
  • 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
  • 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.
  • Bertalanffy, L., Problems of organic growth, Nature, 163 (1949), 156-158.
  • Richards, F.A., Flexible growth function for empirical use, Journal of Experimental Botany, 10 (1959), 280-300.
  • 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
  • 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
  • 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
  • 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.
  • Reddingius, J., Gambling for existence: A discussion of some theoretical problems in animal population ecology, Acta Biotheoretica, 20 (1971), 1-208.
  • 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
  • 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
  • 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
  • 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
  • Jazwinski, A.H., Stochastic Processes and Filtering Theory, Academic Press, 1970.
  • Anderson, B.D.O., Moore, J.B., Optimal Filtering, Prentice Hall, 1979.
  • Chui, C.K., Chen, G., Kalman Filtering with Real-time Applications, Springer Verlag, 1991.
  • Ljung, L., Söderström T., Theory and Practice of Recursive Identification, The MIT Press, 1993.
  • Chen, G., Approximate Kalman Filtering, World Scientific, 1993.
  • Grewal, S.M., Andrews, A.P., Kalman Filtering: Theory and Practice, Prentice Hall, 1993.
  • Özbek, L., Kalman Filtresi, Akademisyen Kitabevi, 2017.
  • 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
  • Özbek, L., Aliev, F.A., Comments on adaptive Fading Kalman filter with an application, Automatica, 34(12) (1998), 1663-1664.
  • Efe, M., Özbek, L., Fading Kalman filter for manoeuvring target tracking, Journal of the Turkish Statistical Assocation, 2(3) (1999), 193-206.
There are 33 citations in total.

Details

Primary Language English
Subjects Applied Mathematics
Journal Section Research Articles
Authors

Levent Özbek 0000-0003-1018-3114

Publication Date December 30, 2022
Submission Date April 12, 2021
Acceptance Date July 7, 2022
Published in Issue Year 2022 Volume: 71 Issue: 4

Cite

APA Özbek, L. (2022). A study on modeling of rat tumours with the discrete-time Gompertz model. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 71(4), 1169-1179. https://doi.org/10.31801/cfsuasmas.914887
AMA Özbek L. A study on modeling of rat tumours with the discrete-time Gompertz model. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. December 2022;71(4):1169-1179. doi:10.31801/cfsuasmas.914887
Chicago Özbek, Levent. “A Study on Modeling of Rat Tumours With the Discrete-Time Gompertz Model”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 71, no. 4 (December 2022): 1169-79. https://doi.org/10.31801/cfsuasmas.914887.
EndNote Özbek L (December 1, 2022) A study on modeling of rat tumours with the discrete-time Gompertz model. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 71 4 1169–1179.
IEEE L. Özbek, “A study on modeling of rat tumours with the discrete-time Gompertz model”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 71, no. 4, pp. 1169–1179, 2022, doi: 10.31801/cfsuasmas.914887.
ISNAD Özbek, Levent. “A Study on Modeling of Rat Tumours With the Discrete-Time Gompertz Model”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 71/4 (December 2022), 1169-1179. https://doi.org/10.31801/cfsuasmas.914887.
JAMA Özbek L. A study on modeling of rat tumours with the discrete-time Gompertz model. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2022;71:1169–1179.
MLA Özbek, Levent. “A Study on Modeling of Rat Tumours With the Discrete-Time Gompertz Model”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 71, no. 4, 2022, pp. 1169-7, doi:10.31801/cfsuasmas.914887.
Vancouver Özbek L. A study on modeling of rat tumours with the discrete-time Gompertz model. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2022;71(4):1169-7.

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics.

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