American Institute of Mathematical Sciences

Advanced
October  2018, 15(5): 1203-1224. doi: 10.3934/mbe.2018055

A dynamic model of CT scans for quantifying doubling time of ground glass opacities using histogram analysis

József Z. Farkas 1, , Gary T. Smith 2, and  Glenn F. Webb 3,,

1. 

Division of Computing Science and Mathematics, University of Stirling, Stirling, FK9 4LA, UK
2. 

Department of Medical Imaging, Department of Veterans Affairs Hospital, Tennessee Valley Healthcare System, Nashville, Tennessee, 37212, USA
3. 

Department of Mathematics, Vanderbilt University, Nashville, TN 37240, USA

* Corresponding author: Glenn F. Webb

Received  October 09, 2017 Revised  January 20, 2018 Published  May 2018

We quantify a recent five-category CT histogram based classification of ground glass opacities using a dynamic mathematical model for the spatial-temporal evolution of malignant nodules. Our mathematical model takes the form of a spatially structured partial differential equation with a logistic crowding term. We present the results of extensive simulations and validate our model using patient data obtained from clinical CT images from patients with benign and malignant lesions.

Keywords: Medical imaging, CT scan, logistic partial differential equation.
Mathematics Subject Classification: Primary: 92C50, 92C55; Secondary: 35K58.
Citation: József Z. Farkas, Gary T. Smith, Glenn F. Webb. A dynamic model of CT scans for quantifying doubling time of ground glass opacities using histogram analysis. Mathematical Biosciences & Engineering, 2018, 15 (5) : 1203-1224. doi: 10.3934/mbe.2018055
References:
[1]

D. Ambrosi and F. Mollica, On the mechanics of a growing tumor, Int. J. Eng. Sci., 40 (2002), 1297-1316.  doi: 10.1016/S0020-7225(02)00014-9.

[2]

H. Ammari, Mathematical Modeling in Biomedical Imaging 1. Electrical and Ultrasound Tomographies, Anomaly Detection, and Brain Imaging, Springer Science and Business Media, New York, 2009. doi: 10.1007/978-3-642-03444-2.

[3]

A. R. A. Anderson, A. M. Weaver, P. T. Cummings and V. Quaranta, Tumor morphology and phenotypic evolution driven by selective pressure from the microenvironment, Cell, 127 (2006), 905-915. 

[4]

F. R. Balkwill, M. Capasso and T. Hagemann, The tumor microenvironment at a glance, J. Cell Sci., 125 (2012), 5591-5596.  doi: 10.1242/jcs.116392.

[5]

T. M. Buzug, Computed Tomography, from Photon Statistics to Modern Cone-beam CT, Springer-Verlag, Berlin-Heidelberg-New York, 2008.

[6]

Á. Calsina and J. Z. Farkas, Positive steady states of nonlinear evolution equations with finite dimensional nonlinearities, SIAM J. Math. Anal., 46 (2014), 1406-1426.  doi: 10.1137/130931199.

[7]

R. S. Cantrell and C. Cosner, Diffusive logistic equations with indefinite weights: Population models in disrupted environments Ⅱ, SIAM J. Math. Anal., 22 (1991), 1043-1064.  doi: 10.1137/0522068.

[8]

O. Clatz, M. Sermesant, P.-Y. Bondiau, H. Delingette, S. K. Warfield, G. Malandain and N. Ayache, Realistic simulation of the 3-D growth of brain tumors in MR images coupling diffusion with biomechanical deformation, IEEE Trans. Med. Imag., 24 (2005), 1334-1346.  doi: 10.1109/TMI.2005.857217.

[9]

F. Cornelis, O. Saut, P. Cumsille, D. Lombardi, A. Iollo, J. Palussiere and T. Colin, In vivo mathematical modeling of tumor growth from imaging data: Soon to come in the future?, Diag. Inter. Imag., 94 (2013), 593-600. 

[10]

H. Enderling and M. A. J. Chaplain, Mathematical modeling of tumor growth and treatment, Curr. Pharm. Des., 20 (2014), 4934-4940.  doi: 10.2174/1381612819666131125150434.

[11]

R. A. Gatenby, P. K. Maini and E. T. Gawlinski, Analysis of a tumor as an inverse problem provides a novel theoretical framework for understanding tumor biology and therapy, Appl. Math. Lett., 15 (2002), 339-345.  doi: 10.1016/S0893-9659(01)00141-0.

[12]

C. I. Henschke, D. F. Yankelevitz, R. Yip, A. P. Reeves, D. Xu, J. P. Smith, D. M. Libby, M. W. Pasmantier and O. S. Miettinen, Lung cancers diagnosed at annual CT screening: Volume doubling times, Radiology, 263 (2012), 578-583.  doi: 10.1148/radiol.12102489.

[13]

C. I. Henschke, R. Yip, J. P. Smith, A. S. Wolf, R. M. Flores, M. Liang, M. M. Salvatore, Y. Liu, D. M. Xu and D. F. Yankelevitz, CT screening for lung cancer: Part-solid nodules in baseline and annual repeat rounds, Am. J. Roentgenol, 207 (2016), 1176-1184.  doi: 10.2214/AJR.16.16043.

[14]

G. N. Hounsfield, Computed medical imaging, Nobel Lecture, J. Comput. Assist. Tomogr., 4 (1980), 665-674. 

[15]

Y. Kawata, N. Niki, H. Ohmatsu, M. Kusumoto, T. Tsuchida, K. Eguchi, M. Kaneko and N. Moriyama, Quantitative classification based on CT histogram analysis of non-small cell lung cancer: Correlation with histopathological characteristics and recurrence-free survival, Med. Phys., 39 (2012), 988-1000.  doi: 10.1118/1.3679017.

[16]

E. Konukoglu, O. Clatz, B. H. Menze, B. Stieltjes, M-A. Weber, E. Mandonnet, H. Delingette and N. Ayache, Image guided personalization of reaction-diffusion type tumor growth models using modified anisotropic eikonal equations, IEEE Trans. Med. Imag., 29 (2010), 77-95.  doi: 10.1109/TMI.2009.2026413.

[17]

Y. Kuang, J. D. Nagy and S. E. Eikenberry, Introduction to Mathematical Oncology, Mathematical and Computational Biology Series, Taylor & Francis Group, Boca Raton-London-New York, 2016.

[18]

J. S. Lowengrub, H. B. Feiboes, F. Jin, Y.-I. Chuang, X. Li, P. Macklin, S. M. Wise and V. Cristini, Nonlinear modelling of cancer: Bridging the gap between cells and tumors, Nonlinearity, 23 (2010), R1-R91.  doi: 10.1088/0951-7715/23/1/R01.

[19]

R. H. Martin, Nonlinear Operators and Differential Equations in Banach Spaces, Pure and Applied Mathematics, Wiley-Interscience, John Wiley & Sons, New York-London-Sydney, 1976.

[20]

D. Morgensztern, K. Politi and R. S. Herbst, EGFR Mutations in non-small-cell lung cancer: Find, divide, and conquer, JAMA Oncol., 1 (2015), 146-148. 

[21]

D. P. Naidich, A. A. Bankier, H. MacMahon, C. M. Schaefer-Prokop, M. Pistolesi, J. M. Goo, P. Macchiarini, J. D. Crapo, C. J. Herold, J. H. Austin and W. D. Travis, Recommendations for the management of subsolid pulmonary nodules detected at CT: A statement from the Fleischner Society, Radiology, 266 (2013), 304-317.  doi: 10.1148/radiol.12120628.

[22]

National lung screening trial research team, Reduced lung-cancer mortality with low-dose computed tomographic screening, N. Engl. J. Med., 365 (2011), 395–409.

[23]

J. Prüss, Equilibrium solutions of age-specific population dynamics of several species, J. Math. Biol., 11 (1981), 65-84.  doi: 10.1007/BF00275825.

[24]

R. Rockne, E. C. Alvord, Jr., M. Szeto, S. Gu, G. Chakraborty and K. R. Swanson, Modeling diffusely invading brain tumors: An individualized approach to quantifying glioma evolution and response to therapy, in Selected Topics in Cancer Modeling: Genesis, Evolution, Immune Competition and Therapy, Modeling and Simulation in Science, Engineering and Technology Series, Birkh¨auserBoston, Boston, MA, 2008,207–221.

[25]

K. R. Swanson, C. Bridge, J. D. Murray and E. C. Alvord Jr., Virtual and real brain tumors: Using mathematical modeling to quantify glioma growth and invasion, J. Neurol. Sci., 216 (2003), 1-10.  doi: 10.1016/j.jns.2003.06.001.

[26]

C. H. Wang, J. K. Rockhill, M. Mrugala, D. L. Peacock, A. Lai, K. Jusenius, J. M. Wardlaw, T. Cloughesy, A. M. Spence, R. Rockne, E. C. Alvord Jr. and K. R. Swanson, Prognostic significance of growth kinetics in newly diagnosed glioblastomas revealed by combining serial imaging with a novel biomathematical model, Can. Res., 69 (2009), 9133-9140.  doi: 10.1158/0008-5472.CAN-08-3863.

[27]

A. Y. Yakovlev, A. V. Zorin and B. I. Grudinko, Computer Simulation in Cell Radiobiology, Lecture Notes in Biomathematics, 74, Springer-Verlag, Berlin-Heidelberg-New York, 1988. doi: 10.1007/978-3-642-51716-7.

show all references

References:
[1]

D. Ambrosi and F. Mollica, On the mechanics of a growing tumor, Int. J. Eng. Sci., 40 (2002), 1297-1316.  doi: 10.1016/S0020-7225(02)00014-9.

[2]

H. Ammari, Mathematical Modeling in Biomedical Imaging 1. Electrical and Ultrasound Tomographies, Anomaly Detection, and Brain Imaging, Springer Science and Business Media, New York, 2009. doi: 10.1007/978-3-642-03444-2.

[3]

A. R. A. Anderson, A. M. Weaver, P. T. Cummings and V. Quaranta, Tumor morphology and phenotypic evolution driven by selective pressure from the microenvironment, Cell, 127 (2006), 905-915. 

[4]

F. R. Balkwill, M. Capasso and T. Hagemann, The tumor microenvironment at a glance, J. Cell Sci., 125 (2012), 5591-5596.  doi: 10.1242/jcs.116392.

[5]

T. M. Buzug, Computed Tomography, from Photon Statistics to Modern Cone-beam CT, Springer-Verlag, Berlin-Heidelberg-New York, 2008.

[6]

Á. Calsina and J. Z. Farkas, Positive steady states of nonlinear evolution equations with finite dimensional nonlinearities, SIAM J. Math. Anal., 46 (2014), 1406-1426.  doi: 10.1137/130931199.

[7]

R. S. Cantrell and C. Cosner, Diffusive logistic equations with indefinite weights: Population models in disrupted environments Ⅱ, SIAM J. Math. Anal., 22 (1991), 1043-1064.  doi: 10.1137/0522068.

[8]

O. Clatz, M. Sermesant, P.-Y. Bondiau, H. Delingette, S. K. Warfield, G. Malandain and N. Ayache, Realistic simulation of the 3-D growth of brain tumors in MR images coupling diffusion with biomechanical deformation, IEEE Trans. Med. Imag., 24 (2005), 1334-1346.  doi: 10.1109/TMI.2005.857217.

[9]

F. Cornelis, O. Saut, P. Cumsille, D. Lombardi, A. Iollo, J. Palussiere and T. Colin, In vivo mathematical modeling of tumor growth from imaging data: Soon to come in the future?, Diag. Inter. Imag., 94 (2013), 593-600. 

[10]

H. Enderling and M. A. J. Chaplain, Mathematical modeling of tumor growth and treatment, Curr. Pharm. Des., 20 (2014), 4934-4940.  doi: 10.2174/1381612819666131125150434.

[11]

R. A. Gatenby, P. K. Maini and E. T. Gawlinski, Analysis of a tumor as an inverse problem provides a novel theoretical framework for understanding tumor biology and therapy, Appl. Math. Lett., 15 (2002), 339-345.  doi: 10.1016/S0893-9659(01)00141-0.

[12]

C. I. Henschke, D. F. Yankelevitz, R. Yip, A. P. Reeves, D. Xu, J. P. Smith, D. M. Libby, M. W. Pasmantier and O. S. Miettinen, Lung cancers diagnosed at annual CT screening: Volume doubling times, Radiology, 263 (2012), 578-583.  doi: 10.1148/radiol.12102489.

[13]

C. I. Henschke, R. Yip, J. P. Smith, A. S. Wolf, R. M. Flores, M. Liang, M. M. Salvatore, Y. Liu, D. M. Xu and D. F. Yankelevitz, CT screening for lung cancer: Part-solid nodules in baseline and annual repeat rounds, Am. J. Roentgenol, 207 (2016), 1176-1184.  doi: 10.2214/AJR.16.16043.

[14]

G. N. Hounsfield, Computed medical imaging, Nobel Lecture, J. Comput. Assist. Tomogr., 4 (1980), 665-674. 

[15]

Y. Kawata, N. Niki, H. Ohmatsu, M. Kusumoto, T. Tsuchida, K. Eguchi, M. Kaneko and N. Moriyama, Quantitative classification based on CT histogram analysis of non-small cell lung cancer: Correlation with histopathological characteristics and recurrence-free survival, Med. Phys., 39 (2012), 988-1000.  doi: 10.1118/1.3679017.

[16]

E. Konukoglu, O. Clatz, B. H. Menze, B. Stieltjes, M-A. Weber, E. Mandonnet, H. Delingette and N. Ayache, Image guided personalization of reaction-diffusion type tumor growth models using modified anisotropic eikonal equations, IEEE Trans. Med. Imag., 29 (2010), 77-95.  doi: 10.1109/TMI.2009.2026413.

[17]

Y. Kuang, J. D. Nagy and S. E. Eikenberry, Introduction to Mathematical Oncology, Mathematical and Computational Biology Series, Taylor & Francis Group, Boca Raton-London-New York, 2016.

[18]

J. S. Lowengrub, H. B. Feiboes, F. Jin, Y.-I. Chuang, X. Li, P. Macklin, S. M. Wise and V. Cristini, Nonlinear modelling of cancer: Bridging the gap between cells and tumors, Nonlinearity, 23 (2010), R1-R91.  doi: 10.1088/0951-7715/23/1/R01.

[19]

R. H. Martin, Nonlinear Operators and Differential Equations in Banach Spaces, Pure and Applied Mathematics, Wiley-Interscience, John Wiley & Sons, New York-London-Sydney, 1976.

[20]

D. Morgensztern, K. Politi and R. S. Herbst, EGFR Mutations in non-small-cell lung cancer: Find, divide, and conquer, JAMA Oncol., 1 (2015), 146-148. 

[21]

D. P. Naidich, A. A. Bankier, H. MacMahon, C. M. Schaefer-Prokop, M. Pistolesi, J. M. Goo, P. Macchiarini, J. D. Crapo, C. J. Herold, J. H. Austin and W. D. Travis, Recommendations for the management of subsolid pulmonary nodules detected at CT: A statement from the Fleischner Society, Radiology, 266 (2013), 304-317.  doi: 10.1148/radiol.12120628.

[22]

National lung screening trial research team, Reduced lung-cancer mortality with low-dose computed tomographic screening, N. Engl. J. Med., 365 (2011), 395–409.

[23]

J. Prüss, Equilibrium solutions of age-specific population dynamics of several species, J. Math. Biol., 11 (1981), 65-84.  doi: 10.1007/BF00275825.

[24]

R. Rockne, E. C. Alvord, Jr., M. Szeto, S. Gu, G. Chakraborty and K. R. Swanson, Modeling diffusely invading brain tumors: An individualized approach to quantifying glioma evolution and response to therapy, in Selected Topics in Cancer Modeling: Genesis, Evolution, Immune Competition and Therapy, Modeling and Simulation in Science, Engineering and Technology Series, Birkh¨auserBoston, Boston, MA, 2008,207–221.

[25]

K. R. Swanson, C. Bridge, J. D. Murray and E. C. Alvord Jr., Virtual and real brain tumors: Using mathematical modeling to quantify glioma growth and invasion, J. Neurol. Sci., 216 (2003), 1-10.  doi: 10.1016/j.jns.2003.06.001.

[26]

C. H. Wang, J. K. Rockhill, M. Mrugala, D. L. Peacock, A. Lai, K. Jusenius, J. M. Wardlaw, T. Cloughesy, A. M. Spence, R. Rockne, E. C. Alvord Jr. and K. R. Swanson, Prognostic significance of growth kinetics in newly diagnosed glioblastomas revealed by combining serial imaging with a novel biomathematical model, Can. Res., 69 (2009), 9133-9140.  doi: 10.1158/0008-5472.CAN-08-3863.

[27]

A. Y. Yakovlev, A. V. Zorin and B. I. Grudinko, Computer Simulation in Cell Radiobiology, Lecture Notes in Biomathematics, 74, Springer-Verlag, Berlin-Heidelberg-New York, 1988. doi: 10.1007/978-3-642-51716-7.

Figure 1.  Photomicrograph showing a small lung area at the microscopic level. Lighter pink areas are representing the thickened alveolar walls and the darker purple ones are cancer cells lining up along the walls. As the tumor grows further, it will fill the white air spaces between the alveolar walls, thereby shifting the density histogram closer to water
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Figure 2.  Patient 1: Five serial CT images spanning $826$ days (as detailed in the text) for a biopsy proven benign GGO (arrow)
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Figure 3.  CT scan histograms of Patient 1. A: 9/21/12, type $\beta$, $f_1 = 0.94$, $f_2 = 0.06$, $f_3 = 0.0$. B: 12/17/12, type $\beta$, $f_1 = 0.92$, $f_2 = 0.08$, $f_3 = 0.0$. C: 5/7/13, type $\beta$, $f_1 = 0.94$, $f_2 = 0.06$, $f_3 = 0.0$. D: 6/26/14, type $\beta$, $f_1 = 0.92$, $f_2 = 0.08$, $f_3 = 0.0$. E: 8/14/14, type $\beta$, $f_1 = 0.95$, $f_2 = 0.05$, $f_3 = 0.0$
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Figure 4.  Patient 1 model simulation. A: the initial tumor spatial density $u(0, x, y)$. B: the initial spatial density of the tumor plus the background spatial density $u(0, x, y) + u_b(x, y)$. C: the tumor spatial density $u(692, x, y)$ at time $t = 692$ days
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Figure 5.  Model simulation histograms of Patient 1. A: 9/21/12, type $\beta$, $f_1 = 0.97$, $f_2 = 0.03$, $f_3 = 0.0$. B: 12/17/12, type $\beta$, $f_1 = 0.98$, $f_2 = 0.02$, $f_3 = 0.0$. C: 5/7/13, type $\beta$, $f_1 = 0.98$, $f_2 = 0.02$, $f_3 = 0.0$. D: 6/26/14, type $\beta$, $f_1 = 0.95$, $f_2 = 0.05$, $f_3 = 0.0$. E: 8/14/14, type $\beta$, $f_1 = 0.94$, $f_2 = 0.06$, $f_3 = 0.0$
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Figure 6.  Histogram fractions $f_1, f_2, f_3$ of Patient 1 for CT scan data and model output
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Figure 7.  Patient 2: Six serial CT images over a span of $932$ days for a stable GGO (arrow), clinically considered benign due to lack of change in size or density
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Figure 8.  CT scan histograms of Patient 2. A: 5/22/12, type $\beta$, $f_1 = 0.94$, $f_2 = 0.06$, $f_3 = 0.0$. B: 9/6/12, type $\beta$, $f_1 = 0.91$, $f_2 = 0.09$, $f_3 = 0.0$. C: 12/6/12, type $\beta$, $f_1 = 0.93$, $f_2 = 0.07$, $f_3 = 0.0$. D: 6/12/13, type $\beta$, $f_1 = 0.92$, $f_2 = 0.08$, $f_3 = 0.0$. E: 12/11/13, type $\beta$, $f_1 = 0.89$, $f_2 = 0.11$, $f_3 = 0.0$. F: 12/10/14, type $\beta$, $f_1 = 0.90$, $f_2 = 0.10$, $f_3 = 0.0$
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Figure 9.  Patient 2 model simulation: A: The initial spatial density of the tumor plus the background spatial density $u(0, x, y) + u_b(x, y)$. B: The initial tumor spatial density $u(0, x, y)$. C: The tumor spatial density $u(932, x, y)$ at time $t = 932$ days
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Figure 10.  Model simulation histograms of Patient 2. A: 5/22/12, type $\beta$, $f_1 = 0.92$, $f_2 = 0.08$, $f_3 = 0.0$. B: 9/6/12, type $\beta$, $f_1 = 0.91$, $f_2 = 0.09$, $f_3 = 0.0$. C: 12/6/12, type $\beta$, $f_1 = 0.91$, $f_2 = 0.09$, $f_3 = 0.0$. D: 6/12/13, type $\beta$, $f_1 = 0.89$, $f_2 = 0.11$, $f_3 = 0.0$. E: 12/11/13, type $\beta$, $f_1 = 0.87$, $f_2 = 0.13$, $f_3 = 0.0$. F: 12/10/14, type $\beta$, $f_1 = 0.84$, $f_2 = 0.16$, $f_3 = 0.0$
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Figure 11.  Histogram fractions $f_1, f_2, f_3$ of Patient 2 for CT scan data and model output
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Figure 12.  Patient 3: Four serial CT images spanning $917$ days for atypical cells (arrow) highly suspicious for adenocarcinoma by biopsy
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Figure 13.  CT scan histograms of Patient 3. A: 10/20/10, type $\beta$, $f_1 = 0.74$, $f_2 = 0.23$, $f_3 = 0.03$. B: 5/16/11, type $\beta$, $f_1 = 0.69$, $f_2 = 0.24$, $f_3 = 0.07$. C: 1/23/13, type $\gamma$, $f_1 = 0.69$, $f_2 = 0.22$, $f_3 = 0.09$. D: 4/24/13, type $\gamma$, $f_1 = 0.63$, $f_2 = 0.24$, $f_3 = 0.13$
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Figure 14.  Patient 3 model simulation: A: The initial spatial density of the tumor plus the background spatial density $u(0, x, y) + u_b(x, y)$. B: The initial tumor spatial density $u(0, x, y)$. C: The tumor spatial density $u(917, x, y)$ at time $t = 917$ days
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Figure 15.  Model simulation histograms of Patient 3. A: 10/20/10, type $\beta$, $f_1 = 0.78$, $f_2 = 0.22$, $f_3 = 0.0$. B: 5/16/12, type $\beta$, $f_1 = 0.62$, $f_2 = 0.38$, $f_3 = 0.0$. C: 1/23/13, type $\gamma$, $f_1 = 0.55$, $f_2 = 0.42$, $f_3 = 0.03$. D: 4/24/13, type $\gamma$, $f_1 = 0.52$, $f_2 = 0.43$, $f_3 = 0.05$. E: 4/24/13 + 300 days, type $\delta$, $f_1 = 0.44$, $f_2 = 0.43$, $f_3 = 0.13$. F: 4/24/13 + 600 days, type $\delta$, $f_1 = 0.37$, $f_2 = 0.40$, $f_3 = 0.23$
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Figure 16.  Histogram fractions $f_1, f_2, f_3$ of Patient 3 for CT scan data and model output
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Figure 17.  Patient 4: Four CT image recordings of a suspicious nodule spanning $471$ days (arrow)
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Figure 18.  CT scan histograms of Patient 4. A: 10/2/13, type $\beta$, $f_1 = 0.72$, $f_2 = 0.24$, $f_3 = 0.04$. B: 5/28/14, type $\gamma$, $f_1 = 0.54$, $f_2 = 0.35$, $f_3 = 0.11$. C: 11/28/14, type $\gamma$, $f_1 = 0.56$, $f_2 = 0.33$, $f_3 = 0.11$. D:1/15/15, type $\gamma$, $f_1 = 0.46$, $f_2 = 0.36$, $f_3 = 0.18$
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Figure 19.  Patient 4 model simulation: A: The initial spatial density of the tumor plus the background spatial density $u(0, x, y) + u_b(x, y)$. B: The initial tumor spatial density $u(0, x, y)$. C: The tumor spatial density $u(471, x, y)$ at time $t = 471$ days
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Figure 20.  Model simulation histograms of Patient 4. A: 9/21/12, type $\beta$, $f_1 = .90$, $f_2 = 0.10$, $f_3 = 0.0$. B: 12/17/12, type $\gamma$, $f_1 = 0.66$, $f_2 = 0.34$, $f_3 = 0.0$. C: 5/7/13, type $\gamma$, $f_1 = 0.47$, $f_2 = 0.40$, $f_3 = 0.13$. D: 6/26/14, type $\gamma$, $f_1 = 0.43$, $f_2 = 0.39$, $f_3 = 0.17$. E: 8/14/14, type $\delta$, $f_1 = 0.33$, $f_2 = 0.37$, $f_3 = 0.30$. F: 10/20/15, type $\epsilon$, $f_1 = 0.23$, $f_2 = 0.33$, $f_3 = 0.44$
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Figure 21.  Histogram fractions $f_1, f_2, f_3$ of Patient 4 for CT scan data and model output
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Figure 22.  Total tumor mass growth curves from model simulations. Black dots are time points corresponding to CT scan data for patients 1, 2, 3, 4. Red dots are for two additional time points for Patients 3 and 4. The values are scaled to 1.0 at time 0
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Table 1.  The five CT scan histogram categories
TypeDescription
$\alpha$high peak at low $HU$ values and no peak at high $HU$ values
$\beta$medium peak at low $HU$ values and no peak at high $HU$ values
$\gamma$low peak at low $HU$ values and lower peak at high $HU$ values
$\delta$low peak at low $HU$ values and higher peak at high $HU$ values
$\epsilon$low peak at low $HU$ values and very high peak at high $HU$ values
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TypeDescription
$\alpha$high peak at low $HU$ values and no peak at high $HU$ values
$\beta$medium peak at low $HU$ values and no peak at high $HU$ values
$\gamma$low peak at low $HU$ values and lower peak at high $HU$ values
$\delta$low peak at low $HU$ values and higher peak at high $HU$ values
$\epsilon$low peak at low $HU$ values and very high peak at high $HU$ values
Table 2.  The three output fractions
fractionCT scan histogram output at time $t$model output $u(t, \mathbf{x})$
$f_1$ $ < -600$ $HU$ $ < 500$
$f_2$between $-600$ $HU$ and $-100$ $HU$between $500$ and $1000$
$f_3$ $ > -100$ $HU$ $ > 1000$
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fractionCT scan histogram output at time $t$model output $u(t, \mathbf{x})$
$f_1$ $ < -600$ $HU$ $ < 500$
$f_2$between $-600$ $HU$ and $-100$ $HU$between $500$ and $1000$
$f_3$ $ > -100$ $HU$ $ > 1000$
Table 3.  Model parameters and simulation doubling times. Units of $a$ are $1/$ time units and units of $b$ are area units$^2$/time units
Patient $a$ $b$Doubling time from baseline
1 $0.003$ $0.02$$353$ days
2 $0.002$ $0.006$ $687$ days
3 $0.004$ $0.001$ $380$ days
4 $0.012$ $0.001$ $115$ days
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Patient $a$ $b$Doubling time from baseline
1 $0.003$ $0.02$$353$ days
2 $0.002$ $0.006$ $687$ days
3 $0.004$ $0.001$ $380$ days
4 $0.012$ $0.001$ $115$ days
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