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

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.

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.  Google Scholar

[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.  Google Scholar

[3]

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

[4]

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

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[8]

O. ClatzM. SermesantP.-Y. BondiauH. DelingetteS. K. WarfieldG. 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.  Google Scholar

[9]

F. CornelisO. SautP. CumsilleD. LombardiA. IolloJ. 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.   Google Scholar

[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.  Google Scholar

[11]

R. A. GatenbyP. 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.  Google Scholar

[12]

C. I. HenschkeD. F. YankelevitzR. YipA. P. ReevesD. XuJ. P. SmithD. M. LibbyM. 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.  Google Scholar

[13]

C. I. HenschkeR. YipJ. P. SmithA. S. WolfR. M. FloresM. LiangM. M. SalvatoreY. LiuD. 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.  Google Scholar

[14]

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

[15]

Y. KawataN. NikiH. OhmatsuM. KusumotoT. TsuchidaK. EguchiM. 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.  Google Scholar

[16]

E. KonukogluO. ClatzB. H. MenzeB. StieltjesM-A. WeberE. MandonnetH. 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.  Google Scholar

[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.  Google Scholar

[18]

J. S. LowengrubH. B. FeiboesF. JinY.-I. ChuangX. LiP. MacklinS. 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.  Google Scholar

[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.  Google Scholar

[20]

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

[21]

D. P. NaidichA. A. BankierH. MacMahonC. M. Schaefer-ProkopM. PistolesiJ. M. GooP. MacchiariniJ. D. CrapoC. J. HeroldJ. 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.  Google Scholar

[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. Google Scholar

[23]

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

[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.  Google Scholar

[25]

K. R. SwansonC. BridgeJ. 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.  Google Scholar

[26]

C. H. WangJ. K. RockhillM. MrugalaD. L. PeacockA. LaiK. JuseniusJ. M. WardlawT. CloughesyA. M. SpenceR. RockneE. 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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

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

[4]

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

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[8]

O. ClatzM. SermesantP.-Y. BondiauH. DelingetteS. K. WarfieldG. 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.  Google Scholar

[9]

F. CornelisO. SautP. CumsilleD. LombardiA. IolloJ. 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.   Google Scholar

[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.  Google Scholar

[11]

R. A. GatenbyP. 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.  Google Scholar

[12]

C. I. HenschkeD. F. YankelevitzR. YipA. P. ReevesD. XuJ. P. SmithD. M. LibbyM. 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.  Google Scholar

[13]

C. I. HenschkeR. YipJ. P. SmithA. S. WolfR. M. FloresM. LiangM. M. SalvatoreY. LiuD. 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.  Google Scholar

[14]

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

[15]

Y. KawataN. NikiH. OhmatsuM. KusumotoT. TsuchidaK. EguchiM. 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.  Google Scholar

[16]

E. KonukogluO. ClatzB. H. MenzeB. StieltjesM-A. WeberE. MandonnetH. 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.  Google Scholar

[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.  Google Scholar

[18]

J. S. LowengrubH. B. FeiboesF. JinY.-I. ChuangX. LiP. MacklinS. 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.  Google Scholar

[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.  Google Scholar

[20]

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

[21]

D. P. NaidichA. A. BankierH. MacMahonC. M. Schaefer-ProkopM. PistolesiJ. M. GooP. MacchiariniJ. D. CrapoC. J. HeroldJ. 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.  Google Scholar

[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. Google Scholar

[23]

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

[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.  Google Scholar

[25]

K. R. SwansonC. BridgeJ. 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.  Google Scholar

[26]

C. H. WangJ. K. RockhillM. MrugalaD. L. PeacockA. LaiK. JuseniusJ. M. WardlawT. CloughesyA. M. SpenceR. RockneE. 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.  Google Scholar

[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.  Google Scholar

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
Figure 2.  Patient 1: Five serial CT images spanning $826$ days (as detailed in the text) for a biopsy proven benign GGO (arrow)
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 $\beta$, $f_1 = 0.95$, $f_2 = 0.05$, $f_3 = 0.0$
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
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 $\beta$, $f_1 = 0.94$, $f_2 = 0.06$, $f_3 = 0.0$
Figure 6.  Histogram fractions $f_1, f_2, f_3$ of Patient 1 for CT scan data and model output
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
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 $\beta$, $f_1 = 0.90$, $f_2 = 0.10$, $f_3 = 0.0$
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
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 $\beta$, $f_1 = 0.84$, $f_2 = 0.16$, $f_3 = 0.0$
Figure 11.  Histogram fractions $f_1, f_2, f_3$ of Patient 2 for CT scan data and model output
Figure 12.  Patient 3: Four serial CT images spanning $917$ days for atypical cells (arrow) highly suspicious for adenocarcinoma by biopsy
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 $\gamma$, $f_1 = 0.63$, $f_2 = 0.24$, $f_3 = 0.13$
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
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 $\delta$, $f_1 = 0.37$, $f_2 = 0.40$, $f_3 = 0.23$
Figure 16.  Histogram fractions $f_1, f_2, f_3$ of Patient 3 for CT scan data and model output
Figure 17.  Patient 4: Four CT image recordings of a suspicious nodule spanning $471$ days (arrow)
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 $\gamma$, $f_1 = 0.46$, $f_2 = 0.36$, $f_3 = 0.18$
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
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 $\epsilon$, $f_1 = 0.23$, $f_2 = 0.33$, $f_3 = 0.44$
Figure 21.  Histogram fractions $f_1, f_2, f_3$ of Patient 4 for CT scan data and model output
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
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
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$
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$$353$ days
2 $0.002$ $0.006$ $687$ days
3 $0.004$ $0.001$ $380$ days
4 $0.012$ $0.001$ $115$ days
Patient $a$ $b$Doubling time from baseline
1 $0.003$ $0.02$$353$ days
2 $0.002$ $0.006$ $687$ days
3 $0.004$ $0.001$ $380$ days
4 $0.012$ $0.001$ $115$ days
[1]

Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264

[2]

Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020047

[3]

Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317

[4]

Andy Hammerlindl, Jana Rodriguez Hertz, Raúl Ures. Ergodicity and partial hyperbolicity on Seifert manifolds. Journal of Modern Dynamics, 2020, 16: 331-348. doi: 10.3934/jmd.2020012

[5]

Hua Qiu, Zheng-An Yao. The regularized Boussinesq equations with partial dissipations in dimension two. Electronic Research Archive, 2020, 28 (4) : 1375-1393. doi: 10.3934/era.2020073

[6]

Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020050

[7]

Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020107

[8]

Stefan Ruschel, Serhiy Yanchuk. The spectrum of delay differential equations with multiple hierarchical large delays. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 151-175. doi: 10.3934/dcdss.2020321

[9]

Marc Homs-Dones. A generalization of the Babbage functional equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 899-919. doi: 10.3934/dcds.2020303

[10]

Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440

[11]

Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020468

[12]

Peter Poláčik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 413-438. doi: 10.3934/dcds.2020136

[13]

Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345

[14]

Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384

[15]

Anh Tuan Duong, Phuong Le, Nhu Thang Nguyen. Symmetry and nonexistence results for a fractional Choquard equation with weights. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 489-505. doi: 10.3934/dcds.2020265

[16]

Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079

[17]

Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081

[18]

Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364

[19]

Jiaquan Liu, Xiangqing Liu, Zhi-Qiang Wang. Sign-changing solutions for a parameter-dependent quasilinear equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020454

[20]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020448

2018 Impact Factor: 1.313

Article outline

Figures and Tables

[Back to Top]