# American Institute of Mathematical Sciences

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
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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
Patient 1: Five serial CT images spanning $826$ days (as detailed in the text) for a biopsy proven benign GGO (arrow)
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 $\beta$, $f_1 = 0.95$, $f_2 = 0.05$, $f_3 = 0.0$
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
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 $\beta$, $f_1 = 0.94$, $f_2 = 0.06$, $f_3 = 0.0$
Histogram fractions $f_1, f_2, f_3$ of Patient 1 for CT scan data and model output
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
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 $\beta$, $f_1 = 0.90$, $f_2 = 0.10$, $f_3 = 0.0$
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
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 $\beta$, $f_1 = 0.84$, $f_2 = 0.16$, $f_3 = 0.0$
Histogram fractions $f_1, f_2, f_3$ of Patient 2 for CT scan data and model output
Patient 3: Four serial CT images spanning $917$ days for atypical cells (arrow) highly suspicious for adenocarcinoma by biopsy
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 $\gamma$, $f_1 = 0.63$, $f_2 = 0.24$, $f_3 = 0.13$
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
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 $\delta$, $f_1 = 0.37$, $f_2 = 0.40$, $f_3 = 0.23$
Histogram fractions $f_1, f_2, f_3$ of Patient 3 for CT scan data and model output
Patient 4: Four CT image recordings of a suspicious nodule spanning $471$ days (arrow)
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 $\gamma$, $f_1 = 0.46$, $f_2 = 0.36$, $f_3 = 0.18$
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
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 $\epsilon$, $f_1 = 0.23$, $f_2 = 0.33$, $f_3 = 0.44$
Histogram fractions $f_1, f_2, f_3$ of Patient 4 for CT scan data and model output
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
The five CT scan histogram categories
 Type Description $\alpha$ 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
 Type Description $\alpha$ 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
The three output fractions
 fraction CT 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 $1000$ $f_3$ $> -100$ $HU$ $> 1000$
 fraction CT 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 $1000$ $f_3$ $> -100$ $HU$ $> 1000$
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
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