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[J]. Mathematical Biosciences and Engineering, 2018, 15(5): 1203-1224. doi: 10.3934/mbe.2018055
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Non-small-cell lung carcinomas (NSCLC) are the most common epithelial lung cancers. The development of thin slice CT (computed tomography) scans, coupled with new recommendations for lung cancer screening in high risk patients, has led to increased detection of subtle pulmonary subsolid or nonsolid nodules in the lungs [13]. CT scan x-rays measure these nodules, also known as ground glass opacities (GGOs), as the partial filling of air spaces in the lungs by exuded fluids. Published recommendations [4], [20], [21], [22] for how to follow GGOs over time depends only on nodule size and the presence or absence of a solid component. Recent work has demonstrated the utility of volumetric CT (vCT) for diagnosis of cancer in solid nodules by measuring growth rate over time. For these cancers, which include adenocarcinoma, a growth rate given by a volume doubling time (DT) less than 400 days is predictive of malignancy [12]. However, GGOs often grow slowly in size, thus giving a high false negative rate when using nodule volume as the imaging parameter. Additionally, GGOs can be difficult to segment on CT, making assessment of growth using vCT problematic. In this work, we investigate the potential to assess GGO growth based on a quantitative change in its
A recent report correlated five categories of CT histogram with histopathological characteristics and recurrence-free survival times [15]. Our objective is to model these five qualitative GGO measurement histogram categories, and their interpretations of tumor progression, to a quantitative dynamic mathematical model of tumor growth, which also allows estimation of tumor DT.
The mathematical model we use for the spatial-temporal evolution of a GGO is a diffusive logistic partial differential equation. We assume cell mass grows almost exponentially in an early time phase from an initial condition consisting of a small nodule, but ultimately slows in growth as time advances. CT scans are quantified in Hounsfield units (
We identify the five histogram categories formulated in [15], which are based on qualitative
The Tennessee Valley Healthcare System VA Hospital Institutional Review Board approved the analysis of the anonymized CT scan data used in this paper and waived the need for informed consent.
It has been recently documented that spatial intra-tumor heterogeneity plays an important role in lung cancer development at both the micro-molecular and at the macro-visible level [4], [20]. At the microscopic level and at the early stages of pulmonary adenocarcinoma in situ (previously bronchioloalveolar cell carcinoma), cancer cells align along alveolar walls in a so-called lepidic pattern. As the tumor invades the air spaces, it becomes more dense on CT.
Mathematical models of tumor growth in spatial regions have been developed by many researchers, including [1], [3], [10], [11], [17], [18], [24], [25]. Many mathematical models have been designed specifically to connect to CT scan imaging, including [2], [5], [8], [9], [16], [26], [27]. Our goal is to develop a mathematical model that aids lung CT scan analysis, and therefore our model captures tumor spatial growth dynamics at the macro-visible level. Our model has the following form of a diffusion partial differential equation with a growth-limiting logistic term:
∂u∂t(t,x)=∇(b∇u(t,x))+au(t,x)(1−u(t,x)+ub(x)um),t>0,x∈Ω; | (1) |
u(t,x)|∂Ω=0,t≥0;u(0,x)=u0(x),x∈Ω. | (2) |
In the model above
The parameter
The maximum of
In the subsequent section when we present our simulation results, the cell density
The spatial growth of the tumor in model (1)-(2) is limited by the normal background lung cell distribution, denoted by the time-independent background density function
The proof of existence of unique solutions of model (1)-(2) is provided in the Appendix. Note that, here we are mainly interested in the early transient behavior of solutions of the model, and not the long-term asymptotic behavior.
The five CT scan histogram categories presented in [15] are summarised in Table 1 below. The classification of these categories is qualitative and subject to interpretation. The classifications of patient examples in [15] were each constructed by visual assessment of two expert observers, using a decision tree algorithm, with disagreements resolved by consensus. The histograms in the study in [15] were given in terms of continuous smoothed-out renderings of the histogram bar graphs, which allowed easier determination of category type. In our study we use actual histogram bar graphs, which preserve more information. In general, the classification of category for a given patient data set is necessarily subjective, and in fact, some patient data in our database do not readily fit any of the classifications. Our main goal is to construct a model that fits patient CT scan histogram data, rather than a model that fits the interpretation of these data according to the classification scheme in Kawata et al. We believe that our model simulations will aid in the designation of these categories for individual patients.
Type | Description |
| high peak at low |
| medium peak at low |
| low peak at low |
| low peak at low |
| low peak at low |
To compare model output to patient data for a time series of CT scan histograms for a given patient, we will use a quantitative determination of the fractions of both CT scan histogram outputs and model (1)-(2) outputs. The CT scan fractional histogram outputs are the fractions of histogram bar heights in a given range of
fraction | CT scan histogram output at time | model output |
| | |
| between | between |
| | |
We use the output fractions
We provide here the results of simulations for four case studies, all compared to patient data. Our patient data and model simulation codes (developed in MATHEMATICA) are available upon request to the authors. All histograms, for both CT scan data and model simulations, are constructed with binning width of 10
Patient 1 is an example of a biopsy proven benign GGO. In Figure 2 we show CT scan images for Patient 1 at five time points. Patient 1 data consists of CT scan histograms in a series of five time points over approximately two years. These five histograms, with their category type and fractional values
In Figure 5 we graph the histogram plots (with bin width 10) of the model simulation of Patient 1 at the five time points as in Figure 3, where the values of
Patient | | | Doubling time from baseline |
1 | | | |
2 | | | |
3 | | | |
4 | | | |
Patient 2 is an example of a benign GGO nodule. In Figure 7 we show CT scan images for Patient 2 at six time points. Patient 2 CT scan histograms at six time points, their category type, and fractional values
In Figure 10 we show the histogram plots (with bin width 10) of the model simulation of Patient 2 at the six time points as in Figure 8, where the values of
Patient 3 is an example of atypical cells highly suspicious for adenocarcinoma by biopsy. In Figure 12 we show CT scan images for Patient 3 at four time points. Patient 3 CT scan histograms (with bin width 10
In Figure 15 we show the histogram plots of the model simulation for Patient 3 at the six time points as shown in Figure 13, where the values of
Patient 4 is an example of a proven adenocarcinoma that started as a GGO that increased in density on CT over time. In Figure 17 we show CT scan images for Patient 4 at four time points. Patient 4 CT scan histograms (with bin width 10
In Figure 20 we show the histogram plots of the model simulation for Patient 4 at the four time points as shown in Figure 18, where the values of
In Figure 22 we graph the total tumor mass from the model simulations for each patient over time, where mass is scaled to 1.0 at time 0. Patients 1 and 2 have smaller growth than Patients 3 and 4, corresponding to their smaller growth parameter
In a recent paper [15], a qualitative five-category classification method was proposed for analyzing NSCLC, and its utility justified using statistical tools. The results indicated a satisfactory inter-observer agreement simply through visual assessment of CT histograms. Our goal here has been to quantify the five categories in [15] in terms of a dynamic spatial model of tumor growth; and to connect the temporal dynamics of the categories to tumor DT. We have compared CT scan data and model outputs for four patient studies. For each patient, we see good agreement between these data and model outputs, in terms histogram categories and
In the current work we hypothesized that the five categories identified in [15] actually correspond to temporal tumor progression. Indeed, Kawata [15] already speculated that change from type
Our results show that model (1)-(2) supports the five category classification in adenocarcinoma in situ. Further, these five categories can be viewed as a hypothesized 5-step lung cancer progression theory. Moreover, since it takes into account the spatial heterogeneity of the tumor, which is particularly important for irregular nodules investigated here, the model gives us a tool to estimate tumor mass doubling times using CT histogram data only.
Major challenges for application of the model (1)-(2)are the identifications of the initial tumor nodule characteristics, the background non-tumor bias parameter
Our model already shows very good agreement with patient data, and the
● Full
● Systematically analyze the simulation outcomes as functions of the model parameters and initial condition (transient vs asymptotic behavior, is there a globally stable steady state?).
● Inclusion of nonlinear diffusion to account for a more realistic description of tumor spatial growth (in particular to model competition effects).
● To include different type of placement processes for the tumor cells (other than diffusion) to account for the complex spatial structure of the lung.
Estimation of tumor doubling time in GGOs has not been described. This work offers a method to compute growth rate of GGOs as a predictive biomarker of malignancy, similar to that used for solid nodules using volumetric CT. Further work is needed to investigate the impact of different reconstruction algorithms and reconstructed image quality on the estimate of GGO growth rate.
Global behaviour of solutions. The basic mathematical theory of general classes of nonlinear reaction diffusion equations of the type (1)-(2) is well understood. However, for completeness, here we provide a concise proof of the global existence and positivity of solutions of our model in the biologically relevant state space of Lebesgue integrable functions
We set
dudt=Au+F(u),u(0)=u0∈K, | (3) |
where
Au=∇(b∇u)+au(1−ubum), | (4) |
D(A)={v∈W2,1(Ω)|v(x)=0x∈∂Ω}, | (5) |
F(u)=−au2um. | (6) |
We say that the abstract semilinear problem (3) satisfies the sub-tangential condition (see e.g. [23]) with respect to
limh→0+d(K,T(h)u+hF(u))h=0, | (7) |
where
(u,v)−:=minv∗∈X∗{(u,v∗)|||v∗||=||v||,(v,v∗)=||v||2}. |
We recall the following result from [23], see also [19].
Let
We now apply this result to our model (1)-(2).
Theorem 6.1. Assume that
Proof. It follows from the assumptions that the densely defined operator
limh→0+d(K,u+hF(u))h=0, | (8) |
which is easily seen to hold true, as for all
Next note that in our setting we have
(F(u),u)−=minu∗∈L∞(Ω){−aum∫Ωu2u∗|||u||1=||u∗||∞,∫Ωuu∗=(∫Ω|u|)2}. | (9) |
Hence for every
Our model (1)-(2) always admits the trivial steady state
Φvu=∇(b∇u)+au(1−ubum)−auumv, | (10) |
D(Φv)={u∈W2,1(Ω)|u(x)=0x∈∂Ω},∀v∈K. | (11) |
It is then shown that for
We also note that applying earlier results by Cantrell and Cosner from [7] (see in particular Theorem 3.1 in [7]) would also allow us to obtain sufficient conditions for the existence of a globally stable unique positive steady state for
We acknowledge support provided by the National Cancer Institute U01CA152662. We also thank the Royal Society, which supported visits of József Farkas to Vanderbilt University.
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Type | Description |
| high peak at low |
| medium peak at low |
| low peak at low |
| low peak at low |
| low peak at low |
fraction | CT scan histogram output at time | model output |
| | |
| between | between |
| | |
Patient | | | Doubling time from baseline |
1 | | | |
2 | | | |
3 | | | |
4 | | | |
Type | Description |
| high peak at low |
| medium peak at low |
| low peak at low |
| low peak at low |
| low peak at low |
fraction | CT scan histogram output at time | model output |
| | |
| between | between |
| | |
Patient | | | Doubling time from baseline |
1 | | | |
2 | | | |
3 | | | |
4 | | | |