Interactive Modeling and Evaluation of Tumor Growth · 2016-06-08 · Interactive Modeling and Evaluation of Tumor Growth Jacob Scharcanski,1 Luciano Silva da Silva,1 David Koff,2

Interactive Modeling and Evaluation of Tumor Growth

Jacob Scharcanski,1 Luciano Silva da Silva,1 David Koff,2 and Alexander Wong3

This paper addresses the need to quantify tumor growthand detect changes as this information is relevant tomanage the patient treatment and to aid biotechnolog-ical efforts to cure cancer (Silva et al. 2008). Aninteractive tumor segmentation technique is used torecover the shape and size of tumors without imposingshape constraints. This segmentation algorithm providesgood convergence, is robust to the initialization con-ditions, and requires simple and intuitive user interac-tions. A parametric approach to model tumor growthanalytically is proposed in this paper. The preliminaryexperimental results are encouraging. The segmentationmethod is shown to be robust and simple to use, even insituations where the tumor boundary definition ischallenging. Also, the experiments indicate that theproposed model potentially can be used to extrapolatethe available data and help predict the tumor size(assuming unconstrained growth). Additionally, the pro-posed method potentially can provide a quantitativereference to compare the tumor shrinkage rate in cancertreatments.

KEY WORDS: Computed tomography (CT), imagesegmentation, active contours, lung cancer, computer-assisted diagnosis, chest radiographs, computeranalysis, computer vision, diagnostic imaging, digitalimage processing, cancer detection

INTRODUCTION

T he quantitative measurement of tumor growthprovides relevant information for the man-

agement of the patient treatment and to aidbiotechnological efforts to cure cancer throughearly assessment of growth pattern and response tochemotherapy.1 One of the difficulties faced inquantifying tumors is that the process is specific tothe individual event and to the particular organ.Another important quantification difficulty is thedose–response relationships existing for varioustherapeutic agents. Mathematical models for tumorgrowth have been studied for over a century, andat this stage, there is a lack of uniform opinion in

the literature regarding the appropriate model forthe growth rate of human cancers.The necessity for intensive follow-up of patients

in the primary stage of the cancer has beendebated. When growth rate is estimated frommeasurements, inaccuracies in measurement becomemore important. Therefore, measurement uncertain-ties can be reduced by adopting image processingmethods specific for image segmentation and usingimage comparison approaches that do not deform theoriginal data while being robust to noise and theinherent variability of computed tomography (CT)image properties. This work presents a method tobuild data-driven tumor growth models on samplecases of lung cancer using as inputs the changes insize measured at different stages of tumor develop-ment. At each stage of tumor development, the tumorsize is measured by an interactive segmentationalgorithm.Many methods for tumor segmentation and

registration rely on deformation models.2,3 In thework of Clatz et al.,4 the patient image is registeredwith an anatomical atlas, and the finite elementmethod is used to simulate tumor growth. Zacharaki

1From the Instituto de Informática, Universidade Federal doRio Grande do Sul, Caixa Postal 15064, 91501-970, PortoAlegre, RS, Brazil.

2From the Department of Radiology, McMaster University,Hamilton, ON, L8N 3Z5, Canada.

3From the Systems Design Engineering, University ofWaterloo, Waterloo, N2L 3G1, Canada.

Correspondence to: Jacob Scharcanski, Instituto deInformática, Universidade Federal do Rio Grande do Sul,Caixa Postal 15064, 91501-970, Porto Alegre, RS, Brazil;tel: +55-51-33087128; fax: +55-51-33087308; e-mail: [email protected]

Copyright * 2009 by Society for Imaging Informatics inMedicine

Online publication 19 September 2009doi: 10.1007/s10278-009-9234-4

Journal of Digital Imaging, Vol 23, No 6 (December), 2010: pp 755Y768 755

et al.5 applied a multimodal registration and fusionapproach to simulate 3D tumor growth, but resultsare reported only for registered images with nosignificant temporal gap. A hybrid approach forsemi-automated measurement of lung tumor thick-ness was proposed by Armato et al.6 In theirapproach, the user indicates initial endpoints alongthe outer tumor margin; these user-defined endpointsare refined by further processing, and an estimate ofthe maximum tumor diameter is obtained. However,the maximum tumor diameter is a limited measure toquantify the tumor growth, since the shape of a tumorcan be complex and contain concavities. Thresh-olding techniques and morphological operators havebeen used to segment lung tumors,7 leading tomeasures such as the greatest diameter (1D), theproduct of greatest diameter and greatest perpendic-ular diameter (2D), and volumetric measures (3D).These measures have been proposed to analyze CTscans obtained before and after patient treatment, andthe detected measurement changes were comparedwith ground truth data. The comparison resultsindicate that 2D and 3Dmeasurements are preferablecompared to 1D measurements; however, Zhao etal.7 observed that thin-section CT series are impor-tant to guarantee reliable volume measurements.Additionally, the segmentation algorithm employedby them may require manual delimitation of theregion of interest, especially when the region ofinterest and the tumor present similar attenuations(e.g., when a lesion is attached to the mediastinum orto the combined right diaphragm and liver). Haneyet al.8 evaluated methods for tumor growth rateestimation and compared two 3D image analysisalgorithms, namely, nearest-neighbor tissue segmen-tation and surface modeling, both applied to mag-netic resonance images of patients with gliobastomamultiform. In their comparison of the tumor growthrates computed using both segmentation methodswith ground truth data, the nearest-neighborsegmentation algorithm presented better results.Nevertheless, this method requires complex userinteractions, since a high number of tags (160) has tobe selected by the user.Our main goals in this work are: (a) to quantify

tumor growth and changes in tumor size; and (b) tobuild an analytic tumor growth model as areference to assess the tumor development. In thepresent work, a method for interactive segmenta-tion and modeling of the development of lungtumors on CT scans is presented. The only

interaction needed is the selection, by the specialist,of a point within the lung tumor in the more centralslice of a CT scan image stack. The tumor outline isthen detected by an active-contour method (i.e.,snake) proposed by Xu and Prince,9 which has twoimportant properties for our purposes: robustness toinitialization and convergence to concave bounda-ries. The segmentation results is used to recoverautomatic measurements, such as tumor shapechanges and growth rate. Based on measurementson the segmentation results, an analytic tumor growthmodel is built. This model can be used in applicationssuch as tumor growth prognosis or for evaluatingquantitatively the effectiveness of a cancer treatment,comparing the measured tumor shrinkage rate withthe prediction generated by the model.This paper is organized as follows. Initially, the

interactive tumor segmentation and area measure-ment is outlined in “Tumor Segmentation UsingActive Contours”. The tumor growth model ispresented in “Modeling Unconstrained TumorGrowth” and our experimental results in “Exper-imental Results”. Finally, the conclusions arepresented in “Conclusions.”

TUMOR SEGMENTATION USING ACTIVECONTOURS

Active contours, or snakes, are curves defined inthe image domain used to locate object bounda-ries.10 These curves can move under the influenceof internal forces and external forces. Externalforces are based on image measurements and aredesigned to move points of the curve towards thedesired features (usually edges). Internal forces arerelated to properties of the curve itself and aredesigned to hold the curve together (i.e., elasticityforces) and to keep it from bending too much (i.e.,bending forces).Usually, a snake is a curve that evolves by

minimizing an energy functional E in the imagespatial domain:

E ¼Z 1

0

1

2a x0 sð Þj j2 þ b x00 sð Þj j2

� �þ Eext x sð Þð Þds;

ð1Þwhere, x(s)=[x(s), y(s)], s∈ [0,1], represent aspatial location in the curve, and α and β areweighting parameters that control the snake tension

756 SCHARCANSKI ET AL.

and rigidity, respectively. The first and secondderivatives of x(s) in relation to s are x′ and x″.The energy function Eext represents the externalforces derived from image measurements and hassmaller values in image locations where the featuresof interest occur.In lung tumor segmentation, the image gradient

is a feature of interest, since the margins of lungtumors often correspond to sharp edges in CTimages. Then, the external force function can bedefined as the negative of the local image gradient▿f(x, y):

Eext x; yð Þ ¼ � rf x; yð Þj j2;

and a snake that minimizes E must satisfy theEuler equation:

ax00 sð Þ � bx000 sð Þ � rEext ¼ 0: ð2Þ

Considering that x is a function of time t, thepartial derivative of x with respect to t is given bythe left hand side of Eq. 2:9

xt s; tð Þ ¼ ax00 s; tð Þ � bx000 s; tð Þ � rEext: ð3Þ

A solution for the snake x(s,t) is found when theterm xt(s,t) reaches steady state. This result impliesthat the external force is proportional to the intensityof edges and only acts near the edge locations.However, despite its simplicity, this formulation hastwo important drawbacks: (1) makes it difficult toapproximate concave regions,11 posing difficultiesfor detecting the boundaries of lung tumors ifthey present irregular shapes, which often is the casein such tumors; (2) the initial values of xt(s,t) (i.e.,x0(s,t)) must be very close to the actual tumormargins (i.e., the tumor boundaries), and thisinitialization scheme may be impractical. Therefore,a different approach should be used.Xu et al.9 proposed a new external function,

called Gradient Vector Flow, that overcomes theabove mentioned problems by assuming that, inthe absence of other local evidences in the image,local information vary smoothly in all directions.Therefore, the information available near theimage edges can propagate across adjacent homo-geneous regions, allowing to develop the snakeconvergence process away from the image edges(if the image boundaries are avoided). This is

achieved by defining a vector field v(x,y)=(u(x, y),v(x, y)) that minimizes the energy function:

" ¼ZZ

� u2x þ u2y þ v2x þ v2y

� �þ rfj j2 v�rfj j2dxdy;

for the sake of notation simplicity, the coordinates(x,y) have been omitted, and f≡f (x, y). Note thatwhen |▿f | is small, the energy is dominated by thepartial derivatives of the vector field, and when|▿f | is large, the energy is dominated by thesecond term and is minimized by making v=▿f.The parameter μ is a tradeoff between the firstterm and the second term and is set according tothe noise level in the image (we used μ=0.05 in allour experiments).As demonstrated by Xu et al., the gradient

vector flow field v(x,y) can be found by solvingthe following Euler equations:

�r2u� u� fxð Þ f 2x þ f 2y

� �¼ 0

�r2v� v� fy� �

f 2x þ f 2y

� �¼ 0;

where, ▿2 is the Laplacian operator.After v(x,y) is computed, it replaces the negative

of the external force (−▿Eext) in Eq. 3, and thevalue of xt(s,t) is obtained at time t:

xt s; tð Þ ¼ ax00 s; tð Þ � bx000 s; tð Þ þ v: ð4ÞIn all our experiments, we used α=5 and β=50.Figure 1 shows an example of gradient vector

flow field. Figure 1a displays the tumor regioncropped from an original CT image, and Figure 1bshows the corresponding negative of the gradient(−|▿f (x, y)|2), which is obtained with the Prewittoperator.12 The gradient vector flow field obtainedis showed in Figure 1c, and a zoom of the sameflow is displayed in Figure 1d.The tumor segmentation process is semi-auto-

matic, in the sense that it uses a simple userinteraction. All the user have to do is to select anypoint inside the tumor region. Once this internalpoint is selected, the curve x0(s,t) (Eq. 4) isinitialized with a small circle (radius=12 pixels),and the selected point is the circle center, asshowed in Figure 2a. The iterative active contourapproximation is then performed (Fig. 2b, c) untilconvergence ( xtþ1 s; t þ 1ð Þ � xt s; tð Þj j2 � 0:5 forall s∈[0,1]), as shown in Figure 2d.

INTERACTIVE MODELING AND EVALUATION OF TUMOR GROWTH 757

The tumor perimeter in a given CT slice isestimated based on the calculated active contourboundaries, after convergence, and the tumor areais estimated based on the number of pixels withinthe area delimited by active contour boundaries.As mentioned before, this active contour method

is able to converge even for irregularly shapedtumors, as illustrated in Figure 3. Besides, theactive contour boundaries tend to be robust to theselection of initial points within the tumor (see“Experimental Results”).

MODELING UNCONSTRAINED TUMOR GROWTH

In this section, an analytical model for tumortemporal evolution is discussed. This model can beused to predict tumor growth (i.e., prognosis for agiven number of time units, e.g., days), or, in theopposite way, to predict tumor shrinkage, whichcan be an interesting application.Assuming that tumors as masses are approx-

imately dense and that each tumor cell generates

continuously new tumorous cells (i.e., the tumorgrowth is not constrained), in a time increment dt,the number of tumorous cells Nc grows by:

Nc t þ dtð Þ � Nc tð Þ � dNc

dtdt þ . . . ; ð5Þ

where the right hand side comes from a Taylorseries expansion. Consequently, based on theseassumptions, the initial number of cells Nc(t)grows by a rate r in time increment dt:

dNc

dtdt ¼ rNc tð Þdt: ð6Þ

Dividing Eq. 6 by dt, an ordinary differentialequation is obtained:

dNc

dt¼ rNc tð Þ; ð7Þ

which has the following solution:

Nc t þ dtð Þ ¼ erdtNc tð Þ: ð8ÞNow, considering that each tumor cell occupies

Ac area units in a computerized tomography slice,

Fig 1. Example of gradient vector flow field: a original image; b negative of the gradient; c gradient vector flow field down sampled bya factor of 0.5; and d zoom of area of the flow field indicated by the rectangle in c.


Nc cells occupy NcAc area units, and the corre-sponding area grows proportionally to the tumorgrowth rate r. Therefore, assuming unconstrainedtumor growth, the following exponential tumorarea growth model is justifiable:

AcNc t þ dtð Þ ¼ erdtAcNc tð Þ; ð9Þor yet,

A t þ dtð Þ ¼ erdtA tð Þ; ð10Þwhere, A(t) is the tumor area measured at time t,and A(t+dt) is the area the tumor shall occupy attime t+dt.In order to fit the generic exponential model (see

Eq. 11) to a set of tumor areas measured at discretetimes, namely At1 ;At2 ; . . . ;Atn � fAng, the expo-nential model can be linearized, and then least-squares techniques can be used to find the modelparameters. Without loss of generality, we can writeEq. 11 as:

A tð Þ ¼ ertA 0ð Þ ¼ erta: ð11Þ

Taking the logarithm of both sides, we obtain:

lnA tð Þ ¼ ln aþ rt; ð12Þor yet,

y ¼ c0 þ c1t; ð13Þwhere, y=ln A(t), c0=ln a, and c1=r. Now, theproblem of fitting the exponential model to the set

Fig 2. Active contour convergence: a curve initialization, based on the seed point; b corresponding curve after 25 iterations—onecurve is shown for every five iterations; c corresponding curve after 50 iterations; and d final curve (after convergence).

Fig 3. Complex-shaped tumor with corresponding snakeoverlayed.


{An}, has been reduced to the simpler problem offitting a linear model (see Eq. 13) to ln{An}, usingleast-squares techniques.In order to calculate the least squared error

estimates for the linear model parameters, let themodel errors ei be:

13

e1 ¼ A t1ð Þ � c0 � c1t1;e2 ¼ A t2ð Þ � c0 � c1t2;� � �en ¼ A tnð Þ � c0 � c1tn:

ð14Þ

Let the sum of squared errors be ε:

" ¼Xni¼1

e2i ¼Xni¼1

A tið Þ � c0 � c1tið Þ2: ð15Þ

Now, the least-squares criterion is used tochoose c0 and c1 while minimizing the fitting errorε. Setting the derivatives to zero, we obtain:

@"

@c0¼ �2

Xni¼1

A tið Þ � c0 � c1tið Þ ¼ 0; ð16Þ

and,

Xni¼1

A tið Þ ¼ nc0 þ c1Xni¼1

ti: ð17Þ

Also, the derivative of ε with respect to c1 gives:

@"

@c1¼ �2ti

Xni¼1

A tið Þ � c0 � c1tið Þ ¼ 0; ð18Þ

and

Xni¼1

tiA tið Þ ¼ c0Xnti

ti þ c1Xni¼1

t2i : ð19Þ

Observing that Eqs. 17 and 19 are two inde-pendent equations with two unknowns c0 and c1,these equations can be written in matrix form as:

nPni¼1

ti

Pni¼1

tiPni¼1

t2i

26664

37775

c0c1

� �¼

Pni¼1

A tið Þ

Pni¼1

tiA tið Þ

26664

37775: ð20Þ

By solving Eq. 20 for c0 and c1, we obtain theparameters a and r of the exponential growth

model such that the squared adjustment error isminimized:

a ¼ ec0 ;r ¼ c1:

ð21Þ

Therefore, the exponential growth model shownin Eq. 11 becomes:

A tð Þ ¼ ec1tþc0 : ð22Þ

EXPERIMENTAL RESULTS

The segmentation method and proposed modelingapproach have been tested using MATLAB 7.0 andIBM-PC-based computers, with 2 GB of RAM, andclock of 2 GHz. Considering that our goal in thispaper is to propose an interactive tool for assessingtumor growth patterns, a prototype was assembled asa proof of concept, and its performance is illustratedusing sets of thorax CT scans. As a next step in ourresearch, we plan to investigate the growth patternsof specific types of tumors in clinical trials, and theresults shall be reported separately.In order to segment the tumor boundaries and

estimate the tumor size, we use the methodologyknown as Response Evaluation Criteria in SolidTumours (RECIST).14 According to RECIST, thelongest diameter of the target lesion should beobtained in the axial plane only. Therefore, weillustrate our tumor segmentation based on theslice showing the largest tumor diameter.In order to assess the tumor growth potential

and evolution, several studies on a period of timeare compared. The radiologist must make sure thatthe lesion he/she points is the same on differentstudies, as often those patients have multiplemetastases. Since the radiologist may wish tocompare several studies, he/she often has to pointthe same tumor in different studies; therefore, thetumor segmentation process must be repeatable(i.e., even if the radiologist points the tumor at adifferent pixel, the computation should converge tosimilar results). As mentioned before, the activecontour method based on gradient vector flowtends to be robust to initialization (i.e., to the initialpoint selected inside the tumor to start thesegmentation process). To measure the sensitivityof the proposed approach with respect to itsinitialization, we compared several initializations


in the segmentation of the same tumor andobserved the resulting segmentation maps (i.e.,given an image, we marked different initial pointsinside a tumor region and observed the segmenta-tion results obtained for that tumor). Experimen-tally, we observed that the segmentation mapsobtained differ in at most 7 pixels (i.e., in the worstcase). The tumor regions tested ranged fromapproximately 5,000 to 30,000 pixels. Therefore,provided different initializations for the sametumor segmentation process, a variation of lessthan 0.2% was experimentally observed. Recallthat the tumor segmentation process is semi-automatic, and the user simply needs to selectany point inside the tumor region.The proposed method can be specially efficient

for comparing tumors at distinct developmentstages. Figures 4 and 5 show examples of the finaltumor segmentation, each resulting from theevolution of an active contours from a pointselected in the interior of the tumor region (dottedcurves). These figures show the segmentations ofthe same tumor developing in time, as they appearin subsequent CT scans.The temporal evolution of tumors (as appearing

in subsequent CT scans), the exponential model fitto the measurements, and the tumor size prognosesafter 90 days of unconstrained growth are shownin Figures 6 and 7 for cases 1 and 2, respectively.The numerical results are reported in Table 1. Itshould be observed that tumors present growth

modes, and these growth modes can change withthe tumor evolution, causing changes in angles andeigenvalue ratios of the largest eigenvectors.Figure 8a–c illustrates segmented tumor boundaries(dotted curves) in subsequent CT scans, and Figure8d illustrates the evolution of a tumor with ellipseapproximations (continuous curves), as well as the90 days prognosis (dotted curve), all generated usingthe exponential model.Measuring the tumor evolution in subsequent

CT scans is not trivial, since implicit measurementinaccuracies exist. Often, the same tumor sectionsare not captured in corresponding tomographicslices of subsequent CT scans; consequently,comparative area (or volume) measurements arejust approximations. In order to compare tumorsections in subsequent scans, nonlinear imageregistration techniques have been employed (seedetails in the study of Wong and Bishop15).However, even state-of-the-art nonlinear registrationtechniques can introduce tumor shape size deforma-tions. For example, using the nonlinear registrationmethod described in Wong and Bishop15 in case 2,the tumor areas measured in slices 1, 2, and 3 are10381, 12093, and 29494, respectively, indicatinggrowth rates of 16.49% and 143.89%, for slices 2/1and 3/2, respectively. Nevertheless, applying onlylinear transforms to the images (avoiding tumorshape and size deformations, which is our proposedapproach), the measured tumor areas in slices 1, 2,and 3 are 10308, 12088, and 29429, respectively,

Fig 4. Segmented tumor boundaries (dotted curve) illustrated for the central slice of case 1 and tumor growth: a initial observation(June 04, 2007); b October 04, 2007; c November 11, 2007.


indicating growth rates of 17.2% and 143.46% forslices 2/1 and 3/2, respectively. Figure 9a–d illus-trates the area growth in case 2, using the linear andthe nonlinear registration approaches. The relativetumor sizes and shape differences are noticeable inthe comparison of panels a and c, and panels b and dof Figure 9.Figure 10 illustrates our analytic tumor growth

model applied to case 3, where it provides areference to assess tumor shrinkage (as a tumorresponse to treatments). The tumor area shrinkagebased on the different observations, approximated by

ellipses, and the 90 days prognosis obtained usingthe exponential model are shown in Figure 10a. Theexponential model fit to the measured tumor areasand the 90 days prognosis based on the tumorshrinkage rate according to the exponential modelare shown in Figure 10b.

Discussion

Tumors do not share the same growth pattern,according to the nature of the disease, and it isimportant to assess the growth trend. Most of the

Fig 5. Segmented tumor boundaries (dotted curve) illustrated for the central slice of case 2 and tumor growth: a initial observation(April 09, 2007); b August 30, 2007; c December 27, 2007.


time, tumor measurements are taken in two orthree directions, taking usually the largest diame-ters. Having the previous studies is important asthe reader will try to reproduce the measurementsat the same level of the CT scan, but it may bedifficult if the tumor shape has changed. There areinter-observer differences; sometimes in the waythey put the calipers, some will always takeorthogonal measurements, with lines parallel tothe axis of the CT slice, while others will look forthe largest dimensions. Some will measure from

inside border to inside border, others to outsidelimits. Some may even include the peritumoralinfiltration, which makes the comparison evenmore complex. Therefore, a mathematical repre-sentation is needed to ensure better reproducibilityand more scientific assessment of the tumorpotential. Our main goals of this work are: (a) toreduce the inter-observer variability in tumor sizemeasurements and (b) to build an analytic tumorgrowth model as a reference to assess the tumordevelopment. The expected benefits are: (a) assess

Fig 6. Measured tumor area growth based on the central slice (case 1). Tumors are registered using linear transformations (centroidadjustment): a area growth from June 04, 2007 to October 04, 2007; b area growth from October 04, 2007 to November 11, 2007; cexponential model fit to the measured tumor areas (solid line), and the 90 days prognosis considering unconstrained tumor growth(dotted line).


the tumor aggressiveness in evaluating the speed atwhich the tumor is progressing; (b) assess theresponse of the tumor to treatments and the rate atwhich it decreases in size and also to extractreproducible values over the period of time whichcan be compared to the evolution of the same typeof tumor with similar treatment; (c) create a growthpattern, and this is the most innovative part of ourstudy, as the shape of the tumor is usually over-looked. This will allow us to predict the harmfulpotential to neighboring organs and anticipate moreaccurately the complications and maybe the need

for more aggressive treatment orientation. Mainly,it will be interesting to assess potential invasion orrupture of adjacent vessels and extract trends as alltumors may not have the same pattern.The proposed tumor segmentation method

potentially can be used to estimate the size oftumors based on the number of pixels in thesegmented area14 (since the area corresponding toits pixel is known in a calibrated CT equipment).This segmentation method has been designed to berobust to initialization, flexible to handle irregu-larly shaped tumors, simple to use, and to reduce

Fig 7. Measured tumor area growth based on the central slice (case 2). Tumors are registered using linear transformations (centroidadjustment): a area growth from April 09, 2007 to August 30, 2007; b area growth from August 30, 2007 to December 27, 2007; cexponential model fit to the measured tumor areas (solid line), and the 90 days prognosis considering unconstrained tumor growth(dotted line).


the inter-observer variability in measurements; andthese features were verified experimentally. Also,it shall be observed that nonlinear image registrationtechniques tend to produce tumor shape distortions,which interferes with the size measurements (thisoften occurs when comparing tumor shapes in two or

three dimensions). Therefore, we avoid shape dis-tortions using linear methods to compare area (orvolume) measurements.The proposed analytic tumor growth model

provides a reference to assess tumor growth ortumor shrinkage, measure the response of the

Table 1. Numerical Experimental Results

Measured features 1st observation 2nd observation 3rd observation 1st/2nd observation 2nd/3rd observation

Measurements in case 1Perimeter (pixels) 234 357 439Area (pixels) 5,103 11,637 17,984Angle of largest eigenvector (degrees) −144.55 −145.70 47.05Eigenvectors ratio (largest/smallest) 1.45 1.31 1.41Perimeter growth rate (pixels) 52.56 22.97Area growth rate (pixels) 128.04 54,54

Measurements in case 2Perimeter (pixels) 347 370 612Area (pixels) 10,308 12,088 29,429Angle of largest eigenvector (degrees) 73.94 164.66 140.58Eigenvectors ratio (largest/smallest) 1.50 1.19 1.18Perimeter growth rate (pixels) 6.63 65.41Area growth rate (pixels) 17.27 143.46

Fig 8. Segmented tumor boundaries (dotted line) illustrated for the central slice of case 1, and tumor growth: a initial observation (June04, 2007); b October 04, 2007; c November 11, 2007; d tumor development based on the different observations, approximated by solidline ellipses and the 90 days prognosis obtained using the exponential model (dotted line ellipsis).


tumor to treatments and the rate at which itdecreases in size, and/or to create a growth patternthat can be compared to the evolution of the sametype of tumor with similar treatment. The proposedmodel fits a given data set while minimizing theleast squared error criterion (it is optimal, see“Modeling Unconstrained Tumor Growth”). Theproblem of fitting an exponential model is reduced

to fitting a linear model (see Eq. 13), and since theminimum number of data points to define a line istwo, at least two data points are required to findthe optimal model parameters in Eq. 22. Despitethat more data points tend to produce even betterresults, it may not be practical to obtain many CTscans of the same patient in a short time span. Weshowed in our experimental results that only three

Fig 9. Comparing tumor area growth in case 2 (based on the central slice), using linear transformations (a–b), and nonlineartransformations (c–d). Tumor areas registered using linear transformations (centroid adjustment): a area growth from April 09, 2007 toAugust 30, 2007; b area growth from August 30, 2007 to December 27, 2007; tumor areas registered using a typical nonlinearregistration method (see Zhao et al.8): c area growth from April 09, 2007 to August 30, 2007; d area growth from August 30, 2007 toDecember 27, 2007.


data points have been sufficient to find the optimalmodel parameters, since these parameters producedan adequate model for available data.

CONCLUSIONS

The present work introduces a new method forassessing the growth of lung tumors quantitatively.An interactive tumor segmentation technique

based on active contours is used to recover theshape and size of a tumor. The segmentationmethod is efficient for locating the tumor margins,even if the tumor shape is irregular and containsconcavities. The method presents good convergenceand robustness to the initialization conditions andrequires a simple and intuitive user interaction.This approach can be used to evaluate the tumor

growth rate without imposing shape constraints.The model presented potentially can be used toassess the tumor aggressiveness and the responseof the tumor to treatments and to estimate a growthpattern. To estimate the growth pattern can beimportant for predicting the harmful potential toneighboring organs and anticipating more accu-rately the complications and maybe the need formore aggressive treatment orientation. Our focus ison the assessment of a potential invasion orrupture of adjacent vessels and extract trends asall tumors may not have the same pattern.As a future work, we plan to evaluate the

proposed technique in clinical trials and test itsperformance as a quantitative measure of tumorregression in response to treatment.

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Fig 10. Measured tumor area shrinkage based on the centralslice (case 3). Tumors are registered using linear transformations(centroid adjustment): a tumor shrinkage based on the differentobservations, approximated by solid line ellipses, and the 90 daysprognosis obtained using the exponential model (dotted lineellipsis); b exponential model fit to the measured tumor areas(solid line), and the 90 days prognosis based on the tumorshrinkage rate obtained with the exponential model (dotted line).


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