Stochastic modeling of the tumor volume assessment and growth patterns in hepatocellular carcinoma

ARTICLE

Auteur(s) : Adrian Sãftoiu, Tudorel Ciurea, Florin Gorunescu, Ion Rogoveanu, Marina Gorunescu, Claudia Georgescu

Department of Internal Medicine, University of Medecine and Pharmacy Craiova, Craiova, Romania
E-mail : gorun@umfcv.ro

The growth pattern of hepatocellular carcinoma (HCC) arising from cirrhosis is variable and depends on the degree of differentiation and vascularization. Thus, the growth rate of moderately to poorly-differentiated HCC is usually faster than that of well-differentiated HCC [1]. Well-differentiated HCC has different growth rates related to the hemodynamic features of the tumor : tumors with portal blood flow grow slowly, while hypervascular tumors with arterial blood flow grow rapidly [2].

Tumor growth is not constant in the natural history of HCC. Assessment of subsequent growth rate as a function of histological and ultrasound characteristics, is generally not possible, at the moment of initial diagnosis. However, histological type, grade and ultrasound characteristics (the hyperechoic pattern) may help identifying the cases with long tumor volume doubling time (TVDT) [3]. Recently, MIB-1 labeling index was proposed as the most effective parameter significantly correlated with small HCC growth rate, as compared to the conventional indexes of histological atypism (i.e. nucleo-cytoplasmatic ratio, cellularity, nuclear form factor, the degree of HCC differentiation) [4]. 

The prognosis of HCC patients depends on many other factors which interact in a complex manner at the patients with liver cirrhosis : these include liver failure and gastrointestinal bleeding. However, TVDT is usually used as an index of HCC growth because it is correlated with the survival. Furthermore, in HCC patients with the same severity of the underlying disease, TVDT appears to be a possible determinant of prognosis [1].

Material and Methods

The aim of our study was to assess tumor volume and growth patterns of HCC with the aid of computer-aided ultrasound imaging. Furthermore, starting from initial tumor volumes we applied different forecasting models to determine future tumor volumes. Thus, we included 27 patients with histologically proven HCC, which had multiple (more than three) follow-up ultrasound (US) studies in a six months interval.
HCC was visualized by computer aided US imaging using a GE Logiq 500 MD ultrasound system and the diagnosis of HCC was confirmed in all the patients by ultrasound guided liver biopsy. Currently, the role of liver biopsy for HCC diagnosis is reevaluated, in view of clear progresses of imaging (contrast-enhanced spiral computed tomography or magnetic resonance imaging) and increased values of tumoral markers (serum alpha fetoprotein and des γ carboxiprothrombin) [5]. However, spiral computed tomography (CT) or magnetic resonance imaging (MRI), were not available for all the patients. HCC diagnosis was thus consequently confirmed histologically based on Edmonson and Steiner classification [6].
The patients did not receive any treatment during the observation period, because of : refusal of treatment (n = 10), advanced age (n = 11) or severe functional impairment of the liver (n = 6).

Volume assessment

We obtained initially the primary size quantification using the calipers of the ultrasound system software (figures 1 and 2), followed by the edge detection enhancement.
Tumor volume was calculated as primary size quantification from the largest section plane containing the major axis (a) and minor axis (b). The probe was then rotated at 90 degrees and the third axis was obtained (c). The ellipsoid formula 4/3 π a/2 b/2 c/2 was used [4]. In cases of multiple nodules in the same patient, the total tumor volume was calculated by the sum of each nodule volume. Edge detection techniques were used on a commercial PC (HP Brio, Pentium 3, 1 GHz, 128 Mb RAM) to establish points on the edge of the surface tumor.
By a bi-cubic B-spline smoothing of the edges interpolating points we approximated the surfaces shapes (figures 3 and 4) and we obtained a three dimensional (3-D) reconstruction of the tumor by the corresponding 3-D Bezier polyhedron [8].
Technically speaking,we have assumed a tri-dimensional domain T (the tumor) with its boundary not well-defined. The fact that the frontier is not completely known is compensated by some information, such as points on it – knots together with the corresponding directions of the normal. We have estimated the boundary of T by the bi-cubic B-spline interpolation. Thus, a Bezier surface (side) is obtained by repeated bilinear interpolations of the parameters. The bicubic B-spline surface is seen as a collection of bi-cubic B-sides (B-surfaces with 3 times differentiable joints corresponding to the boundary knots).
Finally, using the hit or miss Monte Carlo method [7], we have estimated the tumor volume. A simple software (in Java) is also available.

Tumor volume doubling time

The growth rate of the tumors was estimated by measuring the TVDT, using a formula developed by Schwartz [9] :
FORMULE

Time series and forecasting

One of the main goals of medical research is to highlight diagnostic/treatment mechanism and decision making starting from a number of observations or data on the relevant variable. The sequence of these observations constitutes the so-called time series. Most time series patterns can be described in terms of two basic classes of components : trend and seasonality. There is neither proven definitive science nor automatic techniques to identify trend components or seasonality in the time series data.
Taking into account these facts and the observed behavior of the tumors evolution for both cases (initial small size and initial large size), we chose a linear, a logarithmic and an exponential model of smoothing. Starting from the previous tumor volumes time series we have applied the above smoothing techniques to forecast the tumor volume for the next 6 months [10]. Let us note that, for a small number of observation, it is possible to use again the bi-cubic B-spline interpolation to increase in a natural way the number of observed data.

Results

We included in our study 27 patients with histologically proven HCC, [19 men and 8 women, age (mean ± SD) 43.2 ± 11.5], which had multiple (more than three) follow-up ultrasound (US) studies in a six months interval. HCC nodules varied between 1.5 and 12.7 cm (mean ± SD = 6.8 ± 2.3 cm), with most of the patients being in an advanced stage. Thus, 15 patients had 1 nodule, 7 patients had two nodules and 5 patients had three or more nodules.
At the moment of initial diagnosis HCC arose from liver cirrhosis in 25 patients (89.28 %). According to the Child-Pugh classification, 5 patients were class A, 6 patients class B and 14 patients class C. The etiology of underlying liver cirrhosis was viral in 19 patients (13 with chronic viral hepatitis C and 6 with chronic viral hepatitis B) and alcoholic in 8 patients. Because the lot of patients was relatively small, we did not perform statistical analysis on separate groups as a function of etiology or Child Pugh’s class.
The difference between the tumor volume estimated by primary size quantification was not significantly different than the tumor volume estimated by Monte Carlo algorithm (p = NS). Thus, the tumor volume was variable between 1.95 and 980.34 cm³ (mean ± SD = 330.45 ± 112.91 cm ³).
TVDT calculated by Schwartz formula from two discrete volume measurements was variable between 28.45 and 580.23 days (mean ± S.D. = 140.4 ± 122 days ; median = 160.1 days). However, TVDT was variable at the same patient during the growth of the tumor, as a function of volumes used.
We choose as example two extreme cases : a small tumor and a large tumor. For the small tumor in the right liver lobe, the initial volume in the moment of detection was 2.3 ml (figure 5), after 92 days it increased to 16 ml (figure 6) and after 159 days from the time of detection it increased to 22.6 ml (figure 7).
However the TVDT as calculated from Schwartz formula was initially 32.87 days (using the first two volumes calculated) and subsequently 134.47 days (using the last two volumes calculated). For the large tumor in the left liver lobe, the initial volume in the moment of detection was 98.3 ml (figure 8), after 153 days it increased to 132.6 ml (figure 9) and after 239 days from the time of detection it increased to 520.4 ml (figure 10).
However the TVDT as calculated from Schwartz formula was initially 350.73 days (using the first two volumes calculated) and subsequently 43.59 days (using the last two volumes calculated).

Discussion

Precise volume estimation of hepatocellular carcinoma is very important in clinical practice, for the assessment of the growth rate of inoperable tumors or the volume of normal liver tissue remaining after resection. There are multiple methods for the estimation of organ and tumor volumes, but there is not a universally accepted way of doing this [11-16]. The volume of liver tumors is frequently estimated on two dimensional images using the ellipsoid formula, i.e. 4/3 π a/2 b/2 c/2 [4], but this can lead to important errors because of the irregularities of liver tumors. Manual tracing of the outline of a tumor is considered improper, time-consuming, and therefore not widely used. Edge detection techniques were recently used to improve and automate the recognition of tumor shape. Stereological methods were also used recently in a study, which found that estimated volumes on cross-sectional computed tomography (CT) images varied from the reference by up to 4.5 % [12]. However this is difficult to accomplish with two dimensional (2-D) ultrasound probes, and can be achieved only with three dimensional (3-D) systems, either by free-hand or 3-D real-time probes. Different methods can be used with 3-D ultrasound or CT systems which include planimetry [13-15], 3-D reconstruction and voxel counting [16]. However, none of these methods can be successfully used on 2-D ultrasound systems, which are widely spread in the medical practice. This is why we tested the Monte Carlo algorithm for the estimation of volume, coupled with edge-detection techniques, using two perpendicular ultrasound planes. Our approach yielded similar values (p = NS) with the classic ellipsoid formula (primary size quantification).
The hit or miss Monte Carlo method is thus a very simple and efficient mathematical tool that can estimate the tumor volume, especially for irregular tumors. It can be easily implemented on 2-D and 3-D ultrasound systems, and combined with edge detection techniques. Furthermore, there is no need for manually placing the calipers and manually selecting the largest diameters of the tumor, both of which are highly operator dependent. Tumor volume can thus be accurately and rapidly assessed, without complicated and time-consuming techniques like voxel counting.
HCCs less than 10 mm are usually well-differentiated tumors [17, 18]. Large HCCs have moderately or poorly-differentiated tissue surrounded by well- differentiated tissue. The most advanced HCCs consist only of moderately to poorly-differentiated tissues. Furthermore, the growth rate of moderately to poorly-differentiated HCCs is usually faster than that of well-differentiated HCCs [1]. Well-differentiated HCC has different growth rates related to the hemodynamic features of the tumor : tumors with portal blood flow grow slowly, while hypervascular tumors with arterial blood flow grow rapidly [2]. Moreover, as small HCCs increase in size and become increasingly dedifferentiated, the number of portal tracts decreases, while intratumoral arterioles develop [19].
Although small HCCs have the tendency to grow slowly and large HCCs tend to grow rapidly [1-3], we found between our patients completely opposite examples : small tumors with rapid initial growth and large tumors with slow initial growth. Furthermore, TVDT was variable in time at the same patient as a function of volumes used. Thus, the growth rate pattern of HCC cannot be accurately assessed from TVDT calculated by Schwartz formula because of the marked variability of growth patterns.
Forecasting of a time series is usually not desirable when we do not have a thorough understanding of the biological phenomenon, but it may be an efficient way for assessing the prognosis and for counseling the patients about survival. The assessment of growth rate and subsequent forecasting are more accurate, if the initial growth pattern is recognized. The growth patterns of small and large HCC are variable, both between different patients, but also at the same patient during HCC evolution. Assessment of TVDT is variable as a function of the timing of tumor detection and subsequent volume measurements. Because HCC has increasing or decreasing growth rates we decided to apply different forecasting models as a function of initial trend : linear, exponential and logarithmic smoothing. Comparing the three types of forecasting techniques applied to our data, the logarithmic smoothing has been proven to be the most accurate (maximum deviation less than 8 % for small tumors and 12 % for large tumors). Anyway, further work is needed to determine the most appropriate forecasting model.
Our conclusion was that a dynamic forecasting model of HCC volumes is very useful for an accurate assessment of the TVDT. Consequently, the growth pattern and evolution of HCC can be better predicted. Furthermore, non-treated patients can be counseled about the natural history of HCC and about the risk-benefits of eventual treatments.

Acknowledgment. These results were partially presented as oral communication at the XVI^th Congress of the European Association of Cancer Research, Halkidiki, 2000.

References

1. Sheu JC, Sung JL, Chen DS Yang PM, Lai MY, Lee CS et al. Growth rate of asymptomatic hepatocellular carcinoma and its clinical implications. Gastroenterology 1985 ; 89 : 259-66.

2. Saitoh S, Ikeda K, Koida I Tsubota A, Arase Y, Chayama K et al. Serial hemodynamic measurements in well-differentiated hepatocellular carcinomas. Hepatology 1995 ; 21 : 1530-4.

3. Barbara L, Benzi G, Gaiani S Fusconi F, Zironi G, Siringo S et al. Natural history of small untreated hepatocellular carcinoma in cirrhosis : a multivariate analysis of prognostic factors of tumor growth rate and patient survival. Hepatology 1992 ; 16 : 132-7.

4. Saito Y, Matsuzaki Y, Doi M Sugitani T, Chiba T, Abei M et al. Multiple regression analysis for assessing the growth of small hepatocellular carcinoma : the MIB-1 labeling index is the most effective parameter. J Gastroenterology 1998 ; 33 : 229-35.

5. Torzilli G, Minagawa M, Takayama T, Inoue K, Hui A, Kubota K et al. Accurate preoperative evaluation of liver mass lesions without fine-needle biopsy. Hepatology 1999 ; 30 : 889-93.

6. Edmonson H, Steiner P. Primary carcinoma of the liver : a study of 100 cases among 48 900 necropsies. Cancer 1954 ; 462-503.

7. Yakowitz S. J. Computational probability and simulation. Addison-Wesley, 1977.

8. Farin G. Courbes et surfaces pour la CGOA (conception géométrique assistée par ordinateur). Masson, 1992.

9. Schwartz M. A biomathematical approach to clinical tumor growth. Cancer 1964 ; 14 : 1272-94.

10. Gourieroux Ch, Monfort A. Time series and Dynamic models. Cambridge University Press, 1997.

11. Treece G, Prager R, Gee A, Berman L. 3D ultrasound measurement of large organ volume. Med Image Anal 2001 ; 5 : 41-54.

12. Kao SC, Wang G. Radiologic volumetry on a personal computer with a stereologic method. Acad Radiol 1998 ; 5 : 665-9.

13. Liess H, Roth C, Umgelter A, Zoller WG. Improvements in volumetric quantification of circumscribed hepatic lesions by three dimensional sonography. J Gastroenterol 1994 ; 32 : 488-92.

14. Lang H, Wolf GK, Prokop M, Weimann A, Pichlmayr R, Zoller WG. Volumetry of circumscribed liver changes with 3-D ultrasound in comparison with 3-D computerized tomography [abstract] ; Langenbecks Arch Chir Suppl Kongressbd 1998 ; 115 : 1478-80.

15. Wolf G, Lang H, Prokop M, Schreiber M, Zoller W. Volume measurements of localized hepatic lesions using three-dimensional sonography in comparison with three-dimensional computed tomography. Eur J Med Res 1998 ; 23 : 157-64.

16. Lin XZ, Chang T, Tsai HM, Sun YN, Sheu BS, Jen C. Liver, spleen and tumor volume measured by personal computer. Hepatogastroenterology 1999 ; 46 : 838-42.

17. Kenmochi K, Sugihara S, Kojiro M. Relationship of histologic grade of hepatocellular carcinoma (HCC) to tumor size, and demonstration of tumor cells of multiple different grades in single small HCC. Liver 1987 ; 7 : 18-26.

18. Sugihara S, Nakashima O, Kojiro M, Majima Y, Tanaka M, Tanikawa K. The morphologic transition in hepatocellular carcinoma : A comparison of the individual histologic features disclosed by ultrasound-guided fine-needle biopsy with those of autopsy. Cancer 1992 ; 70 : 1488-92.

19. Nakashima Y, Nakashima O, Hsia CC, Kojiro M, Tabor E. Vascularization of small hepatocellular carcinomas : correlation with differentiation. Liver 1999 ; 19 : 12-8.