Bio-Medical Materials and Engineering 24 (2014) 163–171 DOI 10.3233/BME-130796 IOS Press
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Evaluation of breast cancer chemotherapy efficacy with multifractal spectrum analysis of magnetic resonance image Li LIa,b , Wen-yong HUa, Li-zhi LIUb, Ya-chun PANGa and Yuan-zhi SHAO a,* a
School of Physics and Engineering, Sun Yat-sen University, Guangzhou 510275, China State Key Laboratory of Oncology in Southern China, Imaging Diagnostic and Interventional Center, Cancer Center, Sun Yat-sen University, Guangzhou 510060, China b
Abstract. Multifractal spectrum analysis of dynamic contrast enhanced (DCE) breast MR images was used to establish a new quantitative analysis method for solid tumor blood perfusion and to explore its applicability in evaluating efficacy of breast cancer chemotherapy. Five randomly selected patients suffering from newly diagnosed malignant breast nodule lesions were enrolled in this study, and four of them were treated with neoadjuvant chemotherapy. Their DCE breast MR images were collected before and after treatment. Chemotherapeutic efficacy was analyzed using international response evaluation criteria for solid tumors (RECIST). Sandbox method for statistical number density was employed to measure and calculate multifractal spectra of DCE breast MR images with spatiotemporal characteristics. Multifractal spectral data of malignant lesions before and after chemotherapy were compared. Multifractal spectra of malignant lesions show an asymmetric bell-shape. Chemotherapy efficacy was assessed to be partial remission (PR) for three patients and their multifractal spectral width significantly increased after chemotherapy while to be stable disease (SD) for other patient and of her changed slightly. Multifractal spectral width correlates with blood-supply condition of tumor lesion before and after chemotherapy, providing a potential suitable characteristic parameter for evaluating chemotherapeutic efficacy quantitatively. Key words: Breast cancer, magnetic resonance imaging, multifractal spectrum, image processing
1. Introduction Multifractal spectrum analysis is an important method and tool in nonlinear science, mainly used to study a class of irregular, chaotic, complex systems with local and global similarities in multi-scale analysis [4]. Multifractal spectrum analysis reveals the characteristics of self-similar scaling of heterogeneous geometric objects or tissues in biomedicine on multiple scales. As a new method for data-mining and medical-info-processing, fractal analysis has been finding its application in medical image analysis [1-3, 6, 8, 12-16] to quantitatively evaluate the correlation of such changes in MRI as: (1) brain gray matter and white matter with atherosclerosis and Alzheimer diseases [12,13,15]; (2) tumor growth of epithelial ovarian cancer and endometrial adenocarcinoma [2, 3]; (3) maternal surface of placenta [1,8]. *
Corresponding author. Email: [email protected], Tel: +86-20-84110399, Fax: +86-20-87343295
0959-2989/14/$27.50 © 2014 – IOS Press and the authors. All rights reserved
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L. Li et al. / Evaluation of breast cancer chemotherapy efficacy with multifractal spectrum analysis
Fractal approach has been reported in studies on breast image analysis. Velanovich applied simple fractal method, for the first time, to analyzing mammographic lesions and figured out that the mean composite fractal dimension of malignant lesions was higher than that of benign lesions [14]. Derado et al. have reported research of multi-fractal image analysis on breast MR images of healthy human body and patients using wavelet processing approach [6]. However, Derado et al merely compared MR images at different times before and after injecting contrast medium, instead of before and after treatment. Compared with conventional X-ray mammography, MRI is more appropriate for diagnosing and staging breast cancer because of its higher resolution for soft tissues and much higher detection rate for breast cancer. After injecting contrast medium, dynamic contrast enhanced (DCE) breast MRI can reflect the blood perfusion of breast lesions, and thus can be used for evaluating early breast responses to chemotherapy [11]. This study analyzed and compared the variance of multi-fractal spectra of DCE breast MRI of malignant lesions before and after chemotherapy, establishing a new method to quantitatively analyze DCE MRI images and evaluate micro-circular distribution heterogeneity, and extended its application in evaluating the responsiveness of breast cancer to chemotherapy. 2. Materials and Methods This study was performed with approval from the institutional ethics committee. 2.1. Clinical and image data Five randomly selected patients who were admitted to the Cancer Center of Sun Yat-Sen University from December, 2007 to October, 2009 for DCE breast MRI and diagnosed pathologically with breast cancer were enrolled in our study, aged from 32 to 59. The maximum diameter of their lesions was of 11-75 mm. Four of them underwent preoperative neoadjuvant chemotherapy and accepted DCE MRI examination within 6 weeks. A surgical resection was operated on all patients. Three were treated with radical mastectomy or modified radical mastectomy, and the others were treated with segment resection. Clinical data of all patients are listed in Table 1. Four patients received neoadjuvant chemotherapy follow-up MRI for 4-8 weeks after the completion of chemotherapy to assess chemotherapy efficacy using international RECIST [9], and the efficacy was classified as complete remission (CR), partial remission (PR), stable disease (SD) or progression disease (PD). 2.2. Image data processing A four-dimensional (3 spatial dimension plus 1 temporal dimension) database is composed of a large number of tomographic images constructed using sagittal vibrant multi-temporal DCE MRI. Its image array consisted of sagittal slice images with 512512 pixels and slice thickness of 3.4 mm. Fig. 1 shows DCE breast MRI array of a certain patient before and after injecting contrast medium into lesion, which clearly demonstrates variation in signal intensity of tumor images at different time T with an interval of 20 s.
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2.3. Algorithm to measure and calculate multifractal spectrum Images were segmented on multiple scales using a conventional sandbox method. The length of sandbox is set as . Under different subset scales, the number-density distribution probability of pixels Pi() is assumed to satisfy the following power-law function:
Pi (ε ) ~ ε α
(1)
Table 1 Patient’s clinical profile No. 1
Age 32
TNM Stage T2N0M0
Pathological Grade Grade II Invasive ductal carcinoma
2
48
T4BN1M0
3
46
T4N2M0
4
59
T3N0M0
Grade III Invasive ductal carcinoma Grade III Invasive ductal carcinoma Grade II-III Invasive ductal carcinoma
5*
46
T1NOM0
Chemotherapy Program Two courses of TEC followed by surgery and four courses of TEC MMC + CBP + THP + Taxotere FAC plus Herceptin chemotherapy Six courses of FAC plus Herceptin + Doalaxol chemotherapy
Drug Administration Intravenously
Artery perfusion Intravenously Intravenously
Grade I Invasive ductal carcinoma * The patient did not accept chemotherapy.
Figure 1. Breast MRI array diagram (a) before and (b) after treatment. Sequence number of sagittal spatial scanning series was 30 and time sequence number was 12.
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where is a subset-related singular index representing heterogeneity degree of number density distribution. If intra-subset density number N() and the length of sandbox subset scale meet
N (ε ) ~ ε − f (α ) |ε →0
(2)
then f()~ constitutes a multifractal spectrum function, which can be used to quantitatively describe growth characteristics of a fractal object at different levels and stages. Multifractal spectrum is commonly worked out using conventional partition function q(), which can be calculated from the weighted sum of q power of density distribution probability Pi(), i.e.
χ q (ε ) = ¦ i Pi (ε ) = ε τ ( q ) q
(3)
where (q) is density index and q usually takes a series of integers ranging from -50 to 50 for a variety of weighted sums. Generalized fractal dimension Dq is defined with (q) by
Dq = τ (q) /(q − 1) = ln χ q (ε ) /[(q − 1) ln ε ] |ε →0
(4)
Dq at q=0 and q=1 correspond to a simple fractal dimension D0 and information dimension D1 respectively. If system is of multifractality, it can be derived from Eqs. (3) and (4) that , q, (q) and f() satisfy the following relationship
τ (q) = α q − f (α )
(5)
Derivation of Eq. (5) leads to the Legendre transformation, which is expressed as
α = dτ (q) / dq
(6)
When (q) and q are known, multifractal spectrum f()~ can be figured out from the combination of Eqs. (5) and (6).
Figure 2. Diagram of multifractal spectrum f()~.
Figure 3. Determined tumor lesion multifractal spectrum of patient No. 5.
Fig. 2 is a typical profile of multifractal spectrum f()~, basically in the shape of bell, which contains the following significant parameters: 1) spectral width =2 – 1, reflecting subset singularity; 2) fractal dimension difference of the minimum and maximum probability subsets f=f(2) – f(1); 3) singular factor H, corresponding to Hurst index H; and 4) left slope, left tangent,
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right slope and right tangent. Slope and tangent parameters are not completely independent, related to spectral width and Hurst index H. Therefore, only the impacts of different cases on , f, H, and H are concerned in this paper. Generally speaking, according to mass multifractality theory, positive and negative q values correspond to high-density and low-density region respectively. For DCE MRI of tumorous lesion, density region closely correlates to tumor.angiogenesis around lesion. Therefore, a large positive q corresponds to low signal region in MR images due to insufficient contrast enhancement, meaning that less blood-supply region is adjacent to 1 at the left end of spectrum after contrast enhancement, while a large negative q indicating high-signal region owing to adequate contrast enhancement. Namely, significant blood-supply region is close to 2 at the right end of spectrum. The shape and parameters of multi-fractal spectrum can serve as indicators for the conditions of lesion blood-supply, especially before and after therapy. 3. Results 3.1. Breast MR images Seven tumor lesions of 5 patients, 3 in left breast and 4 in right breast, were observed and presented as nodular or mass lesions in MRI. The maximum diameter of 11-75 mm, with an average of 35.4 mm, was determined in transverse T2-weighted MRI before treatment. Neoadjuvant chemotherapy for cases of four patients was followed up using MRI. Two of them were conducted once and the others twice. The chemotherapy efficacy was assessed to be PR of three patients and SD of the rest one. No new lesions were found in all cases after chemotherapy. All initial diagnoses and follow-ups are shown in Table 2. 3.2. Multifractal spectrum analysis Multifractal spectrum f()~ of malignant tumor lesion in patient No. 5 was determined and shown in Fig. 3. The asymmetric bell-shape indicates that the blood-supply around lesion exhibits mass multifractal characteristics. Spectral profiles at different time T were more dispersed at right-end than at left-end, implying that T has less effect on low signal region (left end 1, region devoid of blood supply) of MR image than on high signal region after injecting contrast medium. Table 2 MRI initial diagnosis and follow-up No. 1 2
3 4
5
MRI Examination * 1 2 1 2 3 1 2 1 2 3 1
Maximum tumor diameter (mm) 27 16 23 24 25 41 23 75 50 45 11
Efficacy evaluation* PR
* 1: Initial MRI; 2: First MRI follow-up; 3: Second MRI follow-up
SD
PR PR
PR
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Figure 4. Multifractal spectra f()~ of malignant tumor before and after chemotherapy.
Figure 5. Boxplot-data comparison of spectral width among all patients with malignant tumors.
(right end 2, region rich in blood supply). This is probably related to the transient concentration of contrast medium in blood-supply-rich region due to the high permeability of contrast medium caused by rich blood supply.
Figure 6. Comparison of time-DCE-MRI signal intensity curves of a patient with malignant tumor before and after chemotherapy. Inset indicates the location of lesion.
Fig. 4 exhibits the multifractal spectra f()~ of patients No. 2 and No. 4 before and after chemotherapy. The spectral shape and width of No. 4 significantly increased after treatment, while the spectral width of No. 2 hardly varied, although spectral shape had changed significantly. Fig. 5 shows the boxplot data of spectral width distribution of all lesions. As it is shown, among the four patients having received neoadjuvant chemotherapy, three had received chemotherapy efficacy PR and a wider spectral width after chemotherapy, and one had received chemotherapy efficacy SD with slight variation of spectral width .
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In addition to , the other three parameters f, H, and H were also measured and calculated. f of all patients had changed irregularly and drastically. H was 2 constantly and H fluctuated slightly between 1.96 and 2.09. Therefore, the variation of patients who received neoadjuvant chemotherapy is preferred in current investigation. It is demonstrated in Fig. 6 that time-DCE-MRI signal intensity curves of a patient with malignant tumor before and after chemotherapy. The slopes of DCE-MRI curves decreased from 7.75 (before treatment) to 3.5 (after treatment), indicating that tumor blood supply had been significantly reduced after treatment. 4. Discussion Biological tissues display heterogeneous mass density distribution during growth process such as angiogenesis. Generally, the heterogeneity of tissue is related with general features, e.g. heterogeneity and texture of tissue. Concerning the concept of mass-fractality, the heterogeneity has been addressed from another aspect in this preliminary study where heterogeneity is correlated to some extent with the distribution of computed fractal dimensions based on the transmittance intensity of incident ray through observed tissues [5, 6, 10]. Simple fractal measurement and analysis mainly deal with the fractal dimension of tissue boundaries with the aid of average approximate analytical approaches, which describes a sort of self-similarity of different homogeneous geometric tissues that satisfy both scale invariance in terms of homogeneous mass distribution and averaging in spatial scale. Thus, partial valuable microscopic information inside tissue has been ignored. Multifractal spectrum analysis is a complex of numerous scaling indices of fractal structures, which presents heterogeneous or singular self-similarity of objects or tissues consisting of heterogeneously distributed mass in multi-scale analysis. It could be used to study features of entire system based on local information and reveal scale features of self-similarity of heterogeneous geometric tissues on multi-scales. There is only one parameter involved in simple fractal spectrum analysis-fractal dimension DF to identify systematic characteristics while multifractal spectrum analysis adopts a spectrum of fractal dimensions. Selection of parameters to describe heterogeneous characteristics of subjects is relevant to specific research and algorithm. For general biomedical issues, spectral width is usually selected as characteristic parameter because it is directly related to mass density of local tissue (signal intensity, contrast medium concentration, etc.) and the amplitude varies evidently and regularly. In terms of biological metrology, single fractal character caused by reduction of singularity difference inside organisms is considered to be disease indicator. Preliminary work of Derado et al. [6] has shown that normal breast tissues tend to be more singular than tumor tissues, which means normal tissue has a greater spectral width than tumor tissue does. Spectral width of lesions assessed to be PR increased after chemotherapy while that of SD changed only slightly, indicating that variation of spectral width before and after treatment is in accordance with responsiveness of tumor to chemotherapy. Multifractal singularity actually indicates the heterogeneous difference inside a tumor tissue on different scales, and it differs from heterogeneity of a tumor tissue in common sense because the latter only reflects the inhomogeneity assessed on single over-all scale. It is well known that tumor microcirculation has typical fractal features and comprehensively represents the self-similarity and singularity of blood supply distribution on different scales [6, 10]. After treatment, blood vessel system inside and around tumor has been damaged, which directly results in slope declination of time-signal intensity curves as shown in Fig. 6 and variance of tissue homogeneity and singularity closely related to blood supply, causing the alteration of spectrum
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width before and after treatment. The pathological mechanism of this phenomenon probably includes destruction of tumor microcirculation, reduction in blood perfusion, apoptosis and necrosis of tumor cells. RECIST is a crucial evaluation standard for chemotherapeutic efficacy of solid tumors based on tumor size (maximum diameter or volume). Because solid tumor size usually changes after internal alteration, it is necessary to find an accurate method to detect early variation of interior tumor. It is observed in our current preliminary study that variation of is in correspondence with RECIST, implying spectral width serving as an indicator to evaluate chemotherapeutical efficiency. Limitation of current study. (1) As a pilot study, there are only five cases for investigation so that it is insufficient for statistical significance. (2) Because the variation of spectral width during chemotherapy has not been investigated in this paper, the fact that whether could be applied to evaluate chemotherapy efficacy prior to RECIST and serve as a new early evaluation indicator still needs to be further studied. 5. Conclusion Temporospatial multifractal spectrum analysis of breast MR images (3D space plus time lapse) can reveal the interior growth characteristics of lesions more appropriate than conventional heterogeneity and texture analysis does. Especially, it is effective in observing the dynamic effects of contrast medium and assisting tumor blood supply distribution analysis at different treatment stages. Spectral width of multifractal spectrum is closely correlated to the reduction of tumor malignant level after chemotherapy, and may serve as a hopeful quantitative indicator in evaluating therapeutic performance. 6. Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant No. 11274394), the Natural Science Foundation of Guangdong Province (Grant No. S2012010010542), the Fundamental Research Funds for the Central Universities (Grant No. 11lgjc12), and Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 20110171110023). References [1] [2] [3] [4] [5] [6] [7] [8]
A. Kikuchi, N. Unno, M. Shiba, S. Sunagawa, Y. Ogiso and Y. Taketani, Multifractal Description of the Maternal Surface of the Placenta, Gynecologic and obstetric investigation, 66 (2008), 127-133. A. Kikuchi, S. Kozuma and Y. Taketani, 3-D. fractal tumor growth of epithelial ovarian cancer, European journal of gynaecological oncology 27 (2006),561. A. Yasugib and Y. Taketanib, Fractal analysis of surface growth patterns in endometrioid endometrial adenocarcinoma, Gynecol Obstet Invest, 58 (2004), 61-67. B.B. Mandelbrot, The fractal geometry of nature, W.H. Freeman and Co, New York, 1982. Fratacls improve detection of breast cancer.(2006) http//www,mtbeurope,info/news/2006/608007.html G. Derado, K. Lee, O. Nicolis, F. Bowman, M. Newell and B. Vidakovic, Wavelet-based 3-D multifractal spectrum with applications in breast MRI images, Bioinformatics Research and Applications 4983 (2008), 281-292. J.W. Baish and R.K. Jain, Fractals and cancer, Cancer research, 60 (2000), 3683. M. Haraa, S. Sunagawaa and S. Ogisob, Fractal analysis of the maternal surface of the placenta preliminary report, Gynecol Obstet Invest, 63 (2007), 229-233.
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P. Therasse, S.G. Arbuck, E.A. Eisenhauer, J. Wanders, R.S. Kaplan, L. Rubinstein, J. Verweij, Van M. Glabbeke, Van and M.C. Christian, New guidelines to evaluate the response to treatment in solid tumors, Journal of the National Cancer Institute, 92 (2000), 205. R. Lopes and N. Betrouni, Fractal and multifractal analysis: A review, Medical Image Analysis, 13 (2009), 634-649. T.E. Yankeelov, M. Lepagea, A. Chakravarthy, E.E. Broome, K.J. Niermann, M.C. Kelley, I. Meszoely, I.A. Mayer, C.R. Herman, K. McManus, R.R. Price and J.C. Gore, Integration of quantitative DCE-MRI and ADC mapping to monitor treatment response in human breast cancer initial results, Magnetic resonance imaging 25 (2007), 1-13. T. Takahashi, T. Murata, K. Narita, T. Hamada, H. Kosaka, M. Omori, K. Takahashi, H. Kimura and Y. Wada, Multifractal analysis of deep white matter microstructural changes on MRI in relation to early-stage atherosclerosis, NeuroImage 32 (2006), 1158-1166. T. Takahashi, T. Murata, M. Omori, H. Kimura, H. Kado, H. Kosaka, K. Takahashi and Y. Wada, Quantitative evaluation of magnetic resonance imaging of deep white matter hyperintensity in geriatric patients by multifractal analysis, Neuroscience letters, 314 (2001), 143-146. V. Velanovich, Fractal Analysis of Mammographic Lesions: A feasibility study quantifying the difference between benign and malignant masses, Breast cancer research and treatment 49 (1998), 245-249. Y. Shimizu, M. Barth, C. Windischberger and S. Thurner, Wavelet-based multifractal analysis of fMRI time series, NeuroImage 22 (2004), 1195-1202. Y. Xia and R. Zhao, Morphology-based multifractal estimation for texture segmentation, Image Processing, IEEE. Transactions 15 (2006), 614-623.
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Evaluation of breast cancer chemotherapy efficacy with multifractal spectrum analysis of magnetic resonance image Li LIa,b , Wen-yong HUa, Li-zhi LIUb, Ya-chun PANGa and Yuan-zhi SHAO a,* a
School of Physics and Engineering, Sun Yat-sen University, Guangzhou 510275, China State Key Laboratory of Oncology in Southern China, Imaging Diagnostic and Interventional Center, Cancer Center, Sun Yat-sen University, Guangzhou 510060, China b
Abstract. Multifractal spectrum analysis of dynamic contrast enhanced (DCE) breast MR images was used to establish a new quantitative analysis method for solid tumor blood perfusion and to explore its applicability in evaluating efficacy of breast cancer chemotherapy. Five randomly selected patients suffering from newly diagnosed malignant breast nodule lesions were enrolled in this study, and four of them were treated with neoadjuvant chemotherapy. Their DCE breast MR images were collected before and after treatment. Chemotherapeutic efficacy was analyzed using international response evaluation criteria for solid tumors (RECIST). Sandbox method for statistical number density was employed to measure and calculate multifractal spectra of DCE breast MR images with spatiotemporal characteristics. Multifractal spectral data of malignant lesions before and after chemotherapy were compared. Multifractal spectra of malignant lesions show an asymmetric bell-shape. Chemotherapy efficacy was assessed to be partial remission (PR) for three patients and their multifractal spectral width significantly increased after chemotherapy while to be stable disease (SD) for other patient and of her changed slightly. Multifractal spectral width correlates with blood-supply condition of tumor lesion before and after chemotherapy, providing a potential suitable characteristic parameter for evaluating chemotherapeutic efficacy quantitatively. Key words: Breast cancer, magnetic resonance imaging, multifractal spectrum, image processing
1. Introduction Multifractal spectrum analysis is an important method and tool in nonlinear science, mainly used to study a class of irregular, chaotic, complex systems with local and global similarities in multi-scale analysis [4]. Multifractal spectrum analysis reveals the characteristics of self-similar scaling of heterogeneous geometric objects or tissues in biomedicine on multiple scales. As a new method for data-mining and medical-info-processing, fractal analysis has been finding its application in medical image analysis [1-3, 6, 8, 12-16] to quantitatively evaluate the correlation of such changes in MRI as: (1) brain gray matter and white matter with atherosclerosis and Alzheimer diseases [12,13,15]; (2) tumor growth of epithelial ovarian cancer and endometrial adenocarcinoma [2, 3]; (3) maternal surface of placenta [1,8]. *
Corresponding author. Email: [email protected], Tel: +86-20-84110399, Fax: +86-20-87343295
0959-2989/14/$27.50 © 2014 – IOS Press and the authors. All rights reserved
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L. Li et al. / Evaluation of breast cancer chemotherapy efficacy with multifractal spectrum analysis
Fractal approach has been reported in studies on breast image analysis. Velanovich applied simple fractal method, for the first time, to analyzing mammographic lesions and figured out that the mean composite fractal dimension of malignant lesions was higher than that of benign lesions [14]. Derado et al. have reported research of multi-fractal image analysis on breast MR images of healthy human body and patients using wavelet processing approach [6]. However, Derado et al merely compared MR images at different times before and after injecting contrast medium, instead of before and after treatment. Compared with conventional X-ray mammography, MRI is more appropriate for diagnosing and staging breast cancer because of its higher resolution for soft tissues and much higher detection rate for breast cancer. After injecting contrast medium, dynamic contrast enhanced (DCE) breast MRI can reflect the blood perfusion of breast lesions, and thus can be used for evaluating early breast responses to chemotherapy [11]. This study analyzed and compared the variance of multi-fractal spectra of DCE breast MRI of malignant lesions before and after chemotherapy, establishing a new method to quantitatively analyze DCE MRI images and evaluate micro-circular distribution heterogeneity, and extended its application in evaluating the responsiveness of breast cancer to chemotherapy. 2. Materials and Methods This study was performed with approval from the institutional ethics committee. 2.1. Clinical and image data Five randomly selected patients who were admitted to the Cancer Center of Sun Yat-Sen University from December, 2007 to October, 2009 for DCE breast MRI and diagnosed pathologically with breast cancer were enrolled in our study, aged from 32 to 59. The maximum diameter of their lesions was of 11-75 mm. Four of them underwent preoperative neoadjuvant chemotherapy and accepted DCE MRI examination within 6 weeks. A surgical resection was operated on all patients. Three were treated with radical mastectomy or modified radical mastectomy, and the others were treated with segment resection. Clinical data of all patients are listed in Table 1. Four patients received neoadjuvant chemotherapy follow-up MRI for 4-8 weeks after the completion of chemotherapy to assess chemotherapy efficacy using international RECIST [9], and the efficacy was classified as complete remission (CR), partial remission (PR), stable disease (SD) or progression disease (PD). 2.2. Image data processing A four-dimensional (3 spatial dimension plus 1 temporal dimension) database is composed of a large number of tomographic images constructed using sagittal vibrant multi-temporal DCE MRI. Its image array consisted of sagittal slice images with 512512 pixels and slice thickness of 3.4 mm. Fig. 1 shows DCE breast MRI array of a certain patient before and after injecting contrast medium into lesion, which clearly demonstrates variation in signal intensity of tumor images at different time T with an interval of 20 s.
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2.3. Algorithm to measure and calculate multifractal spectrum Images were segmented on multiple scales using a conventional sandbox method. The length of sandbox is set as . Under different subset scales, the number-density distribution probability of pixels Pi() is assumed to satisfy the following power-law function:
Pi (ε ) ~ ε α
(1)
Table 1 Patient’s clinical profile No. 1
Age 32
TNM Stage T2N0M0
Pathological Grade Grade II Invasive ductal carcinoma
2
48
T4BN1M0
3
46
T4N2M0
4
59
T3N0M0
Grade III Invasive ductal carcinoma Grade III Invasive ductal carcinoma Grade II-III Invasive ductal carcinoma
5*
46
T1NOM0
Chemotherapy Program Two courses of TEC followed by surgery and four courses of TEC MMC + CBP + THP + Taxotere FAC plus Herceptin chemotherapy Six courses of FAC plus Herceptin + Doalaxol chemotherapy
Drug Administration Intravenously
Artery perfusion Intravenously Intravenously
Grade I Invasive ductal carcinoma * The patient did not accept chemotherapy.
Figure 1. Breast MRI array diagram (a) before and (b) after treatment. Sequence number of sagittal spatial scanning series was 30 and time sequence number was 12.
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where is a subset-related singular index representing heterogeneity degree of number density distribution. If intra-subset density number N() and the length of sandbox subset scale meet
N (ε ) ~ ε − f (α ) |ε →0
(2)
then f()~ constitutes a multifractal spectrum function, which can be used to quantitatively describe growth characteristics of a fractal object at different levels and stages. Multifractal spectrum is commonly worked out using conventional partition function q(), which can be calculated from the weighted sum of q power of density distribution probability Pi(), i.e.
χ q (ε ) = ¦ i Pi (ε ) = ε τ ( q ) q
(3)
where (q) is density index and q usually takes a series of integers ranging from -50 to 50 for a variety of weighted sums. Generalized fractal dimension Dq is defined with (q) by
Dq = τ (q) /(q − 1) = ln χ q (ε ) /[(q − 1) ln ε ] |ε →0
(4)
Dq at q=0 and q=1 correspond to a simple fractal dimension D0 and information dimension D1 respectively. If system is of multifractality, it can be derived from Eqs. (3) and (4) that , q, (q) and f() satisfy the following relationship
τ (q) = α q − f (α )
(5)
Derivation of Eq. (5) leads to the Legendre transformation, which is expressed as
α = dτ (q) / dq
(6)
When (q) and q are known, multifractal spectrum f()~ can be figured out from the combination of Eqs. (5) and (6).
Figure 2. Diagram of multifractal spectrum f()~.
Figure 3. Determined tumor lesion multifractal spectrum of patient No. 5.
Fig. 2 is a typical profile of multifractal spectrum f()~, basically in the shape of bell, which contains the following significant parameters: 1) spectral width =2 – 1, reflecting subset singularity; 2) fractal dimension difference of the minimum and maximum probability subsets f=f(2) – f(1); 3) singular factor H, corresponding to Hurst index H; and 4) left slope, left tangent,
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167
right slope and right tangent. Slope and tangent parameters are not completely independent, related to spectral width and Hurst index H. Therefore, only the impacts of different cases on , f, H, and H are concerned in this paper. Generally speaking, according to mass multifractality theory, positive and negative q values correspond to high-density and low-density region respectively. For DCE MRI of tumorous lesion, density region closely correlates to tumor.angiogenesis around lesion. Therefore, a large positive q corresponds to low signal region in MR images due to insufficient contrast enhancement, meaning that less blood-supply region is adjacent to 1 at the left end of spectrum after contrast enhancement, while a large negative q indicating high-signal region owing to adequate contrast enhancement. Namely, significant blood-supply region is close to 2 at the right end of spectrum. The shape and parameters of multi-fractal spectrum can serve as indicators for the conditions of lesion blood-supply, especially before and after therapy. 3. Results 3.1. Breast MR images Seven tumor lesions of 5 patients, 3 in left breast and 4 in right breast, were observed and presented as nodular or mass lesions in MRI. The maximum diameter of 11-75 mm, with an average of 35.4 mm, was determined in transverse T2-weighted MRI before treatment. Neoadjuvant chemotherapy for cases of four patients was followed up using MRI. Two of them were conducted once and the others twice. The chemotherapy efficacy was assessed to be PR of three patients and SD of the rest one. No new lesions were found in all cases after chemotherapy. All initial diagnoses and follow-ups are shown in Table 2. 3.2. Multifractal spectrum analysis Multifractal spectrum f()~ of malignant tumor lesion in patient No. 5 was determined and shown in Fig. 3. The asymmetric bell-shape indicates that the blood-supply around lesion exhibits mass multifractal characteristics. Spectral profiles at different time T were more dispersed at right-end than at left-end, implying that T has less effect on low signal region (left end 1, region devoid of blood supply) of MR image than on high signal region after injecting contrast medium. Table 2 MRI initial diagnosis and follow-up No. 1 2
3 4
5
MRI Examination * 1 2 1 2 3 1 2 1 2 3 1
Maximum tumor diameter (mm) 27 16 23 24 25 41 23 75 50 45 11
Efficacy evaluation* PR
* 1: Initial MRI; 2: First MRI follow-up; 3: Second MRI follow-up
SD
PR PR
PR
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Figure 4. Multifractal spectra f()~ of malignant tumor before and after chemotherapy.
Figure 5. Boxplot-data comparison of spectral width among all patients with malignant tumors.
(right end 2, region rich in blood supply). This is probably related to the transient concentration of contrast medium in blood-supply-rich region due to the high permeability of contrast medium caused by rich blood supply.
Figure 6. Comparison of time-DCE-MRI signal intensity curves of a patient with malignant tumor before and after chemotherapy. Inset indicates the location of lesion.
Fig. 4 exhibits the multifractal spectra f()~ of patients No. 2 and No. 4 before and after chemotherapy. The spectral shape and width of No. 4 significantly increased after treatment, while the spectral width of No. 2 hardly varied, although spectral shape had changed significantly. Fig. 5 shows the boxplot data of spectral width distribution of all lesions. As it is shown, among the four patients having received neoadjuvant chemotherapy, three had received chemotherapy efficacy PR and a wider spectral width after chemotherapy, and one had received chemotherapy efficacy SD with slight variation of spectral width .
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In addition to , the other three parameters f, H, and H were also measured and calculated. f of all patients had changed irregularly and drastically. H was 2 constantly and H fluctuated slightly between 1.96 and 2.09. Therefore, the variation of patients who received neoadjuvant chemotherapy is preferred in current investigation. It is demonstrated in Fig. 6 that time-DCE-MRI signal intensity curves of a patient with malignant tumor before and after chemotherapy. The slopes of DCE-MRI curves decreased from 7.75 (before treatment) to 3.5 (after treatment), indicating that tumor blood supply had been significantly reduced after treatment. 4. Discussion Biological tissues display heterogeneous mass density distribution during growth process such as angiogenesis. Generally, the heterogeneity of tissue is related with general features, e.g. heterogeneity and texture of tissue. Concerning the concept of mass-fractality, the heterogeneity has been addressed from another aspect in this preliminary study where heterogeneity is correlated to some extent with the distribution of computed fractal dimensions based on the transmittance intensity of incident ray through observed tissues [5, 6, 10]. Simple fractal measurement and analysis mainly deal with the fractal dimension of tissue boundaries with the aid of average approximate analytical approaches, which describes a sort of self-similarity of different homogeneous geometric tissues that satisfy both scale invariance in terms of homogeneous mass distribution and averaging in spatial scale. Thus, partial valuable microscopic information inside tissue has been ignored. Multifractal spectrum analysis is a complex of numerous scaling indices of fractal structures, which presents heterogeneous or singular self-similarity of objects or tissues consisting of heterogeneously distributed mass in multi-scale analysis. It could be used to study features of entire system based on local information and reveal scale features of self-similarity of heterogeneous geometric tissues on multi-scales. There is only one parameter involved in simple fractal spectrum analysis-fractal dimension DF to identify systematic characteristics while multifractal spectrum analysis adopts a spectrum of fractal dimensions. Selection of parameters to describe heterogeneous characteristics of subjects is relevant to specific research and algorithm. For general biomedical issues, spectral width is usually selected as characteristic parameter because it is directly related to mass density of local tissue (signal intensity, contrast medium concentration, etc.) and the amplitude varies evidently and regularly. In terms of biological metrology, single fractal character caused by reduction of singularity difference inside organisms is considered to be disease indicator. Preliminary work of Derado et al. [6] has shown that normal breast tissues tend to be more singular than tumor tissues, which means normal tissue has a greater spectral width than tumor tissue does. Spectral width of lesions assessed to be PR increased after chemotherapy while that of SD changed only slightly, indicating that variation of spectral width before and after treatment is in accordance with responsiveness of tumor to chemotherapy. Multifractal singularity actually indicates the heterogeneous difference inside a tumor tissue on different scales, and it differs from heterogeneity of a tumor tissue in common sense because the latter only reflects the inhomogeneity assessed on single over-all scale. It is well known that tumor microcirculation has typical fractal features and comprehensively represents the self-similarity and singularity of blood supply distribution on different scales [6, 10]. After treatment, blood vessel system inside and around tumor has been damaged, which directly results in slope declination of time-signal intensity curves as shown in Fig. 6 and variance of tissue homogeneity and singularity closely related to blood supply, causing the alteration of spectrum
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width before and after treatment. The pathological mechanism of this phenomenon probably includes destruction of tumor microcirculation, reduction in blood perfusion, apoptosis and necrosis of tumor cells. RECIST is a crucial evaluation standard for chemotherapeutic efficacy of solid tumors based on tumor size (maximum diameter or volume). Because solid tumor size usually changes after internal alteration, it is necessary to find an accurate method to detect early variation of interior tumor. It is observed in our current preliminary study that variation of is in correspondence with RECIST, implying spectral width serving as an indicator to evaluate chemotherapeutical efficiency. Limitation of current study. (1) As a pilot study, there are only five cases for investigation so that it is insufficient for statistical significance. (2) Because the variation of spectral width during chemotherapy has not been investigated in this paper, the fact that whether could be applied to evaluate chemotherapy efficacy prior to RECIST and serve as a new early evaluation indicator still needs to be further studied. 5. Conclusion Temporospatial multifractal spectrum analysis of breast MR images (3D space plus time lapse) can reveal the interior growth characteristics of lesions more appropriate than conventional heterogeneity and texture analysis does. Especially, it is effective in observing the dynamic effects of contrast medium and assisting tumor blood supply distribution analysis at different treatment stages. Spectral width of multifractal spectrum is closely correlated to the reduction of tumor malignant level after chemotherapy, and may serve as a hopeful quantitative indicator in evaluating therapeutic performance. 6. Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant No. 11274394), the Natural Science Foundation of Guangdong Province (Grant No. S2012010010542), the Fundamental Research Funds for the Central Universities (Grant No. 11lgjc12), and Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 20110171110023). References [1] [2] [3] [4] [5] [6] [7] [8]
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