Multiscale Nakagami parametric imaging for improved liver tumor localization

MUL TISCALE NAKA GAMI P ARAMETRIC IMA GING FOR IMPR O VED LIVER TUMOR LOCALIZA TION Omar S. Al-Kadi King Abdullah II School for IT Uni versity of Jordan Amman 11942, Jordan ABSTRA CT Effecti ve ultrasound tissue characterization is usually hin- dered by complex tissue structures. The interlacing of speckle patterns complicates the correct estimation of backscatter distribution parameters. Nakagami parametric imaging based on localized shape parameter mapping can model different backscattering conditions. Howe ver , performance of the con- structed Nakagami image depends on the sensitivity of the estimation method to the backscattered statistics and scale of analysis. Using a fixed focal region of interest in estimating the Nakagami parametric image would increase estimation variance. In this work, localized Nakagami parameters are es- timated adaptively by means of maximum likelihood estima- tion on a multiscale basis. The v arying size k ernel inte grates the goodness-of-fit of the backscattering distrib ution param- eters at multiple scales for more stable parameter estimation. Results sho w improved quantitative visualization of changes in tissue specular reflections, suggesting a potential approach for impro ving tumor localization in lo w contrast ultrasound images. Index T erms — Nakagami imaging, tumor detection, maximum likelihood estimation, li ver tumor , RF en velope 1. INTRODUCTION Ultrasound parametric imaging is gaining increased inter- est as an effectiv e way for quantitativ e tumor characteriza- tion. Changes in properties of soft tissue texture, e.g. liv er parenchyma, can be reflected in the radio-frequency (RF) backscattered statistics as different Rayleigh distributions [1]. Howe ver parametric estimation is not a straightforward process and is generally faced with increased estimation v ari- ance, especially for complex speckle patterns. This may obscure abnormal tissue structures, e.g. tumors and fibrosis, which are deemed important for early diagnosis [2]. The analytical simplicity of the bi-parametric Nakagami distribution model, along with its goodness-of-fit with the en- velope histogram of the ultrasound-backscattered signal [3], can be attractiv e for tissue characterization [1, 4, 5]. The shape of the Nakagami distribution is specified by the µ pa- rameter corresponding to the local concentration of scatterers, and the amount of spread (i.e. the local backscattered en- ergy) is represented by the scale parameter ω . Different con- ditions of the RF env elope statistics can be achiev ed by vary- ing the µ parameter . V alues of µ between 0 and 1 yield pre- Rayleigh and Rayleigh distrib utions. The Rayleigh distribu- tion case ( µ = 1) resembles of having a lar ge number of ran- domly distrib uted scatterers, and in the case of high de gree of variance the distribution conforms to pre-Rayleigh ( µ < 1) . For a mixture of random and periodically located scatterers, the RF en velope statistics becomes a post-Rayleigh distrib u- tion ( µ > 1) . The map of local µ parameter v alues – that correspond to tissue properties – is normally considered in constructing the Nakagami parametric image. The estimated Nakagami parameters as a function of the backscattered en ve- lope statistics has sho wn to be a reliable tool for quantitative visualization of tissue structure changes [4, 5, 6]. Previous work has improved local window-based µ pa- rameter estimation to generate the Nakagami parameter map from en velopes of raw ultrasound signals [7, 8, 9]. As using a gamma k ernel density estimation to achie ve a smooth estima- tion of the distribution from small fixed-size windows [7], or by using a number of windows ha ving a size 3 times the pulse length of the ultrasound [8], or by summing and averaging multiple Nakag ami parametric images generated using dif fer- ent sliding square window sizes (7-10 times the transducer pulse length) [9]. Howe ver challenges persist with fix ed-size window approaches. A focal region of interest (i.e. using small windows) enables enhanced resolution of the Nakagami parametric image, but large tissue structures require a large spatial scale to achie ve stable parameter estimation. There- fore parameter smoothing might not suffice when prominent parts of the examined tissue structure is truncated or located outside the window borders. On the other hand, using large window sizes for summation and av eraging may affect the re- sults when compounded with windows of smaller-sizes, and hence af fecting the reliability of the constructed parametric image resolution. In this work an alternati ve approach of employing a multi- scale kernel-based technique to model the backscattering dis- tribution statistics is proposed. The focal region of inter - est should be large enough to hav e suf ficient tissue variation, while being also as small to a void inclusion of irrelev ant tex- tures from the nearby regions. The backscattered en velope from tissue was estimated v oxel-by-v oxel via Nakagami dis- tribution and subsequently used to generate optimized local parametric images for improved liv er tumor detection. The assumption is based on that tumor regions tend to have differ - ent backscattered distribution than normal tissue [10], and a localized approach based on a varying-size kernel can assist in better identifying the tumor speckle patterns. 2. METHODOLOGY The speckle pattern is dependent on the ultrasound wav e- length and underlying tissue structure. As the former factor is fixed in this case and the latter varies across the ultrasound image, a multiscale approach that can localize the different tissue structures would best suit the modeling of the backscat - tered en velope. A non-linear kernel is applied adaptively to characterize the tissue speckle patterns. The estimation of the best parametric Nakagami maps from v arying size kernels is described as follows: 2.1. Multiscale k ernel localization Let V be an order set of constructed en velope detected RF images I i ( x, y ) , where i is a certain slice in the acquired v ol- ume, and P µ,ω ( x, y ) are the corresponding µ and ω paramet- ric images. The RF images are calculated from the en velope of the ultrasound-backscattered signal just before performing any intensity mapping and post processing filtering. This rep- resentation preserv es the ultrasound data unaltered while pro- viding better quantitativ e analysis, i.e. without the risk of los- ing information due to RF signal shaping. Then a set of v ary- ing size kernels K can be defined for each I i ( x, y ) , where K = { v 1 , v 2 , . . . , v k } , v j ∈ V and k = 1 / 8 of the size of I i ( x, y ) . Different focal regions are in vestigated in a multi- scale manner as in (1) by v arying the size of two non-negati ve integer v ariables a and b , which are used to center each local- ized kernel v j ( s, t ) with a different size of m × n on each vox el l in I i ( x, y ) of size M × N . P µ,ω ( x, y ) = a X s = − a b X t = − b v j ( s, t ) I i ( x + s, y + t )  k j  2 (1) where a = d m +2 2 e , b = d n +2 2 e , and m , n = 1 , 2 , . . . , k . 2.2. Modeling backscattered statistics The Nakag ami distrib ution N ( x ) is a gamma related distrib u- tion which is known for its analytical simplicity [5], and has Fig. 1 . Samples of simulated ultrasound speckle images rep- resenting: [left-right] fine texture (dense scatterers), coarse texture (sparse scatterers), heterogeneous texture (random scatterers), and homogeneous texture (periodic scatterers), re- ferring to, respectiv ely , phantoms D57, D30, D5, D37 in T a- ble 1. been proposed as a general model for ultrasonic backscatter- ing under different scattering conditions and scatterer densi- ties [3]. This distribution has the density function N ( x | µ, ω ) = 2  µ ω  µ 1 Γ ( µ ) x (2 µ − 1) e − µ ω x 2 , ∀ x ∈ R ≥ 0 (2) where x is the en velope of the RF signal and Γ ( · ) is the gamma function. If x has a Nakag ami distrib ution N ( x ) with parameters µ and ω , then x 2 has a gamma distribution Γ with shape µ and scale (ener gy) parameter ω / µ . Although there are other distributions exist in the literature for modeling ultra- sonic backscattering, the Nakagami probabilistic distribution was chosen for its simplicity and ability to characterize differ - ent scattering conditions ranging from pre- to post-Rayleigh [6]. Each voxel l in I i is adaptively transformed via K at different scales to its corresponding parametric Nakagami parameters by means of maximum likelihood estimation (MLE) forming a set of parametric vectors for each vox el l . The MLE ˆ θ ( v ) for a density function f  v l 1 , . . . , v l k | θ  when θ is a vector of parameters for the Nakagami distribution family Θ , estimates the most probable parameters ˆ θ ( v ) = ar g max θ D  θ | v l 1 , . . . , v l k  , where D ( θ | v ) = f ( v | θ ) ,θ ∈ Θ is the score function. Finally the goodness-of-fit is estimated via root mean square error for the calculated Nakagami pa- rameters θ m at different scales, giving the localized paramet- ric Nakagami images P µ,ω as summarized in Algorithm 1. The shape parametric image P µ is used for subsequent tissue characterization. Algorithm 1 Multiscale kernel localization Input: Set of ultrasound backscattered en velope images I i = { ( x 1 , y 1 . . . , x j , y j ) } Output: Localized ultrasound Nakagami shape and scale parametric images P µ , P ω for all vox els l in I i do for all localized kernels ν l 1 → ν l k do { Step1 } // Fit with a Nakagami distribution N ( x | µ, ω ) = 2  µ ω  µ 1 Γ( µ ) x (2 µ − 1) e − µ ω x 2 { Step2 } // Calculate Nakagami shape µ and scale ω parameters using maximum likelihood estimation as: ˆ θ ( ν ) = ar g max θ D  θ /ν l 1 , . . . , ν l k  where θ is a vector of parameters for the Nakagami distri- bution f amily f  ν l 1 , . . . , ν l k /θ  end for { Step3 } // Estimate goodness-of-fit of the determined Nakagami parameters θ m with the av erage RF signal θ α within set of localized kernels K  P µ 1 ,ω 1 , . . . , P µ j ,ω j  = argmin      s n P s =2 ( θ m − θ α ) 2 n      end for retur n P µ , P ω 3. RESUL TS 3.1. Simulated ultrasound speckle images Simulation experiments were performed on 11 different ultra- sound speckle images generated from corresponding texture images adopted from the Brodatz texture alb um [11]. The ul- trasound speckle images were synthesized with given te xtures as the initial point scatterer image, gi ving clinical echo alike images that resemble tissue scatterers in appearance. V ari- ous specular reflection conditions of tissue texture boundaries are synthesized ranging from fine to coarse (i.e. high density to lo w density scatterers) and from heterogeneous to homo- geneous (i.e. random to periodic scatterers alignment), c.f. Fig. 1. The windo w-based µ parameter estimation methods: gamma k ernel function (GKF) [7], windows-modulated com- pounding (WMC) [9] and the proposed multiscale kernel localization (MKL) methods where applied to the simulated ultrasound speckle images, and performance quantitatively compared with the µ parameters estimated from the original synthetic texture images (ground-truth), as shown in T able 1. Results show that the MKL method gi ves more stable µ parameter estimation in nearly all cases. 3.2. Liver tumor detection In order to quantitativ ely e v aluate the robustness of the 3 different Nakagami parametric image estimation methods, T able 1 . Mean absolute dif ference comparison of estimated Nakagami shape parametric images against ground-truth. Nakagami shape estimation methods Phantom GKF WMC MKL D5 0.29 0.10 0.04 D11 0.18 0.09 0.04 D13 0.40 0.17 0.01 D30 0.51 0.33 0.07 D37 0.29 0.09 0.05 D57 0.30 0.03 0.07 D71 0.16 0.09 0.02 D88 0.64 0.46 0.12 D91 0.55 0.47 0.19 D99 0.49 0.28 0.04 D101 0.62 0.13 0.11 they were applied to real ultrasound li ver tumor images ob- tained using a diagnostic ultrasound system (z.one, Zonare Medical Systems, Mountain V iew , CA, USA) with a 4 MHz curvilinear transducer and 11 MHz sampling. The whole RF ultrasound image (without log-compression and filtering) was used in generating the Nakagami parametric images, so the sensitivity of methods to various tissue scatterers can be in- vestigated. Fig. 2 shows an ultrasound liv er tumor image and corresponding Nakagami parametric images via the 3 differ - ent methods. T umor tissue specular reflections tend to appear more prominent from the background tissue using the MKL approach as compared to GFK and MWC window-based µ parameter estimation methods. The dif ferent kernel sizes used in generating the Nakagami parametric image using the MKL method is shown in Fig. 3. 4. DISCUSSION T umor texture tends to be more heterogeneous as compared to normal tissue [12]. This property has been reported to be useful in tumor grading [12, 13, 14, 15] and assessing aggres- siv eness [16]. Howe ver tumor spatial and contrast resolution in ultrasound images is lo w as compared to other modalities. Modeling the RF backscattered en velope from li ver tissue re- quires an adaptive method that can effecti vely in vestigate tis- sue heterogeneity while reliably estimate the distribution pa- rameters. Different spatial v ariations exist in speckle patterns across the ultrasound image due to the Rayleigh scattering be- havior [1], and man y tissue structures are prone to lo w spatial contrast and displacement during successiv e image acquisi- tion. This makes the use of constant focal regions in estimat- ing the backscattering distribution parameters very limiting. Such approach may result in missing parts of the analyzed speckle pattern if the focal region was too small, or possibly (a) (b) (c) (d) Fig. 2 . (a) Clinical ultrasound B-mode image showing a li ver tumor (indicated by a yello w arro w), and corresponding Nak- agami shape parametric image using (b) WMC, (c) GKF , and (d) MKL methods. interlacing of irrele vant patterns from surrounding regions if the focal region was too large. Thereby subtle tissue structure (e.g. tumor regions in its early stages) could be obscured due to the presence of mixture of patterns. Different window sizes hav e di verse ef fects on the forma- tion of the Nakagami parametric image. The experiments on simulated and real ultrasound images demonstrated the need for an adaptive approach that can enhance image resolution without degrading smoothness, i.e. having stable parameter estimation. Stable performance was achiev ed using the MKL method when applied to div erse speckle patterns simulating different soft tissue conditions in clinical practice. An ex- ceptional case of D57 in T able 1 – which had a fine tissue structure – did not give the best stable µ estimation. The uni- form speckle pattern appearance across the D57 image texture would reduce the sensiti vity to texture variations of the adap- tiv e approach employed by MKL. Thus a variant spatial res- olution throughout the entire imaging field of vie w would not be best for this particular case [17]. Howe ver in clinical prac- tice, liv er tissue characterization in volv es analyzing the whole ultrasound image before the tumor is localized (cf. Fig. 2(a)), which means encountering regions with different tissue char- acteristics; thus a non-v arying window size may reduced the reliability of Nakagami imaging. The ability to rapidly and accurately identify tumor loca- Fig. 3 . Localized adaptiv e kernel sizes for Fig. 2(d). tion in ultrasound images is limited due to inherent low con- trast. Therefore ultrasound parametric imaging is normally applied to analyze the RF en velope statistics to giv e an indi- cation of the properties of tissue scatterers. Fig. 2(d) sho ws visual improv ement in the contrast between the specular re- flections of tissue boundaries (c.f. Fig. 2(b) and Fig. 2(c)), with a stronger parametric response in the localized tumor region from surrounding tissue. This could be attributed to the adaptiv e approach of the MKL method that integrates the goodness-of-fit of the backscattering distribution parameters at multiple scales before parameter estimation. Examining how the focal regions v ary in size as shown in Fig. 3, the MKL allows for the aggre gation of suf ficient vox els that w ould bet- ter represent the en velope statistics in order to highlight dif- ferences in tissue properties. Such localized multiscale neigh- borhood around each voxel contributes for the best resolution and improv ed smoothness. Finally , a number of challenges may arise with the em- ployment of Nakagami imaging in tumor segmentation, such as the presence of blood vessels, ducts and other connective tissues. Although these small areas might give signs on liver inflammation, they would rather degrade the local image res- olution and hence affect the smoothness of the parameter esti- mation. Also tumor spatial contrast varies according to depth and lev el of speckle artifacts. Such challenges would serve as future work for improving accurate se gmentation of tumor boundaries in Nakagami parametric images. 5. CONCLUSION Nakagami parametric imaging based on localized shape pa- rameter maps can model different backscattering conditions. Results sho w more stable estimation of the backscattering dis- tribution parameters within a varying size kernel by means of MLE. 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