Characterization of Tissue-specific Pre-log Bayesian CT Reconstruction by Texture-Dose Relationship

Abstract

Purpose:

Tissue texture has been recognized as biomarkers for various clinical tasks. In computed tomography (CT) image reconstruction, it is important but challenging to preserve the texture when lowering X-ray exposure from full- toward low-/ultralow-dose level. Therefore, this paper aims to explore the texture-dose relationship by one tissue-specific pre-log Bayesian CT reconstruction algorithm.

Methods:

To enhance the texture in ultralow-dose CT (ULdCT) reconstruction, this paper presents a Bayesian type algorithm, where shifted Poisson model is adapted to describe the statistical properties of pre-log data and a tissue-specific Markov random field (MRF) prior is used to incorporate tissue texture from previous full dose CT, thus called SP-MRFt algorithm. Utilizing the SP-MRFt algorithm, we investigated tissue texture degradation as a function of dose levels from full dose (100mAs/120kVp) to ultralow dose (1mAs/120kVp) by introducing a quantitative texture-based evaluation metrics.

Results:

Experimental results show the SP-MRFt algorithm outperforms conventional filtered back projection (FBP) and post-log domain penalized weighted least square MRFt (PWLS-MRFt) in terms of noise suppression and texture preservation. Comparable results are also obtained with shifted Poisson model with 7×7 Huber MRF weights (SP-Huber7). The SP-MRFt is a reasonably good tool to investigate the texture-dose relationship approaching the ultra-low dose end. The investigation on texture-dose relationship shows that the quantified texture measures drop monotonically as dose level decreases, and interestingly a turning point is observed on the texture-dose response curve.

Conclusions:

This important observation implies that there exists a minimum dose level, at which a given CT scanner (hardware configuration and image reconstruction software) can achieve without compromising clinical tasks. Moreover, the experiment results show that the variance of electronic noise has higher impact than the mean to the texture-dose relationship.

1. Introduction

As application of computed tomography (CT) becomes more popular for various clinical tasks, including screening early cancers, diagnosing abnormalities, quantifying treatment responses, optimizing interventions, etc., the concern about the associated ionizing radiation exposure also grows^1,2. This concern would not stop until the benefit vs. risk of using CT for a specific clinical task fully understood^3,4. Therefore, tremendous efforts are demanded to minimize the radiation exposure to subjects through hardware advancement and software development⁴. In the software effort, particularly algorithms developed for low-mAs (or low-dose) acquisition data, model based iterative reconstruction methods (MBIR) based on Bayesian theory has made a great progress^5,6 in recent years. Radiation could be reduced up to 80% using MBIR compared with traditional filtered back projection (FBP) reconstruction as reported in Aberle et al.⁷ and Liang et al.⁸.

Among MBIR algorithms, the pre-log data fidelity term under the Bayesian theory is currently considered as the most accurate model to deal with photon starvation problem^9–20, particularly for ultralow-mAs acquisition data (at a few mAs level), hereafter called ultralow-dose CT (ULdCT) imaging. It directly uses the transmission data from detector and can accurately model the statistical property of both X-ray quanta and electronic background noise in the Bayesian cost function. It also overcomes difficulties associated with log transformation of the post-log domain algorithms^21–26 in the photon starvation situation, like undefined negative values^21,22, weight estimation from the noisy data^23–26, etc. A study of Fu et al.¹⁸ based on both simulated data and clinical data has shown that pre-log data fidelity model can gain more quantitative accuracy than post-log data fidelity model in ULdCT reconstructions.

There are many types of a priori models explored for the MBIR algorithms under the Bayesian theory in order to suppress the low-mAs data noise through regional noise smoothing and edge sharpening^27–30. As the tissue textures have been recognized as important imaging biomarkers for various clinical tasks and playing an important role in computer aided detection and diagnosis^31–35, modeling the tissue textures as the a priori term for the Bayesian cost function has attracted great interests for us^36–40. We believe that it is clinically important to preserve tissue textures when lowering the X-ray dose level from currently accepted full-dose toward ultralow-dose which could be approached by sophisticated MBIR-type ULdCT Bayesian image reconstructions. To investigate the potential of tissue texture prior for Bayesian reconstruction of post-log low-mAs or low-dose CT (LdCT) imaging, we constructed a tissue-specific Markov random field prior (MRFt) extracted from previous full-dose CT (FdCT) scans consisting of textures of four major tissue types (i.e. lung, bone, fat and muscle across the chest) for lung imaging^36–40. Compared with many other MRF priors for regional noise smoothing and edge sharpening types, e.g. MRF with Huber weight for tradeoff between noise smoothing and edge sharpening²⁷, the MRFt takes into account tissue-specific inhomogeneity across the field of view (FOV) and has shown certain advantages in texture related studies^38,39. Targeting ULdCT texture enhancement, one main contribution of this work is the integration of tissue specific texture prior model with pre-log data fidelity model, denoted as SP-MRFt algorithm.

In addition to the above integration contribution, in this work we also introduced texture-based evaluation metrics for the dose characterization task based on the widely used Haralick features³¹. While many image quality measures have been used to quantify image degradation when dose level decreases, texture-based measures have not been widely explored although it has been recognized as a footprint of lesion evolution, ecology and treatment response^32,33,35,41. These texture metrics enable us to quantitatively investigate the relationship between dose level and texture change, which could provide insights for dosage optimization in clinical operations. In other words, the texture-dose relationship may reflect a minimal dose that a given CT scanner and image reconstruction software may achieve with clinically meaningful image quality. Some preliminary results are reported in Gao et al.⁴². The conference paper was an extended abstract submitted for the 2018 IEEE NSSMIC conference. This full paper provides a full description of method, experimental design and results analysis with more comprehensive evaluation.

The remainder of this paper is organized as follows. Section 2 describes the mathematical foundation and implementation of the SP-MRFt algorithms, followed by presentation of the texture-dose characterization strategy and experimental design. Experimental Results are reported in section 3. Discussion and conclusion are drawn in Section 4 and 5.

2. Materials and Methods

A. Bayesian Image Reconstruction Framework in Pre-log Domain

Given a set of acquired transmission data, denoted by a vector Y ∈ R^I×1, where I is the number of data elements, we are interested in a solution, denoted by a vector μ ∈ R^J×1 with J number of image voxels, which maximizes the posterior probability p(μ|Y). From the Bayesian theorem, we have:

$$p\left(\mathit{\mu }|Y\right)=\frac{p\left(Y|\mu \right)p\left(\mu \right)}{p\left(Y\right)},$$ (1)

where p(Y) becomes a constant when maximizing the posterior probability. The solution for Eq. (1) can be obtained by minimizing the objective function Φ, which is:

$$\Phi \left(\mu \right)=-L\left(\mu \right)+\beta \Theta \left(\mu \right)$$ (2)

where μ is the attenuation map, L(·) is the data fidelity likelihood function, Θ(·) is a prior function, which can be described by a MRF model, β is a weighting factor to balance the data fidelity term and the texture prior term. We will describe each term with more details in the following subsections.

B. Shifted Poisson Data Fidelity Model

Raw data from CT scanners could be modeled as combination of X-ray quanta and system electronic background noise. Because of the polychromatic nature of X-ray interaction and the energy integration in the detection, the X-ray quanta is a complicated mixture of compound Poisson distribution^6,43. In some study, e.g. Elbakri et al.¹⁶, X-ray spectrum was considered in the cost function. For simplicity, in this work, the X-ray quanta are approximated to follow Poisson statistics under mono-energetic assumption and the noise follows the Gaussian statistics. The measurement of X-ray detector can be expressed as:

$$Y_i^M=\mathit{\text{Poisson}}\left\{I_i^0\times \text{exp}\left(-l_i\right)\right\}+\mathit{\text{Gaussian}}\left\{m_{b,i},{\sigma }_{b,i}^2\right\}$$ (3)

where $Y_i^M$ is the measurement from CT detector bin i, $I_i^0$ represents the mean number of X-ray flux from the X-ray tube toward detector bin i before reaching the subject, l_i is the line integral of the attenuation coefficients along the projection line i, m_b,i and ${\sigma }_{b,i}^2$ are the mean and variance of the system electronic background noise. In general, m_b,i and ${\sigma }_{b,i}^2$ are calibrated so that all detector bins share the same response to some input energy and thus their subscript i may be ignored.

The shifted Poisson (SP) model was proposed to approximate data statistics in Eq. (1). Assuming the above-mentioned Poisson and Gaussian variables are statistically independent from each other, the mean and variance of the measurement at each bin are:

$$E\left(Y_i^M\right)=I_i+m_b;Var\left(Y_i^M\right)=I_i+{\sigma }_b^2,$$ (4)

where $I_i=I_i^0\times \text{exp}\left(-l_i\right)$. Following the description in Erdogan et al.¹⁴ and Whiting et al.⁴³, we introduce an artificial quantity $Y_i^A:Y_i^A=Y_i^M-m_b+{\sigma }_b^2$, of which the mean and variance can be easily proved identical. It could be assumed that if a random variable has the same mean and variance, it could be described as Poisson distribution approximately. Mathematically, this can be expressed as:

$${\left[Y_i^A\right]}_{+}\approx \mathit{\text{Poisson}}\left\{I_i+{\sigma }_b^2\right\},$$ (5)

where the annotation []₊ is to constraint the variable to be nonnegative and will be ignored for simplicity hereafter. Therefore, the fidelity term −L(μ) in Eq.(2) can be expressed as log likelihood of this artificial quantity:

$$-L\left(\mu \right)=-\sum _i\left\{Y_i^A\text{ ln}\left(I_i^0{e}^{-{[A\mathit{\mu }]}_i}+{\sigma }_b^2\right)-\left(I_i^0{e}^{-{[A\mathit{\mu }]}_i}+{\sigma }_b^2\right)\right\}$$ (6)

where, A is the projection matrix, Aμ is the line integral.

C. Tissue-specific MRF (MRFt) Prior Model

In previous studies^35–39, we proposed an MRF type tissue-specific texture prior to enhance texture reconstruction in LdCT imaging. Traditionally, the Huber MRF prior²⁷ was widely applied, of which the MRF weights are inversely proportional to the Euclidean distance between two neighboring pixels and are spatially invariant. In our texture MRF model, we consider the regional texture specific prior and extract the MRF weights from four tissue types for chest imaging, i.e. lung, bone, fat and muscle, respectively. Given an FdCT image and a fixed MRF window size, the MRF texture corresponding to a tissue region can be determined by a linear regression strategy such that every image pixel inside the MRF window can be constrained by linear combination of its neighboring pixels. Among all the linear regression estimation algorithms, the least squares algorithm is adapted in this study because of its computational efficiency. The least-squares predicted MRF coefficients can be formulated as:

$$w_r^{FD}=\text{arg min}{|}_{w_r} \sum _{j\in \mathit{\text{Region}}\left(r\right)}{\left({\mu }_j^{FD}- w_r^T{\mu }_{{\Omega }_j}^{FD}\right)}^2,$$ (7)

where, $w_r^{FD}$ refers to the MRF weights extracted from previous FdCT, and r refers to the tissue type, i.e. lung, bone, fat and muscle in this paper. Vector ${\mu }_{{\Omega }_j}^{FD}$ refers to the attenuation coefficients within the neighboring window Ω of jth pixel, where FD indicates the full dose image. In this study, the MRF window size was selected as 7×7 in a two-dimensional (2D) presentation³⁶. In the end, the texture MRF prior can be expressed as:

$$\Theta \left(\mu \right)=\sum _{r=1}^R\sum _{j\in \mathit{\text{Region}}\left(r\right)}\sum _{k\in \Omega \left(j\right)}{w}_{jk}{\left({\mu }_j^r-{\mu }_k^r\right)}^2,$$ (8)

where w_jk is the regional MRF weights between the jth pixel and kth pixel, and R stands for the number of regions, which equals to four in this work considering the chest scan. The MRF-t prior is does not require strict image registration. Once we extracted MRF weights from FdCT, we can apply it to the LdCT reconstruction. Fig. 1 shows an example of the extracted MRFt weights of subject #57 for the four tissue types, i.e. lung, bone, fat and muscle. The data information will be described in detail later.

Fig. 1.

The extracted four sets of MRFt model coefficients for the four tissue regions of subject #57.

D. The SP-MRFt Model

Integrating the SP data fidelity model with tissue-specific texture (MRFt) prior model, the overall objective function can be expressed as:

$$\begin{gathered} \Phi \left(\mu \right)=-\sum _i\left\{Y_i^A\text{ ln}\left(I_i^0{e}^{-{\left[A\mathit{\mu }\right]}_i}+{\sigma }_b^2\right)-\left(I_i^0{e}^{-{\left[A\mathit{\mu }\right]}_i}+{\sigma }_b^2\right)\right\} \\ +\beta \sum _{r=1}^R\sum _{j\in \mathit{\text{Region}}\left(r\right)}\sum _{k\in \Omega \left(j\right)}{w}_{jk}{\left({\mu }_j^r-{\mu }_k^r\right)}^2. \end{gathered}$$ (9)

For simple annotation, the tissue indicator r is dropped from hereafter. We denote this presented method as SP-MRFt.

The objective function of Eq. (9) is of a complex form that has no directly analytical minimization. Additionally, it is also a non-separable function, which makes it impossible for parallel computing. Therefore, we would like to derive a surrogate function with similar strategy used in Erdogan et al.¹⁴ and Xing et al.⁴⁴ to simplify and parallelize the objective function. For each ray path, we define the likelihood function as h_i(l_i) and a parabolic function $q_i\left(l;l_i^n\right)$ expressed as:

$$h_i\left(l_i\right)=\left(I_i^0{e}^{-l_i}+{\sigma }_b^2\right)-N_i^A\text{ ln}\left(I_i^0{e}^{-l_i}+{\sigma }_b^2\right),$$ (10)

$$q_i\left(l;l_i^n\right)=h_i\left(l_i^n\right)+{\dot{h}}_i\left(l_i^n\right)\left(l-l_i^n\right)+\frac{1}{2}{c}_i\left(l_i^n\right){\left(l-l_i^n\right)}^2,$$ (11)

where $l_i^n$ is the line integral at the n-th iteration, ${\dot{h}}_i\left(l_i^n\right)$ is the first derivative of the maximum likelihood function, $c_i(.)$ is coefficient and must be designed to ensure the objective function monotonically descending. Based on the analysis in Erdogan et al.¹⁴ and Xing et al.⁴⁴, in this study, we updated the coefficient $c_i(.)$ as follows:

$$c_i\left(l_i^n\right)=\left\{\begin{array}{c}\hfill {\left[\left(2\frac{h_i\left(0\right)-h_i\left(l_i^n\right)+{\dot{h}}_i\left(l_i^n\right)\left(l_i^n\right)}{{\left(l_i^n\right)}^2}\right)\right]}_{+},l_i^n>0\\ \hfill {\left[\ddot{h}(0)\right]}_{+},l_i^n=0\end{array}\right.,$$ (12)

where, $\ddot{h}(\cdot )$ is the second order derivative of the likelihood function. Then the global surrogate function below (Eq.13) can be monotonically minimized to find the optimal solution to Eq. (9).

$$\Psi \left(\mu ;{\mu }^n\right)=\sum _i{q}_i\left(l;l_i^n\right)+\frac{1}{2}\beta \sum _{r=1}^R\sum _{k\in \Omega (j)}{w}_{jk}{\left({\mu }_j-{\mu }_k\right)}^2.$$ (13)

We then take advantage of the convexity twice to make the fidelity and prior term separable respectively. The first trick is similar with that used in Erdogan et al.¹⁴. The line integral for each ray can be represented as:

$$\begin{array}{cc}\begin{array}{c}{\left[A\mu \right]}_i=\sum _j{A}_{ij}{\mu }_j=\sum _j\left\{A_{ij}\left({\mu }_j-{\mu }_j^n\right)+A_{ij}{\mu }_j^n\right\}\\ =\sum _j\left\{A_{ij}\left({\mu }_j-{\mu }_j^n\right)\right\}+l_i^n\\ =\sum _j{\alpha }_{ij}\left[\frac{A_{ij}}{{\alpha }_{ij}}\left({\mu }_j-{\mu }_j^n\right)+l_i^n\right]\end{array}& ,\sum _j{\alpha }_{ij}=1,\end{array}$$ (14)

where, A_ij is the element of the projection matrix, ${\mu }_j^n$ is the value of the attenuation after the n-th iteration. Here α_ij is any arbitrary positive value and satisfies their summation through all the pixels are unit. One simple choice is to set ${\alpha }_{ij}=\frac{A_{ij}}{\sum _j{A}_{ij}}$. Then the parabolic function satisfies:

$$q_i\left(l;l_i^n\right)\le \sum _j{\alpha }_{ij}{q}_i\left(\frac{A_{ij}}{{\alpha }_{ij}}\left({\mu }_j-{\mu }_j^n\right)+l_i^n;l_i^n\right).$$ (15)

For the prior term, we can re-formulate it as:

$$\Theta \left({\mu }_j-{\mu }_k\right)=\Theta \left(\frac{1}{2}\left(2{\mu }_j-{\mu }_j^n-{\mu }_k^n\right)+\frac{1}{2}\left(-2{\mu }_k+{\mu }_j^n+{\mu }_k^n\right)\right)$$ (16)

where, ${\mu }_k^n$ is the attenuation of pixel k after the n-th iteration. Applying the convexity of the parabolic form prior, we can obtain that

$$\Theta \left({\mu }_j-{\mu }_k\right)\le \frac{1}{2}\Theta \left(2{\mu }_j-{\mu }_j^n-{\mu }_k^n\right)+\frac{1}{2}\Theta \left(2{\mu }_k-{\mu }_j^n-{\mu }_k^n\right)$$ (17)

Thus, the overall separable surrogate function is:

$$\begin{gathered} \Psi \left(\mu ;{\mu }^n\right)=\sum _j\sum _i\frac{A_{ij}}{\sum _j{A}_{ij}}{q}_i\left(\left(\sum _j{A}_{ij}\right)\left({\mu }_j-{\mu }_j^n\right)+l_i^n;l_i^n\right)+ \\ \frac{1}{2}\beta \sum _{r=1}^R\sum _{k\in \Omega (j)}\frac{1}{2}{w}_{jk}\left({\left(2{\mu }_j-{\mu }_j^n-{\mu }_k^n\right)}^2+{\left(2{\mu }_k-{\mu }_j^n-{\mu }_k^n\right)}^2\right). \end{gathered}$$ (18)

Appling Newton’s method to minimize Eq. (18), the final updated formula of the SP-MRFt method is:

$$\begin{array}{c}{\mu }_j^{n+1}={\left[\begin{array}{c}{\mu }_j^n-\\ \frac{\sum _i\frac{A_{ij}}{\sum _j{A}_{ij}}{I}_i^0{e}^{-l_i^n}\left(\frac{Y_i^A}{I_i^0{e}^{-l_i^n}+{\sigma }_e^2}-1\right)+2\beta \left(\sum _{k\in \Omega (j)}{w}_{jk}\left({\mu }_j^n-{\mu }_k^n\right)+\sum _{j\in \Omega (k)}{w}_{kj}\left({\mu }_k^n-{\mu }_j^n\right)\right)}{\sum _i{A}_{ij}{c}_i\left(l_i^n\right)\left(\sum _j{A}_{ij}\right)+2\beta \left(\sum _{k\in \Omega (j)}{w}_{jk}+\sum _{j\in \Omega (k)}{w}_{kj}\right)}\end{array}\right]}_{+}.\\ \sum _{k\in \Omega (j)}{w}_{jk}=\sum _{j\in \Omega (k)}{w}_{kj}=1.\end{array}$$ (19)

where the annotation []₊is to constraint the variable to be nonnegative.

Table I shows the pseudo code of this SP-MRFt algorithm.

TABLE I:

PSEUDO CODE OF THE PRESENTED SP-MRFT ALGORITHM.

Algorithm: SP-MRFt
# Determine the tissue specific MRF weights from FdCT
Segment μ of full dose FBP image into four tissue types
determine the MRE weights for each tissue type
# Pre-log SP-MRFt for ULdCT
# Initialization:
for each pixel j:
μjold = 0;
update the μ by Eq. (19) with β=0
iteration number > criteria (i.e. the image can be segmented)
end
# pixel iterated with prior knowledge
for each iteration
image segmentation
for pixel j inside body mask
determine tissue type for pixel j;
update μj by Eq. (19)
end
Update ci by Eq. (12)
end

E. Implementation Optimization of SP-MRFt

Although the pre-log domain algorithm has advantage in ULdCT image reconstruction, it suffers from slow convergence in practice because of the non-linear transformation, which means it requires longer computation time. The time cost is sometimes unacceptable without careful optimization. To address the large computation time cost, we further optimized our implementation from two aspects (1) reducing the iteration times (2) reduces the computing cost per iteration.

One novel optimization is that we used strategies of dynamic memory assignment and some data management to reduce the time for each iteration cycle. We profiled the time consumption of the SP-MRFt reconstruction program and found the projection matrix calculation took majority of the computing time. Since the matrix only needs to be calculated once and will be accessed in each iteration cycle, we can save it to the memory instead of calculating it every time. However, the matrix consumes 200GB memory if we allocate statically for elements. In common scenario, only a small portion of rays contributes to one pixel, thus the projection matrix is very sparse when we update one pixel. So we designed a dynamic memory allocation to save only non-zero projection matrix elements and implemented this strategy using std::vector and std::map from the C++ Standard Template Library (STL). For our experiments, this strategy reduced the memory usage to 5GB and accelerates the reconstruction efficiency by more than 18 times. Moreover, we applied the ordered subset strategy^{13, 45} to reduce the iteration times to speed up the convergence. We also applied the body mask to reduce the necessary pixels that are needed to be updated. Parallel computing is also applied to maximize the computing resources. These optimizations and their performance for speeding up have been summarized in Table II.

TABLE II:

SUMMARY OF THE STRATEGIES FOR SPPEDUP FOR PRE-LOG IMAGE RECONSTRUCTION.

Strategy	Techniques	Performance
Ordered subset updating	Ordered subset	Only 1000 iterations needed
Save projection matrix A to memory	std::vector and std::map from the C++ Standard Template Library (STL).	single iteration 18 times faster
Body mask to reduce pixels	VQ segmentation	Around 1/3 pixels are reduced
Parallel computing	Open MP	speed approximately scale with number of cores.

F. Texture-dose Characterization Strategy

Tissue texture reflects the spatial distribution of contrasts of image voxel gray levels (i.e., tissue heterogeneity), has been treated as important biomarker or reference to differentiate malignant from benign lesions (i.e. computer-aided diagnosis) and also abnormal from normal tissues (i.e. computer-aided detection)^32,33,35,41. It is important to preserve tissue texture reflected in the CT imaging when lowering the X-ray dose levels. Texture based image assessment is of clinical value to quantitatively evaluate a reconstruction algorithm, performance and, therefore, is explored in this paper as a clinically relevant task-based evaluation metrics. Using texture-based evaluation metrics, this paper aims to characterize the dose level vs. the quantified image texture descriptor. It is conjectured that the quantified image texture descriptor will decreases as the dose level decreases and the plot of the relationship between texture and dose will reflect the performance of a reconstruction algorithm vs. the dose reduction.

While there are several texture descriptors available for this exploratory study, we choose the Haralick texture descriptor³¹ based on our experience in adapting the model for computer-aided diagnosis of polyps and nodules^46–47. The Haralick texture model generates a total of 28 texture features from a region of interest (ROI) in a reconstructed image, e.g. a nodule within the lung volume³¹. By treating the 28 Haralick texture features as a vector, the normalized Euclidean distance of the Haralick features between the reconstructed image and that of reference is selected as the texture metric to evaluate the texture quality, which can be expressed as:

$$\text{NRMSE}=\sqrt{\frac{{\sum }_{n=1}^{28}{\left(x\left(n\right)-x_{ref}\left(n\right)\right)}^2}{{\sum }_{n=1}^N{\left(x_{ref}\left(n\right)\right)}^2}}$$ (20)

where x(n) and x_ref(n) represent the Haralick features of LdCT and FdCT images at voxel n, respectively. Afterwards, the response curve, or the plot of the relationship between texture and dose, can be obtained.

In addition to the counts from the X-ray source, i.e. the mAs level, there is another factor that may affect the response curve: background electronic noise. For example, under the same mAs, if the noise is stronger, the texture quality is expected poorer in theory. There are many sources causing the background noise during the cascade random processes like X-ray photon generation, transmission, signal collection and data read-out system and so on¹⁸. According to Eq. (3), the noise can be modeled as Gaussian distribution, which can be represented by two parameters, i.e. mean and variance. Hence, we can manipulate these two parameters to mimic the noise distribution and evaluate their effect on the texture quality and on the texture-dose characterization. Details will be presented in the following experiments section.

G. Experimental Design

Based on the description in method section above, we implemented the presented SP-MRFt algorithm to reconstruct ULdCT images using transmission data. We implemented FBP and PWLS-MRFt as reference to evaluate the performance of SP-MRFt qualitatively and quantitatively. We further characterized the texture-dose relationship by the SP-MRFt method. We reconstructed CT images at 12 dose levels from the FdCT level of 100mAs/120kVp down to ULdCT level of 1mAs/120kVp. Comprehensive studies on the texture-dose characterization as well as the noise effect were then preformed.

Three subjects, who were scheduled for CT-guided lung nodule needle biopsy, were recruited to this study under informed consent after approval by the Institutional Review Board. These three subjects were selected to represent three different types of nodules and scanning positions to ensure diversity of the experimental data. The subjects were scanned by the same clinical CT scanner. The X-ray tube voltage was set to be 120kVp, and the tube current was set to be 100mAs for the FdCT scan. Assisted by the vendor, we obtained raw sinogram data from the CT scanner.

Starting from the acquired FdCT sinogram, we simulated other mAs scans at lower dose levels at {20, 18, 15, 12, 10, 8, 6, 5, 4, 3, 2, 1} mAs respectively. We started from 20mAs based on our observations that the texture from MBIR reconstruction above 20mAs level does not show noticeable changes compared to FBP results at 100mAs level (i.e. the baseline or reference) for the clinical tasks like nodule detection and characterization⁴⁰. The simulation design is based on the CT transmission data statistical properties introduced in the method section, which follows a combination of Poisson plus Gaussian distributions. The mean of the Poisson distribution for a given mAs level can be obtained from semi-experimental model in Ma et al.²⁶ and Zeng et al.⁴⁸, where photon counts per ray is linearly dependent on the current settings at a fixed voltage setting. We then added Poisson noise to the X-ray quanta and combined it with Gaussian distributed background noise. For the Gaussian, we varied its mean value from zero to 20 and variance from 225 to 1,225 or standard deviation from 15 to 35. By manipulating these two parameters, we can evaluate the effect of noise on textures from the FdCT towards to ULdCT.

3. Results

A. Image Reconstruction

We segmented the FdCT images into four tissue types (i.e. lung, bone, fat and muscle) and extracted their corresponding MRF texture prior respectively. The extracted MRF weights and the simulated CT transmission data were fed into the presented SP-MRFt reconstruction model for image reconstruction.

In the surrogate function (Eq. 13), the hyper parameter β is used to tune the strength of the prior. i.e. to balance between detail preservation and noise suppression. We scanned β from 10 to 100,000 and found a proper value should be between 100 and 10000. It is also observed that the optimal β depends on the dose level or noise level. Lower dosage images need smaller β to match the weights of the fidelity term. For variable control purpose, we used β = 500 for all dose levels to study the texture-dose curve characterization. The effect of beta was discussed in the discussion section.

The overall surrogate function (Eq. 13) consists of two parabolic functions for log-likelihood and prior respectively. The former is similar with that used in Erdogan et al.¹⁴, which transforms the problem into a simple quadratic optimization problem and guarantees the monotonicity in each iteration. We have also shown in Gao et al.⁴⁰ that the cost function with quadratic form for the log likelihood and the presented texture prior converged once the hyper parameter was chosen properly because the presented texture prior is bounded between [−1, 1]. The derivation can be found in the reference Gao et al.⁴⁰. The pre-log domain SP-MRFt needed around 1,000 iterations to reach convergence. Fig.2 presents the tissue texture quantified by Haralick measures along with iteration times. Good convergence is also observed. With our optimized implementation, total computing time is around 20 minutes on a 16-core computer. This could be further scaled down with more cores.

Fig. 2:

Tissue texture quantified by Haralick measures along with iteration times.

B. Reconstruction Results Comparison

The ULdCT images at the 13 different X-ray dose levels (from 100mAs to 1mAs) were reconstructed with the SP-MRFt algorithm. The widely used filtered back projection (FBP), penalized weighted least square with the MRF texture prior (PWLS-MRFt) and the shifted Poisson with the Huber MRF model were also implemented and used as reference. To make the results comparable, the MRF window for Huber weight is also chosen 7×7, which is the same size with the MRFt. To address this, we denote this model as SP-Huber7. For all the MBIR models, their hyper parameters are all empirically optimized. In the work Zhang et al.³⁷, it is observed that, as long as the β is in a reasonable range (within an order of magnitude), the image texture does not change noticeably. The optimal β is at the order of ~1E5. In this experiment, we tried the β 2E5. It does give us good eye-appealing results. Therefore, we used 2E5 for PWLS-MRFt in this work. For SP-Huber7 model, we also found the optimal β in the trail-error manner, which is 3E3. The hyper parameter effect will be further explored in the discussion section.

All the models are implemented in the same desktop of 16-cores, 3.1GHz and 128 RAM. The PWLS-MRFt model takes around 1 minute for one iteration and 15 iterations to reach convergence. The total time to reconstruct one 512×512 image is around 15 minutes. After optimization, both SP-Huber7 has the same speed with SP-MRFt model, which takes 38.4 second per CPU core per iteration on average. Due to slow convergence, both models take around ~1000 iterations to reach convergence. it will take around 20 minutes to reconstruct one 512×512 image, which is comparable with PWLS-MRFt.

Figure 3 shows an example of images reconstructed by FBP with “ramp” filter, PWLS-MRFt, SP-Huber7 and SP-MRFt reconstruction at 5mAs ultralow-dose level for the three subjects, indexed as #10, #75 and # 185. As mentioned above, to ensure the diversity of the experimental data, three subjects with different nodule types and positions were used. As shown in Fig. 3, nodule of subject #10 is on the wall near spine and in the middle of the lung volume. Nodule of subject #57 is in the middle region of the lung and in the upper lung volume. Nodule of subject #185 is on the wall near body side and at bottom of the lung volume. In Fig. 3, the nodule regions marked by yellow boxes are our “region of our interest” or ROI’s in this study. Compared with FBP, three iterative models have suppressed the noise effectively. It is much easier to identify the nodule around the wall for #10 from the PWLS-MRFt, SP-Huber7 and SP-MRFt images than the FBP one. From visual inspection, we could clearly see the two SP models outperforms the other two in terms of noise suppression and texture preservation. For example, the heterogeneity of nodule for #185 can be clearly observed in SP-MRFt and SP-Huber7, but not in the other two algorithms. There are some streak-like artifacts in the PWLS-MRFt images. This might be due to the data variance weight in the least-square model is not sufficiently accurate or efficient for the PWLS model when the noise counts are comparable to the signal counts. Comparing SP-Huber7 with SP-MRFt, their reconstruction results are comparable by visual judgement. We can further compare them through the quantitative measures.

Fig. 3:

Reconstructed images of three subject data using different algorithms. Top row: The FdCT of three subjects, where the region of interest (ROI) are marked by yellow box, and the subject index are marked in red label. ULdCT at 5mAs reconstructed by FBP with “ramp” filter (2^nd row), PWLS-MRFt (3^rd row), SP-Huber7 (4^th row) and SP-MRFt (5^th row). The display window is [0 0.035] mm⁻¹.

The ROI-based quantitative measurements are presented in Table III. We employed the conventional mean-squared error (MSE), peak of signal-to-noise ratio (PSNR), structure similarity index (SSIM) and correlation coefficient (C.C.) to assess the image quality comparing to the FdCT. The SSIM and C.C. are two common metrics to evaluate how close the two images to each other from the point of structure similarity. In addition, we used the Haralick features (HF) metric of Eq. (20) mentioned in the method section to quantify the texture quality for each image. For PSNR, SIIM and C.C., higher value means better image quality. For MSE and HF, lower value means better image quality. For each subject, the best measure has been highlighted in bold.

TABLE III:

QUANTIFIED IMAGE QUALITY COMPARISON AMONG DIFFERENT RECONSTRUCTION METHODS.

Case	Methods	MSE	PSNR	SSIM	C.C.	HF
#10	FBP	0.2989	35.24	0.6809	0.2549	8540.71
	PWLS-MRFt	0.0183	47.38	0.9697	0.7505	56.75
	SP-Huber7	0.0076	51.17	0.9884	0.8913	0.97
	SP-MRFt	0.0053	52.73	0.9915	0.9285	0.78
#57	FBP	0.7918	41.01	0.8359	0.5811	2824.08
	PWLS-MRFt	0.1248	49.04	0.9737	0.8525	23.71
	SP-Huber7	0.0060	53.15	0.9943	0.9348	2.60
	SP-MRFt	0.0634	51.98	0.9852	0.9373	0.64
#185	FBP	0.1062	39.74	0.8807	0.4068	3471.41
	PWLS-MRFt	0.0084	50.76	0.9877	0.8515	6.00
	SP-Huber7	0.0098	50.09	0.9889	0.7687	0.93
	SP-MRFt	0.0073	51.36	0.9880	0.8695	0.57

According to Table III, the performance of each algorithm is consistent among three subjects and agrees well with our visual inspection. The SP based two models outperforms the other two algorithms in terms of not only noise reduction but also texture preservation. The SP-MRFt and SP-Huber7 have comparable image quality. For the HF texture measure, the SP-MRFt scores about 10 times better than PWLS-MRFt and 5000 times better than FBP. Even though there is no obvious visual difference between SP-MRFt and SP-Huber7, their texture measures are different. MRFt can preserve the texture better than Huber7. Although we incorporated the tissue-specific texture as prior in the post-log PWLS-MRFt, the image texture is also dependent on the fidelity data. In this ultralow-dose case, the post-log line-integral data may bring in too much bias in its data fidelity model. It also agrees with some previous study¹⁶ that pre-log domain methods outperform post-log domain methods. As observed, as the dose level decreases, the reconstructions for both the pre-log and post-log domain algorithms degrade the image quality due to the X-ray quanta and electronic noise. In the following subsection, we will quantitatively investigate the texture change using the HF measure at different dose levels with various noise models.

To evaluate the proposed SP-MRFt model in a more realistic situation, the real low-dose sinogram data (120kVp/20mAs) of one subject #128 was used to perform the study, which has corresponding FdCT. The reconstructed results by SP-MRFt and three comparison models are shown in Fig. 4. It is observed that the three MBIR models can suppress the noise levels significantly comparing to FBP. From Fig.3 and Table III, the pre-log domain methods can give much better results than post-log domain method in the ultra-low dose scenario (5mAs). However, there is no obvious difference between the pre-log and post-log domain models at higher dose level according to Fig. 4. For comparison of MRFt and Huber7 weights, there is no obvious visual difference. However, their texture measures may vary according to Table III.

Fig. 4:

Reconstructed images by FBP with “ramp filter” (a), SP-Huber7 (b), PWLS-MRFt (c) and SP-MRFt. The display window is [0 0.035] mm⁻¹.

C. Texture-Dose Characterization

Figure 5 shows the ROI regions of reconstructed images at each dose level. As dose level goes down, borders of the nodules become blurry gradually. From visual inspection of #10 and #57, some details like vessels around the nodules can still be recognized down to 6mAs. Below that, it is hard to distinguish real vessels and noise. For #185, a dark triangle pattern in the center can be clearly recognized down to 12mAs. The boundary of that triangle is blurred from 12 to 6mAs, and it cannot be clearly identified below 6mAs. In addition, noise becomes more prominent in the bottom left corner of #185 when dose level goes below 6mAs.

Fig. 5:

Magnified SP-MRFt reconstructed CT images of the three subjects with different body anatomies and nodule locations.

To quantify the texture quality, Haralick features were used. Specifically, ROIs were outlined by an experienced radiologist in each image, and HF measures within those ROIs were calculated and normalized by the reference image, i.e. the full-does image, using Eq. (20). Smaller relative texture measure change indicates better tissue texture preservation. Doing this for each dose level, we obtained a curve of relative texture measure change as a function of X-ray current, i.e. the texture-dose response curve.

Figure 6 shows a set of the texture-dose response curves for subjects #10, #57 and #185. As discussed in section II, in the pre-log simulation model, two parameters, the mean and variance, are used to describe noise statistical properties as regarding to dose level. In the following text, we denote the mean and standard deviation (square root of variance) as Emean and Estd. In Fig. 6, Emean and Estd are 10 and 25 respectively, which are reasonable empirical estimations. With Emean and Estd fixed, the dose level would be the only parameter controlling the noise level of simulated sinograms. The investigation of the dose effect on reconstructed texture will be described in the following text.

Fig. 6:

Plots of Haralick texture measure vs. the X-ray current mAs level for the three subjects scaled to the same window. Left panel, 100 mAs to 1 mAs; right panel, zoom in of left panel, 20 mAs to 1 mAs.

In Fig. 6, all three curves follow a similar trend, that from 100mAs going down, initial changes are slow but drops rapidly around 8mAs. This phenomenon of an existing turning point was expected before experiments. It indicates that the tissue textures could be preserved well above the turning point and degraded significantly below that. The performance of SP-MRFt at lower dose levels implies effective balance between fidelity and prior terms in the cost function. At relative high dose, signal to noise ratio is good, the details are preserved by the fidelity term. As noise level increases, the prior term imposes regularization, aiming relative clean image while preserving the texture. From the X-ray quanta generation model³⁹, with fixed X-ray tube voltage, the radiation exposure, i.e. the incident photon flux, should be linear with regarding to the current. While the reconstructed texture by SP-MRFt follows a stable-turning-point-collapse pattern. This feature could be used to reduce radiating dose while retaining the useful tissue texture. Fig. 6 shows the SP-MRFt works well down to its turning point around 8mAs.

We then investigated the effect of noise model on the texture-dose response curve. As mentioned above, Emean and Estd were manipulated to simulate different noise models. Emean was scanned from 0 to 20; Estd was scanned from 15 to 35. We reconstructed images using the SP-MRFt method for this simulated dataset and calculated HF measures within ROIs with normalization from reference images. Results are shown in Fig. 7 and Fig. 8. Fig. 7 (8) shows the HF measure change as a function of Estd (Emean) with fixed Emean’s (Estd’s). It shall be noted that the axis is also not linear for display purpose. We could see the turning-point phenomena persists in all scanned Emean and Estd values. In Fig. 7, the texture degrades more when Estd gets higher, while curves from different Emean values largely overlap showing the impact of Emean on texture is weak with the SP-MRFt method. Comparing Fig.7 and Fig.8, the SP-MRFt reconstructed texture quality is observed to be more sensitive to the noise’s variance than its mean. Results are similar for subject #57 and #185. Due to limited paper length, results for these two subjects are attached in the APPENDIX.

Fig. 7:

Plots of Haralick texture measure vs. the X-ray current mAs level with background noise at fixed Emean but various Estd for subject #10. Note: for display purpose, the x-axis is not linear.

Fig. 8:

Plots of Haralick texture measure vs. the X-ray current mAs level with background noise at fixed Estd but various Emean for subject #10. Note: for display purpose, the x-axis is not linear.

4. Discussion

To address the slow convergence and long computing time issue, multiple techniques were adapted and developed in implementation of the SP-MRFt algorithm, e.g. separable surrogate functions, ordered subset iteration strategy, dynamic memory allocation, multiple threading and etc. After optimization in the single thread mode, the program ran 18 times faster. In addition, in the multi-threading mode, good scalability between speed and number of CPU cores was observed. The fast implementation made it possible to perform a comprehensive study on the texture-dose relationship within a reasonable time. In this study, 780 experimental points (13 dosages×5 Emean×4 Estd×3 subjects) needed to be tested. It would take years to complete the reconstruction before optimization, but only 10 days after the optimization on a 16-core computer.

We studied effect of noise models on the texture-dose relationship based on the shifted Poisson data fidelity model. The texture-dose response curves are not sensitive to Emean of noise models suggesting effective baseline subtraction of the SP-MRFt method. There are other models considering data statistical properties in ultralow-dose CT scenario, e.g. introducing the X-ray energy spectrum¹⁶. Bring in energy spectrum will model the physics process more accurately. It is expected to improve the reconstruction performance, especially in the ultra-low dose scenario. This is one of our future research interests to improve the model. More studies are needed to study the noise effect as well as the response curve by using different reconstruction algorithms. The hyper parameter beta will also affect the reconstructed image quality. Fig.9 presents the normalized Haralick measure and PSNR of reconstructed images at 5mAs along with different beta. The smaller measurement, the better tissue texture is preserved. As we can see when an optimal beta can preserve the tissue texture better. However, how to obtain the optimal beta in accordance with noise level is still an open question. In this study, we use the beta 500 for texture-dose study. Considering optimal beta into the texture-dose study is also one of our future research interests.

Fig. 9:

The normalized Haralick measure and PSNR of reconstructed images at 5mAs along with different beta.

One limitation of the MRFt prior in this paper is full dose scan needs to be done at least once to provide accurate texture information. To remove this limitation and expand application of MRFt, we proposed building an MRF coefficient database to assist low dose CT scans without a previous full dose prior³⁸. In the work of Gao et al.³⁸, we quantitively demonstrated the reconstructed tissue textures could be largely enhanced (higher fidelity with regarding to full dose textures) using MRF coefficients from another subject from the database. To apply this texture database method, one important thing is how to find the proper texture from the database. The preliminary results of matching one proper texture prior from database using physiological, geometry factors and image features are reported in Gao et al.⁴⁸.

Although the observed turning point indicates the limit of a CT hardware configuration and software architecture, which reflects the minimal radiation that CT system may be able to achieve reasonable image quality, more detailed quantitative studies may be needed to fully understand the limit for specific clinical tasks. Possible future studies could include (1) what clinical tasks can be achieved by the reasonable image quality around the turning point dosage and (2) how much additional dose would be needed beyond the minimum dose for other clinical tasks.

5. Conclusions

In this study, we presented the SP-MRFt algorithm, which combines shifted Poisson pre-log fidelity data model and tissue-specific texture prior term, to enhance the textures in reconstruction of ULdCT images. We further introduced texture-based metrics to assess the performance of the presented algorithm at various dosage levels. Experimental results on the datasets with diverse body conditions demonstrated the presented SP-MRFt outperforms the baseline algorithms (FBP, PWLS-MRFt) in terms of noise suppression and texture preservation through both visual inspection and quantitative metrics, see Table III. Using the SP-MRFt algorithm and texture metrics, we studied the texture and dosage relationship using different noise parameters in the pre-log domain. Comparison studies among different noise parameters demonstrated that the texture measure is more sensitive to the noise variance than the noise mean in the pre-log domain. From the obtained texture and dosage curves, we noticed a striking turning point, after which the texture quality breaks down quickly. This was observed in all datasets with different subject conditions (including nodule position and types). Depending on subject conditions and noise levels, this turning point occurred around 10 to 5 mAs. We believe this observation on the texture and dosage relationship could provide important guidance for evaluation of both hardware configurations and software image reconstruction for various clinical tasks.

APPENDIX: Haralick Texture Measure Response to the X-ray Current of #57 AND #185

Fig. 10:

Plots of Haralick texture measure vs. the X-ray current with various noise parameters. Top (bottom) images are for subject #75 (#185). Note: for display purpose, the x-axis is not linear.