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Phys Med Biol. Author manuscript; available in PMC 2010 May 19.

Published in final edited form as:

Published online 2008 October 20. doi: 10.1088/0031-9155/53/22/004

PMCID: PMC2872939

NIHMSID: NIHMS201911

Xuanqin Mou: nc.ude.utjx.liam@uomqx

The publisher's final edited version of this article is available at Phys Med Biol

See other articles in PMC that cite the published article.

Microcalcification is one of the earliest and main indicators of breast cancer. Because dual-energy digital mammography could suppress the contrast between the adipose and glandular tissues of the breast, it is considered a promising technique that will improve the detection of microcalcification. In dual-energy digital mammography, the imaged object is a human breast, while in calibration measurements only the phantoms of breast tissue equivalent materials are available. Consequently, the differences between phantoms and breast tissues will lead to calibration phantom errors. Based on the dual-energy imaging model, formulae of calibration phantom errors are derived in this paper. Then, this type of error is quantitatively analyzed using publicly available data and compared with other types of error. The results demonstrate that the calibration phantom error is large and dominant in dual-energy mammography, seriously decreasing calculation precision. Further investigations on the physical meaning of calibration phantom error reveal that the imaged objects with the same glandular ratio have identical calibration phantom error. Finally, an error correction method is proposed based on our findings.

Breast cancer is a deadly disease that adversely affects the lives of many people, primarily women. Microcalcification is one of the earliest and main indicators of breast cancer, and mammography is the gold standard for breast cancer screening. Thus, the visualization and detection of microcalcifications in mammography play a crucial role in reducing the rate of mortality from breast cancer. Microcalcifications are usually smaller than 1.0 mm and mainly composed of calcium compounds (Fandos-Morera *et al* 1988) such as apatite, calcium oxalate and calcium carbonate. Microcalcifications have greater x-ray attenuation coefficients than the surrounding breast tissues, so they are more visible on homogeneous soft-tissue backgrounds. However, the visualization of microcalcifications could be obscured in mammograms because of overlapping of tissue structures. Tissue structures in mammograms arise from the differences of the x-ray attenuation coefficients between adipose tissue, glandular tissue, ducts, vessels and soft-tissue masses. Microcalcifications, especially smaller ones, are extremely difficult to discriminate even if the signal-to-noise ratio is high (Brettle and Cowen 1994). Therefore, microcalcification detection in mammography suffers from a high false-negative rate (Soo *et al* 2005).

Dual-energy digital mammography is considered as a prospective technique to improve the detection of microcalcification. The healthy breast is mainly composed of adipose, epithelial and stroma tissues which can be grouped into two attenuating types: adipose tissue and glandular tissue. In dual-energy digital mammography, high- and low-energy images of the breast are acquired using two different x-ray spectra. By exploiting the difference in x-ray attenuation between different materials at different x-ray energies, the high- and low-energy images can be synthesized to suppress the contrast between adipose and glandular tissues and subsequently generate the dual-energy calcification image. Under ideal imaging conditions, when the imaging data are free of scatter and other biases, dual-energy digital mammography could be used to determine the thickness of microcalcification and the breast glandular ratio (Lemacks *et al* 2002). This quantitative information can be incorporated into computer-aided detection algorithms to enhance mammographic interpretation.

However, a lot of research must be carried out prior to applying dual-energy digital mammography in clinics. Early research on the feasibility of dual-energy mammography includes optimal high- and low-energy spectra selection (Johns and Yaffe 1985), selection of inverse-map functions (Cardinal and Fenster 1990), influence of detectors (Chakraborty and Barnes 1989, Boone *et al* 1990, Brettle and Cowen 1994) and experimental studies (Johns *et al* 1985, Brettle and Cowen 1994). Lemacks *et al* (2002) presented a theoretical framework to calculate the (quantum) noise in dual-energy calcification image and made numerical analysis as well. Their results were presented in terms of contrast-to-noise ratio under various imaging conditions, including the x-ray spectra, microcalcification size, tissue composition and breast thickness. Based on the work of Lemacks *et al* (2002), Kappadath *et al* made a series of investigations on dual-energy digital mammography (Kappadath and Shaw 2003, 2004, 2005, Kappadath *et al* 2004, 2005). Kappadath and Shaw (2003) investigated various inverse-map functions for dual-energy digital mammography, and concluded that the mean fitting error is ~50 *μ*m and the max fitting error is ~150 *μ*m for the microcalcification thickness when using the cubic or conic functions. Kappadath and Shaw (2005) developed a scatter and nonuniformity correction technique for dual-energy digital mammography. In their implementation on the clinical equipment, microcalcifications in 300–355 *μ*m range were clearly visible in dual-energy calcification images. The microcalcification size threshold decreased to 250–280 *μ*m when the visible criteria were lowered to barely eyesight. Dual-energy digital mammography has also been investigated and evaluated by other investigators (Bliznakova *et al* 2006, Brandan and Ramirez-R 2006, Taibi *et al* 2003).

In dual-energy digital mammography, there are two necessary steps: the first step is calibration, which estimates the polynomial coefficients using calibration data, while the second step is the calculation of the component information of the imaged object. When dual-energy digital mammography is used clinically, the imaged object is a human breast. In calibration, however, only phantoms of a breast tissue equivalent material can be used. The composition and density differences between the phantoms and breast tissues cause differences in linear attenuation coefficients, which consequently lead to so-called calibration phantom errors. The error was first indicated and investigated by Mou and Chen (2007) and consisted of two parts: microcalcification thickness error and glandular ratio error. The small differences in linear attenuation coefficients between calibration phantoms and breast tissues lead to huge calibration phantom errors in the final results. The phantom materials, like PMMA, polyethylene and RMI phantoms, are not suitable for calibration of dual-energy digital mammography in clinics. Comparably, CIRS phantoms are a better choice.

In reality, the elemental composition ratio of human breast varies greatly (Hammerstein *et al* 1979). The differences in linear attenuation coefficients between calibration phantoms and breast tissues are inevitable. Therefore, calibration phantom errors cannot be removed solely by redesigning calibration phantoms. This study systematically analyzes the calibration phantom errors, investigates their physical meaning and develops possible error correction methods.

This section first presents the imaging principles and the physical model of dual-energy digital mammography and then develops the formulae for calibration phantom errors.

The principles of dual-energy imaging have been discussed extensively in the literature (Alvarez and Macovski 1976, Lehmann *et al* 1981, Brody *et al* 1981, Johns and Yaffe 1985, Gingold and Hasegawa 1992). In the medical diagnostic x-ray imaging modality, x-ray attenuation includes photoemission and Compton scattering. The linear attenuation coefficient *μ*(*E*) is (Alvarez and Macovski 1976)

$$\mu (E)={b}_{pu}{E}^{-3}+{b}_{du}{f}_{\text{KN}}(E),$$

(1)

where *f*_{KN}(*E*) is the Klein–Nishina function,

$${f}_{\text{KN}}(E)=\frac{1+\alpha}{\alpha}\left[\frac{2(1+\alpha )}{1+2\alpha}-\frac{1}{\alpha}ln(1+2\alpha )\right]+\frac{1}{2\alpha}ln(1+2\alpha )-\frac{(1+3\alpha )}{{(1+2\alpha )}^{2}},$$

(2)

and *α* = *E*/510.975. The attenuation coefficient of any material above the k-edge can be approximated by equation (1). Because the atomic numbers of the elements in human body are small and their k-edge energies are low, equation (1) is valid for medical diagnostic x-ray imaging. The term *E*^{−3} approximates the energy dependence of the photoelectric interaction, and *f*_{KN}(*E*) gives the energy dependence of the total cross-section for Compton scattering. The dependences of *b _{pu}* and

$${b}_{pu}\approx {K}_{1}\frac{\rho}{A}{Z}^{n},\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}{b}_{du}\approx {K}_{2}\frac{\rho}{A}Z,$$

(3)

where *K*_{1} and *K*_{2} are constants, *ρ* is the mass density, *n* ≈ 4, *A* is the atomic weight and *Z* is the atomic number. If the imaged object is composed of two known materials, A with thickness *L _{A}* and B with thickness

$$M(E)={b}_{p}{E}^{-3}+{b}_{d}{f}_{\text{KN}}(E),$$

(4)

where

$${b}_{p}={b}_{\mathit{puA}}{L}_{A}+{b}_{\mathit{puB}}{L}_{B},\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}{b}_{d}={b}_{\mathit{duA}}{L}_{A}+{b}_{\mathit{duB}}{L}_{B}$$

(5)

and *b _{puA}*,

An object can be imaged under high- and low-energy spectra:

$${f}_{j}=-ln\left(\int {P}_{0j}(E)exp[-{b}_{p}{E}^{-3}-{b}_{d}{f}_{\text{KN}}(E)]Q(E)\phantom{\rule{0.16667em}{0ex}}\text{d}E\right),\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}j=h,l,$$

(6)

where *P*_{0}* _{h}*(

$$\{\begin{array}{l}{L}_{A}={k}_{A0}+{k}_{A1}{f}_{h}+{k}_{A2}{f}_{l}+{k}_{A3}{f}_{h}^{2}+{k}_{A4}{f}_{l}^{2}+{k}_{A5}{f}_{h}{f}_{l}+{k}_{A6}{f}_{h}^{3}+{k}_{A7}{f}_{l}^{3}\hfill \\ {L}_{B}={k}_{B0}+{k}_{B1}{f}_{h}+{k}_{B2}{f}_{l}+{k}_{B3}{f}_{h}^{2}+{k}_{B4}{f}_{l}^{2}+{k}_{B5}{f}_{h}{f}_{l}+{k}_{B6}{f}_{h}^{3}+{k}_{B7}{f}_{l}^{3}.\hfill \end{array}$$

(7)

The coefficients *k _{Ai}* and

Lemacks *et al* (2002) proposed a numerical framework to perform the dual-energy digital mammography calculation. They assumed that there are three attenuating materials in the breast: adipose tissue (thickness *t _{a}*), glandular tissue (thickness

$$P(E)={P}_{0}(E)exp[-{\mu}_{a}(E){t}_{a}-{\mu}_{g}(E){t}_{g}-{\mu}_{c}(E){t}_{c}],$$

(8)

where *P*_{0}(*E*) and *P*(*E*) are the incident photon fluence on the surface of the breast and the transmitted fluence, respectively; *μ _{a}*(

Because of the characteristics of diagnostic x-ray imaging physics shown in equation (1), only two unknowns can be solved for dual-energy digital mammography. During mammography, the breast is usually compressed to a uniform thickness *T* that can be easily measured. The contribution of microcalcifications to the total breast thickness can be ignored because the microcalcifications are small in size and sparsely present, i.e. *T* ≈ *t _{a}* +

$$P(E)={P}_{0}(E)exp[-{\mu}_{a}(E)T-g({\mu}_{g}(E)-{\mu}_{a}(E))T-{\mu}_{c}(E){t}_{c}]$$

(9)

In dual-energy imaging calculations, a reference signal *I _{r}* is needed in order to increase the dynamic range of the intensity values. The reference signal can be measured by a certain breast phantom. The exposure data

$$\begin{array}{l}{f}_{j}({t}_{c},g)=ln({I}_{rj}/{I}_{j})=ln({I}_{rj})-ln\left(\int {P}_{0j}(E)exp[-{\mu}_{a}(E)T-g({\mu}_{g}(E)-{\mu}_{a}(E))T-{\mu}_{c}(E){t}_{c}]Q(E)\phantom{\rule{0.16667em}{0ex}}\text{d}E\right),\\ j=h,l.\end{array}$$

(10)

In this paper, we assume that there is no scatter because it can be well corrected (Kappadath and Shaw 2005). Therefore, *g* and *t _{c}* can be solved by a dual-energy imaging method.

In dual-energy digital mammography, phantoms for calibration are always different from real breast tissues that will be imaged. These differences lead to calibration phantom errors Δ*t _{c}* of the microcalcification thickness and Δ

When phantoms are used for calibration, equation (10) can be expressed as

$${f}_{j}({t}_{c},g)=ln({I}_{rj}/{I}_{j})=ln({I}_{rj})-ln\left(\int {P}_{0j}(E)exp[-{\mu}_{a}^{hu}(E)T-g({\mu}_{g}^{hu}(E)-{\mu}_{a}^{hu}(E))T-{\mu}_{c}^{hu}(E){t}_{c}]Q(E)\phantom{\rule{0.16667em}{0ex}}\text{d}E\right)=ln({I}_{rj})-ln\left(\int {P}_{0j}(E)exp[-{\mu}_{a}^{ph}(E)T-{g}^{\prime}({\mu}_{g}^{ph}(E)-{\mu}_{a}^{ph}(E))T-{\mu}_{c}^{ph}(E){t}_{c}^{\prime}]Q(E)\phantom{\rule{0.16667em}{0ex}}\text{d}E\right),j=h,l$$

(11)

where the second row of equation (11) represents the imaging of breast tissues and the third row represents the imaging of phantoms. The superscript ‘*hu*’ indicates breast tissues while ‘*ph*’ indicates phantoms. *g* and *t _{c}* represent the true glandular ratio and microcalcification thickness of the breast, respectively. Let

$${g}^{\prime}=g+\mathrm{\Delta}g,$$

(12)

$${t}_{c}^{\prime}={t}_{c}+\mathrm{\Delta}{t}_{c},$$

(13)

where Δ*g* and Δ*t _{c}* are calibration phantom errors. The third row of equation (11) can be approximated by the first-order Taylor series expansion of variables

$$ln({I}_{rj})-ln\left(\int {P}_{0j}(E)exp[-{\mu}_{a}^{hu}(E)T-g({\mu}_{g}^{hu}(E)-{\mu}_{a}^{hu}(E))T-{\mu}_{c}^{hu}(E){t}_{c}]Q(E)\phantom{\rule{0.16667em}{0ex}}\text{d}E\right)=ln({I}_{rj})-ln\left(\int {P}_{0j}(E)exp[-{\mu}_{a}^{ph}(E)T-g({\mu}_{g}^{ph}(E)-{\mu}_{a}^{ph}(E))T-{\mu}_{c}^{ph}(E){t}_{c}]Q(E)\phantom{\rule{0.16667em}{0ex}}\text{d}E\right)+{\beta}_{j}^{c}\times \mathrm{\Delta}{t}_{c}+{\beta}_{j}^{g}\times T\times \mathrm{\Delta}g,\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}j=h,l,$$

(14)

where ${\beta}_{j}^{c}$ and ${\beta}_{j}^{g}$ are defined as

$$\begin{array}{l}{\beta}_{j}^{c}=\frac{\int {P}_{0j}(E){\mu}_{c}^{ph}(E)exp[-{\mu}_{a}^{ph}(E)T-g({\mu}_{g}^{ph}(E)-{\mu}_{a}^{ph}(E))T-{\mu}_{c}^{ph}(E){t}_{c}]Q(E)\phantom{\rule{0.16667em}{0ex}}\text{d}E}{\int {P}_{0j}(E)exp[-{\mu}_{a}^{ph}(E)T-g({\mu}_{g}^{ph}(E)-{\mu}_{a}^{ph}(E))T-{\mu}_{c}^{ph}(E){t}_{c}]Q(E)\phantom{\rule{0.16667em}{0ex}}\text{d}E},\\ j=h,l\end{array}$$

(15)

and

$$\begin{array}{l}{\beta}_{j}^{g}=\frac{\int {P}_{0j}(E)({\mu}_{g}^{ph}(E)-{\mu}_{a}^{ph}(E))exp[-{\mu}_{a}^{ph}(E)T-g({\mu}_{g}^{ph}(E)-{\mu}_{a}^{ph}(E))T-{\mu}_{c}^{ph}(E){t}_{c}]Q(E)\phantom{\rule{0.16667em}{0ex}}\text{d}E}{\int {P}_{0j}(E)exp[-{\mu}_{a}^{ph}(E)T-g({\mu}_{g}^{ph}(E)-{\mu}_{a}^{ph}(E))T-{\mu}_{c}^{ph}(E){t}_{c}]Q(E)\phantom{\rule{0.16667em}{0ex}}\text{d}E},\\ j=h,l.\end{array}$$

(16)

Let ${\mu}_{c}^{hu}(E)={\mu}_{c}^{ph}(E)+\mathrm{\Delta}{\mu}_{c}(E),\phantom{\rule{0.38889em}{0ex}}{\mu}_{g}^{hu}(E)={\mu}_{g}^{ph}(E)+\mathrm{\Delta}{\mu}_{g}(E),\phantom{\rule{0.38889em}{0ex}}{\mu}_{a}^{hu}(E)={\mu}_{a}^{ph}(E)+\mathrm{\Delta}{\mu}_{a}(E)$. By substituting the attenuation coefficients of phantoms for those of human breast on the left side of equation (14), expanding the left side, and eliminating the same terms on both sides, we have

$$\frac{\int {P}_{0j}(E)\mathrm{\Delta}{\mu}_{c}(E)exp[-{\mu}_{a}^{ph}(E)T-g({\mu}_{g}^{ph}(E)-{\mu}_{a}^{ph}(E))T-{\mu}_{c}^{ph}(E){t}_{c}]Q(E)\phantom{\rule{0.16667em}{0ex}}\text{d}E}{\int {P}_{0j}(E)exp[-{\mu}_{a}^{ph}(E)T-g({\mu}_{g}^{ph}(E)-{\mu}_{a}^{ph}(E))T-{\mu}_{c}^{ph}(E){t}_{c}]Q(E)\phantom{\rule{0.16667em}{0ex}}\text{d}E}\times {t}_{c}+\frac{\int {P}_{0j}(E)\mathrm{\Delta}{\mu}_{g}(E)exp[-{\mu}_{a}^{ph}(E)T-g({\mu}_{g}^{ph}(E)-{\mu}_{a}^{ph}(E))T-{\mu}_{c}^{ph}(E){t}_{c}]Q(E)\phantom{\rule{0.16667em}{0ex}}\text{d}E}{\int {P}_{0j}(E)exp[-{\mu}_{a}^{ph}(E)T-g({\mu}_{g}^{ph}(E)-{\mu}_{a}^{ph}(E))T-{\mu}_{c}^{ph}(E){t}_{c}]Q(E)\phantom{\rule{0.16667em}{0ex}}\text{d}E}\times T\times g+\frac{\int {P}_{0j}(E)\mathrm{\Delta}{\mu}_{a}(E)exp[-{\mu}_{a}^{ph}(E)T-g({\mu}_{g}^{ph}(E)-{\mu}_{a}^{ph}(E))T-{\mu}_{c}^{ph}(E){t}_{c}]Q(E)\phantom{\rule{0.16667em}{0ex}}\text{d}E}{\int {P}_{0j}(E)exp[-{\mu}_{a}^{ph}(E)T-g({\mu}_{g}^{ph}(E)-{\mu}_{a}^{ph}(E))T-{\mu}_{c}^{ph}(E){t}_{c}]Q(E)\phantom{\rule{0.16667em}{0ex}}\text{d}E}\times T\times (1-g)={\beta}_{j}^{c}\times \mathrm{\Delta}{t}_{c}+{\beta}_{j}^{g}\times T\times \mathrm{\Delta}g,\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}j=h,l.$$

(17)

Define the symbols

$$\begin{array}{l}{\alpha}_{j}^{m}=\frac{\int {P}_{0j}(E)\mathrm{\Delta}{\mu}_{m}(E)exp[-{\mu}_{a}^{ph}(E)T-g({\mu}_{g}^{ph}(E)-{\mu}_{a}^{ph}(E))T-{\mu}_{c}^{ph}(E){t}_{c}]Q(E)\phantom{\rule{0.16667em}{0ex}}\text{d}E}{\int {P}_{0j}(E)exp[-{\mu}_{a}^{ph}(E)T-g({\mu}_{g}^{ph}(E)-{\mu}_{a}^{ph}(E))T-{\mu}_{c}^{ph}(E){t}_{c}]Q(E)\phantom{\rule{0.16667em}{0ex}}\text{d}E}\\ m=c,g,a;\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}j=h,l,\end{array}$$

(18)

and there is

$${\beta}_{j}^{c}\times \mathrm{\Delta}{t}_{c}+{\beta}_{j}^{g}\times T\times \mathrm{\Delta}g={\alpha}_{j}^{c}\times {t}_{c}+{\alpha}_{j}^{g}\times T\times g+{\alpha}_{j}^{a}\times T\times (1-g)={\alpha}_{j}^{c}\times {t}_{c}+({\alpha}_{j}^{g}-{\alpha}_{j}^{a})\times T\times g+{\alpha}_{j}^{a}\times T,\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}j=h,l,$$

(19)

where
${\beta}_{h}^{c}$ and
${\beta}_{l}^{c}$ are the average attenuation coefficients of microcalcification calibration phantom weighted by transmitted high- and low-spectra.
${\beta}_{h}^{g}$ and
${\beta}_{l}^{g}$ are the average attenuation coefficient differences between the glandular calibration phantom and the adipose calibration phantom weighted by transmitted high- and low-spectra.
${\alpha}_{j}^{m}$ (*m* = *c, g, a; j* = *h, l*) is the average attenuation coefficient difference between calibration phantoms and human breast for microcalcifications, glandular and adipose weighted by transmitted high- and low-spectra. In this paper, we focus on the calibration phantom errors caused by breast calibration phantoms and assume that
${\alpha}_{h}^{c}$ and
${\alpha}_{l}^{c}$ are zero. From equation (19), the calibration phantom errors Δ*t _{c}* and Δ

$$\mathrm{\Delta}{t}_{c}=\frac{[({\alpha}_{l}^{g}-{\alpha}_{l}^{a})g+{\alpha}_{l}^{a}]\times T\times {\beta}_{h}^{g}-[({\alpha}_{h}^{g}-{\alpha}_{h}^{a})g+{\alpha}_{h}^{a}]\times T\times {\beta}_{l}^{g}}{{\beta}_{l}^{c}\times {\beta}_{h}^{g}-{\beta}_{h}^{c}\times {\beta}_{l}^{g}},$$

(20)

$$\mathrm{\Delta}g=\frac{[({\alpha}_{h}^{g}-{\alpha}_{h}^{a})g+{\alpha}_{h}^{a}]\times {\beta}_{l}^{c}-[({\alpha}_{l}^{g}-{\alpha}_{l}^{a})g+{\alpha}_{l}^{a}]\times {\beta}_{h}^{c}}{{\beta}_{l}^{c}\times {\beta}_{h}^{g}-{\beta}_{h}^{c}\times {\beta}_{l}^{g}}.$$

(21)

Using the formulae derived in section 2, the calibration phantom errors can be calculated using publicly available data. The imaging conditions are similar to those in Kappadath and Shaw (2003, 2005) and Lemacks *et al* (2002), which agree with the clinical mammography system. The x-ray spectra are 25 kVp and 50 kVp with a Mo anode and a 0.03 mm Mo filter. The spectra data are obtained from the classical handbook (Fewell and Shuping 1978). The detector consists of a CsI:Tl converter layer coupled with an aSi:H + TFT flat-panel detector. In our calculation, the scintillator thickness *t _{s}* is 45 mg cm

$$Q(E)=A(E)E,$$

(22)

$$A(E)=1-exp(-{\mu}_{s}(E){t}_{s}),$$

(23)

where *μ _{s}*(

Hammerstein *et al* (1979) have determined the elemental compositions of glandular and adipose tissues for human breast. The composition per weight of H, N and P was well determined. However, the composition per weight of C and O presents a wide range of possible values. Table 1 lists two extreme compositions 1 and 2 and one general composition 3. In this paper, the composition per weight of C and O refers to ‘composition 3’ without specific notification. The densities equal to 0.93 g cm^{−3} for adipose tissue and 1.04 g cm^{−3} for glandular tissue.

Elemental compositions (percentage per weight) of C and O for glandular and adipose tissues (Hammerstein *et al* 1979).

Phantom materials are used for calibration. While polyethylene (CH_{2}) is commonly used for an adipose phantom material, PMMA (C_{5}H_{8}O_{2}) and acrylic (C_{3}H_{4}O_{2}) are often used for a glandular phantom material. Also, commercial breast tissue equivalent materials are available in the market. For example, breast adipose and glandular tissue equivalent materials from CIRS (Computerized Imaging Reference Systems, Inc., Norfolk, VA, USA) are based on Hammerstein *et al* (1979), and those from RMI (Radiation Measurements Inc., Madison, WI, USA) are based on White *et al* (1977). Our previous work (Mou and Chen 2007) showed that CIRS phantoms used in calibration lead to smaller calibration phantom errors compared with other phantoms. Therefore, we use CIRS phantoms in this paper. The elemental compositions and densities of CIRS phantoms and human breast are listed in table 2 (Byng *et al* 1998).

Elemental compositions (percentage per weight) and densities of CIRS phantoms and breast tissues (Hammerstein *et al* 1979, Byng *et al* 1998).

Fandos-Morera *et al* (1988) analyzed the compositions of microcalcifications and setup possible relationships with malignant and benign lesions. Calcium oxalate (CaC_{2}O_{4}) is a common component of microcalcification in human breasts. In our study, calcium oxalate is employed as microcalcifications with a density of 2.20 g cm^{−3}. Based on the elemental compositions listed in table 2, mass attenuation coefficients of CIRS phantoms and breast tissues are calculated via the database of XCOM from NIST (Berger *et al* 2005). The breast thickness is assumed to be 5 cm.

Microcalcifications are relatively small in size, ranging from 1 mm to less than 100 *μ*m, with most of them smaller than 500 *μ*m. Therefore, we will focus on microcalcifications smaller than 500 *μ*m with glandular ratio ranging from 0% to 100%.

CIRS phantoms are adopted for calibration. First, we calculate the average difference of linear attenuation coefficients between CIRS phantoms and breast tissues. According to the elemental compositions listed in table 2, we can obtain the mass attenuation coefficients using the XCOM database, and then calculate the average difference as

$$\text{Average}\phantom{\rule{0.16667em}{0ex}}\text{Difference}=\frac{1}{40}\sum _{E=10\phantom{\rule{0.16667em}{0ex}}\text{kev}}^{50\phantom{\rule{0.16667em}{0ex}}\text{kev}}\frac{\mid {(\mu (E)/\rho )}^{ph}{\rho}^{ph}-{(\mu (E)/\rho )}^{hu}{\rho}^{hu}\mid}{{(\mu (E)/\rho )}^{hu}{\rho}^{hu}}\times 100\%,$$

(24)

where *μ*(*E*)/*ρ* is the mass attenuation coefficient and *ρ* is the mass density. In the energy range 10–50 keV, the average difference between CIRS phantoms and breast tissues is very small: 1.14% for adipose and 0.95% for glandular. Hence, the attenuation characteristics of CIRS phantoms can mimic those of breast tissues well.

Calibration phantom errors are calculated using equations (20) and (21) with the spectra data, detector response function, mass attenuation coefficients and densities of phantoms and breast tissues mentioned in section 3. Tables 3 and and44 list the calculated calibration phantom errors for different combinations of the microcalcification thickness and glandular ratio for composition 3, showing that the calibration phantom errors are large enough to seriously deteriorate the calculation precision.

The breast compositions of different individuals vary greatly. When calibration phantoms are fixed, the variations of different individuals result in different calibration phantom errors. Tables 5 and and66 list the calibration phantom errors Δ*t _{c}* and Δ

Maps of computed pair (
${t}_{c}^{\prime}$, *g*′) to real pair (*t*_{c}, *g*) when breast composition changes.

The data calculation results in section 4 show that the calibration phantom errors are large enough that they greatly decrease the calculation precision of dual-energy digital mammography. Although the used CIRS phantoms have only about a 1% difference in linear attenuation coefficients compared to the general breast tissue composition, the calibration phantom errors could reach tens to hundreds of percents. For example, when a microcalcification is 250 *μ*m, the calibration phantom error Δ*t _{c}* ranges from −27% (−67

Because our imaging conditions are similar to those of existing references (Kappadath and Shaw 2003, 2005, Lemacks *et al* 2002), we can compare our results to the reported results in these papers. Lemacks *et al* (2002) investigated the errors caused by quantum noise of the detector with normal exposure dose. They showed that a 250 *μ*m microcalcification could be subject to random noise of 80 *μ*m for a 5 cm breast with 50% glandular ratio. However, our study shows that the corresponding calibration phantom error of the microcalcification thickness is 187 *μ*m for composition 3 and even bigger for compositions 1 (−270 *μ*m) and 2 (628 *μ*m). Thus, the calibration phantom errors are much greater than errors caused by detector quantum noise. Kappadath and Shaw (2003) concluded that the mean fitting error is ~50 *μ*m and the max fitting error is ~150 *μ*m for the microcalcification thickness when adopting a cubic or conic function. In our study, when the breast is of composition 3, the mean calibration phantom error for the microcalcification thickness is 192 *μ*m and the maximum is 433 *μ*m (see table 3). When the breast is of composition 2, the maximum calibration phantom error will reach 813 *μ*m (see table 5). Obviously, calibration phantom errors are much greater than the fitting errors in dual-energy digital mammography. Kappadath and Shaw (2005) showed that microcalcifications of 300–355 *μ*m can be clearly seen while microcalcifications of 250–280 *μ*m are barely visible in dual-energy calcification images after scatter and nonuniformity corrections. Due to the existence of large calibration phantom errors and the diversity of breast compositions, the microcalcification thickness threshold cannot be determined even if both scatter and nonuniformity corrections are well addressed. In summary, a calibration phantom error is dominant among all kinds of errors in dual-energy digital mammography calculation.

From tables 3 and and4,4, we can see that for a fixed glandular ratio, calibration phantom errors are almost unchanged when the microcalcification thickness changes from 0 *μ*m to 500 *μ*m. For example, Δ*t _{c}* changes from 193

In section 2.1, we showed that the attenuation of an imaged object can be described by equation (4), and the pair (*b _{d}, b_{p}*) can be used to represent the attenuation of the object. In mammography, the imaged object is breast tissues plus microcalcifications. We have

$${b}_{d}={b}_{d\_\text{breast}}+{b}_{d\_mc},\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}{b}_{p}={b}_{p\_\text{breast}}+{b}_{p\_mc}$$

(25)

where (*b _{d_}*

We define a quantity of attenuation gradient (*AG*) as follows:

$$AG={b}_{p}/{b}_{d}.$$

(26)

In our study, the component of microcalcification is calcium oxalate (CaC_{2}O_{4}). The *AG* for microcalcification can be written as

$${AG}_{mc}={b}_{p\_mc}/{b}_{d\_mc}=({b}_{pu\_{\text{CaC}}_{2}{\text{O}}_{4}}\times {t}_{c})/({b}_{du\_{\text{CaC}}_{2}{\text{O}}_{4}}\times {t}_{c})={b}_{pu\_{\text{CaC}}_{2}{\text{O}}_{4}}/{b}_{du\_{\text{CaC}}_{2}{\text{O}}_{4}.}$$

(27)

According to equation (1), *b*_{du_CaC2 O4} and *b*_{pu_CaC2 O4} only relate to the linear attenuation coefficient of calcium oxalate. Therefore, *AG _{mc}* is a fixed value in our study.

Now, we adopt a 250 *μ*m microcalcification with 50% glandular ratio of composition 3 as the imaged object. Recall that table 3 shows that the calibration phantom error Δ*t _{c}* is 187

As can be seen in tables 3 and and4,4, the calibration phantom errors are almost unchanged when the glandular ratio is fixed and the microcalcification thickness varies. This phenomenon can be explained by figure 4. The blue and mauve dotted line segments, point B and point B_{c}, have the same meaning as in figure 3. For a glandular ratio of 50%, point B is fixed on the blue dotted line segment which maps to the attenuation of breast tissues. We can draw a red line that crosses point B with gradient *AG _{mc}*. The vector

However, there still exist small differences among the calibration phantom errors that can be seen in tables 3 and and44 when the glandular ratio is constant. These differences come from the dependence of energy during the calculation of *b _{p}* and

Omitting the small difference caused by energy dependence, the calibration phantom errors will be constant if the imaged objects have the same glandular ratio. This property suggests a correction method for the calibration phantom error Δ*t _{c}*.

In dual-energy digital mammography, for an imaged breast, we can calculate (*g*′,
${t}_{c}^{\prime}$) of each pixel pair based on the high- and low-energy image and set up a coordinate system with the horizontal axis as *g*′ and vertical axis as
${t}_{c}^{\prime}$. (*g*′,
${t}_{c}^{\prime}$) of each pixel pair can be positioned in the coordinate system. Obviously, the computed
${t}_{c}^{\prime}$ of the non-microcalcification pixel pair equals their Δ*t _{c}*.

In full-field digital mammography, there are several millions of pixels in an image. Because microcalcifications are small and sparsely present in breast tissues, most pixels in mammogram are non-microcalcification pixels. With millions of pairs in the coordinate system, a curve can be fitted. (*g*′,
${t}_{c}^{\prime}$) on the fitted curve can be considered as those correspond to non-microcalcification pixel pairs. Therefore,
${t}_{c}^{\prime}$ of (*g*′,
${t}_{c}^{\prime}$) on the fitted curve is the calibration phantom error Δ*t _{c}*.

Recall that the calibration phantom errors Δ*t _{c}* and Δ

Dual-energy digital mammography can be used to suppress the contrast between adipose and glandular tissues to improve the detection of microcalcifications. It is a prospective technique to reduce the high false-negative rate of microcalcification detection in normal mammography. In this paper, the formulae of calibration phantom errors, which are caused by the differences of linear attenuation coefficients between calibration phantoms and breast tissues, were derived based on the imaging physical model. Based on the publicly available composition data of breast tissues and phantom materials, we calculated the calibration phantom errors and compared them with other types of errors in dual-energy digital mammography. Our results show that the attenuation differences between breast tissues and calibration phantoms can lead to significantly large calibration phantom errors in the order of tens to hundreds of percent, even if CIRS phantoms are adopted. A calibration phantom error is the major source of error in dual-energy digital mammography and varies for different breast compositions of different individuals.

After deeply investigating the physical meaning of calibration phantom errors, we found that calibration phantom errors are constant for a fixed glandular ratio. With this property and using all the pixels in the images, we suggested a correction method for the calibration phantom error of the microcalcification thickness. The results show that the large calibration phantom error of the microcalcification thickness can be corrected.

The project is partially supported by the National Science Fund of China (no 60472004 and no 60551003), the fund of the Ministry of Education of China (no 106143 and no NCET-05-0828), the fund of USA National Institutes of Health (EB007288) and the fund of ICRG of Hong Kong Polytechnic University (YG-79). The authors thank Ms Lena Ye for editorial refinements.

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