A model study of 3-dimensional localization of breast tumors using piezoelectric fingers of different probe sizes

Abstract

Mammography is the only Food and Drug Administration approved breast cancer screening method. The drawback of the tumor image in a mammogram is the lack of tumor depth information as it is only a 2-dimensional projection of a 3-dimensional (3D) tumor. In this work, we investigated 3D tumor imaging by assessing tumor depth information using a set of piezoelectric fingers (PEFs) with different probe sizes which were known to be capable of eliciting tissue elastic responses to different depths and tested it on model tumor tissues consisted of gelatin with suspended clay inclusions. The locations of the top and bottom surfaces of an inclusion were resolved by solving a simple spring model using the elastic measurements of the PEFs of different probe sizes as the input. The lateral sizes of an inclusion were determined as the full width at half maximum of the Gaussian fit to the measured lateral tumor elastic modulus profile. The obtained lateral inclusion sizes were in close agreement with the actual values, and the deduced depth profiles of an inclusion also agreed with the actual depth profiles so long as the bottom surface of the inclusion was within the depth sensitivity of the PEF with the largest probe size. This work offers a simple non-invasive method to predict the extent of a tumor in all 3 dimensions. The method is also non-radioactive.

FIG. 1.

(a) A schematic of a piezoelectric finger (PEF) and (b) a photograph of a 6.5-mm wide PEF.

FIG. 2.

(a) A photograph of the 35 cm × 23 cm × 20 cm model breast tissue with 30 suspended clay inclusions that were 15 mm long and 15 mm wide but with various heights and embedded at various depths, (b) a schematic of the cross-sectional view of a suspended inclusion, (c) a schematic of an inclusion whose d₂ was less than h_D, and (d) that of an inclusion whose d₂ was larger than or equal to h_D, where E_n denotes the elastic modulus of the gelatin and E_t denotes the elastic modulus of the inclusion, d₁ and d₂ were the distances from the surface of the model tissue to the top surface and the bottom surface of the inclusion, respectively, and h_D was the depth sensitivity of PEF D, the largest depth sensitivity of all PEFs.

FIG. 3.

An example of the lateral (x direction) inclusion size determination where the elastic modulus of inclusion versus x was fitted to a Gaussian function and the size of the inclusion in the x direction was taken as the full width of the half maximum (FWHM) of the Gaussian fit, where x represented one of the two lateral directions—the width direction.

FIG. 4.

A schematic of (a) regime (i) where the depth sensitivity, h, of the PEF was less than d₁, (b) regime (ii) where the depth sensitivity of the PEF was such that d₁ < h < d₂, and (c) regime (iii) where h was larger than d₂.

FIG. 5.

2-dimensional (2D) elastic modulus maps of the model tissue measured (a) by PEF A which had a probe of 4.1 ± 0.2 mm in width, (b) by PEF B which had a probe of 6.5 ± 0.2 mm in width, (c) by PEF C which had a probe of 8.2 ± 0.2 mm in width, and (d) by PEF D which had a probe of 9.8 ± 0.3 mm in width where x and y represent the horizontal (width) and vertical (length) coordinates in cm, respectively, and green indicates elastic moduli around 10 kPa and yellow and red indicate an increasing elastic modulus.

FIG. 6.

The lateral elastic modulus profiles in the x (width direction) by four PEFs of (a) inclusion D1 and (b) inclusion D2 measured by the four PEFs versus the x (width) direction. The data were fitted to a Gaussian function, and the size of the inclusion was taken as the full width of the half maximum (FWHM) of the Gaussian fit. Note that the FWHM of the Gaussian fits of the lateral elastic modulus profiles as measured by all four PEFs were essentially the same.

FIG. 7.

Four possible distinctive elastic modulus (E) profiles in the thickness direction for an inclusion as measured in the order of PEF A, PEF B, PEF C, and PEF D: (a) E decreased monotonically from PEF A to PEF D as illustrated by inclusion A2, (b) E peaked at PEF B as illustrated by inclusion B2, (c) E peaked at PEF C as illustrated by inclusion C2, and (d) it increased monotonically from PEF A to PEF D as illustrated by inclusion E2. The horizontal shaded area indicated the range of the measured elastic modulus of the gelatin matrix. The inset in each figure depicted how h_A, h_B, h_C, and h_D relate to d₁ and d₂ for each scenario of the elastic moduli as measured by PEFs A, B, C, and D.

FIG. 8.

(a) Measured d₁ versus actual d₁ of all the inclusions and (b) measured d₂ values versus actual d₂ of inclusions A1-A6, B1-B6, C1-C3, D1-D2, and E1 whose d₂ were resolvable. The dashed line with a slope of unity was to guide the eye. These results indicate that the deduced d₁ and d₂ were in close agreement with the actual d₁ and d₂.