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.

I. INTRODUCTION

Breast cancer (BC) has long been the cause of one of the highest cancer fatality rates for women. It is the second most common malignancy diagnosed in women. In the US, it is estimated that 252 710 invasive breast cancers and 63 410 carcinomas in situ would be diagnosed in 2017.¹ It is estimated that 500 000 women in the world will die from breast cancer each year.²

Current BC detection methods include clinical breast examination (CBE), ultrasound, mammography, and magnetic resonance imaging (MRI).^3,4 Once a breast tumor is found, localization of the tumor is needed to guide the biopsy and later the surgery.⁵ If tumors can be localized in 3-dimensional (3D) space, it will much improve the accuracy of the biopsy and the surgery. Furthermore, the tumor volume is often used as a measure to monitor how well a tumor responds to a systemic therapy. The ability to accurately image a tumor in 3D will also improve the accuracy of the tumor volume measurement to better gauge how the tumor responds to the therapy. Among the current BC detection methods, CBE can detect only palpable tumors associated with late stages,^6,7 and it is not quantitative and not ideal for tumor 3D localization. Although mammography, MRI, and ultrasound imaging can localize a tumor and quantify its size, their images represent a two-dimensional (2D) projection of a 3D object, lacking the depth information. As a result, these methods cannot always capture the maximum tumor extent (MTE) when the MTE is in the depth direction.⁸ Mammography and MRI need to flatten a breast between two plates during testing. As a result, the location and extent of a tumor derived from such images can be distorted. For example, it has been well documented that MRI can both underestimate and overestimate the tumor size, and thus the tumor size obtained by MRI alone cannot be relied on for staging.^9–11 The discordance between the MRI tumor size and the pathological tumor size could also contribute to re-excisions in patients who undergo lumpectomy. Although ultrasound is widely available and does not require flattening of the breast during testing, it is known to underestimate the tumor size especially for invasive lobular breast cancers,^12–14 which again may lead to incomplete excisions in lumpectomies.⁵ Although 3D microfilter devices can detect cancer by capturing circulating cancer cells^15,16 in the blood, they do not detect or locate the primary cancer.

Breast tumors are stiffer than the surrounding tissue.^17–19 This very property has motivated many researchers to detect breast tumors by tactile imaging. SureTouch is an early tactile imaging-based breast tumor detection method where a tumor is detected and its location and extent determined through inversion simulations^20–22 and is able to detect model tumors as small as 5 mm in size.²¹ However, the accuracy of its tumor depth prediction or its depth detection limit has not been examined.²² Tactile Sensation Imaging System (TSIS) is another tactile imaging-based method that used inversion simulations along with the neural network analysis to further improve its tumor detection capability.^23,24 Although TSIS detected model tumors as small as 2 mm in size, it, too, has not addressed the accuracy of its tumor depth prediction or its depth detection limit.

Clearly, there is a need for a breast cancer detection method that can accurately localize a tumor and determine its extent, particularly in the depth direction to help guide biopsies and surgeries, or help monitor the response of a tumor to therapies or the progression or remission of a cancer.^25–34 A piezoelectric finger (PEF) (as shown in Fig. 1) is a new type of sensor that can measure the tissue elastic modulus (E) in vivo by directly placing the PEF on the tissue of interest.^35–38 A tumor is detected by contrasting the stiffness of the tissue containing a tumor with those of the surrounding normal tissues. In our previous in vivo study, PEF has been tested on 40 patients and demonstrated that it could detect most types of breast tumors in vivo including fibroadenomas, cysts, invasive carcinomas, and ductal carcinomas in situ.^39,40 The overall sensitivity of the PEF test was 87%. In women who were 40 years old or younger, the overall sensitivity was 100%. The smallest tumor detected by the PEF was 2 mm × 5 mm.^39,40 Furthermore, our previous model tissue studies have also shown that the detectable depth by a PEF was approximately twice that of the probe size of the PEF.^37,40 In other words, with a larger probe size, a PEF can assess the elastic response of the tissue at a larger depth. It is therefore possible to use a set of PEFs each with a different probe size to do the elastic modulus measurements on the same tissue and use the obtained results to deduce the tissue elastic modulus profile in the depth direction without the need of any inversion simulations. In addition to the need of inversion simulations, tactile imaging measures the pressure distribution which could be affected by the variation of the pressure applied to the device by the operator and required an additional algorithm to remove the background pressure profile.^21,22 PEF, on the other hand, measures the elastic modulus map by mN-range forces and μm-scale displacements.⁴¹ As a result, PEF tumor detection and tumor size determination have been shown to be insensitive to the pressures applied on the housing of the PEF,³⁹ thus less likely to be operator dependent.

FIG. 1.

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

The goal of this study is to investigate the feasibility of 3D tumor localization by determining the location and the extent of a tumor in the depth direction using a set of PEFs each with a different probe size. Previously, we have shown that using the elastic modulus measurements from two PEFs each with a different probe size as the input for a two-spring model, we could determine the depth of a bottom-supported inclusion in model breast tissues.³⁷ In this study, we focus on determining the depth profiles of suspended tumors by probing the elastic modulus of a tumor-containing tissue using multiple PEFs each with a different probe size and using these measurements to solve for the tumor depth profile via a simple spring model.

II. EXPERIMENTAL

A. Piezoelectric finger (PEF) and PEF elastic modulus measurement

A piezoelectric finger (PEF)^35–37 is a piezoelectric cantilever consisting of two lead zirconate titanate (PZT) layers (T105-H4E-602, Piezo Systems, Inc.) sandwiching a stainless steel layer (Alfa Aesar) as schematically shown in Fig. 2(a). A probe made of a stainless loop was glued to the tip of the underside of the stainless-steel layer illustrated in Fig. 2(a) for contacting the tissue. The elastic modulus of a tissue can be measured by simply placing the probe of a PEF on top of the tissue and applying a direct current (DC) voltage to the top PZT layer of a PEF to exert a force on the tissue, which in turn induces a piezoelectric voltage across the sensing PZT layer,³⁸ and the elastic modulus, E, of the tissue can be determined as^36,38

$$E=\frac{1}{2}{\left(\frac{\pi }{A}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\left(1-{\nu }^2\right)\frac{K\left(V_{in,0}-V_{in}\right)}{V_{in}},$$ (1)

where V_in,0 and V_in are the induced voltages without and with the tissue, respectively, ν is Poisson’s ratio of the tissue, A is the contact area of the probe, and K is the effective spring constant of the PEF. In this study, four PEFs of different probe sizes were used. The probe widths were 4.1 ± 0.2, 6.5 ± 0.2, 8.2 ± 0.2, and 9.8 ± 0.3 mm. The dimensions of all four PEFs are listed in Table I.

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.

TABLE I.

The dimensions of PEFs used in the experiment.

	Probe width	Length of driving	Length of	Depth sensitivitya
PEF cantilever	(mm)	PZT (mm)	sensing PZT (mm)	(mm)
A	4.1 ± 0.2	22.3 ± 0.3	10.1 ± 0.3	8.2 ± 0.4
B	6.5 ± 0.2	22.6 ± 0.4	10.5 ± 0.3	13.0 ± 0.4
C	8.2 ± 0.2	22.1 ± 0.4	10.3 ± 0.4	16.4 ± 0.4
D	9.8 ± 0.3	22.4 ± 0.4	10.6 ± 0.3	19.6 ± 0.6

^aEstimated as twice the probe width.⁴⁰

B. Breast tumor model

The model breast tissue was prepared by mixing gelatin (Beef Gelatin Powder, Now Foods, Bloomingdale, IL) with water at 80 °C at a concentration of 0.12 g/ml. When cooled to room temperature, this gelatin solution would solidify to yield an elastic modulus of about 10 kPa as measured by using Bose ElectroForce 3100 (Bose Corporation, New Castle, DE), similar to the elastic moduli of the normal breast tissues reported in the literature.^{18,19,42–45} To prepare model tissues with suspended tumors, the gelatin solution was first poured into a 35 cm × 23 cm × 20 cm container and placed in the refrigerator for 10 min to solidify. Rectangular clay inclusions (Modeling Clay, Crayola, Easton, PA) 15 mm long, 15 mm wide, and 5-15 mm in height were then placed on top of the gelatin layer followed by pouring more gelatin solution of the same concentration over the clay inclusions to embed all the clay inclusions. The total height of the gelatin matrix was 34 mm. Note that the clay model was chosen for its elastic modulus of 60 kPa as measured by using Bose ElectroForce 3100 which was similar to the elastic moduli of excised breast tumors measured by PEF^39,40 and by other methods.^18,19,46 A photograph of the final gelatin-clay model breast tissue with suspended tumors is shown in Fig. 2(a). Figure 2(b) is a schematic of the vertical cross-sectional view of a suspended model tumor (inclusion), where d₁ and d₂ denote the distances from the surface of the model tissue to the top and the bottom of the inclusion, respectively. The model tumors were organized such that in each row, all inclusions had the same length (l), width (w), and height (d), where the only differences were their depths, i.e., different d₁ and hence different d₂ = d₁ + d, while in each column all inclusions have the same d₁ but different heights (d). The l, w, d, and d₁ of all the inclusions as measured with a caliper are listed in Table II. There were two types of model tumors. One type of model tumors including A1-A6, B1-B6, C1-C3, D1, D2, and E1 had both d₁ and d₂ smaller than or equal to the largest depth sensitivity of the PEFs, i.e., the depth sensitivity of PEF D, h_D, (19.6 ± 0.6 mm) as schematically shown in Fig. 2(c). Since d₂ was less than h_D, it was expected that the entire depth profiles of this type of model tumors could be resolved by the measurements of the current set of PEFs. Another type of model tumors including C4-C6, D3-D6, and E2-E6 had d₁ smaller than or equal to h_D (19.6 ± 0.6 mm) but d₂ larger than h_D as schematically illustrated in Fig. 2(d). For the latter type of model tumors, d₁ might be resolvable but d₂ might not be since d₂ was larger than h_D. The reason to include the latter type of model tumors was to show all possible patterns of the measured elastic moduli by a set of PEFs of different probe sizes including those with d₂ larger than h_D and to illustrate how to use the patterns of the measured elastic moduli to resolve the depth profiles of tumors.

TABLE II.

The known length, width, height, and top and bottom depths, d₁ and d₂, respectively, of the suspended inclusions in the phantom as derived from actual caliper measurements.

	Inclusion dimensions			Inclusion depths
Inclusion #	Length (mm)	Width (mm)	Height (mm)	d1 (mm)	d2 (mm)
A1	14.5 ± 0.3	15.2 ± 0.4	4.9 ± 0.4	2.2 ± 0.3	7.1 ± 0.5
A2	15.2 ± 0.4	15.3 ± 0.2	5.0 ± 0.3	4.1 ± 0.3	9.1 ± 0.4
A3	15.4 ± 0.3	15.1 ± 0.4	5.2 ± 0.3	5.9 ± 0.3	11.2 ± 0.4
A4	15.9 ± 0.4	15.2 ± 0.2	5.0 ± 0.4	7.2 ± 0.3	12.2 ± 0.5
A5	15.2 ± 0.3	15.2 ± 0.3	5.2 ± 0.3	10.3 ± 0.3	15.6 ± 0.4
A6	15.4 ± 0.3	15.6 ± 0.2	5.3 ± 0.3	11.9 ± 0.2	17.2 ± 0.4
B1	14.5 ± 0.3	14.6 ± 0.4	8.5 ± 0.3	3.0 ± 0.3	11.5 ± 0.4
B2	15.2 ± 0.2	15.2 ± 0.4	8.0 ± 0.3	4.1 ± 0.4	12.0 ± 0.5
B3	14.7 ± 0.2	15.8 ± 0.3	7.2 ± 0.3	6.5 ± 0.3	13.7 ± 0.4
B4	15.4 ± 0.3	15.7 ± 0.4	7.5 ± 0.4	9.6 ± 0.4	17.0 ± 0.5
B5	15.6 ± 0.2	15.5 ± 0.4	7.7 ± 0.3	11.8 ± 0.3	19.5 ± 0.5
B6	15.6 ± 0.4	15.3 ± 0.4	7.7 ± 0.3	12.0 ± 0.4	19.7 ± 0.4
C1	16.2 ± 0.4	16.0 ± 0.4	9.4 ± 0.3	5.3 ± 0.3	14.7 ± 0.4
C2	16.2 ± 0.3	15.3 ± 0.2	9.7 ± 0.3	7.0 ± 0.3	16.7 ± 0.5
C3	15.9 ± 0.3	15.4 ± 0.3	9.5 ± 0.3	8.3 ± 0.3	17.8 ± 0.5
C4	15.7 ± 0.2	16.1 ± 0.3	9.3 ± 0.3	11.2 ± 0.5	20.5 ± 0.6a
C5	16.0 ± 0.3	15.7 ± 0.3	9.8 ± 0.3	12.2 ± 0.3	22.0 ± 0.4a
C6	15.9 ± 0.4	16.2 ± 0.3	9.7 ± 0.3	13.7 ± 0.4	23.4 ± 0.5a
D1	14.7 ± 0.4	15.4 ± 0.2	11.4 ± 0.4	4.2 ± 0.3	15.6 ± 0.5
D2	14.7 ± 0.3	14.9 ± 0.3	11.6 ± 0.3	6.7 ± 0.3	18.3 ± 0.6
D3	14.6 ± 0.2	15.2 ± 0.3	11.7 ± 0.3	10.5 ± 0.4	22.1 ± 0.5a
D4	14.9 ± 0.3	15.0 ± 0.4	11.6 ± 0.3	12.1 ± 0.3	23.7 ± 0.4a
D5	14.8 ± 0.2	15.1 ± 0.3	11.7 ± 0.3	15.1 ± 0.3	26.8 ± 0.4a
D6	15.2 ± 0.4	14.8 ± 0.4	11.7 ± 0.4	16.2 ± 0.3	27.9 ± 0.5
E1	15.8 ± 0.3	15.7 ± 0.3	14.8 ± 0.3	3.2 ± 0.3	18.0 ± 0.4
E2	15.8 ± 0.4	15.8 ± 0.3	15.0 ± 0.3	6.4 ± 0.3	21.5 ± 0.4a
E3	15.8 ± 0.3	16.1 ± 0.3	15.1 ± 0.3	8.6 ± 0.4	23.7 ± 0.5a
E4	15.4 ± 0.4	15.7 ± 0.4	15.1 ± 0.3	10.3 ± 0.3	25.4 ± 0.4a
E5	15.7 ± 0.4	15.7 ± 0.3	15.2 ± 0.4	13.6 ± 0.4	28.8 ± 0.6a
E6	15.7 ± 0.3	15.6 ± 0.4	14.8 ± 0.4	15.6 ± 0.4	30.4 ± 0.5a

^aIndicates d₂ was larger than the largest depth sensitivity, h_D.

C. PEF elastic modulus map

All four PEFs were used to scan the entire model tissue. First, a set of five voltages was applied to the PEF without touching the sample to produce the five corresponding induced voltages without the sample, V_in,0. At each location, care was taken that the probe of the PEF was in complete contact with the sample surface and the set of five voltages was applied to the driving PZT to produce five induced voltages with sample, V_in, at the sensing PZT. Using these five V_in measurements together with the five V_in,0 obtained from the same set of applied voltages, $\frac{1}{2}{\left(\pi /A\right)}^{1/2}\left(1-{\nu }^2\right)K\left(V_{in,0}-V_{in}\right)$ was then computed and plotted versus V_in. The elastic modulus at that location was then obtained as the slope of the plot.^{35–37,39,40} At each location, five independent elastic modulus measurements were made; the average of which was then used to create the elastic modulus map.

D. Lateral size of inclusion

To determine the lateral inclusion size in the x direction at a particular y location, the average elastic moduli at the particular y location was plotted versus x and fitted to a Gaussian form around the elastic modulus maximum. The inclusion size in the x direction at that particular y location was then taken as the width at the half height of the Gaussian fit. As an example, the elastic modulus versus x at y = 3.0 cm of inclusion D1 obtained by PEF D is shown as filled squares in Fig. 3 and the Gaussian fit as the solid line. The size of inclusion at y = 3 cm was then taken as the full width at the half maximum (FWHM) of the solid line which was 1.55 cm, close to the 1.54 cm known inclusion size. The inclusion sizes in the x direction at other y values and those in the y direction at certain x values were obtained in a similar fashion.

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.

E. Inclusion depth profile determination

Consider a suspended inclusion with the depths of the top and bottom surfaces located at d₁ and d₂, respectively, as illustrated in Fig. 2(b). The depth sensitivity of a PEF, h, can fall into one of the three regimes: (i) h < d₁, (ii) d₁ < h < d₂, and (iii) h > d₂. In regime (i), the PEF detected the elastic response of only the gelatin above the inclusion as schematically shown in Fig. 4(a). Thus, the measured elastic modulus, E, should be the same as that of the gelatin, i.e.,

$$E=E_n\mathrm{f}\mathrm{o}\mathrm{r}h<d_1,$$ (2)

where E_n is the elastic modulus of the gelatin matrix.

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.

In regime (ii), d₁ <h < d₂ [see Fig. 4(b)], the elastic response measured by the PEF was the combination of the elastic response of the gelatin above the inclusion and that of the portion of the inclusion. Following our earlier approach for bottom-supported inclusion,³⁷ this could be treated as two springs in series as illustrated in Fig. 4(b) and the measured elastic modulus, E, could be related to E_n, d₁, h, and E_t which is the elastic modulus of the inclusion³⁷

$$\frac{h}{E}=\frac{d_1}{E_n}+\frac{h-d_1}{E_t}.$$ (3)

Note that E_n can be obtained in the inclusion-free regions of the sample. Therefore, there are only two unknowns, namely, d₁ and E_t. In general, this would require measurements with two PEFs of different h values to solve for d₁ and E_t simultaneously. However, we notice that because E_t is generally much larger than E_n and E, the obtained values for d₁ should not vary much for E_t ≥ 40 kPa. Therefore, within the experimental uncertainty, it is possible to use only the E measurement of one PEF to obtain d₁ by assuming E_t ≥ 40 kPa. Earlier studies showed that breast tumors exhibited E_t = 40-60 kPa as measured by PEF^39,40 and by other methods.^18,19,46 We will examine how the assumed E_t values ranging from 40 to 200 kPa affect the obtained d₁.

In regime (iii), the depth sensitivity, h, of the PEF is larger than d₂ as illustrated in Fig. 4(c). The PEF detects the elastic response beyond the bottom of the inclusion. Therefore, the measured elastic modulus, E, could be thought as the result of three springs in series [Fig. 4(c)] and be related to E_n, E_t, d₁, d₂, and h as

$$\frac{h}{E}=\frac{d_1}{E_n}+\frac{h-d_1}{E_t}+\frac{h-d_2}{E_n}.$$ (4)

The first term and the last term of the right-hand side of Eq. (4) can be combined, and Eq. (4) can be rewritten as

$$\frac{h}{E}=\frac{h+d_1-d_2}{E_n}+\frac{h-d_1}{E_t}.$$ (5)

Note that d₁ and E_t can be independently obtained by solving Eq. (3) using measurements from PEFs with h smaller than d₂ as described above. Thus, there is only one unknown left, i.e., d₂, which could be solved using the elastic modulus measurements obtained by at least one PEF with h > d₂ with the same E_t values obtained/used in regime (ii).

III. RESULTS AND DISCUSSIONS

A. Elastic modulus maps

The resultant elastic modulus scans on the model tissue presented as color-coded elastic modulus maps as measured by PEFs A, B, C, and D are shown in Figs. 5(a)–5(d), respectively. Measurements of the gelatin matrix indicated that the gelatin matrix had an elastic modulus of 10.15 ± 0.75 kPa. Therefore, we would deem areas with elastic moduli ranging 9-12 kPa as the gelatin matrix and use green color to represent these areas in Figs. 5(a)–5(d). Areas with the inclusion embedded underneath had higher elastic moduli—yellow indicates elastic moduli 14-16 kPa, and red indicates elastic moduli 18-28 kPa. As can be seen, PEF A—which had the smallest probe size—detected the fewest inclusions. As can be seen from Figs. 5(a)–5(d), more inclusions became detectable as the probe size became larger. This is because a larger probe size allowed a higher depth sensitivity, thus permitting deeper inclusions to be detected. In addition, many of the inclusions became redder as the probe size of the PEFs became larger. This was also a manifestation of the fact that a PEF with a larger probe size had a higher depth sensitivity which meant that a higher proportion of the elastic response was derived from the inclusion, thus a higher measured elastic modulus. However, this was not always the case. There were inclusions that showed just the opposite, i.e., the color of the inclusion became less red or turned from red to yellow with an increasing probe size. This was because these inclusions had a smaller height, d, thus allowing a PEF of a larger probe size to also sense the elastic response of the gelatin matrix beneath the bottom of the inclusion and leading to a smaller measured elastic modulus. Inclusions A1-A6 in the first row with a height (d) of only 5 mm were such examples where the measured elastic moduli by PEF D were smaller than those by PEF B. As a result, d₂ = d₁ + d was less than the h of PEF D {regime (iii) [see Fig. 4(c)]} but larger than the h of PEF B {regime (ii) [see Fig. 4(b)]}. More detailed analyses and discussions about various depth profiles of the suspended inclusion will be given in Sec. III C. It is also worth noting that although the sizes of the yellow and red spots seemed to increase with an increasing probe size and suggest that a larger probe size could give rise to a larger measured lateral size, the truth was quite the contrary—by means of a Gaussian fit we found that the measured lateral sizes of an inclusion remained unchanged irrespective of the probe size (see Sec. III B).

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.

B. The lateral size of inclusions

To determine the lateral sizes of an inclusion, we plot elastic moduli versus the lateral coordinates centered about the inclusion. As examples, we plot elastic moduli versus x measured by all four PEFs for inclusion D1 and inclusion E1 in Figs. 6(a) and 6(b), respectively, where x denotes the lateral coordinate in the width direction. Each curve was fitted to a Gaussian distribution. The lateral sizes (i.e., widths in these two cases) were then determined as the full widths at half maximum (FWHM) of the Gaussian fits. For inclusion D1, the widths as determined by PEFs A, B, C, and D were 1.60 ± 0.25, 1.54 ± 0.13, 1.56 ± 0.19, and 1.55 ± 0.23 cm, respectively. These were essentially identical within the experimental uncertainty and were all in agreement with the known width 15.4 ± 0.2 mm of D1 (see Table II). For inclusion E1, the widths as determined by PEFs A, B, C, and D were 1.55 ± 0.10, 1.59 ± 0.21, 1.58 ± 0.18, and 1.56 ± 0.22 cm, respectively. These again were all identical within the experimental uncertainty and were all consistent with the actual width of E1, 1.57 ± 0.03 cm (see Table II). Clearly, the use of the Gaussian fit was effective to accurately deduce the lateral size despite that the sizes of the yellow and red spots may have appeared larger or smaller with different PEFs. Note that the results for the lateral sizes of all other inclusions though not shown were all similarly accurate and similarly independent of the probe size. As such, in the following, for all inclusions, a measured lateral size of an inclusion was taken as the average of the measurements by all four PEFs.

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.

C. The depth profile of inclusions

Because the four PEFs had different probe sizes (hence different depth sensitivities), their elastic modulus measurements for the same inclusion were quite different. With the depth sensitivities of PEF A, PEF B, PEF C, and PEF D defined as h_A, h_B, h_C, and h_D, respectively, the elastic moduli of a given inclusion as measured in the order of PEF A, PEF B, PEF C to PEF D may fall into one of the four scenarios as illustrated in Figs. 7(a)–7(d). As illustrated in Figs. 2(c) and 2(d), there were two types of inclusions: one with d₂ < h_D and the other with d₂ > h_D. For the type with d₂ < h_D, we will have three scenarios where (1) the measured elastic moduli could decrease monotonically from PEF A to PEF D as illustrated by those of inclusion A2 in Fig. 7(a), (2) the measured elastic moduli could be peaked at PEF B as illustrated by those of B2 in Fig. 7(b), or (3) the measured elastic moduli could be peaked at PEF C as illustrated by those of inclusion C2 in Fig. 7(c). For the type with d₂ > h_D, the measured elastic modulus could increase monotonically from PEF A to PEF D as illustrated by those of inclusion E2 in Fig. 7(d). We would use these measurements together with the analysis described in Sec. II E to determine the depth profile (i.e., the top location d₁ and the bottom location d₂ of an inclusion).

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.

For the scenario illustrated by inclusion A2 in Fig. 7(a), the elastic modulus as measured by PEF A was greater than those measured by PEFs B, C, and D. This indicates that d₁ was smaller than the depth sensitivity of PEF A [regime (ii)] and d₂ was smaller than the depth sensitivity of PEF B [regime (iii)] as well as that of PEF D as illustrated in the schematic shown in the inset in Fig. 7(a). Because PEF A was in regime (ii), d₁ could be deduced by solving Eq. (3) using the measurement by PEF A together with the known elastic modulus of the gelatin matrix. The obtained d₁ was 3.6 ± 1.0 mm, which was consistent with the known value of d₁, 4.1 ± 0.3 mm, for inclusion A2 (see Table II) but also in agreement with the fact that d₁ was smaller than the depth sensitivity of PEF A (8.2 ± 0.4 mm). The value of d₂ was determined to be 8.8 ± 1.3 mm by solving Eq. (5) using the above value for d₁ and the elastic moduli obtained by PEFs B, C, and D. This value was consistent with that the known value of d₂ was 9.1 ± 0.4 mm (see Table II) but also that d₂ was indeed smaller than the depth sensitivity of PEF B (13.0 ± 0.4 mm), that of PEF C (16.4 ± 0.4 mm), and that of PEF D (19.6 ± 0.6 mm).

For the scenario illustrated by inclusion B2 in Fig. 7(b), the elastic modulus as measured by PEF A was greater than E_n, indicating that d₁ was smaller than the depth sensitivity of PEF A (8.2 ± 0.4 mm). In addition, the elastic modulus measured by PEF B was larger than that obtained by PEF A, indicating that both PEF A and PEF B were in regime (ii) [Fig. 4(b)]. Therefore, d₁ could be obtained by solving Eq. (3) using the elastic moduli obtained by PEF A and PEF B as the input. The obtained d₁ was 4.5 ± 0.6 mm, consistent with both the known values of d₁, 4.1 ± 0.4 mm (see Table II), but also in agreement with that d₁ must be smaller than the depth sensitivity of PEF A (8.2 ± 0.4 mm). That the measured elastic moduli by PEFs C and D were smaller than that measured by PEF B indicates that PEFs C and D were in regime (iii) [Fig. 4(c)]. Therefore, d₂ was solved using Eq. (5) with d₁ = 4.5 ± 0.6 mm as obtained with the measurements by PEF A and B. The obtained d₂ = 12.4 ± 1.5 mm was consistent with that the known value of d₂ was 12.0 ± 0.5 mm (see Table II) and that d₂ was indeed smaller than both the depth sensitivities of PEF C (16.4 ± 0.4 mm) and that of PEF D (19.6 ± 0.6 mm). A schematic illustrating how d₁ and d₂ were relating the depth sensitivities of all four PEFs is shown in the inset in Fig. 7(b).

For the scenario illustrated by inclusion C2 shown in Fig. 7(c), the elastic modulus as measured by PEF A was close to E_n, indicating that d₁ was close to the depth sensitivity of PEF A (8.2 ± 1.0 mm). Furthermore, the elastic moduli measured by PEF B and PEF C were both larger than that obtained by PEF A, indicating that PEF B and PEF C were in regime (ii) [Fig. 4(b)]. Therefore, d₁ could be obtained by solving Eq. (3) by using the measurements PEF A, PEF B, and PEF C as the input. The obtained d₁ was 7.2 ± 0.6 mm, consistent with that the known value of d₁ was 7.0 ± 0.3 mm (see Table II) and that d₁ was close to the depth sensitivity of PEF A (8.2 ± 1 mm) and smaller than that of PEF B (13 ± 0.4 mm) and that of PEF C (16.4 ± 0.4 mm). That the measured elastic modulus by PEF D was smaller than that measured by PEF C indicates that PEF D was in regime (iii) [Fig. 4(c)]. Therefore, d₂ was obtained by solving Eq. (5) with d₁ = 7.2 ± 0.6 mm as obtained by PEFs B and C. The obtained d₂ = 17.0 ± 1.5 mm was consistent with that the known value of d₂ was 16.7 ± 0.5 mm (see Table II) and that d₂ was indeed larger than both the depth sensitivity of PEF B (13 ± 1.0 mm) and that of PEF C (16.4 ± 1.6 mm) and smaller than that of PEF D (19.6 ± 0.6 mm). A schematic illustrating how d₁ and d₂ were relating the depth sensitivities of all four PEFs is shown in the inset in Fig. 7(c).

For the scenario illustrated by inclusion E2 shown in Fig. 7(d), the elastic modulus was found to increase monotonically in the order of PEF A to D indicating that d₂ was larger than h_D and might not be resolvable. To solve for d₁, we noted that the elastic modulus measured by PEF A was greater than E_n, indicating that d₁ was smaller than the depth sensitivity of the PEF A (8.2 ± 1.0 mm) and PEF A was in regime (ii) [Fig. 4(b)]. That the elastic moduli measured by PEFs B, C, and D were larger than E_n and all increased with an increasing probe size indicates that PEF B, PEF C, and PEF D were all also in regime (ii). Therefore, d₁ could be obtained by solving Eq. (3) using the measurements by PEFs A, B, C, and D as the input. The obtained d₁ were 6.6 ± 1.7, 6.1 ± 11.9, 6.4 ± 1.5, and 6.7 ± 1.7 mm for PEFs A, B, C, and D, respectively. Clearly, all four values agreed with one another. That the deduced d₁ was independent of the probe size further reaffirmed the validity of the current methodology of deducing d₁. The final measured d₁ = 6.5 ± 1.7 mm was taken as the average of the four values obtained with different PEFs. As can be seen, this was consistent with the actual value of d₁, 6.4 ± 0.3 mm, for D2 (see Table II). As for the determination of d₂, because we did not observe a decrease in the measured elastic moduli with an increasing probe size, d₂ was unresolvable from the current measurements. This was consistent with the fact that the actual value of d₂ was 21.5 ± 0.4 mm (see Table II), which was equal to or larger than the depth sensitivity of PEF D, 19.6 ± 0.6 mm. A schematic illustrating how d₁ and d₂ were relating the depth sensitivities of all four PEFs is shown in the inset in Fig. 7(d).

After analyzing the measurements of all the model tumors, we found that d₁ was resolvable for all the inclusions because d₁ was less than h_D for all the inclusions. On the other hand, d₂ was only resolvable for inclusions A1-A6, B1-B6, C1-C3, D1-D2, and E1—the first type of inclusions whose d₂ were smaller than or equal to h_D but not for inclusions C4-C6, D3-D6, and E2-D6—the second type of inclusions whose d₂ were larger than or equal to h_D.

In Fig. 8(a), we plot the deduced d₁ versus the actual d₁ of all inclusions as full circles for E_t = 60 kPa. As can be seen, all the deduced values of d₁ were in close agreement with the actual values for all the inclusions. In Fig. 8(b), we plot the deduced d₂ versus the actual d₂ (full circles) with E_t = 60 kPa for the inclusions whose d₂ could be resolved, i.e., inclusions A1-A6, B1-B6, C1-C3, D1-D2, and E1. Again, for these inclusions, the deduced values were in close agreement with the actual values. In addition to using E_t = 60 kPa, we also deduced d₁ for all the inclusions and d₂ for inclusions A1-A6, B1-B6, C1-C3, D1-D2, and E1 using E_t = 40, 100, and 200 kPa as well. Also plotted in Figs. 8(a) and 8(b) were the deduced d₁ and d₂ using E_t = 40 kPa (full squares), 100 kPa (open up triangles), and 200 kPa (open down triangles). Clearly, the obtained d₁ and d₂ values were all within the experimental uncertainty of their actual values and all the data points in Figs. 8(a) and 8(b) were close to the dashed line which had a slope of unity irrespective of the values of E_t. The p values of the Chi-square test were much less than 0.05, indicating that the measured d₁ and d₂ agreed very well with their actual values and were insensitive to the choice of E_t within the range of 40-200 kPa. The insensitivity of the deduced values of d₁ and d₂ to the choice of E_t was due to the fact that E_n was much smaller than E_t. As a result, in Eqs. (3) and (5), the term involving E_t was much smaller than the term involving E_n. Consequently, the obtained d₁ and d₂ were all within the experimental uncertainty of the actual values because the values of E_t at 40-200 kPa were all much larger than that of E_n, which was around 10 kPa. Note that the observed insensitivity of deduced d₁ and d₂ to the choice of E_t in the range of 40-200 kPa will also apply in actual breast tumor measurements as the elastic moduli of breast tumors are known to be in the range of 30–72 kPa, while that of normal breast tissues was around 10 kPa.⁴⁷

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.

IV. CONCLUSIONS

In this study, we have investigated the feasibility of 3D tumor profiling using elastic modulus measurements obtained by piezoelectric fingers (PEFs) with different probe sizes, ranging 4.1 ± 0.2, 6.5 ± 0.2, 8.2 ± 0.2, and 9.8 ± 0.3 mm on model tissues consisted of gelatin with suspended clay inclusions. The lateral sizes of the inclusions were determined by fitting the lateral elastic modulus curve of an inclusion to a Gaussian function and taking the full width of half maximum (FWHM) of the Gaussian fit as the lateral size of the inclusion. The obtained lateral sizes agreed well with the actual values and were independent of the probe size of the PEF. The depth profile of an inclusion was obtained by determining the location of the top surface, d₁, and that of the bottom surface, d₂, of the inclusion using an empirical spring model together with the elastic modulus measurements from PEFs with different probe sizes. We found that d₁ and d₂ could be resolved when they were less than the depth sensitivity of the PEF with the largest probe size. The deduced depth profiles agreed well with the actual depth profiles when the bottom of the inclusion was within the depth sensitivity of the PEF with the largest depth sensitivity. Furthermore, the deduced depth values were shown to be insensitive to the choice of the elastic modulus of tumor used in the spring model.