An accurate analytical modeling method for microwave-based breast tumor detection and phantom manufacturing

Abstract

Early detection of breast cancer significantly improves survival rates, with nearly all patients surviving for over five years. Mathematical modeling of cancerous tissue dynamics facilitates the rapid detection of tumors. This study introduces an innovative segmented hemispherical modeling approach for breast tissue, wherein the tissues are modeled as electrical capacitors with unequal plates. The structure and performance of the proposed hemispherical model are thoroughly examined. The effective permittivity, , of both individual breast tissues and the entire breast is computed using their dielectric properties. The proposed closed-form breast model is analyzed and compared with state-of-the-art methods through analytical, simulation-based, and experimental approaches. The proposed segmented hemispherical modeling technique significantly outperforms traditional cubic models, achieving substantially higher discrimination levels of 0.335 compared to 0.001 for fatty breast tissue and 0.412 compared to 0.001 for dense breast tissue. The model accurately replicates real breast anatomy and demonstrates superior efficacy in tumor detection, showing a simulated difference of 3 dB and 7 degrees in the magnitude and phase of the -parameters, respectively. Furthermore, the proposed method holds promising potential for developing an affordable and simple breast phantom fabrication method that, if adopted, could significantly facilitate research in laboratory settings. These phantoms would maintain high accuracy in replicating real breast tissue and contribute to more practical and reliable results in breast cancer detection techniques.

Introduction

Breast cancer (BC) has overtaken lung cancer as the most frequently diagnosed cancer globally, now accounting for 1 in 8 cancer diagnoses, with a total of 2.3 million new cases across both sexes [1]. Early detection of BC is crucial for improving survival rates. Imaging techniques are essential and primarily include three primary methods. First, mammography serves as a screening tool for BC and for detecting mammary gland calcifications. However, it may not be suitable for individuals with high gland density and should not be performed more than twice a year. Second, ultrasonography screens young women and detects mammary gland inflammation. However, its accuracy depends on the examining physician, and its definition and resolution are not always high. Third, magnetic resonance imaging (MRI) is appropriate for screening high-risk groups, such as those with a family history of BC, and for patients who have undergone breast-conserving surgery. However, it may not be suitable for everyone, such as patients with claustrophobia, and it is not typically used for BC staging [2]. Microwave breast imaging (MBI) is non-invasive, avoiding the need for invasive procedures and ionizing radiation, ensuring patient safety. It is also cost-effective, making it suitable for widespread adoption. MBI functions by analyzing and distinguishing changes in backscattered signals due to variations in dielectric properties between cancerous and normal cells. This capability facilitates the detection of malignant cells, which is crucial for precise diagnosis [3].

Mathematical oncology has evolved into a recognized field of cancer research where the mathematical methods can enhance biomedical understanding of the disease and enable the prediction and optimization of patient-specific treatment strategies [4, 5]. In breasts categorized as fatty, nearly all of the tissue consists of fat, with less than 25% composed of fibro-glandular tissues. Breasts characterized by scattered fibro-glandular tissue exhibit 26–50% of the breast volume as fibro-glandular tissue. Heterogeneously dense tissue comprises 51–75% fibro-glandular tissue, while extremely dense breasts comprise more than 75% fibrous connective tissue [6]. Phantoms are critical in medical quality assurance, research, education, and training by providing an appropriate testing environment [7]. In [8], A methodology is proposed for constructing realistic breast phantoms intended for microwave breast cancer detection. Atat et al. highlighted the significance of 3D culture systems in accurately replicating the tumor microenvironment. They emphasized the critical role of cancer cell-microenvironment interactions in influencing cancer behavior, including resistance, proliferation, and metastasis [9]. Researchers in [10] applied various differential analytical models to tumor volume data from patients receiving chemotherapy or cancer immunotherapy for solid tumors. Their findings indicate that early treatment responses moderately correlate with treatment outcomes. In [11], Hormuth and his colleagues identified promising approaches that combine mathematical modeling with experimental data across both in vitro cell scale and in vivo tissue scale. They explored opportunities for bridging models at the tissue levels and highlighted prospects for applying them to optimize therapeutic regimens. Authors of [12] developed approaches to predict responses to fractionated radiation therapy to optimize and adapt radiotherapy treatment plans. To support this goal, they created mathematical models that describe the reaction of both tumors and vasculature to fractionated radiation therapy. Numerous studies have addressed the development of realistic 3D human-body emulation platforms for applications such as brain tumor detection, breast cancer diagnosis, and capsule endoscopy [13]. These platforms are often designed for 3D printing and rely on advanced image processing pipelines—such as image registration, bias correction, and tissue segmentation—to replicate internal anatomical structures accurately [14]. Particular attention has been given to breast phantoms due to the breast tissue's complex, heterogeneous composition. Prior works have introduced anatomical breast phantoms to explore the capabilities of microwave radiometry in monitoring thermal processes within biological tissues [15]. Multi-step fabrication methods have been proposed to create multi-layered breast phantoms, enabling accurate dielectric characterization up to 40 GHz while maintaining the fidelity of tissue-mimicking mixtures [16]. Previous studies have also demonstrated phantom evaluations using advanced imaging systems capable of scanning inclusions with varying sizes to assess localization precision and detection rates [17]. Experimental validations have confirmed the feasibility of heterogeneous phantoms featuring embedded tumor regions, particularly when used with novel antenna designs for enhanced field focusing and resolution [18].

This paper introduces a novel and updated analytical method for modeling various breast tissues as electrical capacitors with unequal plates. The segmentation approach employed in this proposal effectively replicates the structure of the real breast with a high degree of accuracy. Simulation, experimental, and analytical results affirm the effectiveness of this approach in distinguishing between different breast compositions compared to conventional methods. The primary advantages of this proposed methodology include novelty in tissue modeling, enhanced tumor detection in both fatty and dense breasts, and facilitating the production of reproducible, low-cost, and effective heterogeneous breast phantoms.

The following is a breakdown of how the paper is structured. Section 2 presents the anatomy and modeling theory of the human breast. Section 3 describes the proposed and conventional breast modeling methods. Section 4 explains the effective permittivity calculation for the proposed approach. Section 5 presents the phantom construction. Section 6 discusses the analytical, simulation, and experimental results. Finally, a conclusion has been added in Sect. 7 to sum up the article.

Anatomy and modeling theory of the human breast

The primary anatomy of the breast encompasses the skin, adipose tissue (fat), fibro-glandular breast tissue, all situated above the chest wall. The dielectric properties of biological tissues are fundamental parameters necessary for electromagnetic modeling of the human body. These properties, including permittivity and electrical conductivity, are essential for evaluating how the human body interacts with electric, magnetic, and electromagnetic fields. The permittivity and conductivity of the human breast tissues for the frequency range of 1–6 GHz are shown in Table 1 [19].

Table 1.

Dielectric properties of tissues at different frequencies [19]

Frequency (GHz)	Dielectric property	Skin	Fat	Gland	Tumor
1.0	(F/m) (S/m)	40 0.5	15 0.5	35 0.2	59 0.5
1.5	(F/m) (S/m)	39 0.6	14.5 0.1	34 0.5	57 0.7
2.0	(F/m) (S/m)	38 0.8	14.2 0.2	33 0.8	56 1.2
2.5	(F/m) (S/m)	37 1.5	14 0.3	32 1.0	55 2.0
3.0	(F/m) (S/m)	36 1.8	13.8 0.4	31.8 1.5	54 2.2
3.5	(F/m) (S/m)	35 2.3	13.5 0.5	31 2.0	53 2.8
4.0	(F/m) (S/m)	34 2.5	13.2 0.6	30 2.2	52 3.5
4.5	(F/m) (S/m)	33 3.2	13 0.7	29 2.4	50 4.2
5.0	(F/m) (S/m)	32 3.8	12.5 1.0	28.5 2.8	47 5.1
5.5	(F/m) (S/m)	31 4.2	12 1.2	28 3.2	46 5.5
6.0	(F/m) (S/m)	30 4.8	11 1.3	27 3.8	45 6.2

This study adopts two scenarios to anatomically model the breast tissues. In the first scenario, where the incident wave is normal to the interfaces between layers, it was demonstrated that the effective permittivity equals the sum of a normalized weighted average function of the reciprocal permittivity values of individual layers. In the second scenario, where incident waves were parallel to the multiple layers, it was shown that the effective permittivity equals the sum of a normalized weighted average function of the permittivity values of individual layers [20].

The conventional and proposed modeling methods

Conventional segmented cubic modeling

In [19], the authors presented a human breast model as a segmented cube, with each segment represented as parallel capacitors with equal plates. Each segment of the cube corresponds to a specific breast tissue type. This approach has two primary drawbacks. First, it assumes that all layers are parallel, with the electric field component parallel to all interfaces between the breast layers. Second, it assumes that these parallel layers have equal surfaces, representing capacitors with equal plates. The conventional cubic composition with the inclusion of the glands and tumor tissues is depicted in Fig. 1a. This consideration explains how the conventional scheme has weak discrimination for the existence of glands and tumors.

Fig. 1.

Segmented modeling of the human breast; a cubic, b hemispherical

Proposed segmented hemispherical modeling

In this novel modeling approach, an actual modeling is proposed to mimic the real breast. Here, the real breast of a hemispherical shape is segmented into parallel segments. Each segment represents the volume of a specific tissue of the real breast. The proposed segmented hemispherical model is depicted in Fig. 1b which has 88.47% fat and 7.63% glands and the tumor tissue comprises 3.34% of the breast.

To find the approximate volume of the real breast, it is assumed to have a hemispherical shape [21–23]. The volume of the whole breast (skin and fat) with radius of 50 , = = .

As the fat has a radius of 48 , then = = . As a result, = – = 3.0176 × .

The sizes of the tissues define the position of the cutting planes when the real breast is segmented into distinct layers that represent the breast tissues in a segmented hemisphere. The resulting segments will be spherical frustums of volume. = (3 + 3 + ), where a, b, and w are the radius of the first base, the radius of the second base, and the frustum height, respectively.

For the skin frustum:

1

Assume that the interface between the nipple and the skin layers, , is 3.392 [24], then

2

For the fat frustum:

3

where +==50 , =, and =50 , then

4

As (2) and (4) are nonlinear, we utilized a system of nonlinear algebraic equations to find and , we found that, ≈ 35 mm and ≈ 14 mm.

Next, we want to find the height of the immersed gland tissue, , within the fat tissue. The gland tissue has the same area as the fat tissue ( =). A 2D plan view, illustrated in Fig. 2, has been drawn to display the areas of the segments at the breast center. The area of the cut segment of a circle, , is obtained by subtracting the area of the triangle from the area of the sector.

5

Fig. 2.

The segmentation geometry of the immersed gland tissues in the fat

To calculate the area of the fat segment, we use

6

As shown in [25], the area of the skin segment can be calculated as

From (6)

As V_gland = A_{gland segment} × h_{gland segment} and A_{gland segment} = A_{fat segment}, then h_{gland - segment} = .

Following the same manner for drawing the tumor sector, which has a volume = ( of the whole breast). This sector has the same area as the gland sector, . Then, the height of this sector will be

Effective permittivity calculation of the proposed modeling technique

For a given capacitor with unequal plates of areas and . The effective capacitance can be derived as follows:

The capacitance is given by:

7

Q is the maximum stored charge in the capacitor, and the potential difference is

8

Let be the distance between the capacitor plates, then

9

From Maxwell’s equations in dielectrics, the electric displacement is related to the electric field by the constitutive relation [26, 27]:

10

As the electric flux density related to the free surface charge is given by

11

12

13

From (9) and (13)

14

The electrostatic stored energy is

15

16

Now, from (8)

17

18

19

By substituting in (7), we get

20

For the parallel capacitors, the effective capacitance for the breast layers can be calculated as

21

then,

22

23

where

24

From (16),

25

Then the effective permittivity is a function of the electric displacement, volume fraction, electric field strength, and energy distribution.

For the series capacitor model, the effective capacitance is

26

Phantom construction and experimental setup

Based on the findings discussed earlier, a simplified breast model manufacturing process is proposed. In [8], a comprehensive breast model was constructed. The main limitations of that manufacturing method include:

• It requires two hemispherical bowls to pour the liquid skin mixture between them, leading to the formation of air pockets within the liquid.

• Separating the two containers can potentially tear the skin phantom.

• Real glands have a conical shape, making it challenging to pour the liquid gland mixture into the fat.

• Anchoring cylindrical glass rods in the fat for the gland mixture incurs additional costs.

Therefore, the proposed segmented hemispherical modeling method aims to streamline breast phantom manufacturing and address the drawbacks of the process mentioned above. In the proposed method:

• The liquid skin mixture is poured directly into a single hemisphere bowl, eliminating the need for an additional bowl.

• This method efficiently mimics skin tissue, reducing the possibility of air pocket formation.

• The proposed novel model allows direct integration of glandular tissue into the fat layer, eliminating the requirement for anchoring cylindrical glass rods. This simplifies the process and provides a straightforward method for positioning glandular tissue within the fat layer.

• The novel model minimizes the use of tools, thereby reducing overall costs.

The two containers used for molding the cubic and hemispherical phantoms were 3D printed using polylactic acid (PLA) material, and both share the same internal volume. To separate the gelatin-based tissue regions, thin double-sided copper-clad laminates were employed as non-adhesive separators according to the dimensions in Fig. 2.

The phantom compositions for all tissue types were prepared by weighing and volumetric calibration according to the formulations listed in Table 2. After preparation, the tissue-mimicking mixtures were thoroughly stirred and heated to ensure homogeneity. The dielectric properties, specifically permittivity and conductivity, were then measured and validated using the dielectric assessment kit (DAK-TL2). The measured values of the permittivity and conductivity for all four tissues at 2.4 GHz are also included in Table 2.

Table 2.

Percentages of the fabricated phantom tissues and measured dielectric properties

Tissue, reference	Water	Gelatin	Sugar	Oil	Measured permittivity (F/m)	Measured conductivity (S/m)
Tumor, [28]	85%	10%	5%	–	57.9	1.51
Skin, [28]	60%	20%	20%	–	33.1	0.42
Fat, [28]	70%	10%	–	20%	16.8	0.15
Glands, [29]	252 ml	50.5 g	5.25 g	–	59.8	1.72

For the phantom fabrication process, we detail the manufacturing steps of the segmented hemispherical phantom as an example. Initially, two metallic separators are positioned and fixed at the center of the 3D-printed container. The placement of these separators is precisely adjusted using a vernier caliper, ensuring accurate inner and outer thicknesses. Following this, the skin-mimicking tissue is poured into the container until it reaches the lower edge of the fixed separators (Fig. 3a). The assembly is then placed in a freezer for 15 min to allow the skin layer to solidify.

Fig. 3.

The fabrication process of the proposed segmented hemispherical phantom

Next, the fat-mimicking tissue is poured into both side compartments formed by the separators (Fig. 3b). Once the fat layers have solidified, the separators can be gently removed, creating a central cavity for the glandular and tumor tissue layers. The separator is then reinserted and fixed within this cavity to form the boundary for the glandular tissue (colored green for visualization), which is then poured and allowed to solidify (Fig. 3c). Finally, the separators are removed again, and the remaining cavity is filled with the tumor-mimicking tissue (colored red), completing the internal segmentation (Fig. 3d). The same fabrication procedure is applied to construct the segmented cubic phantom, ensuring consistency in internal structure and layer composition.

Results and discussion

Throughout this section, the analytical, simulation-based, and experimental validation results are drawn and analyzed. The displayed results depend basically on collecting the back-scattered signals from the different models of the breast and comparing them. The reflected waves from the breast models rely on the frequency of the incident signals as well as the dielectric properties of the breast models [30].

To set the stage for the following results, it is essential to emphasize that the ability to discriminate between different tissue types, such as normal, glandular, and tumorous, relies primarily on the permittivity contrast among them. This contrast directly affects the scattered electromagnetic field, and consequently, the measured S-parameters. The relationship is governed by the following expression [31, 32]:

27

where is the difference between the permittivity of the targeted object (tumor) and the background (skin, fat, and glands), are the -parameters data, are known constants. is the permittivity contrast, is Green’s vector function, and is the total transmitted field.

This equation highlights the central role of permittivity differences in influencing the scattered response, and thus, the detection capability. Accordingly, the subsequent results focus on how changes in tissue permittivity, due to the presence of tumors or structural variations, affect the measured -parameters, providing the foundation for reliable tissue discrimination.

Analytical results

The previously driven effective permittivity formulas for both the conventional and proposed modeling methods are exploited for the calculation of the transmission and reflection coefficients [33].

The reflection coefficient is given by:

28

The transmission coefficient T is given by:

29

where

30

and

31

For the fatty breast, Fig. 4 reveals the discrimination ability of both cubic and hemispherical modeling. Numerical analysis shows that the cubic segmentation method cannot differentiate between a healthy breast and a cancerous one. In contrast, the proposed hemispherical segmentation method can accurately discriminate the presence of a tumor. The non-monotonic variation in the coefficients is attributed to the analytical nature of the model, which calculates field interactions at discrete frequency points. These fluctuations are further influenced by the effective permittivity of the entire breast structure, reflecting interactions between multiple tissue layers. In Fig. 4a, it is observed that the reflection coefficient remains almost constant across the frequency range, starting at 0.161 at 1 GHz and gradually decreasing to 0.147 at 6 GHz. However, for the proposed approach, in Fig. 4b, the difference between the healthy and tumorous reflection coefficients is clear, reaching 0.335 at 4 GHz.

Fig. 4.

Transmission and reflection coefficients for the conventional and proposed models in fatty and dense breast types

For the dense breast, the traditional cubic segmentation method, in Fig. 4c, behaves the same as on the fatty breast. The backscattered signals for healthy and cancerous breasts are the same. However, the hemispherical segmentation method, in Fig. 4d, keeps its robustness in detecting the immersed tumor in the glands. The difference in reflection coefficients reached 0.412.

Simulation-based results

The two modeling methods were also tested through simulations in Computer Simulation Technology (CST) software. The different breast compositions used in the analytical part were simulated in CST. A planar microstrip antenna at 2.4 GHz was applied to each composition individually to check the backscattered parameters. The standard reference used was the realistic breast composition, which was simulated first. Here, "real breast" refers to a simulation without segmentation that closely mimics the actual composition and shape of a human breast.

The behaviors of the cubic and segmented modeling methods are compared with the real breast as demonstrated in Table 3. The reflection coefficient values for the cubic modeling are higher than those for the real and hemispherical models. It appears that the geometric shape of the modeling affects the amount of reflected power. From the table, it can be concluded that the proposed method does not have high-power absorption and closely approximates the real breast model. Consequently, the hemispherical segmentation method can be more reliable than the traditional cubic method for modeling different breast compositions.

Table 3.

-parameter discrimination for real, segmented cubic, and segmented hemispherical models in both fatty and dense breast types

-parameters	Breast case	Real		Segmented cubic		Segmented hemispherical
		Fatty	Dense	Fatty	Dense	Fatty	Dense
Mag. (dB)	Normal	− 19.97	− 20.40	− 13.71	− 13.51	− 24.64	− 27.13
	Cancerous	− 24.34	− 23.04	− 13.66	− 13.41	− 27.93	− 30.20
Phase (deg)	Normal	99.28	101.40	84.84	84.17	108.68	113.88
	Cancerous	108.05	105.37	84.59	84.04	115.28	118.33

The dielectric and structural heterogeneity of breast tissues critically influences the accuracy of electromagnetic (EM) breast cancer detection. Regarding tumor thickness, thicker tumors present a larger dielectric volume, which enhances both the strength and spatial coherence of the scattered field. This results in more pronounced perturbations in the reflection coefficient. In contrast, thin tumors generate weaker backscattered signals that may become indistinguishable from background noise or glandular reflections, as illustrated in Fig. 5a. Tumors typically exhibit higher relative permittivity than surrounding fatty or glandular tissues. This contrast in dielectric properties creates a more substantial local impedance mismatch, which amplifies the magnitude of the reflected signal. Consequently, tumors with higher permittivity contrast produce clearer detection signatures in the -parameter profile, as shown in Fig. 5b. Finally, the distribution of glandular tissue significantly impacts the background dielectric landscape. For the symmetric case, the glands are equally distributed on the breast sides, while the asymmetric distribution puts 70% of the glands on one side and 30% on the other side. The symmetrically positioned glandular tissues tend to produce balanced and predictable scattering patterns. However, asymmetric glandular arrangements introduce irregular spatial and dielectric variations that can either mimic or obscure actual tumor signatures. These asymmetries induce additional multiple scattering paths and increase the difficulty of accurate image reconstruction, as demonstrated in Fig. 5c.

Fig. 5.

Effect of geometric features, dielectric properties, and glandular tissue distribution on the tumor detection process

Measurement-based results

To evaluate the electromagnetic response of the fabricated breast phantom, an E8361C PNA Microwave Network Analyzer was used to measure the -parameters, specifically the reflection coefficient (). The phantom was placed in contact with a planar array antenna designed and used for breast cancer detection [28, 34].

Figure 6 illustrates that the experimental measurements show strong agreement with simulation-based results. As shown in Fig. 6a, tumor detection in the cubic phantom is weak, with minimal shifts observed in the reflection coefficient amplitudes relative to the standard case. In contrast, Fig. 6b demonstrates improved detection performance in the hemispherical phantom. The segmented hemispherical structure enhances the reflection of EM waves due to the dielectric property variations among breast tissues, thereby making tumor-induced differences more distinguishable.

Fig. 6.

Reflection coefficients for simulated and measured breast phantoms

Comparative study

Table 4 provides a comparative overview of recent studies focused on microwave-based breast tumor detection.

Table 4.

Comparison of existing microwave breast imaging studies concerning detection methods, tissue types, and phantom design

Reference, year	Frequency band (GHz)	Studied breast types	Detection methodology	Reflection coefficient (dB)	Used antenna	Phantom novelty
[35], 2022	3.4–10	Fatty	Simulation	0–30	UWB monopole	No
[36], 2022	3.1–10.6	Fatty	Simulation	0–13	UWB patch	No
[37], 2022	2.48	Dense	Simulation	13–25	microstrip patch rectangular	No
[38], 2022	5.72–5.82	Dense	Simulation	0.4–9	microstrip patch	No
[39], 2023	2.2–8	Dense	Simulation and experimental	2–15	monopole sensors	No
[40], 2024	6.5–12.5	Fatty	Simulation	0.1–1	dielectric resonator-based wide band	No
[41], 2025	2	Dense	Simulation and experimental	5–15	partially reflective surface monopole slotted-antenna	No
This work	1–6 GHz	Fatty and Dense	Analytical, simulation, and experimental	3	2 × 2 planar array	Yes

Conclusion

This study proposes a novel approach for modeling the human breast using a hemispherical representation, where breast tissues are modeled as electrical capacitors with unequal plates. The effective permittivity of each tissue, as well as the entire breast, was determined using the dielectric properties, and a closed-form expression was employed for the analysis of the model. Performance assessments were conducted through both analytical and simulation-based methods to evaluate tumor detection capabilities. Compared to traditional cubic models, the hemispherical approach demonstrated a significant improvement in discrimination levels, with values of 0.335 for fatty breast tissue and 0.412 for dense breast tissue, in contrast to only 0.001 for the cubic model. Simulations validate the model’s anatomical accuracy and its improved tumor detection performance, reflected in notable differences of 3 dB and 7 degrees in -parameters. Additionally, the proposed method offers substantial potential for creating affordable and straightforward breast phantoms that, if implemented, could enhance laboratory research. These phantoms would accurately replicate real breast tissue and provide more reliable, practical results in breast cancer imaging applications. Future research will consider the integration of ultrawideband (UWB) sensor to further capitalize on the anatomical accuracy and multilayered structure of the proposed hemispherical phantom, to achieve higher spatial resolution and enhanced tumor detection over an extended frequency range.

Author contributions

Kyrillos Youssef conceptualized the research idea, designed the study, conducted the practical experiments, and drafted the initial manuscript. Mohammed Abo-Zahhad refined the research idea, conducted manuscript revision, contributed to the discussion section, and offered critical insights into the broader implications of the findings. Haruichi Kanaya contributed to the literature review, significantly shaped the research questions, and refined the overall research focus. Ahmed H. Abd El-Malek provided valuable insights into the framework, played a key role in refining the research methodology, and critically revised the manuscript.

Funding

No funding was received to conduct this study.

Data availability

All data supporting the findings of this research are available within the paper.

Declarations

Conflict of interest

All the authors declare that they have no competing interests.

Ethical approval

Not applicable.