The Added Value of Low-Intensity Focused Ultrasound Treatment on Breast Recovery Following Aspiration of Breast Abscesses in Lactating Women

Abstract

Background:

Low-Intensity Focused Ultrasound (LIFU) may enhance tissue recovery and support breastfeeding post-aspiration.

Methods:

A total of 60 patients were randomly assigned to either a control group or a treatment group. Fifty-six participants completed the study, with two dropouts per group. The control group received ultrasound-guided aspiration and antibiotics for 3 days, while the treatment group additionally received LIFU therapy. Recovery of tissue structure at the abscess site was assessed. The primary outcome was the recovery time of breast tissue via ultrasound imaging. Secondary outcomes included pain scores, time for disappearance of local induration and abscess cavity, breastfeeding self-efficacy scores, breastfeeding rates, and repeat aspiration rates.

Findings:

The treatment group exhibited a significantly shorter recovery time for breast tissue compared to the control group. Induration disappearance time was also reduced in the treatment group. Significant differences in pain scores were noted 1 week post-surgery. Breastfeeding self-efficacy scores improved significantly in the treatment group at 2 weeks and 2 months post-surgery. Higher breastfeeding maintenance rates and lower cessation rates were observed in the treatment group.

Discussion:

The use of LIFU therapy appears to enhance tissue repair and support breastfeeding post-aspiration, offering a promising treatment option for lactating women with breast abscesses.

Conclusion:

LIFU treatment effectively promotes tissue repair, reduces recovery time, and improves breastfeeding outcomes after breast abscess aspiration. The Clinical Trial registration number: ChiCTR2200060973 (registration date: June 14, 2022). Version Date: 22 May 2025.

Keywords

low-intensity focused ultrasound breast abscess breastfeeding ultrasound therapy

Introduction

Lactational mastitis is a common inflammatory condition of the breast that occurs in women during breastfeeding. It can arise at any stage of lactation and is often recurrent, potentially progressing to a breast abscess. This condition poses a significant risk to the continuation of breastfeeding and can have a negative impact on the patient’s ability to maintain breastfeeding.¹ Research indicates that independent risk factors for the progression of acute lactational mastitis to a breast abscess include primiparity, a history of non-medical massage, previous breast surgery or mastitis, Grade II or III nipple inversion, and sleeping positions.^2,3 In the early stages of mastitis, patients often adopt practices such as high-pressure prolonged milk suction and heat applications, largely due to insufficient awareness or outdated medical advice. These practices can inadvertently accelerate the progression of mastitis to a breast abscess. Furthermore, it is important to note that in the United States, despite updated concepts in mastitis management, many health care providers still recommend such potentially harmful practices. They are unaware that these habits, including high-pressure prolonged milk suction, may exacerbate mastitis. This lack of awareness among providers stems from a lag in incorporating the latest research findings into clinical guidelines. The reported incidence of breast abscess ranges from 4.6% to 11%.⁴ After the formation of a breast abscess, the affected local tissue structure is damaged, leading to a loss of lactation function and severely impacting the breastfeeding process.⁵

Traditionally, the common treatment method for an abscess has been surgical incision and drainage. Given the significant local tissue damage associated with surgical incision and drainage for breast abscesses, which may lead to exacerbated postoperative pain and complex nursing requirements, and considering patients’ concerns about potential impacts on wound healing and infant health during breastfeeding, clinical studies have shown that the majority of patients proactively choose to cease lactation prior to deciding to undergo surgical treatment.⁶ Furthermore, the prolonged recovery time could potentially diminish the breastfeeding confidence of lactating women, as the extended period of discomfort and the need for ongoing wound care may discourage continued breastfeeding.⁷ Currently, for the treatment of lactational breast abscesses, ultrasound-guided puncture and lavage is gradually replacing surgical methods in clinical practice. This approach minimizes trauma and avoids the need for lactation cessation, thereby maximizing breastfeeding capability. However, conventional puncture treatment often involves a prolonged recovery period for breast tissue, which is a significant factor affecting breastfeeding. Additionally, the success rate of a single puncture is low, often requiring multiple attempts and carrying a risk of eventual surgical intervention, which also severely impacts breastfeeding.⁸

When considering the safety of breastfeeding for infants, many treatment methods available all present certain limitations. For instance, some medications may be transferred through breast milk, potentially causing adverse effects in the infant. Similarly, surgical treatments may interfere with breastfeeding due to issues related to anesthesia, wound care, and other postoperative concerns.⁵ Ultrasound therapy, as a form of subtle mechanical stimulation, falls under the category of physical factor treatments. It is clinically safe and has no reported side effects. Existing studies have documented its applications in tissue repair, pain relief, and inflammation reduction.^9,10 Research on the use of ultrasound during the breastfeeding period has explored its benefits for conditions such as mastitis, insufficient milk supply, and milk stasis.¹¹ Ultrasound has been shown to effectively promote milk secretion, alleviate breast engorgement, clear blocked milk ducts, and improve mastitis-related inflammation.^12,13 However, there are no current reports on the application of ultrasound for tissue repair in breastfeeding-associated breast abscesses. Low-intensity focused ultrasound (LIFU) can focus ultrasound waves on deep tissues, providing stable and effective treatment doses at the target area with precision and minimal energy attenuation.¹⁴ This study examines the impact of combining LIFU with antibiotics on abscess tissue repair and breastfeeding outcomes following puncture surgery.

Materials and Methods

General information

From July 2022 to March 2024, 60 breastfeeding patients with breast abscesses treated were included. Prior to the commencement of the study, randomization was carried out using SPSS 20 (randomization numbers). Once the subjects met the inclusion criteria and agreed to participate, the assistant reached out to them and assigned them to either the control group or the treatment group. The study was completed with 58 cases: 28 in the control group (2 dropped out) and 28 in the treatment Group (2 dropped out). Inclusion criteria: (1) Breastfeeding patients with a single breast abscess, with ultrasound showing abscesses ≥2 cm in size; (2) Indications for puncture treatment according to the established standard;¹⁵ (3) Normal blood count prior to puncture treatment; (4) No prior abscess puncture treatment at the lesion site before enrollment; (5) Eligible for breastfeeding and willing to continue breastfeeding. Exclusion criteria: (1) Central type breast abscess (abscesses that are located in the central region of the breast, typically around the areola and nipple area); (2) Abscess with skin rupture; (3) Presence of ≥2 breast abscesses. (4) Coagulation disorders; (5) skin with circulatory disturbances (e.g., compromised blood flow) or sensory disturbances (e.g., altered sensation or numbness) in the treatment area; (6) Severe acute pulmonary, liver, kidney, cardiovascular, or endocrine diseases (e.g., diabetes, including gestational diabetes mellitus) at the time of enrollment) at the time of enrollment; (7) Coexisting mental disorders, speech or hearing impairments, or inability to cooperate with treatment; (8) History of malignant tumors in the breast or a malignancy in any other part of the body; (9) History of breast surgery; (10) Presence of a cardiac pacemaker. All participants were informed about the study details and provided written informed consent prior to enrollment by the same researcher. Participants were provided with both written and oral information regarding the study, and they signed informed consent forms. Each participant was carefully instructed not to reveal their treatment allocation to the examiner during follow-up. All treatments were administered by trained assistants who were not involved in the examination and evaluation of the participants. The two investigators were unaware of the intervention trial conducted on the subjects. During follow-up, they conducted the same clinical examination as baseline on the same subject.

This study was approved by the hospital’s ethics committee (R22024) and registered with the Chinese Clinical Trial Registry (ChiCTR2200060973). The study design was summarized as Figure 1.

FIG. 1.

Flowchart of study design. LIFU, Low-Intensity Focused Ultrasound; VAS, Visual Analog Scale.

Intervention method

Both the control group and the treatment group underwent abscess puncture treatment guided by ultrasound (B-mode). The GE Voluson P8 ultrasound diagnostic instrument with a probe frequency of 5–12 MHz was utilized. The ultrasound-guided puncture procedure involved the following steps: (1) Patients were positioned supine with the chest and armpit area adequately exposed. The affected breast and the corresponding axillary region were examined to determine the location, size, and internal liquefaction of the lesion using ultrasound or palpation; (2) The puncture point was identified using ultrasound (avoiding the thinnest part of the abscess wall and staying as far away from the nipple-areola complex as possible); and (3) Standard disinfection procedures were followed, sterile gloves were worn, and a 12-gauge needle was inserted at a 45° angle into the abscess cavity for rapid aspiration. In cases where the abscess cavity had septations, the needle angle and orientation were adjusted appropriately to puncture the septum and aspirate the pus thoroughly; (4) A syringe containing approximately two-thirds of the aspirated pus volume in physiological saline was used to irrigate the abscess cavity, repeating the process until the aspirated fluid became clear; (5) During irrigation, gentle massage techniques were applied along the needle axis to dislodge and dilute viscous pus adhering to the abscess cavity wall, facilitating its aspiration. Both groups received standard anti-inflammatory treatment. Prior to puncture, antibiotics deemed safe for infants, in accordance with the Chinese guidelines for the management of lactational mastitis, were administered. Post-puncture, the same antibiotics were continued for 3 days.

In addition to standard treatments, the treatment group received LIFU therapy. The LIFU therapy utilized the Taiyu Postpartum Rehabilitation Ultrasound Instrument (Sichuan Taiyu Technology Co., Ltd., Model TY-200A), with treatment parameters set to a frequency of 0.84 MHz and an output power of 4.5 W.

LIFU Treatment Procedure: (1)

Preparation: Ensure the environment was prepared, maintaining patient privacy. Instruct the patient to lie supine and expose the breast area. The therapist should conduct a local skin examination, including the closure of the puncture site, confirming that the puncture site was closed and the surrounding skin is intact.

(2)

Application: Once the device was turned on, apply a coupling agent to the intervention area. Position the treatment head’s acoustic window directly on the skin over the intervention area. The treatment was started, and the treatment head was moved at a speed of 1–2 cm/s.

(3)

Treatment Areas: Perform circular movements in the following areas:

Between the Shoulder Blades: 3 minutes

Base of the Breast: 3 minutes

Around the Lesion: Circular treatments were performed around the abscess cavity at a 45° angle toward the center of the lesion for 4 minutes.

(4)

Avoidance and Adjustment: Ensure to avoid the nipple and areola during treatment. The treatment head should not be fixed in place. If the patient experienced any warmth in the skin, additional coupling agent should be applied, or the movement speed should be increased to prevent discomfort.

In the first phase, LIFU treatment was administered at a dose of 1.5 W/cm², starting 24 hours after the procedure (once the puncture site was closed). Treatment was given every other day, with each session lasting 10 minutes. Treatment was conducted twice in the first phase. In the second phase, 1 week post-surgery, the dose was increased to 3 W/cm². This treatment was also given every other day, with each session lasting 10 minutes. Treatment was conducted three times in the second phase. During the treatment period, breastfeeding on the affected side was paused. Instead, milk was expressed using a pump or by hand at normal breastfeeding intervals for both the treatment group and the control group. In the control group, the pumped milk was fed to the infant. Breastfeeding was resumed once the abscess had healed. All patients received standard support and education for breastfeeding during the study.

Evaluation index

(1)

Recovery Time of Breast Abscess: Referring to the criteria for a cured breast abscess as defined in previous studies,^16,17 recovery was defined by the disappearance of local inflammatory symptoms (the subsidence of breast redness and swelling, relief of pain, normalization of skin temperature, and corresponding normal laboratory test results such as blood routine and C-reactive protein.), absence of significant fluid-filled areas on ultrasound (the scope of the liquid—dark area has significantly reduced and shows no increasing trend, with improved echo characteristics), and restoration of normal breast tissue (the glandular layer returns to normal, without abnormal echo and dark areas; during palpation, the breast has a uniform texture, no lumps, and normal lymph nodes). Both groups underwent ultrasound examinations every 7–10 days, and follow-up records should document the number of days required for the resolution of the breast abscess in both groups.

(2)

Breast Pain Visual Analog Scale (VAS) Score: On the day of the breast abscess puncture and at 3 days and 1 week postoperation, patients should be assessed for pain using the VAS during follow-up visits. The VAS ranges from 0 (no pain) to 10 (worst possible pain).

(3)

Breastfeeding Self-Efficacy Scale:¹⁸ The Breastfeeding Self-Efficacy Scale consisted of 14 items. Patients completed the scale based on their own situation, with each item rated from 1 (not confident at all) to 5 (completely confident). Higher scores indicated greater self-efficacy. The scale should be assessed before treatment and at 2 weeks, 1 month, and 2 months postoperation.

(4)

Breastfeeding Assessment: Breastfeeding assessment was categorized into three types: (1) Continued breastfeeding as before the abscess, (2) Reduction in breastfeeding amount by more than 30%, and (3) Cessation of breastfeeding. Breastfeeding status should be evaluated before treatment, and at 2 weeks, 1 month, and 2 months postoperation. The percentage of each assessment category should be calculated for both groups.

(5)

Repeat Aspiration: The number of repeat aspiration procedures for each group was recorded, and the probability of repeat aspirations between the two groups was compared.

Sample size calculation

This study an conducted a priori power analysis using PASS (version 2021) software, setting the significance level at α = 0.05 (two-tailed test) and the power at 1-β = 0.80. Based on clinical experience, the anticipated effect size (Cohen’s d) between the intervention and control groups was set at 0.8 (defined as the difference in group means divided by the pooled standard deviation). This effect size was referenced from the researchers’ historical observations of treatment cycles for similar diseases (for example, a reduction of approximately 30 days is considered a clinically significant improvement), corresponding to a “large effect” as classified by Cohen. An independent samples t-test (assuming equal variances) indicated that 26 participants per group (total sample size of 52) were required. Considering the potential for dropouts due to the extended treatment duration, 30 participants were recruited per group (total sample size of 60) in practice. Ultimately, 28 participants per group completed the study.

Statistical analysis

Data analysis, performing baseline descriptive analysis, was performed using SPSS 25.0. Variables with normal distribution were analyzed using an independent sample t-test, while the nonparametric Wilcoxon paired observation and chi-square test were used for the remaining variables. A 95% confidence interval (CI) was applied. The significance level should be set at α = 0.05.

Results

Baseline characteristics of the two patient groups

The baseline characteristics of the two patient groups include age, postpartum days, and baseline measurement indicators. Statistical analysis of these characteristics reveals no significant differences between the two groups in terms of age, postpartum days, induration size, pain level, volume of pus extracted during the initial treatment, self-efficacy in breastfeeding, and breastfeeding assessment before treatment (see Table 1).

Table 1.

Baseline Characteristics of Subjects in Both Groups (Mean ± SD)

	Control group (n = 28)	Treatment group (n = 28)	p
Age (years), Mean ± SD	30.50 ± 4.30	31.80 ± 3.10	0.215
Postpartum days, Mean ± SD	81.20 ± 69.80	82.7 ± 95.1	0.952
Induration size, Mean ± SD
Length (mm)	56.31 ± 19.46	58.36 ± 24.29	0.735
Width (mm)	40.38 ± 16.08	44.04 ± 16.22	0.733
VAS, Mean ± SD	4.70 ± 2.80	4.50 ± 1.80	0.734
Volume of pus extracted (first aspiration treatment) (mL), Mean ± SD	43.40 ± 29.70	38.00 ± 27.10	0.514
Breastfeeding self-efficacy, Mean ± SD	32.61 ± 13.09	37.04 ± 13.40	0.216
Breastfeeding evaluation
Exclusive breastfeeding, n (%)	12 (42.9)	11 (39.3)
Breastfeeding + Formula, n (%)	9 (32.1)	10 (35.7)	0.953
Stopped breastfeeding, n (%)	7 (25)	7 (25)

The data were presented as the mean ± standard deviation.

Abscess recovery time and pain assessment

In terms of abscess recovery duration, the treatment group demonstrated significantly shorter recovery times compared to the control group for induration resolution (47.8 ± 16.6 days vs. 69.6 ± 26.2 days, p = 0.001) and abscess cavity resolution (42.5 ± 21.1 days vs. 69.6 ± 25.7 days, p < 0.001), indicating a more rapid recovery. Additionally, the treatment group also exhibited a significantly shorter tissue recovery time (51.0 ± 16.6 days vs. 82.0 ± 26.2 days, p < 0.001). These findings suggest that the treatment was associated with an accelerated healing process.

Regarding postoperative pain assessment using the VAS, there were no significant differences in pain scores between the treatment and control groups on the day of surgery (Mean ± SD: 4.7 ± 2.8 for control vs. 4.5 ± 1.8 for treatment, p = 0.734). However, at 3 days post-surgery, the treatment group experienced significantly greater pain relief compared to the control group, as evidenced by a lower median VAS score (1.0 vs. 1.5, p = 0.041). By 1 week post-surgery, both groups had a median VAS score of 0 (median [P25, P75]: 0 [0,1] for control vs. 0 [0.0] for treatment), indicating no pain; however, this difference was statistically significant (p < 0.001), suggesting that the treatment group reached this pain-free state more rapidly than the control group.

The results of the breastfeeding self-efficacy assessment

The Breastfeeding Self-Efficacy Questionnaire evaluated the differences in score changes between the two groups of patients. The results showed that the scores in the treatment Group increased more than those in the control group at 2 weeks, 1 month, and 2 months post-surgery. There were statistically significant differences at 2 weeks and 2 months post-surgery (see Table 2).

Table 2.

Comparison of Breastfeeding Self-Efficacy Between the Two Groups During Follow-Up Within 2 Months Post-Surgery (Mean ± SD)

	Control group (n = 19)	Treatment group (n = 22)	t	p
Day of surgery	32.6 ± 13.1	37.0 ± 13.4	−1.251	0.216
2 Weeks post-surgery	30.8 ± 13.3	41.4 ± 16.4	−2.66	0.010
1 Month post-surgery	35.6 ± 14.6	43.0 ± 16.7	−1.766	0.083
2 Months post-surgery	36.0 ± 14.8	45.4 ± 17.1	−2.200	0.032

The data were presented as the mean ± standard deviation.

Results of the breastfeeding assessment

The results of the breastfeeding assessment showed that following treatment, the rate of subjects in the treatment group that could continue the same breastfeeding protocol as before the breast abscess significantly increased at 2 months post-surgery. The rate of subjects in the treatment group that experienced reduction in breastfeeding volume by more than 30% (evaluated by the volume of pumped milk) and cessation of breastfeeding significantly decreased, respectively. In contrast, there were no significant changes in the above-mentioned two parameters for the subjects of the control group before and after the assessment. There were statistically significant differences in the beforementioned two parameters between the two groups at 1 month and 2 months post-surgery (see Table 3).

Table 3.

Comparison of Changes in Breastfeeding Between the Two Groups During Follow-Up Within 2 Months Post-Surgery (n, %)

Time post-surgery	Group	Continued Pre-Abscess breastfeeding(n,%)	Reduced breastfeeding volume by ≥30% (n,%)	Stopped breastfeeding(n,%)	Chi-Square value	p
2 Weeks	Control group	10 (35.7)	12 (42.9)	6 (21.4)	2.560	0.297
	TreatmentgGroup	16 (57.1)	8 (28.6)	4 (14.3)
1 Month	Control group	10 (35.7)	12 (42.9)	6 (21.4)	9.674	0.007
	Treatment group	21 (75)	4 (14.3)	3 (10.7)
2 Months	Control group	9 (32.1)	11 (39.3)	8 (28.6)	10.588	0.006
	Treatment group	21 (75)	3 (10.7)	4 (14.3)

In the study, there were a total of four cases of repeated puncture. Among them, three cases were in the control group and one case in the treatment Group, with incidence rates of 10.71% and 3.57%, respectively.

Discussion

Breast abscess tissue damage, pain, and psychological stress are significant factors leading to reduced lactation ability and cessation of breastfeeding during the lactation period. However, clinical methods for postoperative tissue repair are limited, resulting in prolonged tissue healing time, which severely affects breastfeeding rates. The study’s evaluation data on LIFU treatment showed that it could shorten the recovery time for patients with lactation-related breast abscesses by about 1 month compared to the control group. It also helped alleviate pain and restore confidence in breastfeeding more quickly. Ultrasound, as a crucial component of physical factor therapy, relies on three major effects for its clinical efficacy: mechanical effects, thermal effects, and physicochemical effects. It is applied in tissue repair, pain relief, and inflammation regulation. Previous research has primarily focused on the repair of muscles, ligaments, tendons, and cartilage. By addressing structural-pathological changes and functional recovery in the body, ultrasound accelerates the overall rehabilitation process, achieving the goal of recovery.¹⁹ Literature reports suggest that the mechanisms by which ultrasound promotes repair may include: (1) Regulation of Inflammatory Response: Ultrasound can modulate inflammatory responses at the cellular level by affecting multiple signaling pathways. This promotes circulation, creating an environment unfavorable for bacterial growth, thereby controlling the further development of inflammation. Additionally, it helps to soften and dissipate indurations.²⁰ (2) Enhanced Cellular Permeability and Metabolism: As a mechanical wave, ultrasound increases cell membrane permeability, facilitating substance exchange, accelerating local tissue fluid and blood circulation, and improving lymphatic drainage. This promotes the rapid removal of local metabolic waste, enhances local tissue nutrition and metabolism, and boosts tissue regeneration capacity.^21,22 (3) Pain Reduction: Therapeutic doses of ultrasound can reduce the excitability of pain nerves and stimulate the secretion of analgesic substances.²³ The study employed LIFU, which focuses ultrasound energy and controls the energy and depth to treat deep tissues. Therefore, LIFU may become one of the clinical methods to promote the repair of breast abscess tissue.

Breast abscesses can lead to reduced confidence in breastfeeding and a decrease in the number of patients who continue to breastfeed.¹ In this study, LIFU treatment significantly increased the proportion of patients continuing breastfeeding. The likely reason for this is that the treatment group experienced better tissue recovery and pain relief, which boosted their confidence in breastfeeding. Although there was a statistically significant difference in the improvement of breastfeeding efficacy between the two groups, this could also be influenced by baseline inconsistencies. For example, some patients in the control group might have shown slower improvement in scores due to a lack of confidence in breastfeeding. Existing studies have shown that ultrasound therapy can effectively enhance lactation function. It does so by accelerating the recovery of lactating cells, increasing their milk production capacity, and improving lactation efficiency. The mechanical stimulation from ultrasound can effectively enhance the milk ejection reflex and increase prolactin levels, among other effects.²⁴ The improvement in breastfeeding observed in this study may be related to two main effects of ultrasound therapy: the acceleration of tissue repair, allowing damaged tissues to recover quickly and resume lactation, and the promotion of lactation function in the normal tissues surrounding the abscess. Ultrasound waves are mechanical waves, which are relatively safe and free from radiation and side effects. Clinically, there have been reports of using ultrasound for the intervention of lactation insufficiency and milk stasis during the breastfeeding period.^25,26 This study is the first to report the clinical effects of ultrasound treatment post-aspiration for mastitis and breast abscess.

In our study, only 4 out of the 60 subjects (6.7%) required a repeat puncture, a proportion significantly lower than the repeat puncture rates reported in other studies within the literature.²⁷ We believe this discrepancy may be attributed to several factors, including the use of high-resolution ultrasound guidance, an experienced procedural team, stringent patient selection criteria, and an optimized treatment protocol. Our experience suggests that these measures substantially enhance the success rate of the initial puncture and reduce the need for repeat procedures.

The preliminary research results on the post-surgical treatment of breast abscesses using LIFU have shown positive effects on tissue repair and breastfeeding. This study is the first to apply the reparative effects of ultrasound to breast abscess tissue. However, due to the widespread clinical guidance and education on breastfeeding, cases of mastitis leading to abscess formation are relatively rare. Therefore, further analysis with a larger sample size and randomized controlled trials are needed to further validate the clinical effects and value of ultrasound in post-surgical abscess repair. Additionally, future research should explore the use of LIFU for the treatment of nonsurgical abscesses during breastfeeding and the repair of breast tissue damage caused by other diseases.

Conclusion

The use of LIFU technology in the treatment following abscess aspiration can better promote the repair of local breast tissue after abscess aspiration. It helps reduce local pain, accelerates the absorption of residual pus after aspiration, and aids in the repair of the abscess cavity walls. Compared to using medication alone, LIFU can shorten recovery time, increase breastfeeding rates, and enhance the ability to maintain breastfeeding. This expands the application of ultrasound in tissue repair and provides a new method for the clinical rehabilitation treatment of postpartum breast abscesses.

Ethical Approval

This study was approved by the Ethics Committee of the Third Xiangya Hospital, Central South University (R22024), and registered with the Chinese Clinical Trial Registry (ChiCTR2200060973).

Authors’ Contributions

Conceptualization: L.D.; Funding acquisition: S.Y. and Y.Z.; Software: L.Z.; Supervision: L.J.; Writing—original draft: J.L.; Writing—review and editing: L.D. and L.Z.

Footnotes

Author Disclosure Statement

No competing financial interests exist.

Funding Information

This work was supported by Hunan Provincial Natural Science Foundation (grant number 2024JJ8121 and 2024JJ6626).

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