Stability of vancomycin during prolonged infusion: seeking evidence to support nurses’ practices

Abstract

Objective  To evaluate the behavior of vancomycin solutions in different concentrations (5 mg/mL and 10 mg/mL) and under various temperature conditions (22ºC and 37ºC) at infusion times of zero (T0), two(T2) and four(T4) hours.

Methods  The experiment simulated clinical practice of vancomycin infusion. Antimicrobials were diluted at 5 and 10mg/mL, studied according to temperature variations (22°C and 37°C), and submitted to analyses immediately after preparation (T0), T2 and T4 hours of environmental exposure. The high-performance liquid chromatography method determined concentration and stability. A total of 108 concentrations and 36 pH measurements were analyzed for mean, standard deviation, ANOVA, and p≤0.05.

Results  The pH values did not show statistically significant variations during the four hours maintaining the acidic characteristic. There were no significant variations of vancomycin 5mg/mL at 22°C at T0 and T2, but there was a statistically significant concentration reduction at T4(<0.001). For vancomycin at 10mg/mL and 22°C, the active compound had an increase in concentration from T0 to T2, with a further decrease at T4(p=0.006); in 37°C there was a significant in concentration in T2 and T4(p<0.001).

Conclusion  Vancomycin at 5mg/mL showed significant concentration reduction in the fourth infusion hour. For vancomycin at 10mg/ml, there was an increase in concentration in the first two hours of analysis. There were statistically significant variations in vancomycin concentrations over time and in the experimental situations studied. Despite these variations, all changes remained within the 7% threshold accepted by the European Pharmacopeia, indicating that vancomycin stability is adequate for clinical use under the tested conditions.

Vancomycin; Drug stability; Liquid chromatography; Chromatography, high pressure liquid; Temperature; Intravenous infusions

Resumo

Objetivo  Avaliar o comportamento de soluções de vancomicina em diferentes concentrações (5 mg/mL e 10 mg/mL) e sob diversas condições de temperatura (22ºC e 37ºC) nos tempos de infusão de zero (T0), duas (T2) e quatro (T4) horas.

Métodos  O experimento simulou a prática clínica de infusão de vancomicina. Os antimicrobianos foram diluídos em 5 e 10mg/mL, estudados conforme variações de temperatura (22°C e 37°C) e submetidos a análises imediatamente após o preparo (T0), T2 e T4 horas de exposição ambiental. O método de cromatografia líquida de alta eficiência determinou a concentração e a estabilidade. Um total de 108 concentrações e 36 medições de pH foram analisadas para a média, desvio padrão, ANOVA e p≤0,05.

Resultados  Os valores de pH não apresentaram variações estatisticamente significativas durante as quatro horas, mantendo a característica ácida. Não houve variações significativas da vancomicina 5mg/mL a 22°C em T0 e T2, mas houve redução estatisticamente significativa da concentração em T4 (<0,001). Para a vancomicina a 10mg/mL e 22°C, o composto ativo teve aumento da concentração entre T0 e T2, com nova diminuição em T4 (p=0,006); a 37°C houve redução significativa da concentração em T2 e T4 (p<0,001)).

Conclusão  Houve redução significativa da concentração da vancomicina a 5mg/mL na quarta hora de infusão. A concentração da vancomicina a 10mg/mL aumentou nas duas primeiras horas de análise. Houve variações estatisticamente significativas nas concentrações de vancomicina ao longo do tempo e nas situações experimentais estudadas. Apesar dessas variações, todas as alterações permaneceram dentro do limite de 7% aceito pela Farmacopeia Europeia, indicando que a estabilidade da vancomicina é adequada para uso clínico nas condições testadas.

Vancomicina; Estabilidade de medicamentos; Cromatografia líquida; Cromatografia líquida de alta pressão; Temperatura; Infusões intravenosas

Resumen

Objetivo  Evaluar el comportamiento de soluciones de vancomicina en diferentes concentraciones (5 mg/mL y 10 mg/mL) y bajo diversas condiciones de temperatura (22 °C y 37 °C) en los tiempos de infusión de cero (T0), dos (T2) y cuatro (T4) horas.

Métodos  El experimento simuló la práctica clínica de infusión de vancomicina. Los antimicrobianos fueron diluidos en 5 y 10 mg/mL, estudiados de acuerdo con variaciones de temperatura (22 °C y 37 °C) y sometidos a análisis inmediatamente después de la preparación (T0), T2 y T4 horas de exposición ambiental. El método de cromatografía líquida de alta eficacia determinó la concentración y la estabilidad. Se analizó el promedio, desviación típica, ANOVA y p≤0,05 de un total de 108 concentraciones y 36 mediciones de pH.

Resultados  Los valores de pH no presentaron variaciones estadísticamente significativas durante las cuatro horas y se mantuvo la característica ácida. No hubo variaciones significativas de la vancomicina 5 mg/mL a 22 °C en T0 y T2, pero hubo una reducción estadísticamente significativa de la concentración en T4 (<0,001). En la vancomicina a 10 mg/mL y 22 °C, el compuesto activo tuvo un aumento de la concentración entre T0 y T2, con una nueva reducción en T4 (p=0,006); a 37 °C hubo una reducción significativa de la concentración en T2 y T4 (p<0,001)).

Conclusión  Se observó una reducción significativa de la concentración de vancomicina a 5 mg/mL en la cuarta hora de infusión. La concentración de vancomicina a 10 mg/mL aumentó en las dos primeras horas de análisis. Hubo variaciones estadísticamente significativas en las concentraciones de vancomicina a lo largo del tiempo y en las situaciones experimentales estudiadas. A pesar de estas variaciones, todas las modificaciones se mantuvieron dentro del límite del 7 % aceptado por la Farmacopea Europea, lo que indica que la estabilidad de la vancomicina es adecuada para el uso clínico en las condiciones estudiadas.

Vancomicina; Estabilidad de medicamentos; Cromatografía liquida; Cromatografía liquida de alta presión; Temperatura; Infusiones intravenosas

Introduction

In recent decades, treating bacterial infections has become a challenge for multi-professional teams, owing to the increased number of multidrug-resistant pathogens. Among the drugs of choice, vancomycin continues to be widely used in critically ill patients, including adults, children and newborns.(¹-³)

Vancomycin is low-cost, simple to manipulate and easy to monitor plasma levels. To maintain drug stability and ensure desired therapeutics the evidence-based protocols recommend dilution with sodium chloride 0.9% or dextrose 5% in water solution, diluted to 5 milligrams per milliliter (mg/mL) and administered within one hour.(⁴,⁵)

However, particularly for patients with critical cardiac or renal disorders, as well as for premature newborns vancomycin is increasingly used for periods longer than one hour of infusion.(²,⁶-⁸)

In clinical practice, different infusions are found, with variations in drug concentration and prolonged exposure time. And, as far as the authors are aware, there are no robust studies that support the stability of this antimicrobial in different infusion situations.

Additionally, in some countries, units are not acclimatized, with extreme environmental variations of temperature. It can be supposed that such a situation may compromise the stability of the drug, and that reverse translation research is needed to describe and support practices.(²,⁷,⁹)

Drug stability studies performed by the pharmaceutical industry are developed to fulfill licensing requirements. After medicines are licensed, less attention is given to the variations across time that occur during the practical use of the drugs and there is no recognition that drugs start a new perspective of use once they are infused according to the patient’s needs and differently from the initial clinical trials protocols that sustain the use of the drug.

The stability of medicines may be affected by degradation processes, thus forming products with different chemical compositions. These reactions may be caused by inappropriate drug concentrations during infusion, exposure to longer infusion periods, inadequate dilutions and use under environmental variations of light or temperature.(¹⁰,¹¹)

Safe clinical practice demands that drug infusion in nursing clinical practice can be performed with scientific support, where the professionals involved can obtain answers to these challenges, aiming at the best development of the proposed therapy for patients, and with the availability of solutions in appropriate and safe fractions and dilutions. To achieve this, presuppose translation research embraces transdisciplinary studies giving opportunities to elucidate clinical research problems and increasing collaboration between clinical and basic research teams, with the possibility of investigating beside research questions on the bench.(¹²)

The aim was to evaluate the behavior of vancomycin solutions in different concentrations (5 mg/mL and 10 mg/mL) and under various temperature conditions (22ºC and 37ºC) at infusion times of zero (T0), two (T2) and four (T4) hours.

Methods

Experimental research conducted at the Laboratory of Nursing Experiments (LEEnf) of the Paulista Nursing School, Federal University of São Paulo, Brazil, prepared to allow the simulation of the clinical practice of antimicrobial infusion in a temperature and luminosity-controlled environment.

The study sample was composed of 12 solutions of vancomycin, six at 5 mg/mL and six at 10 mg/mL, analyzed in time T0, T2, and T4 of infusion. These analyses were performed at two temperature variations of 22ºC and 37ºC.

To ensure greater reliability of the results, the experiments were planned under conditions that allowed for the verification of intra-assay precision. This was achieved by preparing the solutions and conducting measurements by the same operator on alternate days, following a Latin square randomization obtained through a draw. Three solutions were prepared for each of the four experimental situations to promote the reproducibility of the proposed method through triplicate analysis. For the analysis of the vancomycin content, three aliquots of each solution were collected three exposure times. Distinct vials were prepared to ensure sample homogeneity and equipment efficiency, totaling 108 analyses. For pH verification, one aliquot of each solution was collected at each time point, resulting in 36 measurements.

Vancomycin in a 500 mg lyophilized powder (Vancocin^®, ABL-Antibiotics of Brazil, Cosmópolis, Brazil), after the reconstitution, was diluted in a 0.9% sodium chloride solution in a flexible bag to a concentration of 5 mg/mL or 10 mg/mL. To verify whether drug administration over a period longer than one-hour results in stable solutions, measurements were accomplished at T0, T2 and T4.

For the experiments using a temperature of 37ºC, solutions were kept within a neonatal incubator (Fanem^® - Vision 2186, Guarulhos, Brazil). This equipment was chosen because it can maintain air that is constantly heated inside the acrylic dome, enabling it to verify whether the temperature is an agent that causes the instability of the drug.

To check the pH, an aliquot of each, 5 mg/mL and 10 mg/mL, vancomycin solution was collected, at T0, T2 and T4. The pH values were verified using a benchtop digital pH-meter (Kasvi^® K39-2014B, Curitiba, Brazil).

To check the concentration of the drug, the difference between the analysis at the beginning (T0) with those of the following times (T2 and T4) was considered, expressed as a percentage. Pharmacological stability was defined as retention of at least 93% (i.e., degradation of up to 7% of the labeled concentration), following the provisions of the European Pharmacopoeia(¹³) concerning the acceptable content limit of vancomycin preparation.

To determine the vancomycin concentration, the HPLC-UV analysis method was employed and conducted using Modular Agilent Technologies^® equipment (1260 Infinity, Waldbronn, Germany) in isocratic mode using a C18-type reverse-phase analytical column (Waters^® Spherisorb ODS-Hypersil, Dublin, Ireland), temperature at 30ºC, and detection by diode array at 220 nanometers (nm).

Before checking the concentration of vancomycin solutions the analytical method was validated for selectivity, linearity, matrix effect, robustness, precision, and accuracy.(¹⁴)

After the validation method, we prepared samples for chromatographic analysis. From vancomycin diluted in 0.9% sodium chloride solution, 2 mL (5 mg/mL) and 1 mL (10 mg/mL) were aspirated and transferred to a 10 mL volumetric flask. The volumetric flask was filled with a mobile phase.

The mobile phase was composed of 92% buffer solution ammonium phosphate monobasic (Sigma-Aldrich, Vetec^®, Duque de Caxias, Brazil), and 8% organic solvent acetonitrile (Carlo Erba^®, Val-de-Reuil, France), with pH=4.

After mixing the vancomycin solution with the buffer solution, 1 mL of this new solution was aspirated and transferred to the second 10 mL volumetric flask. It was completed with 9 mL of mobile phase, resulting in solutions containing 0.1 mg/mL, a linear range suitable for chromatographic analysis. A 1.5 mL aliquot of this solution was filtered into the chromatographic analysis vials. The chromatographic conditions were a flow rate of 1 milliliter per minute (mL/min), an injection volume of 20 microliters (µL), and an analytical run time of 15 minutes.(¹⁴) This procedure was repeated at all time points established for analysis.

United States Pharmacopeia reference vancomycin standard considered primary reference standard (USP, United States of America, vancomycin content 98.8% USP Catalog No.1709007) was used throughout the experiment. The chromatographic data were recorded and integrated with OpenLab software (Agilent Technologies^®).

The data were deposited in an electronic database and analyzed using standard software (SPSS 20.0 and Stata 12). For pH values, an analysis of variance ANOVA was applied, and for vancomycin concentration, a linear mixed mode was used to compare means with ANOVA according to variations in dilution, temperature, and time. A significance level of 5% was considered.

Results

Thirty-six aliquots of the solutions were collected for pH analysis. The values did not show statistically significant variations during the four hours of solution infusion in all proposed variations of concentration and temperature, maintaining the acidic characteristic (Table 1).

Table 1
Influence of time of infusion (T0, T2, and T4), in the vancomycin pH stability, according to concentration (5 mg/mL and 10 mg/mL) and temperature (22ºC and 37ºC) variations

pH analysis time	Vancomycin solution
	5 mg/mL Temperature		10 mg/mL Temperature
	22oC	37oC	22oC	37oC
	Mean (±SD) Min. - Max.	Mean (±SD) Min. - Max.	Mean (±SD) Min. - Max.	Mean (±SD) Min. - Max.
T0	4.06± 0.23	3.95± 0.14	3.69± 0.01	3.95± 0.14
	3.86 - 4.31	3.83 - 4.10	3.68 - 3.70	3.81 - 4.10
T2	3.97± 0.13	3.96± 0.14	3.67± 0.02	3.96± 0.14
	3.82 - 4.06	3.81 - 4.08	3.65 - 3.69	3.83 - 4.09
T4	3.99± 0.15	3.92± 0.15	3.68± 0.02	3.92± 0.15
	3.82 - 4.10	3.80 - 4.08	3.66 - 3.70	3.80 - 4.08
p (ANOVA)	0.581	0.346	0.408	0.346

• T0 - immediately after solution preparation; T2 - two hours after solution exposure; T4 - four hours after solution exposure

Table 2 presents the concentration means at T0 and T2 of vancomycin infused at 5 mg/mL (22°C), they were similar, but there was a reduction in the concentration at T4 (p<0.001). The experiment observed the same trend at 37°C (p<0.001). Regarding the experimental condition of the drug at 10 mg/mL and exposed to a temperature of 22°C, the active compound increased in concentration from T0 to T2, with a further decrease at T4 (p=0.006). For the vancomycin at the same dilution and exposed to 37°C, the mean at T0 was lower than the other time points, thus considering an increase in concentration (p<0.001) over the evaluated period. Considering the solutions at T0, there was variation in the proposed and actual vancomycin concentrations (Table 2).

Table 2
Concentration (%) of vancomycin according to time (T0, T2, and T4), dilution (5 mg/mL and 10 mg/mL), and temperature (22 °C and 37 °C)

Time	Vancomycin
	5 mg/mL		10 mg/mL
	Temperature		Temperature
	22oC	37oC	22oC	37oC
	Mean (±SD)	Mean (±SD)	Mean (±SD)	Mean (±SD)
	Min. - Max.	Min. - Max.	Min. - Max.	Min. - Max.
T0	101.93 ± 1.66A	106.12 ± 2.60A	92.96± 1.38	96.17± 2.25B
	100.75 - 104.33	101.93 - 108.95	91.11 - 94.58	92.58 - 98.95
T2	101.45 ± 1.46A	106.02 ± 1.43A	94.39 ± 3.69A	97.70 ± 1.50A
	99.54 - 103.58	104.08 - 107.47	89.58 - 98.37	95.65 - 99.71
T4	100.31± 0.87B	104.83± 1.66B	91.86± 1.12B	97.47 ± 0.68A
	99.38 - 101.39	102.63 - 106.47	90.02 - 93.35	96.50 - 98.58
p*	<0.001	<0.001	0.006	<0.001

• T0 - immediately after solution preparation; T2 - two hours after solution exposure; T4 - four hours after solution exposure. (A) and (B) have different meanings according to multiple comparisons for time with Bonferroni correction; p* - a mixed linear mode for comparing the means with ANOVA

In table 2, it can be observed that the concentrations at T0 and T2 at 5 mg/mL (22°C) were similar, but there was a reduction at T4 (p<0.001). The same occurred at 37°C (p<0.001). Regarding the solutions at 10 mg/mL exposed to a temperature of 22°C, the active compound increased in concentration from T0 to T2, with a decrease at T4 (p=0.006). For vancomycin at the same dilution and exposed to 37°C, the mean at T0 was lower than at the other time points, indicating an increase in concentration (p<0.001) over the evaluated period. Considering the solutions at T0, there was a variation between the proposed and actual concentrations of vancomycin (Table 2).

Discussion

In this study, the pH of vancomycin, in all situations analyzed, remained within the expected value, as described in the literature (2.5 and 4.5).(⁵)

In research conducted for stability analysis, pH values were found for vancomycin diluted in sodium chloride 0.9% solution at 5 mg/mL of 4.09±0.3.(¹⁵) A second study identified pH values of 3.76 for vancomycin diluted at 10 mg/mL using the same solvent, and pH values of 3.8 for 5 mg/mL.(¹⁶)

The pH is a relevant factor that interferes with the chemical stability of substances and, as a logarithmic scale, a decrease of one unit scale means a 10-fold increase in proton concentration. Thus, a modification of one or two pH units should not be considered as a ‘‘slight modification in pH values’’ and should be explained.(¹⁷) However, there is no accurate definition of its variation parameter since, to the best of our knowledge, only one study was found in the literature(¹⁸) reporting a change of 0.2 pH units being a statistically significant parameter due to the potential loss of drug stability. It should be noted that the variations found in the current study were lower than this value.

Although the vancomycin solutions studied presented pH values compatible with those described in the literature for the maintenance of stability, it is important to emphasize the care taken with the administration of this antimicrobial, adequate dilution and administration, and highlighting the constant monitoring of the catheter insertion site, mainly in infusions through peripheral intravenous catheters, due to the high risk of extravasation and phlebitis; pH values below 5.0 can be associated with the occurrence these complications, and extravasation is frequently severe vancomycin associated adverse events.(⁹,¹⁹,²⁰)

However, isolated analysis of the solution’s pH is not, in most situations, indicative of drug degradation. From this premise, a more robust analytical method is recommended to analyze the concentration of the solution by employing the analytical technique of HPLC-UV considered the best choice to analyze the stability of vancomycin.(¹²,²¹,²²)

The vancomycin concentration diluted to 5 mg/mL had an initial value of approximately 101-106% of the desired concentration. When vancomycin was diluted to 10 mg/mL, the initial value was approximately 92-96%. Data from the current study corroborated the publication on the preparation of vancomycin solutions by nurses for infusion in children. They were analyzed by the HPLC-UV method, where it was identified that the initial concentrations measured were on average 7% lower than the prescribed concentrations.(²³)

When analyzing the concentration of vancomycin diluted to 5 mg/mL over the four hours of infusion simulation, this investigation found that at both temperatures (22°C and 37°C) there was a loss of the drug over time, but not more than 7%, and there was also no appearance of degradation substances, thus maintaining pharmacological stability. These results are like studies identified in the literature that used the same analysis method, and with similar conditions to the present experiment.(²⁴,²⁵)

In a study that assessed the stability of vancomycin diluted to 5 mg/mL and exposed to extreme temperature (70°C) for five hours, it was observed that the drug remained stable. The authors highlight the accuracy of the HPLC method for this type of analysis.(²⁴)

In an experiment that also used the HPLC method to evaluate the chemical stability of vancomycin and cefepime associated for four hours in simulated intravenous infusion, the researchers prepared 5 mg/mL solutions of vancomycin in sodium chloride 0.9% solution and 5% glucose serum kept at room temperature (22.5°C). They analyzed samples for over four hours and found that vancomycin remained pharmacologically stable, showing 95% of the initial concentration.(²⁵)

In the condition of vancomycin diluted to 10 mg/mL and exposed at 37°C, the chemical behavior was different, there was a statistically significant increase in concentration over time but keeping pharmacological stability less than 7%. The results of the present study corroborate the literature, where it was observed the stability of the antimicrobial at the same concentration after 48 hours of exposure to 37°C temperature.(²⁶)

It is believed that this increase in concentration over time is due to evaporation of the diluent exposed to high temperatures, in addition to the possibility that complete dilution did not occur because of the restricted volume of diluent. However, it is suggested that further studies be conducted to better evaluate this supposed condition.

For about 50 years vancomycin has been one of the most used antimicrobials for the treatment of sepsis in critically ill patients. Its prescription and infusion, however, remain a challenge for professionals working in intensive care units for several reasons, including the high variability of pharmacokinetics and pharmacodynamics, lack of consensus on the dilution, and infusion time of the drug, and the constant control of serum levels in patients.(²⁷,²⁸)

In recent years, the use of antibiotics in a continuous infusion has been evident. However, there is no accurate and safe protocol for such therapy. Studies are varied and sometimes conflicting. In general, they report that continuous infusion of vancomycin has benefits for adults, pediatric and neonatal patients, especially critically ill patients, but they fail to define dose and dilution precisely to ensure the safety and efficacy of the established therapy.(⁶,⁸,²⁹-³¹)

To the best of the authors’ knowledge, no scientific evidence has been found to demonstrate safety regarding drug stability when exposed to prolonged periods. In other studies, the stability was indirectly analyzed according to the serum level, although imprecise due to other possible intrinsic clinical interferences. When it results in values below the appropriate level, the dose is empirically increased, also raising the potential for nephrotoxicity and ototoxicity.(⁶,⁸,²⁷)

Stability studies performed by the pharmaceutical industry are only designed to fulfill licensing requirements. Thus, post-dilution stability data are frequently limited only for bacteriological reasons regardless of the true chemical stability which could, in many cases, be longer. What we find in the literature are studies of the chemical stability of the drug under refrigeration before its dilution for administration to the patient and not during bedside exposure.

For each antimicrobial drug, considering their therapeutic index, the pharmacokinetics/pharmacodynamics variability, specific clinical use and risks related to degradation products, the limit of 7% of degradation can be inappropriate considering the patient’s clinical evolution. Therefore, acceptance limits must be clinically relevant and should be defined for each drug individually. (¹³)

There is a need for studies on the pharmacological stability of vancomycin at different concentrations, with other diluents, and especially in continuous infusion carried out in connection with clinical practice and using HPLC-UV as an analytical method, to promote evidence-based practices. It is important that professionals when using a drug in their clinical practice, perform critical analysis of research, as well as obtain knowledge about pharmacokinetics and pharmacodynamics of drugs, so that these baselines may support safe clinical practice during the administration of antimicrobials. This would promote the patient’s cure considering the correct and stable dose offered and the advancement of science and interventions provided to children.

Another relevant aspect for clinical practice is the fact that the use of prolonged infusion of vancomycin doses can benefit critically ill patients, especially those who are on fluid restriction, have compromised renal function, or are low-weight pediatric patients. Considering these circumstances, nurses should be aware of this type of administration and learn about the appropriate conditions, possible effects and patient safety measures when considering prolonged infusion.

This study was limited by the number of studied analyses and by only containing samples from a single manufacturer. In addition, the authors understand the need for analysis for longer than 4 hours given the applicability of vancomycin in clinical practice.

Conclusion

In vancomycin diluted to 5 or 10 mg/mL, the antimicrobial maintained an acidic pH during the four hours of environmental exposure and a temperature of 37°C. When analyzing solutions at 5 mg/mL there was a significant reduction in drug concentration during the 4 hours. However, in the 10 mg/mL solutions, there was an increase in concentration in the first two hours of analysis. Although the analysis of concentrations during the study periods resulted in statistically significant differences, the variations remain within the quality parameters of the European Pharmacopoeia, indicating pharmacological suitability for use. It suggests that if the prolonged infusion is necessary, it should preferably be carried out at a concentration of 5 mg/mL. More studies are needed to elucidate strategies to avoid minimal variations in antimicrobials, as we don’t know the impact of this condition on clinical practice.

Acknowledgments

This study was supported by grants provided by the National Council for Scientific and Technological Development (Conselho Nacional de Desenvolvimento Científico e Tecnológico, CNPq) [no.311296/2013-0 and 462183/2014-9]. The researchers thank the members of the Safety Technology and Care Research Group (SEGTEC) and the staff of LEEnf for their support during data collection and the Coordination for the Improvement of Higher Education Personnel (CAPES) for providing the doctoral scholarship.

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Edited by

• Associate Editor:
  Juliana de Lima Lopes (https://orcid.org/0000-0001-6915-6781) Escola Paulista de Enfermagem, Universidade Federal de São Paulo, São Paulo, SP, Brazil

Publication Dates

• Publication in this collection
  22 Aug 2025
• Date of issue
  2025

History

• Received
  29 Oct 2024
• Accepted
  17 Mar 2025