Methods Patients The study will enroll patients between the ages of 50 and 80 years with chronic and current knee pain and a clinical and radiographic diagnosis of knee osteoarthritis (OA). Participants will be recruited through the "Treatment of Knee OA with Radial Pressure Wave Therapy" campaign conducted by the Physical Medicine and Rehabilitation Department of the Hospital Civil de Guadalajara "Fray Antonio Alcalde" in México. Recruitment will take place during the months of August and September 2024. Individuals with a history of knee surgery, inflammatory arthritis, septic arthritis, fibromyalgia, cancer, or severe vascular insufficiency will be excluded from participation in this study.

Ethical Statement The participants will be required to sign an informed consent form which will include all details of the study. The patients will be free to decide whether or not to participate or to withdraw from the protocol at any time. The study was approved by the Research Ethics Committee of the Hospital Civil de Guadalajara Fray Antonio Alcalde (number: CEI 157/24) and was conducted in accordance with the World Medical Association's Declaration of Helsinki.

Randomization After obtaining written informed consent, the participants will be randomized to one of three treatment groups: radial pressure wave therapy (RPWT) at the two most painful points (control group), RPWT at the medial interarticular line (medial group), or RPWT at the medial and lateral interarticular lines (medial and lateral group). Randomization will be performed using the paper envelope randomization method. For this purpose, the investigators will prepare a series of opaque papers, each indicating one of the three possible assignments. These papers will be placed in an opaque bag, and during the enrollment process, each patient will randomly draw a paper from the bag. The patient's group assignment will be determined by the label on the paper drawn.

Treatment The three groups will receive RPWT using the BTL-6000 Radial Shockwave Therapy device (BTL Industries, Ltd.). Each patient will receive 3 sessions of RPW, once a week, in the supine position with knee flexion of 30 to 60 degrees, 2,000 pulses per session, 2.0 bar, and frequency of 10 Hz. The air pressure and frequency will be adjusted according to discomfort. The control group will receive 1,000 pulses to the two most painful areas of the treated knee, according to the technique used in most previous studies in this area. The medial group will receive 2,000 pulses at the medial interarticular line of the treated knee, while the medial and lateral groups will receive 1,000 pulses at the medial interarticular line and 1,000 pulses at the lateral interarticular line of the treated knee. All participants will be trained to perform a home exercise program for 4 consecutive weeks, three sessions per week, consisting of superficial thermotherapy for 10 minutes, hamstring and quadriceps stretching, medial gluteus medius and quadriceps strengthening, and propioception exercises. The interventions will be performed at the Physical Medicine and Rehabilitation Department of the Civil Hospital of Guadalajara "Fray Antonio Alalde". Complications, adverse effects and complaints during treatment will be documented.

Outcome measures A questionnaire will be used to assess sociodemographic characteristics, including age, gender, and education level for all patients. In addition, the numerical and categorical forms of body mass index (BMI), physical activity habits, and analgesic use will be assessed after eligibility criteria are established using the aforementioned instrument. The severity of knee OA will also be measured by a board-certified rheumatologist using the Kellgren \& Lawrence scale.

The Spanish version of the Western Ontario and McMaster Universities Osteoarthritis Index (WOMAC) will be used to assess self-reported disability. The WOMAC is a disease-specific, self-administered questionnaire developed for patients with knee OA and takes approximately 5 minutes to complete. It has a multidimensional scale consisting of 24 items grouped into three dimensions: pain (five items), stiffness (two items), and physical function (17 items). We will use the Likert version with five response levels for each item, representing different levels of intensity (none, mild, moderate, severe or extreme) that are scored from 0 to 4. The final score for the WOMAC questionnaire is determined by adding the aggregate scores for pain, stiffness and function. The higher the score, the worse the patient's condition; therefore, improvement is achieved by reducing the total score. The data were standardized to a range of 0 to 100, with 0 representing the best possible health status and 100 representing the worst possible status. Pain intensity will be assessed using the Numeric Pain Rating Scale (NPRS), a world-renowned instrument consisting of a linear scale with numbered intervals from zero (no pain) to ten (the worst pain) that patients use to quantify their current pain intensity. It is well known that pain is the most important symptom of knee OA. Therefore, pain will be the primary outcome measure in the study. Treatment success was defined as an individual improvement in the NPRS of 50% from baseline. Evaluations will be performed at baseline, at the end of the third session, at 2 months, and at 4 months after the first RPWT session.

Blinding This study will use a double-blind design in which both the participants and the clinical evaluator responsible for performing the clinical and radiographic assessments will be blinded to the group assignments. Participants will be informed that they may receive one of the three possible treatment options, but will not know which specific technique will be applied to them. The RPWT will be performed by a final year resident in Physical Medicine and Rehabilitation who will be aware of the group assignments but will not be involved in the outcome assessments. To ensure further blinding, data collection and analysis personnel will also be blinded to the intervention groups. All treatment sessions will be conducted under standardized conditions with identical instructions, setup, and environment for all groups to reduce potential bias.

Sample size calculation Pain intensity, as measured by the NPRS, was considered the primary outcome measure for sample size determination. The investigators used a significance level of 0.05 (two-tailed) and statistical power of 80% (1-β), corresponding to Z values of 1.96 and 0.84, respectively. The calculation was based on an expected effect size of 2.0, considered clinically relevant based on previous studies, and a standard deviation of 2.1 reported in the literature. The initial calculation indicated that approximately 17 participants per group would be required. To account for an expected 20% loss to follow-up, the investigators adjusted the sample size and increased it to 23 participants per group to ensure the robustness and validity of the results. In summary, a total of 23 patients will be recruited for each of the three groups in our study, taking into account both the expected effect size and the adjustment for loss to follow-up.

Statistical Analysis Data will be analyzed using the Statistical Package for the Social Sciences (SPSS) version 22.0 software. Normality of the data distribution was assessed using the Kolmogorov-Smirnov test. For variables with a normal distribution, the mean and standard deviation will be calculated; for variables with a non-normal distribution, the median and interquartile range (IQR) will be reported. Qualitative data are presented as numbers and percentages. In cases of normal distribution, a t-test will be used to analyze differences within and between groups. One-way ANOVA will be used to determine statistical differences in WOMAC questionnaire and NPRS scores over the course of the study. Pearson's correlation coefficient will be used to assess the relationship between the 4th month scores (WOMAC and NPRS), baseline scores, age, BMI, and Kellgren \& Lawrence severity grades. The strength of the correlation is classified as negligible (0.00-0.10), weak (0.10-0.39), moderate (0.40-0.69), strong (0.70-0.89), and very strong (0.90-1.00). All tests are considered statistically significant at a p-value of ≤0.05.