Traumatic dislocation of the tibialis posterior tendon is one of the significantly rare conditions that we might deal with in the emergency department. Approximately only 50 cases have been reported in the literature, and, usually, this condition is misdiagnosed as an ankle sprain in acute settings. It might be neglected in case of improper clinical examination and imaging techniques. 1-meter jump height while playing basketball; the patient was clinically diagnosed primarily as a case of simple ankle sprain in the emergency department and treated conservatively with cold compression. The patient was seen in the clinic after five days with the same complaint, which was medial ankle pain without any improvement. Proper examination and imaging techniques lead us to the accurate diagnoses. The patient was managed surgically and had an excellent outcome postoperatively with a full range of motion of the ankle joint and full weight-bearing without any complaint.We present a case of a 39-year-old patient diagnosed with traumatic dislocation of the tibialis posterior tendon as a result of twisting injury after falling from aTibialis posterior dislocation should be one of the possible differential diagnoses while dealing with any ankle injury even with unremarkable X-rays. History and physical examination, if conducted correctly, are the keys to making an accurate diagnosis. Therefore, we recommend a proper history-taking and precise physical examination with a high index of suspicion for any possible diagnoses. Early surgical intervention for such cases is the preferable method of treatment to avoid further complications and promote early functional recovery. The tibialis posterior tendon (TPT) has a significant role in the plantar flexion of the ankle and inversion of the foot. It is also an essential stabilizer of the medial longitudinal arch. TPT dysfunction can lead to acquired flatfoot deformity in adults. The origin of the TPT is from the lateral part of the posterior surface of the tibia, the medial part of the posterior surface of the fibula, and the interosseous membrane, with insertion in the plantar aspect of the navicular bone, the cuboid, the cuneiforms, and all the metatarsal bases except the first one. The TPT lies immediately posterior to the medial malleolus passing through a fibro-osseous tunnel where it changes direction acutely, like a bowstring. The tunnel consists of the retromalleolar groove in the distal tibia and the overlying flexor retinaculum. The retinaculum keeps the TPT in the groove, preventing its dislocation. The depth and width of the groove vary significantly. In a cadaveric study, Soler et al. reported groove variation in the width ranging from 6 to 15 mm and that in the depth ranging from 1.5 to 4 mm . The TPTA 39-year-old male firefighter presented to the emergency department with left ankle pain and inability to bear weight after falling from a 1-meter jump height while playing basketball. He reported landing on the dorsum of his left foot with the ankle in the plantar flexion position. Clinically, there was moderate ankle swelling in the medial aspect with a limited range of motion of the ankle joint. His X-rays were unremarkable. He was diagnosed with an ankle sprain and was managed with rest, ice, elevation, compression, and analgesia. Five days later, he presented to the orthopedic clinic with the same complaints. On examination, he had localized swelling just anterior to the medial malleolus. Upon palpation, there was a slightly tender cord-like structure that moved back and forth with gentle plantar and dorsiflexion of the ankle Figure -4. The pOperative procedureThe patient was placed in supine position, and to facilitate external rotation of the left leg, a bolster was placed under the contralateral hip. A 6-cm longitudinal incision was made along the anterior aspect of the medial malleolus Figure . The retPostoperative managementThe patient was permitted to touch-down on the back slab using a pair of crutches for six weeks. After the back slab was removed, ankle motion was started and progressive weight-bearing was allowed. The patient had regular follow-up visits for 12 months.OutcomeAfter three months, the patient had no pain or swelling in the ankle. He was able to bear full weight, and he had a full ankle range of motion. He was allowed to play sports after six months. At 12 months, he was asymptomatic and was back to his pre-injury level of activity.Traumatic dislocation of the TPT is an uncommon injury. Approximately only 50 cases have been reported and have reviewed the TPT dislocation in the published literature . The firCT scan is the radiological study of choice to detect hypoplasia of the retromalleolar groove, but it was not performed for our patient since it has limitations in evaluating the soft tissue injuries and it exposes the patient to high level of radiation . MRI is There are two dislocation types reported in the literature: type I, flexor retinaculum tear leading to subcutaneous dislocation, and type II, periosteal avulsion of retinaculum leading to subperiosteal dislocation . Our patOne of the unusual traumatic injuries of the ankle fractures is the interposition of the TPT or flexor digitorum longus tendon which results in irreducible fracture-dislocation in the medial aspect of the ankle. The TPT is an important anatomical structure, which is the most common cause of interposition in irreducible ankle fracture dislocation due to the entrapment of the TPT in the tibiofibular and tibiotalar joints -10.Another unusual traumatic injury to the TPT is a full or partial rupture of the tendon associated with ankle bimalleolar or medial malleolar fracture; these kinds of injuries happen in case of pronation and external rotation ankle fracture causing the most tension to the TPT or in case of a direct trauma to the posterior aspect of the medial malleolus, which may cause a direct injury to the TPT -12.Most authors have reported that surgical intervention is more effective than conservative management . OuzouniTPT dislocation is a very rare condition and can be easily missed in acute settings. However, a detailed history and careful physical examination combined with appropriate imaging techniques can lead to a timely and accurate diagnosis. If TPT dislocation is suspected, the use of ultrasonography or MRI is indicated to confirm the clinical diagnosis. We recommend practicing a high index of suspicion when dealing with ankle injuries to avoid subsequent delay in diagnosing the case and to provide the appropriate management.


The Faraday forcing method in levitated liquid droplets has recently been introduced as a method for measuring surface tension using resonance. By subjecting an electrostatically levitated liquid metal droplet to a continuous, oscillatory, electric field, at a frequency nearing that of the droplet’s first principal mode of oscillation (known as mode 2), the method was previously shown to determine surface tension of materials that would be particularly difficult to process by other means, e.g., liquid metals and alloys. It also offers distinct advantages in future work involving high viscosity samples because of the continuous forcing approach. This work presents (1) a benchmarking experimental method to measure surface tension by excitation of the second principal mode of oscillation (known as mode 3) in a levitated liquid droplet and (2) a more rigorous quantification of droplet excitation using a projection method. Surface tension measurements compare favorably to literature values for Zirconium, Inconel 625, and Rhodium, using both modes 2 and 3. Thus, this new method serves as a credible, self-consistent benchmarking technique for the measurement of surface tension. The reliability of processes like crystal growth, additive manufacturing, and welding depend on precise knowledge of thermophysical phenomena4. However, thermophysical property measurement of high-temperature materials like liquid metals, glasses, and oxides is difficult using conventional means because of high surface reactivity5. The advent of containerless material processing using aerodynamic6, acoustic8, electromagnetic9, and electrostatic11 levitation technologies has provided an avenue for further improvements in the accuracy of thermophysical property measurements and can also be used to provide insights into other important material behavior like phase equilibria and solidification dynamics10. Surface tension, an important thermophysical property, can be measured using levitation technologies by exploiting the natural frequency fn at which a spherical liquid droplet comes to rest, given by12γ is the surface tension, n is the normal mode of oscillation, and M is the mass. By experimentally finding the natural frequency fn, one can solve Eq.  droplets in the investigation of the stability of levitated Zirconium as a function of size and temperature16. The current work aims to include the resonance quantification of mode n = 2 and mode n = 3 as a benchmarking method for determining surface tension of a material using just one levitated sample. The Faraday forcing method was introduced as a method to stimulate resonance in electrostatically levitated droplets using a frequency sweep, where the droplet is forced with an incrementally changing frequency in a range surrounding the predicted natural frequency. The droplet’s deformation is recorded as it is forced. Resonance (increasing amplitude of droplet deformation) occurs as the forcing frequency approaches that of the droplet’s natural frequency. In the earlier work, the magnitude of this deformation  was quantified by the normalized maximum diameter, defined as the maximum prolate diameter in resonance divided by the resting diameter of the droplet. This was due to the fact that only the first principal mode was sought, which corresponds to a prolate–oblate deformation. It is important to note that the normalized maximum diameter is limited to mode n = 2 deformations; an alternative metric presented herein is required to extend this to higher order modes. In Fig. Introduced in this work is a refinement of the Faraday forcing method described in 2018 by Brosius et al.5, where a droplet is forced to oscillate, and the forcing is abruptly stopped. The deformation of the droplet returning to rest is recorded as a function of time and fitted to an exponentially decaying sinusoidal function, as seen in Fig. n = 2, of the droplet. This pulse-decay technique is the current industry standard and has the advantage of being able to measure both surface tension and viscosity. The Faraday forcing method is thought to avoid some of the control system disturbances that may arise with the pulsed perturbation and may provide improved accuracy in surface tension measurement. The two techniques are compared in Table The method stands in contrast to the so-called pulse-decay techniquen = 2) mode frequency and records the result. The sample is then subsequently subject to a second frequency sweep around the expected n = 3 frequency. The frequency at which n = 3 is expected to resonate most strongly is also predicted by Eq. . It was hypothesized in that work that the incorporation of higher-order modes that come from the pulse could cause a shift in natural frequency because of nonlinear interaction between modes. In Table The primary impact of this work is a proof-of-concept for a benchmarked measurement of surface tension using resonance of multiple modes of oscillation in levitated liquid materials. In the earlier work of Brosius et al.In the previous work, the magnitude of resonance was quantified in mode 2 by measuring the normalized maximum diameter of the droplet in resonance. This method, known as the diameter method, was straightforward because the droplet’s shadow oscillates between an oblate and prolate ellipse in mode 2. An important update in the current work is the modified approach to quantifying the resonance of the droplet using the projections of Legendre polynomials . While the prior diameter-based method was more straightforward, it did not lend the generality that the projection method provides – that is, the current method can quantify any modal deformation . As might be expected, the two methods do agree in the characterization for mode 2 resonance, shown in Fig. 17. This setup is known as an area array and is also used to characterize droplet behavior on the Electrostatic Levitation Furnace (ELF) on the International Space Station, also operated by JAXA. The output signal from the receiver is directly proportional to the shadow area and provides a distinct advantage in computation time when compared to imaging-based methods. The strength of oscillations in the droplet’s area is proportional to the magnitude in which the droplet is oscillating in mode 2. The definition below is used to quantify the droplet shadow’s area deviation given time-dependent data.Aavg is the time-averaged shadow area of all instantaneous shadow area values Ai.The magnitude of resonance may also be characterized using indirect (non-image-based) methods. For example, the electrostatic levitator at the Japan Aerospace and Exploration Agency (JAXA) in Tsukuba, Japan uses a He-Ne laser to cast a shadow on a photodetector, which is blocked by the sample, as shown in Fig. 4As shown in Fig. 18. The shadow area A of the deformed droplet is given by the following integral:δr(θ) is the droplet shadow’s deviation from a circle and R is not the physical resting radius R0 but rather a characteristic radius defined such that the volume V is constant, that is:While indirect area arrays may be used to effectively quantify the droplet’s deformation in mode 2, the method is unsuccessful in quantifying resonance of mode 3. The reasoning for this discrepancy lies in the fact that the area of the shadow is negligibly affected by deformation in mode 3. This is proven using an approach akin to that of the application of the Rayleigh work principle in liquid bridge stabilityn = 2 and n = 3, respectively. That is,The mode 2 and mode 3 disturbances are given by the Legendre polynomials for an of each normal mode. Shown in Fig. an. This indicates that, although the droplet may be oscillating strongly, a signal reflecting the area of the droplet’s shadow is insufficient in characterizing the resonance of mode 3. However, the mode n = 4 is included to illustrate that this area array setup can still be used in conjunction with the continuous Faraday forcing method with higher harmonics, provided that n is even.The area of the droplet’s shadow may then be found analytically as a function of the amplitude This conclusion is further confirmed in analyzing experimental data where mode 3 was definitively observed, by attempting to quantify resonance using an artificial area array. Shown in Fig. an (referred to as ϵ in Tsamopolous & Brown19). It may be thought that since the method is reliant on the determination of the most resonant point in a frequency sweep, the frequency at which the droplet most strongly deforms must be slightly shifted by a factor δf/f (given by Eq. (It is well-understood that the oscillation frequency of a droplet is affected by its displacement amplitude n by Eq. ) from it19 of less than 2% and thus within the stated error of 1–5% in the measurement of surface tension. The maximum deviation observed in this work was a2 = 0.1, corresponding to δf/f = 0.008 and an estimated error of 1.6%.While this is a valid concern in principle, all experiments conducted in this work exhibited deformation amplitudes which would correspond to a maximum frequency shift given by ref. 20 and experimental work by Rhim et al.11 established a relationship between the frequency shift and a correction factor which depended on the charge, electric field, and the shape of the droplet’s deformation.ωl is the natural angular frequency given by Eq.  but can be as high as 1.5%. Analogous to the effect of amplitude on the natural oscillation frequency, the shift in mode 3 oscillation from the electric and gravitational fields is much smaller than the shift in mode 2 oscillation.The error stemming from electric and gravitational fields is one of the strong incentives to levitation processing in microgravity. Because of the lack of gravity, the equilibrium shape of a levitated droplet is more perfectly spherical and also requires a significantly weaker electric field to maintain its levitation.fn and thus the error can be estimated as 1–2%. However this error can be reduced to arbitrary precision by lowering the step size of the frequency sweep.The final significant source of error involved in this procedure’s measurement of surface tension involves the frequency step size in the sweep. All frequency step sizes in this work range from 0.5–1.0% of the natural frequency ϕ-symmetric. However, this is an assumption that is based on experimental observation and is not strictly obeyed. The degenerate modes  all ordinarily coexist at the same frequency and all have some probability of being excited. However, in this experiment, the m = 0 mode is most often observed. When one of the degenerate modes are excited, this causes an "off-axis” oscillation and torque from the electric field can cause substantial spinning of the droplet. This spinning splits the degenerate modes into three frequencies. Spinning can also be stimulated without deformation of the droplet due to photon pressure of the heating laser. These dynamics can be minimized by lowering the amplitude of forcing and allowing the droplet to fully relax in between forcing frequencies. To accommodate these challenges, the frequency sweeps garnering highly nonlinear coupling behavior are analyzed but not used in the measurement of surface tension because of the shifting of the natural frequencies.There are several challenges that encompass the practice of this method which either directly or indirectly stem from the presence of gravity. The levitation process on Earth can only successfully levitate samples which have suitable thermophysical properties. For example, very dense materials like Platinum or Iridium are difficult to levitate because of high density and high melting point. The charge of the droplet, needed for levitation control, changes as the sample is heated due to thermionic emission. In some cases, this can cause dramatic changes in surface charge and result in the loss of the sample. Materials that cannot be subcooled sufficiently below melting point (or otherwise have high vapor pressures) present an additional challenge with constant evaporation. This evaporation limits processing time and can cause potential fouling of the sensitive equipment within the chamber. Evaporation can be limited by levitating samples within a pressurized atmosphere, but this is challenging on Earth for liquid metals and alloys because the strength of electric field required for levitation is beyond the threshold for electrical breakdown and results in arcing. There are also challenges that result in the nonlinear dynamics of the spherical droplet being subject to an oscillatory forcing. As described in Methods, the droplet is assumed to be This work confirms that the principle behind using a series of frequency sweeps to identify multiple modes of oscillation in levitated liquid droplets can serve as a self-consistent, benchmarking method for the measurement of surface tension. Three materials – Zirconium, Rhodium, and Inconel 625 – have exhibited consistent, predictable behavior in the resonance of the first two principal modes of oscillation which serves to confirm the surface tension measurement in a fashion superior to repetition alone.n may be sensible in future experiments using area arrays. The difficulty in experimentally attaining higher harmonics lies in the higher energy states and correspondingly high damping rates, which are limited by the forcing voltage and the charge of the droplet. This difficulty may be alleviated by operating on samples in low gravity with low fundamental frequencies given by Eq.  is used to measure the temperature within the range 300 to 3000 ∘C. In surface tension measurements, the sample is melted and then subcooled below the melting point (if possible) to minimize evaporation. Upon reaching the desired temperature, the laser is used to maintain constant temperature. A waveform generator and amplifier are connected to the upper electrode to allow the user to impose an oscillatory electric field. Droplet behavior is characterized with a high-speed camera at 5000fps and 512 by 512 pixels, typically for durations of 1–1.5 s. The sample is weighed both before and after processing to quantify evaporation.All experiments employing the self-consistent measurement method were conducted at the Electrostatic Levitation (ESL) Laboratory at NASA Marshall Space Flight Center (MSFC). Experiments performed at Japan Aerospace Exploration Agency for reasons of comparisons with the aforementioned method are explained in the Discussion. The process of levitating the sample is identical to the methods outlined in Brosius et al.∘C is predicted to be 185 Hz, so the sweep would encompass frequencies ranging between 170–200 Hz, taking steps of 1 Hz. After finishing the mode 2 frequency sweep, the droplet remains levitated but is solidified to suppress evaporation. The videos from the high-speed camera are previewed after the sweep prior to download and resonance is characterized by eye before beginning the mode 3 sweep.The natural frequencies for both modes 2 and 3 are initially predicted using the droplet’s mass and accepted literature surface tension value with equation 1. Modes 2 and 3 are both discovered using a method known as a frequency sweep, following the precise procedure of Brosius et al. For each frequency of a frequency sweep, an oscillatory voltage (known as a forcing frequency) is applied to the top/bottom electrodes for 1–2 s. Upon completion of each forcing frequency, the oscillatory voltage is turned off and the droplet’s deformations can come to rest before starting the next forcing frequency . The sweep is centered about the predicted natural frequency and, depending on confidence level, the step size (difference between one forcing frequency and the next) is 0.5–5 Hz, with 20–30 total frequencies tested. For example, the mode 2 natural frequency of a 35.027 mg Zirconium sample at 1800 If there is no published data for the material of interest, multiple frequency sweeps are performed to find the resonance point of mode 2. First, a broad (100–200 Hz) sweep of large step size is used to force the droplet to resonate in mode 2. Subsequent sweeps follow with narrowing frequency range and decreasing step size until desired accuracy is achieved.fmax), the mode 3 sweep will be centered on its corresponding mode 3 frequency. This corresponding mode 3 frequency, given by Eq. , and the mode 3 frequency sweep is conducted, taking steps of 0.5–5 Hz.The fashion in which the mode 3 sweep is conducted depends on the preliminary results from the mode 2 sweep. If there is conclusive video evidence that the mode 2 sweep yielded strong resonance at a certain frequency  are projected on to the outline of the droplet. While this method returns the same results for mode 2 as the method outlined in Brosius et al., it is more consistently used for both modes 2 and 3 and thus serves as a more robust quantification of resonance, which can be used for any axisymmetric deformation (see the Discussion for a detailed comparison of the projection method to the diameter method). In the absence of external forces and in the limit of low viscosity24, deformations on a spherical surface can be described by25R0 is the resting radius of the sphere, n is a normal mode of oscillation, m represents degenerate modes of n, and fn is given by Eq.  when electrostatically levitated, meaning the droplet could be well-approximated by a simplified version of Eq.  is prefn of Eq. 11\documer, the function δr is defined as the normalized difference between the outline and the resting radius of the droplet.an are now normalized with respect to the resting droplet’s radius, that is an = An/R0. Since the Legendre polynomials obey the relationship:αn(t), by way of projection, that isStarting with an outline of the droplet, which is found through image analysis as a function δr as obtained through image processing and perform a numerical integration scheme to quantify the amplitude of a given mode. The process is given graphically in Fig. Using this definition of the projection amplitudes, one can therefore use the deformation outline T, one may assume the following relationship between the constant an and the time-dependent data given by αn(t) when the number of oscillation cycles is large:Assuming sinusoidal behavior with total acquisition time n = 2 as shown in Fig. n = 3 as shown in Fig. The resonant frequency for a given trial is taken to be the forcing frequency which resulted in the maximum resonance of the droplet in the prescribed modal shape (mode Further information on research design is available in the Mode 2 and Mode 3 Resonance Observation in Levitated Liquid DropletReporting Summary Checklist


Symptomatic breast hypertrophy has a significant impact on the quality of life of women. The amount of tissue to be excised may be preoperatively estimated by an experienced surgeon. However, this remains a subjective assessment. Accurate quantification of the amount of breast tissue to be resected in the preoperative period will be a guide for both patient information and the surgeon during the operation. The aim of this study is to develop a new method based on simple measurements that can accurately estimate the resection weight in the preoperative period in a wide range of patients undergoing reduction mammoplasty.The study was carried out between December 2016 and September 2018. With the determined drawing and measurement methods, a triangle was obtained by measuring the distances among the sternal notch (A) - right nipple areola midpoint (B), sternal notch (A) - left nipple areola midpoint (C) and both internipple areola (B-C). The height of this triangle (h) was found by measuring the distance between the sternal notch and the midpoint of both nipple areola levels. The amount of breast tissue to be resected for each breast was calculated by multiplying the distance between the sternal notch–nipple areola and the height of the large triangle. The formula may be expressed as AB × h for the right breast and AC × h for left breast.When the t values and significance levels of the beta coefficients of the independent variables were examined, the preoperative values were determined to be in accordance with the actual values after surgery (P < 0.05). The values calculated before were calculated as the percentage of the actual values (91%). In other words, the R2 value showed that the calculated values were compatible with the actual values (R2 = 0.910).With the formula described herein, one may accurately estimate the amount of tissue to be resected in a wide range of patients undergoing reduction mammoplasty whose sternal notch–nipple distances are between 28–42 cm. Additionally, because measurements for each breast are performed separately, breast asymmetry does not affect the results. In conclusion, the formula we devised is simple, applicable, and has a high accuracy rate. Symptomatic breast hypertrophy has a significant impact on the quality of life of women. Reduction mammaplasty aims to reduce the symptoms of macromastia and provide the aesthetic appearance of a large ptotic breast. It has been shown that many symptoms such as neck and back pain, shoulder groove, intertrigo and posture disorders have disappeared, and quality of life, self–esteem and even pulmonary function have improved in patients who underwent reduction mammoplasty. Given all these reasons, it has become one of the most commonly performed surgeries by plastic surgeons in recent years [1–3].The weight of the breast tissue excised in reduction mammoplasty may vary from a few hundred grams to 3 kg depending on the original breast size. Reduction mammoplasty techniques are described as standard, and the technique to be administered is preferred based on the size of the breast and the experience of the surgeon. The amount of tissue to be excised may be preoperatively estimated by an experienced surgeon. However, this remains a subjective assessment. Accurate quantification of the amount of breast tissue to be resected in the preoperative period will be a guide for both patient information and the surgeon during the operation .Appropriate preoperative measurements will eliminate such problems. Several studies have been carried out to determine the amount of resection, and various formulae have been proposed . However, widespread use of these formulae could not be achieved due to their complexity, difficulty of applicability, or the lack of sensitivity in measuring the amount of tissue resected at the end points.The aim of this study is to develop a new method based on simple measurements that can accurately estimate the resection weight in the preoperative period in a wide range of patients undergoing reduction mammoplasty.The study was conducted with Ethics Committee approval. It was planned as a prospective study, and 68 patients aged between 25–65 years (136 breasts) who underwent bilateral reduction mammoplasty (inferior pedicle) for symptomatic breast hypertrophy between December 2016 and September 2018 were included in the study.The age, height, weight, and body mass index (BMI) of all patients were recorded before the operation. Following the determined drawing and measurement methods, a triangle was obtained by measuring the distances among the sternal notch (A)–right nipple areola midpoint (B), sternal notch (A)–left nipple areola midpoint (C), and both inter nipple areola (B-C).The height of this triangle (h) was found by measuring the distance between the sternal notch and the midpoint of both nipple areola levels. Afterwards, during the preoperative marking procedures, bilateral new nipple areola points were marked on the level of the inframammary fold.A new triangle was obtained by measuring the distances among the sternal notch (A)–right side new nipple (N), sternal notch (A)–left side new nipple (N’), and the new internipple areolar distance. The distance between the base of the triangle and the sternal notch (A) was accepted as the height of the small triangle (h’) .The amount of breast tissue to be resected for each breast was calculated by multiplying the distance between the sternal notch–nipple areola and the height of the large triangle. The formula may be expressed as AB × h for the right breast and AC × h for the left breast. While measuring the distances from the nipple areolar points, the nipples’ exact midpoints were marked delicately. If the measurements with decimal numbers were 0.4 or smaller, they were rounded to the lower integer. If the decimal numbers were 0.5 or higher, then the measurements were received as one higher integer.In patients whose distances between the sternal notch–nipple areola were shorter than 28 cm or longer than 42 cm, as the difference between the preoperative and postoperative measurements were higher, a new formulation was developed for these patients by including the small triangle in the measurements.Based on the small triangle obtained in the preoperative measurements, the formulae AN × h’ for the right side and AN’ × h’ for the left side were developed. It was observed that, if we added the calculations of the small triangle to the result of the first formulae’s calculations (right breast: [AB × h + (AN × h’)], left breast: [AC × h + (AN’ × h’)]) in the patients whose sternal notch–nipple areola distance was above 42 and subtract the calculations of the small triangle from the results of the first formulae’s calculations (right breast: [AB × h - (AN × h’)], left breast: [AC × h - (AN’ × h’)]) in patients whose sternal notch–nipple areola distance was below 28 cm, the estimated and the actual results were closer to each other. Consideration should be paid to evaluation of patient groups in cases of more extreme endpoints.The drawings and measurements of all patients were performed by the first author, while the operations were carried out by 3 surgeons. After the calculations, all patients were operated on under general anesthesia. Each patient underwent breast reduction with the same technique, inferior pedicle and inverted T scar pattern. The amount of tissue resected from each breast was weighed separately and recorded.All obtained parameters were analyzed by using the SPSS for Windows  software. The prescribed breast tissue was resected in the operating theatre and weighed. For the statistical analyses, Pearson’s correlation coefficient and paired samples t test were utilized. The rate of compatibility between the resected breast tissue amounts and the estimated breast tissue amounts based on the formulae was evaluated by using linear regression analysis. In linear regression analysis, the amount resected in the surgery was the dependent variable, while the amount calculated before the surgery with the specified formula was determined as the independent variable. A P value of < 0.05 was considered statistically significant.Sixty-eight patients aged between 25 and 62 years (mean: 44 years old) were enrolled in the study. The mean BMI of the patients was found to be 32.4, and the mean amount of resected breast tissue was 1112.43 g per breast. The predicted weight of the resection amount based on the formula was 1121.13 g per breast. The mean sternal notch–nipple areola distance was 35 cm (Table 1).). For normality, the Kolmogorov-Smirnov test statistics were as P: 0.20 for the distribution of errors in the right breast  and as P: 0.171 for the distribution of errors in the left breast  .Parametric methods were used since the data satisfied the assumption of normal distribution  (Table 2).Pearson’s correlation analysis was performed to determine the linear relationship between the weight of the resected tissue from the left breast and the estimated weight of the breast tissue to be resected from the left breast before the operation. According to the results, a statistically significant and strong positive correlation was found between the values  (Table 3).The paired samples t test was used to determine the difference between the values calculated from the right and left breasts and the actual weights of the mass removed from the right and left breasts. H01: There is no significant difference between the values calculated for the right breast before the operation and the actual weight of the mass resected from the right breast after the operation. H02: There is no significant difference between the values calculated for the left breast before the operation and the actual weight of the mass resected from the left breast after the operation.According to the test results, the hypothesis was not rejected with 95% confidence that there was no statistically significant difference between the preoperative predicted resection weight values of both the right breast and the left breast and the actual weight values of the resection amounts that were removed as a result of the operation. Accordingly, the difference between the calculated values and the actual values was not statistically significant (P > 0.05) (Table 4).Both the independent and dependent variables were continuous, and their distributions were normal. There was a linear relationship between the independent and dependent variables. A regression analysis was conducted, and the model was found significant. The errors of the model had a normal distribution. There was no problem of autocorrelation in the model. If the DW test statistic is between 1.5–2.5, this means there is no problem of autocorrelation.A simple linear regression analysis was performed to explain the effect of the preoperatively calculated values on the actual values obtained after the study. When the significance level corresponding to the F value was examined, it was seen that the model that was established was statistically significant . When the t values and significance levels of the beta coefficients of the independent variable were examined, the preoperative values were determined to be in accordance with the actual values after surgery (P < 0.05). The values calculated before were calculated as the percentage of the actual values (92%). In other words, R2 showed the compliance of the calculated values with the actual values (adjusted R2 = 0.916) (Table 5). The adjusted R2 value of 0.916 showed that 91.6% of the weight resected from the right breast could be explained by the amount calculated before the operation for the right breast.Normally, there are multiple independent variables that affect the dependent variable. To reduce the complexity of the model and establish comprehensible, interpretable models, variables that do not have an effect on the targeted variable, or have a low or negligible effect are excluded from the study. For this reason, to eliminate the independent variables that are unnecessarily added, the adjusted R2 was used instead of the R2 which is known as a goodness of fit index. A simple linear regression analysis was performed to explain the effect of the preoperatively calculated values on the actual values obtained after the study. When the significance level corresponding to the F value was examined, it was seen that the model that established was statistically significant . When the t values and significance levels of the beta coefficients of the independent variables were examined, the preoperative values were determined to be in accordance with the actual values after surgery (P < 0.05). The values calculated before were calculated as the percentage of the actual values (91%). In other words, the R2 value showed that the calculated values were compatible with the actual values (R2 = 0.910) (Table 6). The adjusted R2 value of 0.910 showed that 91% of the weight resected from the left breast could be explained by the amount calculated before the operation on the left breast.It was shown that the formula used in the statistical analyses was successful in determining the amount of tissue removed, and it could be used in a wide range of patients whose sternal notch–nipple areola distance was between 28–42 cm. Although statistically significant results were obtained in all patients, according to the observations of the authors, the discrepancy between the preoperative values and the postoperative actual values was increasing in the patients whose sternal notch–nipple areola distance was below 28 cm or above 42 cm in comparison to the other patients.Several studies have shown significant improvements in symptoms and quality of life in patients with symptomatic breast hypertrophy following reduction mammaplasty . In the study by Schnur et al., the majority of patients who underwent reduction mammaplasty stated that they required operation for symptomatic relief [7].However, determining the tissue weight to be resected prior to reduction mammoplasty has become an important problem. As experienced surgeons can predict the amount of tissue to be resected in large breasts where the precise weight is not crucial, it is often difficult to predict in border macromastias. Preoperative determination of the amount of tissue to be resected will be helpful for surgeons during the operation. Additionally, insurers will have clear information about the coverage before the operation .There are many equations based on anthropometric measurements and expensive scientific techniques such as the water displacement method (Archimedes), biometric analysis with stereometric cameras, various plastic cup-like devices and 3-dimensional imaging to estimate the weight of the breast tissue to be removed. However, application of the methods described above has not received widespread acceptance in usage because of their limitations such as being difficult or complicated or insufficiency in determining breast sizes at the end points [8–13].Sommer et al. retrospectively examined 263 patients who underwent reduction mammaplasty, and they researched the association between sternal notch–nipple distance and resection weight. Although the generally applied formula was successful in predicting the resected tissue weight in large breasts, the accuracy rate for resections of less than 600 g was calculated as 50%. Additionally, it will not be possible to make accurate estimations with a single measurement due to the varying resection amounts in patients with breast asymmetry [14].Descamps et al. analyzed 214 patients who underwent reduction mammaplasty in South Africa and aimed to determine the resection weight by a single formula. The correlation test between the sternal notch–nipple, inframammary fold-nipple distance, and BMI showed a significant correlation with the resection amount, but the formula created by sternal notch–nipple, inframammary fold-nipple distances could not predict the resection weight. It was also shown that reliability decreased in mild resection quantities and by only using vertical plan measurements as the independent variables [15].On the other hand, Kocak et al. formulated the sternal notch–nipple distance by evaluating the vertical and horizontal measurements of the breast surface. They found that the parameter obtained by multiplying the vertical and horizontal measurements had a relatively high correlation coefficient (r = 0.95) with the resection weight of the breast tissue. They suggested that the sternal notch–nipple distance may be affected by the length of the rib cage, and only the formulae based on breast measurements provide more accurate results [16].Regnault and Daniel developed a formula for estimating the resection weight required to achieve the desired bra size. The formula was based on the measurements of the chest circumference taken on the level of the nipple and from the level of the axillary region under the armpits. Although this method was useful for preoperative planning and estimation of cup size, it contributed little to predicting the required resection weights by the reason that its purpose was to calculate the amount of resection required to achieve the desired bra size. Additionally, it is not possible to estimate different weights from each breast in these cases with significant breast asymmetry [17].Appel et al. examined the relationship of resection weight among sternal notch–nipple distance, inframammary fold to nipple distance and BMI in 349 reduction mammaplasty patients. By inclusion of all 3 parameters, a formula was obtained with a high degree of accuracy to obtain the estimated resection weight, but in the prospective evaluations, the formula was not introduced [18].This study was prospectively performed on 68 patients, with 2 basic anthropometric measurements to define a simple formula that calculates the resection weight with an accuracy of over 90%. The measurements used here consisted of multiplying each lateral leg of the triangle obtained by combining sternal notch–nipple distances with the height of the triangle. The formula can be easily reproduced and calculated without the need for any tools. However, in measurements under 28 cm and over 42 cm, the difference between the resected amount of tissue and the amount calculated by the formula, and additional formulae are required. Additionally, as in other formulae used to estimate resection amounts, BMI is an important parameter.Furthermore, thanks to the formula described herein, one can accurately estimate the amount of tissue to be resected in a wide range of patients undergoing reduction mammoplasty whose sternal notch–nipple distances are between 28–42 cm. Additionally, because each breast’s measurements are performed separately, breast asymmetry does not affect the results.In conclusion, the formula we devised is simple, applicable, and has a high accuracy rate in patients with a BMI greater than 30. We believe that it can guide both patients and surgeons.All of the authors declare that they have all participated in the design, execution, and analysis of the paper, and they have approved the final version. Additionally, there are no conflicts of interest in connection with this paper and the material described is not under publication or consideration for publication elsewhere.


MEtabolic models Reconciled with Gene Expression (MERGE), a computational pipeline that we used to predict tissue‐relevant metabolic function at the network, pathway, reaction, and metabolite levels based on single‐cell RNA‐sequencing (scRNA‐seq) data from the nematode Caenorhabditis elegans. Our analysis recapitulated known tissue functions in C. elegans, captured metabolic properties that are shared with similar tissues in human, and provided predictions for novel metabolic functions. MERGE is versatile and applicable to other systems. We envision this work as a starting point for the development of metabolic network models for individual cells as scRNA‐seq continues to provide higher‐resolution gene expression data.Metabolism is a highly compartmentalized process that provides building blocks for biomass generation during development, homeostasis, and wound healing, and energy to support cellular and organismal processes. In metazoans, different cells and tissues specialize in different aspects of metabolism. However, studying the compartmentalization of metabolism in different cell types in a whole animal and for a particular stage of life is difficult. Here, we present  MERGE is computational pipeline that can convert tissue‐level RNA‐seq data from Caenorhabditis elegans into predicted tissue flux potentials that indicate metabolic function.Monitoring tissue‐ and cell‐type specific metabolic activities by metabolomics remains challenging.  Caenorhabditis elegans is a bacterivore that can be fed individual bacterial strains. Caenorhabditis elegans tissues and metabolism share many functions with mammals. Therefore, it provides a relatively simple model for understanding animal metabolism at a systems level. We have previously reconstructed a C. elegans genome‐scale metabolic network model  , a combined approach Fig A that stC. elegans metabolism  the intestine, which receives, metabolizes, and delivers nutrients to the other tissue; (ii) the intestinal lumen from which bacterial nutrients are imported by the intestine; (iii) the other tissue; and (iv) the extracellular space through which nutrients are exchanged between the intestine and the other tissue  and simultaneously maximized to get Zfitmax  during the first step. Then, while Zfitmax is held constant and the low flux sum is tightly constrained, the total flux in the entire network is minimized based on a parsimonious FBA approach (pFBA)   is defined for each tissue (T) as a function of three sets of variables: (i) the reaction network of the tissue defined as reactions in OFD and ALT , wherein all metabolic genes are expressed at the maximum level found in any tissue. Thus, FPA of the target reaction yields a dimensionless value between 0 and 1 for each tissue, which is named relative flux potential (rFPk) . We describe the concept of FPA by a toy network Fig A. FPA us ALT Fig C, (ii) e ALT Fig  of everyPk) Fig A.FPk, we maximized the flux of the target reaction such that a weighted sum of all fluxes is limited by an arbitrary number called flux allowance (a)  called distance order, which controls how fast the influence of remote reactions decays with distance. In the toy example  Fig A. The wemple Fig A, assumirFP values converged to relative expression levels of ldh‐1 , as the weights, and hence the influence, of other reactions decayed to zero. At lower distance orders, the contribution of other reactions in the pathway became evident, and the difference between LDH flux potential of muscle and neurons was reduced. Importantly, FPA used the presumed pathway , degradation (N = 50), or both (N = 62) based on high‐confidence predictions  and glutamine  is consistent with this molecule being secreted to blood during exercise in humans , extreme values (0 and 10) had a larger impact resulting in the loss of some of the key predictions Fig . In thisC. elegans metabolic network model , intestine (I), other tissue (X), and extracellular space (E)  and extracellular reactions for the intestine were placed in L. Transportable and exchangeable metabolites and extracellular reactions for the X tissue were placed in E. Transport reactions were distributed to exchange metabolites between I and L and between E and X. Exchange reactions were used in L to bring in nutrients from the environment, and in E to secrete by‐products to the environment, with the exception of oxygen, water, orthophosphate, and protons, which could be both taken up and secreted through E.Reactions of iCEL1314 were used to build a dual‐tissue metabolic network model of four compartments: lumen ( (E) Fig A. Internet al, Tissue‐level expression profiles were obtained from the reference study (Cao μ) and standard deviations (σ) that defined these curves were used to set thresholds for high expression (> μHES), low expression (< μLES + σLES), and rare expression (< μLES). Then, for each tissue, genes were categorized accordingly  difference in each interval is separately evaluated. If FC is > 4 in the first interval or 1.5 in others, the gene is labeled as highly expressed in tissues after the interval evaluated. Likewise, if the expression level increases by 4 in the last interval or 1.5 in others, the gene is labeled as lowly expressed in tissues before the interval. If a tissue expression is inconsistently labeled as high and low during the evaluation of different intervals, then it was maintained as moderate. To minimize the effect of noise at low expression levels, the jump in TPM value in an interval was considered significant only if the greater value was higher than a threshold (τrel = μHES − σHES). Also, moderate expression levels were recategorized as high only if they were above this threshold, and as low only if they were below it. The general form of this algorithm is explained in detail in Moderately expressed genes were reevaluated with a heuristic algorithm that categorized some of these genes as highly or lowly expressed depending on their relative expression levels in different tissues. This algorithm evaluates one gene at a time. First, TPM values of the queried gene are listed for all seven tissues in an ascending order. Thus, expression level increases from one tissue to the next in six intervals. The fold change , 243 side nutrients, and three storage molecules . When dea priori  in some parts of the network or a very large bacterial intake flux (≫ 1 unit). This is because some metabolites have very small stoichiometric coefficients in reactions that appear during bacterial digestion or biomass assembly. To avoid this scaling problem, the maximum flux capacity of each reaction was first determined in each available direction (forward or reverse) using FVA  for each reaction that depends on rarely expressed genes, such that yi is 1 if the reaction has no flux and 0 otherwise (ROFF.Define a binary integer variable (therwise . Flux thyif for the forward and yir for the reverse direction), such that yi is 1 if the reaction carries flux greater than ε in the pertaining direction and 0 otherwise  for each highly expressed gene that is associated with at least one reaction in RON. These genes are expected to be associated with flux‐carrying reactions and make the set GON. Constrain zi to be less than or equal to the sum of y values for all RON reactions associated with the gene  for subsequent steps. The δlow value used in this study was 1E‐5.Perform FBA to minimize total flux of reactions that depend on lowly or rarely expressed genes . The totZtotal and its minimum value as Ztotal,min. Constrain Ztotal to be less than or equal to Ztotal,min plus a small tolerance value  for subsequent steps. The Δtotal value used in this study was 0.05Ztotal,min. The flux distribution obtained in this step defines PFD . These are latent reactions.y values (see above) of latent reactions determined  as a percentage of bacterial intake by mass (see above for the method of constraining). An arbitrary upper limit of 1% was used for all stored metabolites combined.Constrain the uptake of side nutrients as a percentage of bacterial intake by mass. The overall uptake of these nutrients was limited by 2% of bacterial intake and the uptake of individual nutrients by 0.5% (see below for the choice of parameters).X and E compartments. Use only rarely and lowly expressed genes in the intestine to constrain reactions in the I and L compartments. Highly expressed genes in the intestine are ignored in this step.Integrate gene expression of the six non‐intestinal tissues one by one with the model. Use all categorized genes in the target tissue to constrain reactions in X compartment and constrain the model with these fluxes for the next step. An efficient method is taking a weighted sum of the fluxes through the metabolites of the E compartment and adding exchange reactions that exchange these metabolites, which are then constrained with the calculated metabolite fluxes  of reactions that carry material to and from the e fluxes . The weis et al,  were uses et al, .I and L compartments, while the flux to and from the X compartment is taken into account by the constraints from the previous step.Finally, integrate all categorized intestine genes with Since all tissues are taken into account in the last step, the integration from that step shows the overall flux load and nutritional requirements. To warrant a bacterially dominated diet, different side nutrient uptake rates were tested for the second step. Because the intake of bacteria itself is not constrained, strict limitation of side nutrients as a percentage of bacterial intake results in the excessive intake of bacteria to allow sufficient uptake of side nutrients for the integration, as a result of which bacterial biomass is wasted and the ratio of side nutrients within the used resources is increased. Based on trials with different ratios , the oveet al, Whether a reaction was part of the metabolic network of a tissue was determined based on OFD (see above) and FVA  the minimum flux the reaction can take, the negative of which shows the flux capacity in the reverse direction .Use the flux of the reaction from integration  and the flux capacity of the reaction calculated by FVA to determine whether the reaction is in the tissue network:oIf the reaction has non‐zero flux in the tissue OFD, then it is part of the tissue metabolic network. These reactions were labeled as OFD .oIf the reaction carries no flux in OFD, but has non‐zero flux capacity based on FVA, then it carries flux in alternate solutions of integration for the tissue of interest and is therefore part of the tissue network. These reactions were labeled ALT.oIf the reaction has zero flux capacity, then it does not carry flux in the solution space (SLNS) of the tissue of interest, and hence not part of the tissue network. These reactions were labeled NONE.I and L compartments for the intestine, and those in the X and E compartments for the other tissues. As an exception, if a reaction in the I or L compartment was labeled ALT for intestine integration but was not in SLNS for any other tissue integration, then it was considered as not part of the intestine network.All FVA and OFD analyses were carried out with the dual‐tissue model. However, tissue networks were eventually defined in a single‐tissue format  using thFlux potential analysis in a tissue is performed with the following steps:Divide reversible reactions into two separate reactions, one representing the forward direction and the other representing the reverse. The reverse reaction has the reactants and products swapped. Therefore, all reactions in the modified model have only a forward direction.Choose a target reaction. If the analysis is targeting a metabolite, choose a demand or export reaction that drains the metabolite to analyze its production potential, or a sink or import reaction that introduces the metabolite to analyze its consumption potential. If these reactions do not exist, insert a demand or sink reaction to the model accordingly.c) for each reaction for the tissue of interest (see below). This calculation is done only once for all FPA performed for a particular tissue. For the dual‐tissue model, expression coefficients in I and L compartments are always those based on intestine gene expression, while the coefficients of X and E reactions are variable.Calculate gene expression coefficients (d) of each reaction in the network from the target reaction (see below).Calculate the metabolic distance (If the target reaction is part of a reversible reaction in the unmodified model (see above), block the reaction that represents the other direction in the original model. This step prevents loops.If the target reaction is a sink, demand, or transport (to or from extracellular space) reaction, block all other reactions of these types that act on the same metabolite. This step prevents short cuts during metabolite‐level analyses.Block all reactions that are not part of the tissue network (see above for the determination of tissue networks).Perform FBA such that:oThe objective function is the maximization of flux in the target reaction .oAll reaction fluxes are constrained between 0 and 1000 .oa)  . The weia) . The allFP) of the target reaction in the tissue of interest.The maximum flux obtained from the FBA step gives the flux potential  of the target reaction in the tissue of interest.The ratio of I and X compartments were not targeted when FPA was performed on non‐intestinal tissues and intestine, respectively. As a specific rule, during FPA for the intestine, all reactions in the X compartment were blocked.In the dual‐tissue model context, reactions in Tissue expression profiles  were used to determine gene expression coefficients of each reaction for each tissue. If a reaction was associated with only one gene, the profile was first normalized by the maximum value, and then the reciprocal of the normalized value for a tissue defined the coefficient of the reaction in that tissue . If multE compartment), which were assigned a zero coefficient for all tissue evaluations. Also, transport reactions between I and E compartments were assigned a zero coefficient when the target reaction is in the X compartment. This rule prevents the penalization of a non‐intestinal tissue for having to transport a metabolite through a longer route than the intestine.Reactions that are not associated with any genes were typically assigned an expression coefficient of 1. Exceptions included bacterial intake and degradation reactions and all exchange reactions .Convert this metabolic network to a reaction tree such that each node represents a particular reaction and is connected to two types of other nodes called “to” nodes and “from” nodes. The “to” nodes represent reactions that consume the products of this reaction and the “from” nodes represent ones that produce its reactants. To prevent creating short cuts in the network, ignore hub metabolites such as atp, nadh, and h2o when linking reaction nodes. In this study, 21 frequent metabolites were ignored.i, first eliminate the reverse of this reaction in the network  and traverse the tree from this reaction to all others that are reachable. When a reaction is reached for the first time, fix the path to that reaction before traversing the tree further to others. Because of this path fixing, the path from reaction i to any reaction j will cover the least number of steps possible, thereby showing the distance from reaction i to reaction j.Given a query reaction i to reaction j. A loop occurs when a reversible reaction of the original metabolic network is traversed twice, i.e., when both the forward and the reverse reactions originating from a reversible reaction are part of the path found. Such paths are not valid and cannot be used for calculating the shortest distance. For all such cases, eliminate the first reaction in the path that creates a loop and find the shortest path from reaction i to reaction j again.Check for loops in the path from reaction i to reaction j. The length of this path defines the distance from reaction i to reaction j.Since the previous step can still produce an invalid path with loops, repeat that step until a valid path is found from reaction i. Use the maximum distance observed in the network for reactions that are not reachable starting from reaction i.Apply the previous three steps to find the distance to all reactions reachable from reaction i to any reaction j in the metabolic network.Repeat the previous four steps using every reaction in the network as the query. The result is a square distance matrix that gives distance from any reaction i to j is not necessarily equal to the distance from j to i. Set the final distance between i and j as the minimum of these two values. This step converts the distance matrix to a symmetric one. During FPA, the row of the distance matrix that represents the target reaction can be used to define the distance between this reaction and all network reactions (columns).Due to the structural properties of metabolic networks, the distance from The loop corrections in the fourth and fifth steps do not warrant shortest paths to be found. However, such corrections were necessary for < 1% of all distances calculated and more than a hundred such cases were manually verified to have yielded the shortest distances. Thus, the compromise made in this algorithm to make the search for the shortest path computationally feasible had a negligible effect on the accuracy of the calculated distances.The exclusion of hub metabolites such as NAD and ATP in the second step means that some redox and energy reactions may appear very distant to a target reaction, and therefore, the weight of their flux may be very low during FPA. This does not mean redox and energy balance can be neglected, as redox, mass, and energy balances are always established by FBA during FPA. However, the influence of the relative expression of redox and energy genes may be diminished by the large distances, depending on the target reaction and the distance order used.https://github.com/WalhoutLab/MERGE), which includes scripts that reproduces the results here, examples of integration of iCEL1314 with whole‐animal datasets, and a preliminary integration of a human model  (Yilmaz & Walhout, et al, Metabolic models: iCEL1314 is available at the WormFlux website (Predictions: The computer code, metabolic models, and datasets produced in this study are available in the following databases: LSY and AJMW conceived the work and wrote the paper. LSY designed and implemented MERGE with help from XL. XL reproduced the results in MATLAB environment, created the GitHub repository, and performed applications with human model. SN helped with metabolic network model updates. BF, XL, and FS reconstructed the ascaroside biosynthesis pathway and helped with metabolic network model updates. All authors carefully reviewed and edited the paper.The authors declare that they have no conflict of interest.AppendixClick here for additional data file.Expanded View Figures PDFClick here for additional data file.Table EV1Click here for additional data file.Table EV2Click here for additional data file.Table EV3Click here for additional data file.Table EV4Click here for additional data file.Table EV5Click here for additional data file.Table EV6Click here for additional data file.Table EV7Click here for additional data file.Table EV8Click here for additional data file.Review Process FileClick here for additional data file.


This paper proposes mobile robot self-localization based on an onboard 2D push-broom (or tilted-down) LIDAR using a reference 2D map previously obtained with a 2D horizontal LIDAR. The hypothesis of this paper is that a 2D reference map created with a 2D horizontal LIDAR mounted on a mobile robot or in another mobile device can be used by another mobile robot to locate its location using the same 2D LIDAR tilted-down. The motivation to tilt-down a 2D LIDAR is the direct detection of holes or small objects placed on the ground that remain undetected for a fixed horizontal 2D LIDAR. The experimental evaluation of this hypothesis has demonstrated that self-localization with a 2D push-broom LIDAR is possible by detecting and deleting the ground and ceiling points from the scan data, and projecting the remaining scan points in the horizontal plane of the 2D reference map before applying a 2D self-location algorithm. Therefore, an onboard 2D push-broom LIDAR offers self-location and accurate ground supervision without requiring an additional motorized device to change the tilt of the LIDAR in order to get these two combined characteristics in a mobile robot. The fundamental tasks for autonomous mobile robots are collision avoidance and self-localization mainly based on the use of onboard ultrasonic sensor systems, vision-based systems or light detection and ranging (LIDAR) sensor systems. One method widely used to implement the self-location task is simultaneous localization and mapping (SLAM)  which usIn general, the use of a fixed horizontal 2D LIDAR in a mobile robot  producesThere are a large number of studies based on developing and using a motorized 2D LIDAR. For example: Lit et al.  proposedThe proposal of this paper is inspired on the work of Takahashi et al.  which prThe paper is structured as follows: The main materials and methods used in this paper are a LIDAR mounted on the mobile robot assistant personal robot (APR), a 2D reference map already available, and the iterative closest point (ICP) self-location algorithm.The mobile robot used to experimentally validate the main hypothesis of this paper is a human-size assistant personal robot (APR) created as a family of versatile research tools for laboratory experimentation. The concept of the APR was firstly presented by Clotet et al.  and was APR mobile robots use a UTM-30LX LIDAR for SLAM and trajectory tracking. These mobile robots can be configured either (1) to apply SLAM from scratch in order to create a new 2D map of a new exploration area, (2) to apply SLAM using as a reference a previously generated 2D map  and (3) to apply SLAM using a previously generated 2D reference map (that can be shared between the mobile robots) without updating this 2D map at the end of the SLAM procedure. The only difference between all these procedures is the final step of updating the current 2D map with new information provided by the 2D LIDAR. In all cases, the SLAM procedure is based on the iterative closest point (ICP) algorithm ,29.The 2D map used in this paper was created by the authors in  and reprThe ICP  is an alThe ICP algorithm has to deal with the problem of having a set of individual scanning point clouds that contain shared information of the scenario without a common coordinate system. This uncertainty is caused due to the fact that each point cloud is unaware of the position of the sensor in the real world during a scanning process, assuming that each scan is taken at the origin of the coordinate system. In general, the ICP algorithm tries to establish the position/orientation difference between an initial scanning point cloud or map and the current scanning point cloud by continuously computing a rotation/transformation matrix that minimizes the distance between the point clouds until the distance between them is lower than a specific threshold, or until a maximum number of iterations is reached. This computation allows the definition of a common coordinate system and origin between all the point clouds in order to describe the scenario and the current position of the mobile robot in it.The ICP algorithm takes advantage of the presence of redundant information between consecutive point clouds in order to calculate the position and orientation difference between them. This redundancy is given by the assumption that two consecutive scanning point clouds will contain common region information of the scenario but observed from a different perspective (position/orientation). The ICP algorithm includes the following internal steps: point selection, neighborhood selection, point matching, weighting, rejection and minimization that can be performed using a wide variety of strategies. Donso et al.  presenteThe ICP algorithm implemented in the SLAM procedure defined for the APR mobile robots use the point to plane strategy. In general, it is well known that point to plane is a noticeably slower metric than point to point due to its complexity . HoweverThe main hypothesis of this paper is that the data gathered with a 2D push-broom LIDAR can be used in a mobile robot for self-location using a 2D reference map previously created with a 2D horizontal LIDAR. This configuration must detect also low-height objects, ramps and holes usually not detected by a horizontal LIDAR  which onA set of detection experiments have been proposed in order to verify the detection capabilities of a 2D push-broom LIDAR and to estimate an optimal tilt angle. In all these experiments some objects have been placed at multiple distances in front of the mobile robot which remains static. The height of the LIDAR was fixed to 380 mm and its tilt angle was manually changed from 0°  to 45° (limited by the structure of the mobile robot) in increments of 5°. The use of a tilted-down 2D LIDAR provides information of the ground plane in front of the mobile robot. In all the experiments the ground appears as a frontal line that has some measurement noise see . Then, aOther obstacle cases fully considered and analyzed but not included on this paper are a medium size box, detected similarly to a small box; a big box, detected as an obstacle; a small paper bin, which is not detected by the 2D LIDAR in its original configuration (only detected when the tilt is 5° or higher); other shoe/leg cases with different textures and colors, detected similarly as the one shoe/leg case; and a static person in front of the robot , detected similarly as the one shoe case.The case of a small box  placed on the ground in front of a mobile robot will never be detected by a horizontal 2D LIDAR with a measurement plane higher than the height of the object. In this experiment the 2D LIDAR has been tilted from 0°  to 45° (limited by the structure of the mobile robot) in increments of 5° but, for simplicity, The case of one shoe/leg  is apparently not a challenging detection because a person is normally tall enough to cross the line of sight of a 2D LIDAR mounted on a mobile robot regardless of its position and orientation. However, the one shoe/leg case analyzed see  includesThe case of stairs going down will never be detected by a horizontal 2D LIDAR. This fact represents a challenge for a fast and heavy mobile robot such as the APR-03 because a floor sensor located under the structure of the mobile robot will not be able to warn and stop the mobile robot in time before falling down the stairs due to its inertia. The detection experiment results shown on The mobile robot APR-03 can go forward at a velocity up to 1 m/s. At the maximum velocity, the obstacles must be detected at around 450 mm in order to stop the mobile robot just before the collision. This minimum detection distance has been increased with an additional safety margin of 250 mm. Therefore, the 2D push-broom LIDAR must detect obstacles at a frontal distance of 700 mm from the mobile robot and the fixed tilt-down angle that guarantees this detection is 25°. The adequacy of this optimal fixed LIDAR tilt angle has been validated by performing pseudo-random repetitive displacements in the area of the stairs of the facility: going forward a random distance between 200 and 2000 mm  plus a rotation with a random angle between −180° and +180°. The development of the validation experiments have been supervised by the members of our research group under controlled and safe conditions, ensuring the absence of people in the experimentation area and also having a remote control device to perform an emergency stop if needed. In all these experiments, the 2D reference map does not include any navigation restrictions or additional annotated information to prevent the robot from attempting to navigate across dangerous areas such as staircases or underneath chairs. In all validation experiments the detection of the stair going up and the stair going down has been successful and the proposed safety margin seemed reasonable and adequate. These verification experiments have been the first experiments in which a mobile robot of the APR family has been configured to operate autonomously in the area of the stairs of the facility. In all of our previous experiments the APR mobile robots used the 2D LIDAR in horizontal for self-location in order to avoid this area. Therefore, this push-broom configuration can represent a huge improvement of the application capabilities of the APR mobile robot in a human compatible scenario in case of validating the utility of this push-broom LIDAR configuration also for self-location.The hypothesis of this paper is that an already available 2D map created with a 2D horizontal LIDAR carried out by a mobile robot or by another mobile device in a specific exploration operation can be used to locate the position of a mobile robot using the same 2D LIDAR tilted down 25°, which is the optimal tilt angle obtained experimentally in the previous section of this paper for the mobile robot APR-03.A specific set of experiments have been carried out to validate the self-location capabilities of the mobile robot APR-03 with the 2D LIDAR tilted down. In these experiments the mobile robot APR-03 will perform pseudo-random displacements and will use the 2D reference map created by the same mobile robot in a previous experiment see . A speciThe push-broom configuration requires pre-processing the 2D scan data gathered by the LIDAR before applying the ICP algorithm for self-location which consist on converting the 2D scan data in a 3D point cloud, removing the ground and ceiling points from the 3D point cloud (because the reference map does not include such information), and the projection of the 3D point cloud in the horizontal 2D plane used by the reference map in order to apply ICP. The complete pre-processing procedure has the following steps: (1) get one complete 2D scan data from the LIDAR, (2) delete the distance points lower than 20 mm as they are measurement artifacts, (3) use the height of the measurement plane of the LIDAR and the tilt angle to convert the 2D scan data in a 3D point cloud see , (4) delAlternatively, in the experiment shown in Finally, The proposal of this paper has been inspired on the work of Takahashi et al.  who propThe use of a 2D LIDAR tilted-down provides information of the ground plane in front of the mobile robot and enables the detection of objects or holes by the application of a threshold to the relative ground height. In this paper a positive threshold of 50 mm over the expected ground height has been used to detect obstacles and a negative threshold of −50 mm to detect holes. In some tilt cases . The successful validation of the hypothesis of this paper has been possible because (1) the large distance range of the 2D LIDAR model used and (2) because the experimentation facility has large walls, plain doors and some big fixed furniture objects that are not prone to environmental changes.The use of a 2D push-broom LIDAR has the drawback of reducing the information gathered from the front of the mobile robot. In the case of the APR-03 using the UTM-30LX the frontal detection range was reduced from 30,000 mm (available with a tilt angle of 0°) to 750 mm (available with a tilt angle of 25°). The availability of a large distance range is ideal to create accurate maps using SLAM but a LIDAR with a tilt angle of 0° does not provide information of small objects placed in the ground and does not detect the hole and the danger that represents a stair going down.Future work will be focused on applying the 2D push-broom LIDAR in domestic scenarios and recognize user activity  in a sma