The novel iSCLBT exhibits a significant improvement over the previous version. Both Lehmann’s theory and the SCLBT variants provide significantly better lower bounds than those obtained from Weinstein’s and Temple’s methods. Compared to each other, the Lehmann and iSCLBT theories exhibit similar performance in terms of the quality and convergence of the lower bounds. By increasing the number of states included in the calculations, the lower bounds are tighter and their quality becomes comparable with that of the Ritz upper bounds. Both methods are suitable for providing lower bounds for low-lying excited states as well. Compared to Lehmann’s theory, one of the advantages of the iSCLBT method is that it does not necessarily require the Weinstein lower bound for its initial input, but Ritz eigenvalue estimates can also be used. Especially owing to this property the iSCLBT method sometimes exhibits improved convergence compared to that of Lehmann’s lower boundsRitz eigenvalues only provide upper bounds for the energy levels, while obtaining lower bounds requires at least the calculation of the variances associated with these eigenvalues. The well-known Weinstein and Temple lower bounds based on the eigenvalues and variances converge very slowly and their quality is considerably worse than that of the Ritz upper bounds. Lehmann presented a method that in principle optimizes Temple’s lower bounds with significantly improved results. We have recently formulated a Self-Consistent Lower Bound Theory (SCLBT), which improves upon Temple’s results. In this paper, we further improve the SCLBT and compare its quality with Lehmann’s theory. The Lánczos algorithm for constructing the Hamiltonian matrix simplifies Lehmann’s theory and is essential for the SCLBT method. Using two lattice Hamiltonians, we compared the improved SCLBT ( While these eigenvalues typically give sufficiently accurate upper estimates for the energy levels, they provide no information on their quality; that is, a tight bound from below the true energy level. The variational principle for eigenvalues is a fundamental theorem in physics and chemistry and not limited to the energy levels in quantum mechanics only. The first lower bound expression was introduced for vibrating systems by Temple in 19283, which is limited if no experimental data is available. The Weinstein lower bound4 is based on the variance associated with the approximate eigenvalues of the Hamiltonian, and Stevenson verified its validity5 and further generalized the method6. These two approaches are the most important lower bound methods based on the variance of the Hamiltonian and several theoretical studies16 and practical implementations19 have improved their performance. A significant step forward was the development of Lehmann’s optimal inclusion intervals, which provided an optimization of the basis set that maximizes Temple’s lower bound23. Another class of lower bound theories is based on the method of intermediate operators25. These approaches were first applied to He energy levels using a special choice by Bazley and Fox27, and motivated several further improvements for Coulombic and other potentials33. Bracketing functions have also been successfully applied to lower bound problems38. Further lower bound calculation strategies are also available for He atoms42. A comprehensive discussion of these methods can be found in Ref.43 and a historical review on lower bound theories can be found in Ref.44.According to the Ritz–MacDonald theorem, the eigenvalues of the Hamiltonian of a quantum mechanical system provide upper bounds for the true energy levels of the system46 have been developed based on the Lánczos construct49, which significantly improved Temple’s lower bound for energy levels, as tested on quartic oscillators. The Lánczos algorithm provides an orthogonalization of states on the Krylov space, and from the resulting tridiagonal matrix, the approximate eigenvalues and the corresponding variances can be readily determined for the original Hamiltonian. In these novel lower bound methods, the energy factor in Temple’s original lower bound formula is replaced by a “residual energy”, for which, based on the Lánczos construct, a well-defined and rigorous algorithm can be provided. The application of this residual energy enables the calculation of lower bounds for ground and low-lying excited states. As an improvement, Pollak and Martinazzo developed a self-consistent lower bound theory (SCLBT)51 and applied it for lattice Hamiltonians50 and quartic51 and double-well52 potentials. Using the concept of the residual energy, the SCLBT can systematically improve the quality of lower bounds based on the information from lower bounds for higher-lying eigenvalues: the higher the maximal considered level, the tighter the lower bounds to the levels below. In the present paper, the SCLBT is further improved by defining tighter bounds for certain parameters used in the theory. The performance of the improved SCLBT (iSCLBT) is tested and compared with that of the Lehmann lower bounds, using two lattice Hamiltonians.Recently, novel lower bound methodsiSCLBT method presented in this paper for the ground-state energy levels of the lattice Hamiltonians. Finally, a comparison of the iSCLBT and Lehmann methods is discussed based on lower bounds calculated up to the fourth excited state of the models considered. The paper ends with a summary and the main conclusions.The remainder of the paper is organized as follows. First, the theory of Weinstein, Temple, SCLBT, and Lehmann lower bound methods is discussed on an equal footing provided by the theoretical framework of the Lánczos construct. Then, the numerical implementation using two lattice Hamiltonians is presented. The lower bounds from the previous implementation of the SCLBT are compared with those from the L states. Let us define a projection operator onto this L-dimensional subspace asIn this section, lower bound theories are considered for the energy levels of quantum mechanical systems and for simplicity, real valued functions are assumed. Let us consider a Hermitian Hamiltonian operator jth eigenfunction in the projected space with the exact kth eigenfunction as5:To discuss lower bound theories on an equal formal footing, let us introduce the square of the overlap of the To obtain the Temple lower bound formula, let us first define a “residual energy” d in Eq. , the resinto Eq. , giving251jth eigenvalue based on the L-dimensional space. This lower bound expression is a significant improvement over Temple’s lower bound. The lower bound provided by these expressions can be further improved by defining tighter bounds to the residual energy By substituting d in Eq. —into thed in Eq.  and usinmentclasspt{minimajth state can be rewritten asiSCLBT, estimates for iSCLBT L values for which 6. According to the Ritz theorem, L. However, the highest possible value, iSCLBT for L values compared to Lehmann’s theory and provides improved lower bound estimates.In order to improve the SCLBT, a better estimate to the residual energy n in Eq.  is satisfies Eq. : this is51, the upper bound for iSCLBT method, Eqs. (For the SCLBTity (Eq. ) using tn in Eq. . Rearran,L∗, Eq.  can be mquation   can be rod, Eqs.  and 38)51, the uj in Eq.  the calcj in Eq. , until tL dimensional space that maximizes the Temple lower bound. The Lehmann eigenvalue equation can be written as22L space. The parameter The Temple lower bound in Eq.  was obtaFor an insightful analysis of the Lehmann equation, let us define a non-normalized staten in Eq.  on the Lying Eq.  by \docuL-space, and the lower bound eigenvalues can be obtained by the diagonalization of Eq.  constant weighting the “exchange term” 55 and available in Ref. N of sites with periodic boundary conditions. The diagonal \documentod, Eqs.  were sol in Eqs.  was detes in Eq.  were estiSCLBT results at iSCLBT at iSCLBT simultaneously calculates lower bounds for the two lowest lying energy levels. It can be seen that the iSCLBT results are a significant improvement over the lower bounds from the SCLBT method50 even at iSCLBT results improve further and converge: the lower bound gaps for 50 are significantly better in the Heisenberg case than those of the Hubbard model case, the iSCLBT results are more accurate for both models, because the variances for the excited states are comparable.A comparison of the SCLBT and entclass1pt{minimaentclass1pt{minimaiSCLBT and the Lehmann theory were compared as well, using both lattice systems. Both methods can provide lower bounds not only for the ground state, but also for higher excited states. However, the number of treatable excited states is limited by the Lánczos construct, which provides reliable eigenvalues for the lowest lying energy levels only. While the iSCLBT does not depend on the Weinstein lower bound explicitly, the Lehmann theory relies on it. Theoretically, the Weinstein lower bound is a monotonically increasing function of L where it satisfies the condition of validity in Eq. , the quality of the roots provided by the Lehmann theory for the first excited state at L. From iSCLBT results are significantly better than those of the Lehmann calculations, while at iSCLBT and 2.113 for the Lehmann method.iSCLBT results are slightly better than the Lehmann bounds. At iSCLBT and 3.071 for the Lehmann calculations. Figure iSCLBT can still provide good quality lower bounds regardless of the quality of the Weinstein lower bound, as shown in the left panel of Fig. iSCLBT result is similar to that of the previous cases, with a final gap ratio of 3.545. As can be seen in Fig. iSCLBT and 6.024 for the Lehmann method.A comparison of the first excited state lower bound gap ratios for the Hubbard model obtained from the mentclass2pt{minimiSCLBT can provide lower bounds for all energy levels at a given L. Generally, the Lehmann method can only provide iSCLBT and Lehmann methods is similar in all cases. The Ritz gaps together with the final iSCLBT and Lehmann gap ratios for the Hubbard model are summarized in Table As can be seen in Figs. mentclass2pt{minimiSCLBT is noticeably better at iSCLBT and 1.250 for the Lehmann method. For the first excited state lower bound shown in Fig. iSCLBT at L values. This does not indicate that the lower bounds are better in this region; it rather shows that the Ritz gaps are still wide at low L values. From iSCLBT is better than the Lehmann result, while at higher iSCLBT and 1.716 for the Lehmann method. The second and third excited state lower bounds can be seen in Figs. iSCLBT and Lehmann results for the second excited state are similar to those of the Hubbard model. The converged lower bound gap ratios are 2.363 and 1.991 for the iSCLBT and Lehmann methods, respectively. In the Heisenberg model, the Lehmann method can provide reasonable lower bounds for the third excited state; however, these values are monotonically increasing with dimensionality L. As there is only one remaining value at The same calculations were performed for the Heisenberg model; the results can be found in the Supplementary Information. As shown in Fig. entclass1pt{minimaiSCLBT. The lower bounds for the Ritz eigenvalues obtained from the iSCLBT were compared with those from the SCLBT and from the Lehmann methods. The framework provided by the Lánczos construct enabled the comparison of these methods on an equal formal footing. For the comparison, two model systems, a 5 iSCLBT was improved over its previous implementation. First, the iSCLBT was compared with the SCLBT method for the ground-state energy. The improved theory provided significantly better lower bounds even at iSCLBT and the Lehmann method up to the fourth excited state. The effect of the highest considered level The definition of tighter bounds for the residual energy and for the diagonal elements of the overlap further improved the SCLBT.iSCLBT and the Lehmann method can provide lower bounds that are significantly better than the Weinstein or Temple bounds.Both the The quality of lower bounds improves with the increase in iSCLBT and Lehmann methods exhibit similar performance and convergence behavior. This is not unexpected, considering that in Ref. iSCLBT are formally different.The iSCLBT, the Lehmann method requires a larger basis set and, as the Lehmann pole is estimated from the Weinstein lower bound, the quality of the Lehmann bounds is strongly affected by the quality of the Weinstein lower bound.Compared to the iSCLBT is simpler than that of the Lehmann method and does not require the Weinstein lower bounds.The numerical implementation of Both the iSCLBT and Lehmann methods are suitable to provide high quality lower bounds for the low-lying energy levels for the studied lattice systems. The number of treatable Ritz eigenvalues is determined by the size of the Lánczos basis set.In this paper, a further improvement of the existing SCLBT method was presented, termed Supplementary Information.


Total knee arthroplasty is an effective treatment for end‐stage knee osteoarthritis. The tibial platform osteotomy must take full account of the coronal plane, the sagittal plane, and the rotational alignment of the tibial prosthesis. During surgery, individual differences in the coronal alignment of the tibia need to be taken into account as poor alignment after surgery can lead to rapid wear of the tibial platform, reducing the longevity of the prosthesis and adversely affecting quality of life. Intraoperative tibial osteotomies are often performed using extramedullary alignment. When an extramedullary alignment approach is used, the proximal tibial osteotomy guide is usually placed in the medial third of the tibial tuberosity. There is no consensus on the most reliable anatomical landmarks or axes for achieving distal tibial coronary alignment. Anatomical points or reference axes that are highly reproducible and precise need to be identified. From available data it appears that most surgeons use the extensor hallucis longus tendon, the second metatarsal, and the anterior tibial cortex to determine the distal localization point. However, its accuracy has not been confirmed in clinical and radiographic data, and the alignment concept and preoperative planning for total knee arthroplasty has paid more attention to rotational alignment, but there are few studies on the coronal alignment of the tibia. This article reviews the recent use of the distal tibial coronal osteotomy reference point in total knee arthroplasty. However, due to there being only a small number of studies available, the evidence collected is insufficient to prove that a certain reference axis has obvious advantages and a combination of different reference points is needed to achieve the ideal lower extremity force line angle. The coronal tibial component angle (CTA) is the medial angle between the mechanical axis of the tibia and the horizontal axis of the tibial tray. Total knee arthroplasty (TKA) can restore knee function, reduce pain, and improve quality of life for patients with end‐stage osteoarthritis of the knee. About 95% of implants survive for more than 15 yearsThere is general agreement among surgeons that the prosthesis should be aligned parallel to the tibial mechanical axis with deviation of no more than ±3°, to provide a consistent load distribution at the prosthesis–bone interface and to prolong survival of the prosthesisFor extramedullary alignment of the tibia, the current method uses an axis connecting the proximal tibial center to the center of the ankle joint. Therefore, correctly determining the center of the ankle joint is crucial. To align the extramedullary guides to the mechanical axis of the tibia, a number of bony and soft tissue marker points have been proposed as distal tibial reference points; these include the second metatarsalThe current clinical use of coronal tibial osteotomies is mostly based on the surgeon's surgical experience and intraoperative empirical judgment, there is no uniform standard for tibial osteotomies, and individual differences in tibial morphology can also have an impact on tibial osteotomies.Computer navigation techniques can help achieve a more precise line of force in the lower limbs. These techniques allow the surgeon to use sensors to determine the position of the mechanical axes during surgery and achieve precise mechanical positioning. However, high cost, long operating time, and the steep learning curve are reasons preventing wider adoption of computer navigationThe purpose of this study was to summarize the advantages and disadvantages of the commonly used coronal anatomical reference points or reference axes in clinical practice, and compare their accuracy, so as to facilitate surgeons to improve the accuracy of postoperative force lines.A literature search was conducted using Medline (PubMed) and SCI for articles published between June 2003 and June 2020 on the coronal alignment of tibial components in TKA. The searches were for descriptors “tibia OR tibial,” “coronal OR frontal,” “alignment OR align OR positioning,” “landmark OR landmarks,” and “total knee arthroplasty OR total knee replacement” in combination with using the Boolean connector “AND.” The inclusion criteria for the literature were as follows: (i) basic and clinical research related to the above keywords; (ii) literature types are monographs, papers, guidelines, or reviews. The exclusion criteria for the literature were as follows: (i) related to computer navigation and patient‐specific guides; (ii) about intramedullary alignment and tibial plateau rotation alignment; (iii) fail to quantify accuracy of one or more landmarks for tibial component alignment; (iv) articles of low quality and low level of evidence. Finally, 13 eligible articles were included , the tibialis anterior tendon (TAT), the extensor digitorum longus tendon (EDLT) Fig. , the linThe tibial mechanical axis is defined as the line connecting the midpoint of the proximal tibia and the midpoint of the talus. Therefore, it is important to accurately identify the center of the talus when applying an extramedullary guide during a TKA tibial osteotomy. Tendons located just beneath the skin are easily identified during the course of the surgery, without need for assistive devices. Therefore, the TAT, the EHLT, the EDLT, and the AT at the ankle could serve as distal reference points Fig. . The TATet al. found the mean distance from the midpoint of the distal tibia to the TAT to be 1.89 mm (1.45–2.33 mm); in 84.4% of ankles, the distance was ≤2 mmet al. examined the horizontal distance of the EHLT and TAT from the midpoint of the talus and also analyzed the anatomical relationship of the two tendons to the internal and external ankle in the neutral position and the pronation and supination positions. The effect of pronation on the distance from the EHLT and TAT to the internal and external ankle was found to be smaller . The deviation of the EHLT was greater than that of the TAT during supination . Using MRI images of healthy volunteers, Schneider et al. showed that extreme supination resulted in a deviation of 7–10 mm both TAT and EHLT. The effect of rotation on EHTL was greater in living volunteers than in the cadaveric samples, probably due to the removal of the support band in the latterIn a study based on an analysis of ankle MRIs, Rajadhyaksha et al. compared the distance of the midpoints of TAT, EHLT, DPA, and AT from the center of the talus at a level 2 mm from the distal tibial articular surface and 2 mm from the talar articular surface. The AT was anatomically closest to the center of the ankle joint. Statistical analysis showed that the AT,Using ankle joint MRI images of 53 volunteers, Tiftikçi the DPA, and the EHLT were significantly closer to the center of the ankle jointThe EHLT begins at the level of the lower third of the tibia and ends at the distal phalanx of the toe. The EHLT, EDLT, and TAT dorsiflex the ankle. The TAT is most obvious in plantar flexion. To avoid confusion between the EHLT and EDLT during the surgery, the EHLT can be identified preoperatively by active and passive dorsiflexion and plantar flexion of the ankle and marked before the patient is anesthetized. The TAT is the easiest to palpate; even in obese patients, the TAT can be palpated under anesthesia. However, it is located too medially. When the distal reference point of the tibial osteotomy is too medial, postoperative tibial prosthesis valgus deformity may result. The AT is located posterior to the ankle, and existing extramedullary alignment devices cannot be applied directly intraoperatively.Patients with end‐stage osteoarthritis of the knee may also have developmental joint deformities such as bunion deformity, subtrochanteric deformity, and forefoot and hindfoot deformities. Such deformities also affect the travel of the tendonset al. examined how TAT, EHLT, Akagi line angle, femoral tibial angle (FTA), hallux valgus angle (HVA), and knee valgus angle (KVA) on MRI images were related to the forefoot deformity index of 61 patients with knee osteoarthritis. EHLT, FTA, HVA, and KVA were not found to be related to the severity of deformity and tibial torsionHino There were no studies on the use of EHLT, TAT, AT, and EDLT as markers for determining postoperative lower extremity force lines, prosthesis alignment, or outcomes of distal tibial osteotomy. The use of tendons for these purposes needs further study.The manual for the use of TKA surgical instruments recommends that the distal end of the tibial cutting block's locating rod should point toward the second metatarsal when extramedullary alignment examination is performed Fig. et al. studied a sample of patients with osteoarthritis and rheumatoid arthritis.Tsukeoka Rotational errors occur mainly in the proximal tibia and increase with the distance between the bone and the locating baret al. used Doppler ultrasound to identify the exact location of the DPA. In patients with osteoarthritis and rheumatoid arthritis, the location of EHLT and DPA may be closer to the center of the ankle: the DPA may be 0.4 ± 3.4 mm more laterally located, the EHLT 0.7 ± 3.5 mm more medially located, the EDLT 5.7 ± 3.1 mm more laterally located, and the TAT 12.0 ± 3.3 mm more medially locatedThe DPA is a continuation of the anterior tibial artery in the front of the ankle. It is located deep to the EHLT and travels along its lateral aspect, adjacent to the approximate center of the ankle joint Fig. . The DPAet al., the distance between the soft tissue center and the true ankle center is only 4.5 mm at the mostThe tibial extramedullary guided bilateral ankle technique is used because the center of the medial and lateral ankle is close to the midpoint of the talus Fig. . The ankThe ankle clip of the extramedullary guide is fixed to the distal tibia, but it does not capture the most medial point of the medial ankle and the most lateral point of the lateral ankle, so it is unclear how much lateral error should be considered in determining the center of the ankle, the disadvantage of which can be difficult during surgery because the sterile sheet is wrapped to locate the medial 3–5 mm. Shigeki analyzed CT data, found that the mean offset distance with reference to the tibial AP axis was 1.8 ± 0.9 mm medial to the intermalleolar midpoint on the coronal plane. The intermalleolar midpoint in the coronal plane is a reliable landmark for the ankle center during TKAAnother article similarly confirmed that the tibial mechanical axis was approximately 2 mm medial to the center of the ankle on anteroposterior view. Tibial mechanical axis marked on the skin could be referred to osteotomize the tibial plateauThe tibia is trigonous in shape. The anterior edge of the tibia and the medial aspect are covered only by skin. It is therefore a convenient anatomical marker for extramedullary alignment. Unlike the bony and soft tissue markers around the ankle, the tibia is not affected by ankle deformity or the presence of excessive soft tissue in obese patients.et al., where they analyzed CT scan data using 3D imaging software and measured the angles made by 10 axes to the mechanical axis in the coronal and sagittal planesThe two main tibial reference lines are the line connecting the proximal third and the distal third of the tibial anterior marginThe median line connecting two points located in the middle of the tibia and 5 mm proximal to the ankle joint, and a line at 58% and 90% of the length of the tibia, are the most significant anatomical references parallel to the mechanical axis of the tibiaet al., the tibial crest cannot be considered as a primary anatomical landmark for the coronal alignment of the tibial component in Caucasian patients. In cases where it is difficult to improve the alignment of a total knee arthroplasty using multiple reference points, the axis of the proximal body projection of the tibial mechanical axis connected to a point 23% of the length of the tibia can be used as a referenceet al. demonstrated that the tibial crest in the coronal planes could be useful guidelines in performing TKA, even if the patient has tibial bowing or severe varus deformityIn a study by Gianluca Cinotti Proximal reference points  are too medial, and so tend to result in internal valgus deformity. Similarly, the distal reference points  are too lateral, and so lead to valgus deformity.Osteotomy performed using only marker points at the anterior tibial margin may be influenced by the physiologic torsion of the tibia and the thickness of the subcutaneous soft tissues, and so the extent of the tibial deformity should be preoperatively determined on tibial orthotopic radiographs.In a study of tibial torsion, the center of the ankle was found to be shifted 9–11 mm laterally relative to the proximal tibial axis. The anteroposterior axis of the ankle  was on average 19°–27° externally rotated compared to the proximal tibial anteroposterior axis Trigonometric analysis of the effect of different rotational alignments of the proximal and distal anteroposterior axes on the coronal plane of the tibial osteotomy showed that a 2 mm external displacement of the distal end resulted in 0.88° of internal rotation of the osteotomy, implying that a 10 mm external displacement would result in 4.4° of internal rotation of the tibial osteotomy. To compensate for the external rotation of the ankle and to avoid tibial osteotomy with internal rotation of the incision, the distal end of the extramedullary guide should be internally relocated; however, it is not clear how much internal displacement is requiredAssuming a tibial length of 160 mm (160 mm × tan (1°) = 5.25 mm), a difference of approximately 5 mm in the horizontal plane would result in only a 1° change in coronal plane alignment, and an error of more than 15 mm in the horizontal plane would result in a 3° deviation in alignment with the mechanical axis of the limb.Because the proximal tibial cut involves coronal and sagittal alignment, the internal rotation is further exacerbated if the extramedullary guide is laterally displaced distally, as the posterior inclination of the tibial osteotomy changes to a posterior medial tibial slope in this case.In TKA surgery, rotational mismatch between the proximal tibia and the ankle has significant impact on the accuracy of the proximal tibial cutThe coronal tibial component angle (CTA) is the medial angle between the mechanical axis of the tibia and the horizontal axis of the tibial tray on the AP radiographs. CTA on the medial side was measured using protractor. Angles of 90 ± 3° were accepted as the normal boundaries; less than 87° was considered varus, and more than 93° was considered valgus.The mechanical axis of the lower extremity is defined as the hip‐knee‐ankle angle (HKA), which is, the medial angle formed by the mechanical axis of the femur and the mechanical axis of the tibia. The mechanical axis of the femur was defined as the connection between the center of the femoral head and the midpoint of the femoral component. The mechanical axis of the tibia are defined as connection between the center of the tibia prosthesis and the center of the ankle joint.et al., the mean value of the medial angle of the tibial prosthesis was 89.4% , there was no significant difference in mean CTA between groups (P = 0.124); however, the authors reported that 69.8% of the cases in the two groups had <2° external and internal valgus, respectively, The number of tibial components with alpha angles in the normal range was significantly higher in the ERT group (P = 0.017)In a study by Bilgen P = 0.096)A study reviewing the alignment of postoperative tibial prostheses in a total of 212 knee replacements in 188 patients with the TAT as the reference point, found that 191 (90.1%) tibial prostheses were well‐aligned. There were five postoperative valgus osteotomies and 16 postoperative internal valgus osteotomies. Of the 196 knees with internal rotation deformity preoperatively, 17 (8.7%) had poor force lines of the prosthesis postoperatively; the caliper test was not statistically correlated after correction . In terms of coronal alignment, the target angle (with ±3° deviation) was achieved in 84% and 88% of cases, respectively. The postoperative tibial prosthesis angle and the lower extremity force line angle were not significantly different in the two methods (P = 0.97)In a clinical trial, Sasanuma et al. compared the line joining the proximal third and distal third of the anterior tibial margin (method A) as a reference line for the coronal plane mechanical axis with the traditional method (method B). The mean coronal tibial prosthesis angle (CTA) was 89.6° ± 1.8° (85.7°–93.8°) for method A vs 89.3° ± 1.6° (84.5°–92.7°) for method B. The mean HKA angle was 179.3° ± 3.6° (168.4°–187.6°) for method A vs 178.5° ± 3.1° (171.8°–185.5°) for method B. The mechanical axis angles did not differ between the two groups, but the proportion of cases achieving ideal alignment by the two methods was significantly different: 87.2% (102/117) with method B vs 95.9% (71/74) with method A; there was a significant difference in ideal CTA between the two groupsIn a similar clinical trial, Nishikawa et al., the author used 3D imaging software to analyze the angle of the tibial crest deviating from the mechanical axis on the coronal plane, and found that the average angle of the tibial crest‐tibial mechanical axis was 0.7° ± 0.9°(1.7° – 3.8° varus) on the coronal plane, and 98%  of the patients had the tibial crest 3° away from the mechanical axisIn a study by Tadashi et al. proved that the proximal one‐third and the distal one‐third of the anterior tibial border were reliable landmarks for the tibial extramedullary alignment guide in TKAet al. proved that the proximal and distal one‐third of the tibia provides good coronal alignment and better accuracy for the posterior tibial slope compared with the conventional methodet al., the EHL tendon is a reliable anatomical landmark to use with extramedullary guide systems and improves coronal alignmentet al. proved that using EHLT results in more accurate lower limb alignment than previously reported reference pointsA study by Nishikawa During TKA, poor alignment leads to uneven load distribution on the tibial platform, increased prosthetic subsidence, and accelerated polyethylene wearet al. found that tibiofemoral prosthesis implantation with internal rotation and femoral prosthesis implantation with >8° external rotation were associated with increased rates of revision surgery. If the femoral implant is poorly aligned, the revision rate is high even when the tibial implant alignment is corrected to achieve neutral alignment of the entire limbRitter During placement of the prosthesis, the surgeon needs to consider alignment in the coronal, sagittal, and axial planes. Coronal alignment is mainly achieved by adjusting the ankle clamp and using the TAT or EHLT to represent the center of the ankle. The method relies on the surgeon's subjective judgment of whether or not the extramedullary positioning bar is aligned coronally with the mechanical axis. Coronal alignment requires the transverse axis of the prosthesis to be perpendicular to the mechanical axes of the femur and tibia. This places the prosthesis in the centre of the lower limb force line.In the sagittal plane, the posterior tibial inclination angle is roughly determined by the surgeon manually adjusting the distance between the extramedullary positioning rod and tibial anterior margin. Reference to the long axis of the fibula and the anterior tibial cortex has also been proposedThe inter‐epicondylar axis of the femur is the mechanical axis of knee flexion and extension. It can be used to determine the internal and external rotation of the femoral prosthesis, and with regard to axial rotational alignment, it is now generally accepted that the line connecting the medial edge of the tibial tuberosity to the midpoint of the posterior cruciate ligament and the curved bend of the anterior edge of the tibia are reliable reference lines for rotational alignmentDespite improvements in surgical techniques and surgical instruments, target alignment is achieved in only 70%–80% of patients with the use of extramedullary guidesComputer navigation techniques are increasingly being used to improve alignment during TKA. Some reports comparing accuracy of postoperative prosthesis alignment with conventional TKA and computer‐assisted surgery have shown smaller deviations from coronal tibiofemoral alignment and fewer outliers (more than ±3° deviation) with the latterBecause the tibia exhibits physiologic torsion—with the degree of torsion varying from individual to individual—many patients with knee deformities also have foot deformities. In patients with ankle deformity, the mechanical axis may not be correctly identified when the second metatarsal or a point 3–5 mm medial to the center of the ankle are used as the reference. Precise positioning of points at the anterior edge of the tibia (one‐third and one‐quarter of the full length of the tibia) is also difficult during surgery.et al. found that tendon position is not related to bunion angle, forefoot deformity index, and knee valgus angle,In patients with end‐stage osteoarthritis in TKA, valgus knee deformity is related to hindfoot inversion position and valgus knee deformity to hindfoot valgus position. Most of the compensation for deformities from the hindfoot to the knee occur in the subtalar joint. Therefore, when TKA is considered in patients with knee arthritis, the surgeon should carefully look for the presence of any hindfoot deformityPrevious studies have tended to focus on the ankle joint and did not consider the knee and ankle joints as a whole. The actual position of the ankle joint during the TKA procedure was not simulated. The actual relationship between the center of the ankle and the tendons during surgery may be inaccurate. The use of the tendon as a distal reference point is also susceptible to differences in patient body mass index and ankle fat thickness.Osteotomy deviations occur due to anatomical differences in the tibia or the soft tissues, or due to inconsistencies in the ankle position, making it impossible to have a standardized osteotomy reference point. Therefore, it is important to find a reliable reference axis for the coronal alignment of the tibial osteotomy applied during surgery. It is also advisable to use multiple anatomical landmarks as references to reduce errors in the alignment of components.Osteoarthritis of the knee can cause changes in the alignment of the lower extremity in all planes. Severe knee deformity results in limited flexion and extension and may lead to other lower extremity deformities. Therefore, the key to successful knee replacement is to ensure proper osteotomy volume and soft tissue balance. The amount of osteotomy directly affects the position of the joint line and the balance of flexion and extension. Excessive osteotomy can result in unstable flexion, while too little osteotomy can result in a limitation of extension. Restoration of the anatomical alignment in various planes, accurate osteotomy volumes in the medial and lateral tibial plateau, and accurate reconstruction of the lower extremity force line are key to maintaining the longevity of the prosthesis. As surgical techniques improve, surgeons seek better post‐replacement knee function. However, studies published to date have not been able to conclusively prove the superiority of any specific alignment method; therefore, multiple alignment methods continue to be used in clinical practice. More clinical trials and imaging data are needed to identify the best alignment method for accurate coronal osteotomy. As the theories of force alignment continue to advance, different distal reference points can be selected based on preoperative measurements of the internal and external valgus angles, and can be verified intraoperatively by force line rods. In our opinion, use of combinations of different reference points on the lower extremity force line will help achieve the ideal lower extremity force line angle.


Ever since the potential of algae in biotechnology was recognized, models describing the growth of algae inside photobioreactors have been proposed. These models are the basis for the optimization of process conditions and reactor designs. Over the last few decades, models became more and more elaborate with the increase of computational capacity. Thus far, these models have been based on light attenuation due to the absorption and scattering effects of the biomass. This manuscript presents a new way of predicting the apparent growth inside photobioreactors using simple models for enzymatic kinetics to describe the reaction between photons and the photosynthetic unit. The proposed model utilizes an inhibition kinetic formula based on the surrounding biomass to describe the average growth rate of a culture, which is determined by the local light intensities inside the reactor. The result is a mixed-inhibition scheme with multiple inhibition sites. The parameters of the new kinetic equation are replaced by empirical regression functions to correlate their dependency on incident light intensity and reactor size. The calibrations of the parameters and the regression functions are based on the numerical solutions of the growth rate computed with a classical Type II model. As a final verification, we apply the new equation in predicting the growth behavior of three phototrophic organisms in reactors of three different sizes. The boom of algal biotechnology in the last decades arises — to a certain extent — from the political pressure driving the search for a green alternative to fossil fuels. In the context of the energy revolution, bioenergy increased the public awareness of the possibilities to utilize microalgae.2, light, and some nutrients. Nutrient supply and mass transfer from the gaseous phase are well-known from traditional bioprocess engineering. However, light supply still poses challenges to the scientific community; it is the missing link to scale-up and efficient process optimization. To overcome this problem, on the one hand, new inventions for efficient light supply have been proposed [Besides bioenergy, microalgae biomass contains various products ranging from little-processed animal feed to dietary supplements and chemical building blocks to active pharmaceutical substances . The genEarly models tried to connect light attenuation to the growth behavior within a reactor system. They were based on calculating the mean light intensity in photobioreactors, correlating it to Monod kinetics , 4. Due Different from the light attenuation dependent kinetic models approaches based on conservation of mass and energy assume a homogeneous thermodynamical system, e.g., a photobioreactor. Therefore, the biomass yield of a system can be correlated to the enthalpy and the Gibbs energy using the stoichiometry of biomass growth and the energy of the absorbed photons . In thisIn perfectly mixed cultures, the growth can be determined by solving the mass balance equation:mentclass2pt{minimAssuming that the photosynthetic response of algae is much faster than the mixing in the reactor, the apparent growth rate is equal to the growth rate averaged over the cross-sectional area of the vessel,Influences of the top and the bottom walls of the cylindrical vessel on the biomass growth rate are neglected. By assuming that light attenuation inside the vessel is mainly caused by cell absorption and neglecting scattering effects, one can compute the light distribution Here, mentclass2pt{minim−3, −1, −2 s−1, −2 s−1, and −3, since it leads to relatively large deviations among the results if an adequate number of points is not chosen for the integration.Equations 0 points . To detementclass2pt{minimThe values of the growth rate decrease as the number of points increases and tend to an asymptotic value, see Table ®. Growth data similar to those of Arthrospira platensis were used with −1, −2 s−1, −2 s−1, and The growth of the culture was calculated by solving numerically Eq. 2 combined with Eq. 14 utilizing a first-order forward Euler method. To justify the choice of the time integration method, the results obtained with the classical first-order forward Euler method were compared with those obtained with the Runge–Kutta method of the subroutine ODE45 implemented in MATLAB 2017−3, where Figure In the previous two sections, we outlined the theoretical and computational strategies deployed when calculating the growth rate in photobioreactors via a standard light attenuation model. We use these calculations later in this paper to calibrate the new model we are proposing. However, the derivation of growth kinetics achieved by our model is independent of those described in “The basic principle of our approach for estimating the apparent growth inside photobioreactors consists of simple enzyme kinetic models. Even though the surrounding biomass mentclass2pt{minimThe left side of the scheme represents the self-shading effect of the biomass which directly interacts with the reaction between t}KD1=1K . However biomass :7\documeThe steady-state conditions at equilibrium can be inserted into the following function of the general reaction kinetics for mixed inhibitions:Equation 12 is a mixed-inhibition kinetic formula, where the inhibitor has cooperative properties, similar to the Langmuir-Hill equation. The main difference to a Langmuir-Hill equation is the use of two different “Hill-coefficients” , as well as the Cyanobacterium Arthrospira platensis (NIES-39), were obtained from culture collections.Three different phototrophic organisms were used to verify the models presented in this paper. The algal strain For cultivation, three cylindrical photobioreactors with different radii entclass1pt{minima2 gas was added to the aeration (3%) whereas a total gas flow rate of 0.5 vvm was used and the gas was filtered before entering the reactor using a 0.22-µm air filter from Sartorius. For the determination of the growth, the biomass concentration was measured indirectly via extinction at 750 nm and directly via lyophilized dry weight.CO, The derived Eq. −2 s−1 were used in combination with different reactor sizes, ranging from a radius of −1.Before performing the calibration, we investigated how different values of incident light intensities The influence of the variation of those parameters on the growth rate is shown in Fig. m and n determine the steepness of the function, see Fig. The parameters of Eq. 12 have two different properties. The values Independent of the parameter settings, the correlation between the numerical data and the fitting function has at least The resulting values show that the exponents By substituting Eq. 15 and the quota mentclasspt{minimaThe values of The parameters The fit of Eq. 18 to the values of The calibration reduces the number of required inputs for the model equation to four. Two are process-related (mentclasspt{minima−3 for the light regime with the highest sensitivity.For estimating the sensitivity of Eq. Variations of the parameter nt}S Eq.  are simiThe parameters of Eqs. −3) is 5.4%.Randomizing the input parameters in the intervals reported in Table Figure The final step in the verification of the new approach is a direct comparison of the model with experimental data using three representatives of photosynthetic organisms from different taxonomic groups. In batch cultivation systems, both biomass and biomass-dependent inhibiting effects are continuously increasing with culture duration. According to our model, this increase in biomass concentration is the only reason for the decrease of culture growth, if other culture parameters are kept constant and no limitation of nutrients occurs. The cultivation took place in three cylindrical reactors with different diameters. The light came from four directions, and both the incident light intensity and the temperature were kept constant during the cultivation. Each batch cultivation was carried out in triplicates and is represented as an average value. A continuous light has been applied during the cultivation time to test the model on the proposed influence of the inhibiting effect of the biomass.The simulations of the growth rate are always performed with the constant parameters listed in Table entclass1pt{minimaHowever, most of the time this resulted in an over-prediction of the biomass concentration in the advanced stage of the cultivation (light limitation due to high biomass concentrations). Hence, after obtaining suitable values of entclass1pt{minimaC. asymmetrica is the fastest of the tested algae, with an apparent growth rate of 2.4 per day . The best match between simulation and experiments was achieved with −1 and −2 s−1. At the beginning of the cultivation, the simulation and the experimental outcome were in good accordance. In the small  [The new model shows an alternative way of approaching the calculation of biomass growth inside photobioreactors, and it can be improved similarly as light attenuation models evolved from Type I to Type III. The parameters of Eq. 14 were calibrated to a growth model, which only considers the adsorption of light Eq. . However kg m−3) . For bioAs far as other limiting factors concern, the new model can also be extended and modified, as it has been done with other models. In so-called uncoupled models, the effect of temperature is seen as independent of light . The infAlthough most culture processes and reactor systems do not operate in photo-inhibited conditions, the effect of photoinhibition can also be included in the new model to widen its applicability to situations where due to high light intensities photoinhibition cannot be avoided. An example is represented by outdoor cultivation systems. However, the present model is based on a kinetic function that already considers the inhibition of the surrounding biomass. Therefore, the presence of an extra term that considers photoinhibition would have added inhibition effect, which would have made this new approach much harder to understand.Moreover, we would like to stress again the difference between the reality and the proposed model. In a biological sense, the biomass growth of phototrophic algae depends on the local light intensity inside the reactor, which is dependent on the surrounding biomass due to light absorption and scattering processes. However, the resulting inhibition kinetic formula, that is, Eq. 14 permits the computation of the growth rate of the culture without the need of calculating the local light intensity. The new model necessitates only the incident light intensity −2 s−1. The final verification of the new approach comprises the growth of three different organisms in three reactors of different diameters, resulting in We proposed a new logistic equation as an alternative model for the calculation of the apparent growth rate in cylindrical photobioreactors. Different from other models, the new approach uses a biomass inhibition concept (mixed inhibition) instead of calculating the light attenuation inside the reactor. Solving for the inhibition kinetics results in an asymmetric sigmoid relation between biomass concentration and apparent growth rate. We were able to express the parameters of the new equation as a function of reactor radius and incident light intensity. The resulting formula fits the numerical data with


Recent advances in 3D imaging technologies provide novel insights to researchers and reveal finer and more detail of examined specimen, especially in the biomedical domain, but also impose huge challenges regarding scalability for automated analysis algorithms due to rapidly increasing dataset sizes. In particular, existing research towards automated vessel network analysis does not always consider memory requirements of proposed algorithms and often generates a large number of spurious branches for structures consisting of many voxels. Additionally, very often these algorithms have further restrictions such as the limitation to tree topologies or relying on the properties of specific image modalities.We propose a scalable iterative pipeline  that extracts an annotated abstract graph representation from the foreground segmentation of vessel networks of arbitrary topology and vessel shape. The novel iterative refinement process is controlled by a single, dimensionless, a-priori determinable parameter.We are able to, for the first time, analyze the topology of volumes of roughly 1 TB on commodity hardware, using the proposed pipeline. We demonstrate improved robustness in terms of surface noise, vessel shape deviation and anisotropic resolution compared to the state of the art. An implementation of the presented pipeline is publicly available in version 5.1 of the volume rendering and processing engine Voreen. As one of the important problems in biomedical imaging, segmentation provides the foundation for tasks like quantification and diagnostics . In partscalable both in terms of memory and runtime, and it should be invariant with regard to increases in image resolution. For application in a wide range of biomedical domains, a pipeline should further not rely on specific imaging conditions, dataset-dependent parameters, a cylindrical vessel shape, isotropic image resolution or a specific network topology.In recent years, improvements in imaging technology and procedures are providing images of increasing resolution (in terms of the number of voxels per unit volume) and thus offer new chances for finer grained analysis, but also pose new challenges for image analysis algorithms. The analysis of fine structures down to the level of capillary networks promises great insights. At the same time it is desirable to analyze large networks at once in order to obtain as much topological information as possible and avoid artifacts at the boundary of the volume. Modern microscopy hardware generates a single volumetric image file of hundreds of GBs or even TBs (the same order of magnitude as hard drive capacity) per sample which existing vessel analysis approaches are not able to process. The problem of rapid advances in technology and increasing data sizes is also present in other domains such as cell type identification , but is main contribution, a novel iterative refinement approach and careful design of all pipeline stages.In this paper we present a pipeline designed to fulfill these requirements while extracting the topology, centerlines and edge associated features from binary volumetric images. This is achieved by our Generally, analyzing ultra large datasets by scalable algorithms is still a huge challenge. For instance, even in the very recent image segmentation tool Biomedisa   or explicitly mention that large datasets pose a problem for their method. In this work we want to fill this gap by presenting a widely applicable pipeline that is designed to handle very large input volumes and fulfil requirements that are outlined below.For the discussion in the remainder of this paper, in this section we will structure and motivate the requirements that a vessel network analysis pipeline should fulfill in order to be applicable to a wide range of datasets and circumstances encountered in practice.runtime (P1). As all voxels of a volume can be expected to be taken into account by the method, the runtime can be expected to be at least linear in the number of voxels n. In order to be applicable to increasingly large datasets, a runtime of The method should be scalable with regard to memory requirements (P2). We can expect the input datasets to fit onto (commodity) non-volatile storage (disk), but not necessarily in main memory. While the price per capacity of both disks and main memory fall exponentially, the rate of  decrease is higher for disks than for memory [roughly grow in a relationship of m to m which also corresponds to the common practice of loading a 2-dimensional slice of a large 3-dimensional volume. Therefore the required main memory size of the method should not exceed The method should be scalable with regard to r memory , 37. Coninvariance with regard to image resolution (P3). For a fixed size specimen, an increase in image resolution can often result in an increase in surface noise-related artifacts. This is not handled well by simple topological thinning-based methods  which iFrom the desire to apply the method in practice we derive a set of further requirements that are not directly related to scalability:unbiased (S1), i.e., not dependent on a set of parameters that are required to be selected carefully and correctly depending on the input image.The method should be robust in terms of deviations from the cylindrical shape (S2) of the analyzed network structure. While blood vessels are cylindrical, lymphatic vasculature as an example is often highly irregular . Volumetric imaging techniques often create datasets with voxel spacing that differs between coordinate axes. As an example, in light sheet microscopy datasets are constructed from a series of (2D) slices for which the spacing is independent from the x-y-resolution. Methods operating on the voxel grid of a volume have to consider this.The method should handle volumes of arbitrary topology (S4). Existing methods often assume a tree structure and either fail to operate on other topologies or drop information [cerebral arterial circle) the assumption of a tree topology is invalid.The method should be able to fully analyze networks of ormation . Howeverindependently of the imaging conditions (S5). Concrete imaging conditions  vary between domains. In order to be widely applicable, the method should not be dependent on concrete models of imaging conditions. Our method achieves this by operating solely on binary input data and leveraging the broad range of other research on vessel image segmentation [The method should be able to analyze images entation , 12.Skeletonization stage the foreground of the binary input volume is reduced to a voxel skeleton using a topological thinning-based algorithm, similar to [bulge size, which defines how far a  branch has to protrude from its parent vessel relative to its and the parent vessel’s radius in order for it not to be removed. Since pruning invalidates the Voxel-Branch Assignment, the first three stages of the pipeline have to be reevaluated. This Extraction-Refinement cycle is repeated until a fixed point is reached. We want to stress that the novel iterative refinement approach is essential to obtain meaningful results from very large datasets, as demonstrated below.The proposed pipeline comprises four stages which are evaluated iteratively see Fig.  which armilar to . We use milar to  which camilar to  this Topmilar to  and addiLymphatic 1/2/3) which were obtained for and used as part of the study in [Below we evaluate and discuss the proposed pipeline in terms of the primary and secondary Requirements for Broad Biomedical Application. Where applicable, we compare the proposed pipeline with a version without iterative refinement, which corresponds to the state-of-the-art method by Drechsler and Laura  except fstudy in , which ustudy in . Additiostudy in . Finallystudy in  is used.study in .Table 1Dscale parameter for a given dataset without changing other characteristics. One possibility would be to use an already large dataset, scaling it down or clipping it to a smaller region. However, besides the problem of availablility of a very large volume (representing the results of future microscopes), this could cause artifacts and loss of detail due to downsampling, making the comparison of generated graphs very difficult. Instead we use a small  volume and artificially increase its size using two strategies  scale is achieved. In order to generate a mostly connected network, in each dimension the The and revealing previously undetected finer networks of larger complexity. When not specified otherwise, the dataset Lymphatic 1  size of previously visible vessels mirror and resample strategies in a log–log plot. As shown, the runtime is only slightly worse than linear, as should be expected in the light of the derived runtime complexity of Figure cf. Fig. .Fig. 4Avresample strategy, an iteration appears to take slightly more time than with the mirror strategy. This appears to be mostly due to more time spent in the Voxel-Branch-Assignment and Feature Extraction steps—likely because larger vessels result in more time spent in searching the k–d-trees in both steps.Figure mirror strategy can expected to be constant for scale 4 and higher , the number of iterations remains at 4 independent of the scale in this experiment. While we do not derive a maximum for the number of iterations for the resample strategy, we can observe the required iterations to increase only moderately with the number of voxels in the experiment. While we acknowledge that the  processing time of longer than 1 week for a 1 TB dataset is certainly far from interactive use, this does not pose a problem in biomedical research since the time required for the preparation, image acquisition and postprocessing of a sample is of the same magnitude, and the presented pipeline is to the best of our knowledge the first to make analysis of these samples possible at all. Furthermore, as our pipeline can be considered unbiased (see Influence of the bulge size Parameter), interactive parameter tuning is not required.The total runtime of the pipeline, however, is not only dependent on a single refinement iteration, but also on the number of iterations required to compute the final result. The number of iterations required to reach a fixed point for an increasing number of voxels is depicted in Fig. single refinement step is not sufficient to obtain meaningful results for very large datasets, as a large number of spurious branches remain. This underlines the necessity of the presented iterative refinement approach.Figure resident set size (RSS), which specifies the portion of the memory map of a process that is actually held in main memory. This excludes allocated memory that has been swapped out to disk, but includes (sections of) files that have been copied to memory.The detailed description of the pipeline in Methods shows that all steps of the pipeline asymptotically allocate mirror strategy reaches the 32GiB limit earlier than the resample strategy. This may be due to the fact that by repeating the graph m-times in each dimension, the total number of centerline points in the graph (and thus the required memory to store it) increases roughly by a factor of resample strategy, the number of centerline points is (roughly) multiplied by m.Figure As a qualitative evaluation, Fig. resample strategy) affects the extracted graph with refinement until fixed point. Since no ground truth graph is available for the dataset, we compare all available graphs with the graph extracted using the lowest and highest scale, respectively. As a quantitative measure we use the edge match ratio [Figure ch ratio  which caAs an additional way of evaluating the image resolution invariance (P3), we want to focus on one of the primary problem of increased image resolution: The increase in surface noise on the individual voxel level. For this we generate 10 datasets using Vascusynth , a tool For each of the volumes, four variants with different random noise seeds were evaluated. As shown in Fig. Note that we chose a bulge size of 1.5 for the iterative refinement, which is unusually low for blood vasculature. However, Vascusynth generates some branches that are very short compared to the radius parent vessel, and in some cases are entirely enclosed, which are only visible as very shallow bumps  in the generated volume . Nevertheless, in Fig. As explained in Refinement, the parameter of the proposed method can be selected a-priori based on the application and the desired resample strategy). In total, we evaluated 10 datasets with resolution scale differences of 1, 2, 4, 8, and 16. We compared the extracted topology using the graph matching method and edge match ratio similarity measure proposed in [The third secondary requirement states that a pipeline should be able to operate on volumes with anisotropic resolution (S3). In order to evaluate our proposed pipeline in terms of that property, we use a volume with known ground truth and isotropic voxel spacing, generated using Vascusynth . We thenposed in  and the posed in . In the Although the ground truth is not matched perfectly and some drift between scales is noticeable , in general no trend towards positive or negative impact on the evaluated score can be observed based on the level of anisotropy. Furthermore, it should be noted that the dynamic axis selection discussed in Skeletonization indeed improves the result for larger anisotropy, although for smaller levels there is next to no difference between both variants. We therefore conclude that the proposed pipeline is in fact able to process data of even highly anisotropic resolution.z-axis, processed using the proposed pipeline, compared to the (non-rotated) template graph. The minimum similarity of all steps denotes the robustness index. The similarity is defined as the product of the edge match ratio and one minus the average normalized difference between property values in the set of matched edges, and thus takes topological differences as well as changes in property value into account. For the synthetic datasets the ground truth graph provided by the tool was used as a template and thus are also measuring the accuracy of the method. For real world datasets, generation of an accurate annotated ground truth graph is virtually impossible [For the evaluation of robustness of the pipeline, especially with regard to deviations from the cylindrical (S2) shape of analyzed vasculature, we use the robustness measure GERoMe introduced in . Additiopossible  and thusThe aggregated results in Table Kidney 1 (176GiB), which shows the complete kidney, the arterial vessel tree was segmented semiautomatically using a hierarchical variant of the random walker method [Kidney 2 (28 GiB) is a subset of Kidney 1 where arterial vessels were segmented down to the capillary level. Both segmentation datasets were postprocessed to remove cavities and noise resulting in occasional small loops on the surface. Then the proposed pipeline was applied with a bulge size of 3.Finally, we demonstrate the applicability of the proposed method on large real datasets. For this purpose we use the scan of an antibody-stained mouse kidney sample obtained via light sheet microscopy as part of the study . In datar method  down to Kidney 1 the method finished after 7 iterations and roughly 1 day and 10 h (2059 min) resulting in a graph with 524 nodes and 566 edges. For processing dataset Kidney 2 roughly 7 h (413 min) of computation time was required. The resulting graph after reaching a fixed point in iteration 7 has contains 685 nodes and 673 edges.For Kidney 1: The colors of the cylinders representing vessel segments between two branching or end points fade from white (indicating a high radius) to black (low radius) for segments farther away from the root of the tree. Figure Kidney 2 that demonstrates the effectiveness of the dimensionless pruning parameter: There are no spurious branches generated, e.g., small bump on the surface of a large vessel, and small branches are preserved as well. Here the bulge size was chosen so that small bulges where the foreground segmentation leaked into the start of otherwise not labeled capillary vessels are not included in the graph.Figure In Results, we have shown by quantitative evaluation that the proposed pipeline fulfills all primary Requirements for Broad Biomedical Application. In particular, it is scalable in terms of run time and memory , which was proven by applying the pipeline to datasets up to a size of more than 800GB on a consumer-grade computer. It was shown that the novel iterative refinement approach is crucial for image resolution invariance (P3) while the state of the art (graph extraction without refinement ) heavilyThe single parameter of our method, the bulge size, is independent of dataset scale, and can be determined a-priori depending on the desired properties and shape of the resulting graph. We therefore consider our method unbiased (secondary requirement S1). Further, we have shown that the dynamic axis selection approach in the thinning step increases accuracy for highly anisotropic resolution datasets (S3) compared to the standard approach removing the same number of voxels in each dimension. The fourth secondary requirement (the ability to process volumes of arbitrary topology) directly follows from how the topology is extracted from the binary skeletonized volume (see Topology Extraction) and, in particular, is not restricted to a tree topology, in contrast to other methods . Furtherbackground without labeling, but removing objects smaller than a specified size. The threshold can typically be chosen very liberally, e.g., as a percentage of the volume diagonal. Removal of loops on the surface is more difficult, but in our experience a (binary) median filter performs well. The filter size should be chosen to not disturb the smallest vessels in the image, but still be able to cover surface loops in the image.It should be noted that the proposed pipeline—like all topological thinning-based methods—by definition suffers from distortions in the binary input image that causes changes to the topology, i.e., cavities in foreground objects and loops on the boundary. While these features are not expected in the actual vessel structure, imaging artifacts, noise or problems with the segmentation procedure can still produce such artifacts in practice. However, careful post-processing of the segmentation can mitigate these effects: Cavities in foreground objects can reliably be removed using a modified variant of  operatinAnother problem in practice is that segmentation procedures sometimes only preserve the vessel wall, but not the lumen, i.e., they create  hollow vessels. If this wall segmentation is of high quality and thus does not have holes, it can be converted to a full segmentation using the streaming variant of  as descrAnalyzing ultra large datasets by scalable algorithms is still a huge challenge. We have presented a pipeline for extracting the topology and various features of vessel networks from large three-dimensional images. We were able to show that our method fulfills previously identified Requirements for Broad Biomedical Application. Our main contribution, a novel iterative refinement approach and careful engineering of all pipeline stages, allowed us to demonstrate the scalability and thus applicability to very large datasets, e.g., generated by modern microscopes (primary requirements). At the same time, our pipeline can be applied to a wide range of problem domains due to its robustness, unbiased nature, and lack of assumptions about the topology and morphology of the analyzed vessel systems (secondary requirements).We will now apply the proposed pipeline and continue previous work  using prIn the future, we would like to explore how even more specific image features  can be iThe presented pipeline is part of version 5.1 of Voreen  a widelyIn the following, the four pipeline stages and the iterative approach are described in more detail, with special attention to the requirements for broad biomedical application. Figure Like other vessel network analysis pipelines , 23, we active surface  by retaining voxels from the previous active surface that are not considered during the current subiteration and adding all foreground voxels in the 26-Neighborhood of a voxel after its deletion. If a voxel was considered, but not deleted during the current subiteration, it will be removed from the active surface even although it is still part of the surface of the object. If one of its neighbors is deleted, it will be re-added to the active surface and reconsidered for deletion in the following iteration. This implementation has a runtime of For an efficient implementation, further considerations are necessary. We model the surface of the object explicitly as a sequence of voxel positions, which is initialized by scanning the volume before the first iteration step. In subsequent subiterations, we build the speed of voxel deletion is equalized on average for highly anisotropic volumes (S3). If the last subiterations of all 6 directions (which may have happened out of order) did not delete any voxels, the active surface is empty and the skeletonization is completed.To account for volumes with anisotropic resolution, we keep track of the real-world depth of voxel layers deleted for each direction and select the direction with the lowest total depth as the direction for the next subiteration. This way, the Previous work , 33 doessingle pass over the volume using a modified version of the streaming connected component finding algorithm by Isenburg and Shewchuk [Proto-Vesselgraph, are each stored in files on the disk and are dynamically mapped to memory. Furthermore, the algorithm of [In constrast, the topology extraction in the proposed pipeline extracts the complete centerline graph in a Shewchuk . Insteadighbors) ) separatighbors) , we do nrithm of   vessels at the boundary of the volume (and thus do not contain any centerline voxels), are not flooded with valid IDs in this step and ignored in subsequent steps.Previously identified cut-off regions are relabeled by flooding all voxels without edge IDs starting from adjacent voxels with valid edge IDs. For the simple case of a region cut-off from a single vessel section, this fills the region with the section’s edge ID. In a more complicated case of a cut-off region adjacent to multiple different vessel sections, the resulting labeling approximates the For efficiency, all identified disconnected regions are processed separately as cuboidal subvolumes and collected during a single pass over the ID-volume. Currently valid bounding boxes of cut-off regions are organized in interval trees. For the z-axis, a single tree references all bounding boxes. For each slice, the active regions can be queried to construct an interval tree for the y-axis. For each line in the slice, using the y-axis tree an interval tree for the x-axis can be constructed. For each voxel in the line, all active regions can be queried from the x-axis tree so that the current voxel value can be written to the corresponding subvolumes. After this collection step, each subvolume is flooded separately. Similar to the skeletonization step, we use the concepts of an active surface, slice-wise processing and memory mapping of files on disk to guarantee linear runtime (P1) and n. Consequently, the construction of the range trees and the querying for each voxel are within The results are written back to the ID volume in a process similar to the collection of the subvolumes. The number of disconnected regions is bounded by length (arc-length of the centerline), distance (euclidean distance of connected nodes) and straightness  is superior to curveness . For efficient lookups, the centerline points are organized in a  k–d tree for each edge. We equally distribute and store the volume occupied by the foreground voxels alongside the corresponding centerline point(s). Surface voxels are assigned to the closest centerline point, for which thus the minimum, maximum and average surface distance is computed.minimum, maximum, average, additional morphological features not discussed in [volume occupied by each vessel segment in the vessel graph. The average cross-section of a vessel is obtained by dividing the volume by the length [Values collected for individual centerline points can be aggregated to compute more per-edge features: For each of the surface distance-based values . Intuitively, the bulge size measures how far a bump, bulge or branch has to extend from a parent vessel in order to be considered a separate vessel. This size is expressed relative to the radius of its parent vessel and itself, making it scale-independent. More formally, the bulge size is an edge feature, that is computed during the Feature Extraction, and is only defined for bulging edges, i.e., edges that connect a leaf node (degree 1) and a branching point  or not (outer points) during the feature extraction. If a point has associated surface points, neither of which are adjacent to surface voxels of another edge, it is condidered an outer point. An inner point is characterized by either not having any associated surface voxels or by having a surface voxel that is a 26-neighbor of a point that belongs to another edge. The inner length of a node is the arc length defined by a run of centerline points starting from that node and ending at the last inner voxel. This corresponds to the length of the piece of centerline that lies within a branching area. The tip radius is defined as the minimum distance to the surface measured from the centerline point closest to the node. Without loss of generality, let For all centerline points of the edge, we decide whether they are within a branching region  cause the bulge size to be overrestimated.While the refined In order to ensure that centerlines and edge properties match the refined graph, we employ an iterative scheme where the previously extracted and refined graph is fed back into the first stage of the pipeline to improve the generated results.fixed foreground during the skeletonization of the input volume. This means that they are never considered for deletion. These voxels mark the beginning/end points of centerlines to be extracted during the skeletonization. The skeletonization algorithm is modified to not preserve voxels at the end of voxel lines . The rek iterations in many cases can be considered a very good approximation of the graph produced by fixed point iteration even for a very small k.The iteration can be stopped if it reaches a fixed point. This is the case if two consecutive iterations generate the same graph. As the number of edges never increases, the check can be reduced to a simple integer comparison. Optionally, an upper limit for the number of iterations can be specified. As shown in the evaluation (section Results), the number of deleted edges in each iteration rapidly decreases, so that the intermediate result after voreentool) which is suitable for offline, headless processing.The presented pipeline was implemented and all experiments were performed in the volume processing and visualization framework Voreen . The algAll volume formats supported by Voreen can be used as input for the graph extraction. Of those, currently volumes in dicom, ome.tiff, hdf5 and nifti format are supported for out-of-core processing.Feature Extraction section of the paper, or as a compressed json file which includes the full extracted information, including centerines. Release 5.2.1 of Voreen will additionally include an option to export the centerline geometry as a Wavefront OBJ file.Currently, the extracted topology can either be saved as a .csv file with one row per edge in the graph, which includes all numerical properties discussed in the


The authors would like to make the following correction to the published paper .There is a mistake in the last sentence of the second paragraph in Section 3.2 Antimicrobial Susceptibility Testing.The sentence should read “ARGs conferring resistance to macrolide (mreA) was found in F49”.This change will not alter the conclusions drawn in the paper.


Staircase cleaning is a crucial and time-consuming task for maintenance of multistory apartments and commercial buildings. There are many commercially available autonomous cleaning robots in the market for building maintenance, but few of them are designed for staircase cleaning. A key challenge for automating staircase cleaning robots involves the design of Environmental Perception Systems (EPS), which assist the robot in determining and navigating staircases. This system also recognizes obstacles and debris for safe navigation and efficient cleaning while climbing the staircase. This work proposes an operational framework leveraging the vision based EPS for the modular re-configurable maintenance robot, called sTetro. The proposed system uses an SSD MobileNet real-time object detection model to recognize staircases, obstacles and debris. Furthermore, the model filters out false detection of staircases by fusion of depth information through the use of a MobileNet and SVM. The system uses a contour detection algorithm to localize the first step of the staircase and depth clustering scheme for obstacle and debris localization. The framework has been deployed on the sTetro robot using the Jetson Nano hardware from NVIDIA and tested with multistory staircases. The experimental results show that the entire framework takes an average of 310 ms to run and achieves an accuracy of  With the advent of the 21st century, autonomous robots have become an integral part of intelligent machines for performing various tasks including inspection, manufacturing, warehouse handling, and maintenance . Among them, autonomous cleaning robots play a vital role in many day to day activities and also in industrial automation applications . VariousComputer vision based environmental perception has been widely deployed for various autonomous robotic platforms ,15,16,17Recently, many works have been reported with depth based fusion to overcome the false detection of objects and improve the model’s classification accuracy ,31,32,33This work proposes an operation framework empowered by the deep learning based environmental perception system for our modular re-configurable staircase cleaning robot sTetro ,13. The The flow of the paper is organized as follows: The architecture of the sTetro platform is detailed in sTetro is a modular re-configurable cleaning and maintenance robot developed by our ROAR lab. The general view and exploded view of sTetro are shown in sTetro hardware architecture is comprised of Jetson Nano hardware module with RGB-D camera, Arduino with DC motor control unit and sensor unit as shown in Here, the Jetson Nano module acts as a Central Control Unit (CCU) which controls the entire operation of the robot. The module is comprised of ARM A57 CPU and 128 Core maxwell GPU with 4GB memory for real-time deep learning inference and running on Ubuntu 18. The processing unit utilizes Rosserial communication interface to enable the communication between ROS nodes, and other modules include environmental perception unit and Arduino control blocks. This bridge is used to communicate the sensor data and trajectory information between Arduino Mega microcontroller and Jetson Nano unit.Arduino mega microcontroller performs the locomotion control and onboard sensor interface. It collects the data from the various sensor modules and sends it CCU. The control signal to the DC motor unit based on the received trajectory information is generated using inbuilt PWM module.Various sensors are integrated to perform the self-reconfiguration and locomotion tasks a. Bump sFour side sweepers are used to accomplish the cleaning task and are mounted on Modules 1 and 3 and propelled by 12v DC with a 100 RPM Metal gear motor. Side sweepers spin inwards to grab the dust particle and push towards the suction mouth; then the suction unit collects to the collection chamber.The previous work suggests the design principles behind the sTetro platform along with a basic sensor based climbing mechanism ,13. ThisEnvironmental Perception System plays a crucial role to enable autonomous control systems in the robot. This system is tasked to recognize the objects in the environment. The EPS system is comprised of the RGB-D vision sensor, SSD MobileNet based object detection module and depth based error correction unit. The EPS addresses the key challenges of staircase cleaning robot including detection of staircases, debris , static and dynamic obstacles present on the stairs  which may obstruct the path of the robot during climbing. In the current study, the vision sensor data was used to identify the human and plant pot on the staircase through a deep learning algorithm. According to obstacle position the robot navigation direction will be controlled by CCU.The SSD-MobileNet v2 object detection framework ,35 is usMobilenet v2 is a lightweight feature extractor, suitable for use in real-time object detection application and low power applications of embedded devices. Its architecture is comprised of multiple residual bottleneck layers. These layers use SSD is an object localizer which runs on top of the MobileNet v2 feature extractor. The network utilizes output from the last few layers (feature map) from the feature extractor to predict the location and class of objects. In the SSD architecture, the input is passed through different convolution layers of different sizes. These layers decrease in size progressively through the network. The purpose of these layers is to enable the network to detect objects of different shapes and sizes. A fixed set of predictions is taken from each one of these convolution layers and merged at the end. Furthermore, the computational cost of the model is reduced by choosing the fixed set of shapes. The model outputs locations and confidence of various objects in the image.t using the gradient of loss L, The loss for each prediction is computed as the combination of confidence loss Equation ). The erlgorithm  is used quations –4)). Th). ThLconThe detection of the staircase through the use of SSD MobileNet model using RGB data leads to false detection of objects with features similar to staircases. This may include objects similar to staircase such as patterned walls, railings, benches, etc. However, the depth data can be used in tandem to correct the classification decision of the SSD MobileNet. The depth information is fed through the MobileNet network architecture which classifies whether a given object is a staircase or not. However, the depth information may be noisy which may also lead to false detection. Hence, the two outcomes are fused using a support vector machine (SVM) to identify staircases with higher accuracy. K on SVM determines a hyperplane which separates the different classes present in the dataset . The posEquation .(5)f(x)Equation .(6)K, with which the robot should not collide. Finally, the robot must move slowly in the slippage region (here we consider liquid spillage debris) and also avoid the slippage region to perform the climbing operation.A combination of the RGB and depth information is used to determine the location of the first step of the staircase and the angle at which the robot is facing the staircase. This information is crucial to guide the robot towards the staircase. A point closer to the center of the step is chosen to allow room for the robot to align itself and start climbing. Common techniques for detection of steps involve the use of edge detection techniques such as Canny  or SobelAlgorithm 1: Contour detection algorithmThis work proposes a contour detection algorithm specifically for the case of first step recognition (Algorithm 1). The algorithm gives a higher preference to points along the gradient. This prevents the contour detected from being highly influenced by noise. Moreover, the algorithm checks for points which have deviated from the gradient. Due to this, the contour detected by this technique can be curved or straight, thereby allowing efficient use in all types of staircases.add added a point to the start of an array. In the algorithm, the gradient A, are also checked but for larger deviation from the gradient as compared to previous iterations  such as debris are retained in the step data through iterations  and dynamic obstacles (humans) are maintained temporarily (represented by ′H′). M) moves the robot in direction M. Sleep (t) pauses the robot operation for t seconds.Algorithm 2: Trajectory planningWhile climbing, the robot generates a zig-zag trajectory a completThis section describes the experimental results of the proposed scheme. To initiate the experiment, the dataset was generated by capturing images of the staircase, obstacles, and debris (stain and liquid spillage) using Intel’s Real Sense RGB-D camera in robot perspective. Each class has about 2600 images. Standard performance metrics (accuracy, precision, recall and To evaluate the proposed scheme, the trained model was tested on three steps. These include staircase detection and localization, first step detection and obstacle and debris detection and localization. The experiment was performed in a staircase present in multistory buildings with the sTetro robot platform as shown in However, only the use of SSD MobileNet model has some false detections e. The faThe next stage of the experiment was the first step detection, which was performed on the detected staircase region. The three steps involved the detection of the staircase, then detection of edges, which was followed by detection of the first step and its center. These steps for determining the first step location are shown in The detection results of obstacles and debris on the climbing stage are shown in Further, timing analysis  shows thThe effectiveness of the proposed model is evaluated with existing staircase recognition and debris detection approach. The staircase detection approach is compared with the Unmesh et al. , Munoz eIn the literature, very few works have been reported for debris detection, such as yang et al.’s  trash clThis work proposed the cascaded machine learning based operational framework for the staircase maintenance robot, sTetro. The SSD MobileNet object detection framework was utilized in the EPS to recognize the staircase, obstacles and debris. The feasibility of the proposed method was verified in real time with Jetson Nano, which is a lightweight and low power hardware. The framework detected the objects in real time and took an average of 200 ms to execute the staircase detection, first step localization and 112 ms for obstacle and debris detection. The statistical analysis shows that the depth based fusion scheme reduces the false detection rate and enhances the classification accuracy of the detection system. Further, the first step detection results ensure that the contour detection algorithm is able to determine the first step of the staircase with varying shapes and structures. Through the depth based clustering scheme, obstacles and debris position are effectively localized in the staircase. In the future, we plan to reduce the detection time of the framework by optimizing the CNN layers and also increase the number of obstacle and debris class for safe navigation and efficient cleaning on staircases.