This paper employs the volume of fluid (VOF) method to numerically investigate the effect of the width, height, and contact angles on burst frequencies of super hydrophilic and hydrophilic capillary valves in centrifugal microfluidic systems. Existing experimental results in the literature have been used to validate the implementation of the numerical method. The performance of capillary valves in the rectangular and the circular microfluidic structures on super hydrophilic centrifugal microfluidic platforms is studied. The numerical results are also compared with the existing theoretical models and the differences are discussed. Our experimental and computed results show a minimum burst frequency occurring at square capillaries and this result is useful for designing and developing more sophisticated networks of capillary valves. It also predicts that in super hydrophilic microfluidics, the fluid leaks consistently from the capillary valve at low pressures which can disrupt the biomedical procedures in centrifugal microfluidic platforms. Conventional clinical diagnostic tasks such as Enzyme-Linked Immunosorbent Assays (ELISAs) consist of a series of sequenced procedures carried out by skillful operators until the final analytical results are obtained Centrifugal microfluidic platform requires minimal external instrumentation to propel and manipulate fluids and its fabrication is a well-established art. It is placed on a rotating system and uses centrifugal force to drive and manipulate fluids. The rotational speed of the disk, the exact shape of the chambers and fluidic conduits together with their positions on the disk are used for flow control and flow sequencing Flow in capillary valves is controlled by the interaction between the centrifugal forces and the capillary forces. The fluid advancement in the capillary is stopped when a capillary meets a sudden change in its geometry. The maximum rotational speed required for overcoming that pressure barrier is referred to as the burst frequency. The effects of capillary dimension, expansion angle and capillary shapes on burst frequencies in hydrophilic and hydrophobic microstructures have been thoroughly studied This paper focuses on flow behaviour in super hydrophilic to less hydrophobic centrifugal microfluidic platforms and attempts to provide a detail view of capillary valves function in such microfluidic platforms. Various capillary structures with the static contact angle range 20° to 90° are investigated. The capillary dimensions are varied from 150 µm to 450 µm to study the effect of dimensions and the contact angle on the burst frequency. The volume of fluid method within version 13.0 of commercial code of ANSYS-Fluent is used for solving the governing equations. In order to validate the implementation of the numerical model our experimental data and existing experimental data in the literature are compared to the present computed data.We used the volume of fluid (VOF) method within the commercial ANSYS-Fluent CFD package, version 13.1 to predict the burst frequency in Geo. 1 and Geo. 2 see . This mev, ρ and µ are the velocity of the mixture, pressure, time, volumetric forces, density and viscosity, respectively. The continuity equation of a multiphase immiscible flow in Fluent is solved solely for the secondary phase qth, which has the following form:pq is the mass transfer from phase p to phase q and m˙pq is the mass transfer in the reverse direction. The primary-phase volume fraction is computed using the following constraint:In the VOF method, the position of the interface between the fluids of interest is tracked in a fixed Eulerian mesh domain. A single set of Navier-Stokes equations is solved for the computational domain and the volume fraction of each fluid is tracked by using an additional transport equation. The volume fraction (a) in each cell in the computational domain is between 0 and 1. For control volume, the sum of the volume fraction of all phases is set to unity. Therefore, any given cell represents either a mixture of phases (0<a<1) or a pure phase (a = 1) flow. The continuity and momentum equations for laminar, incompressible, Newtonian, and isothermal flow employed in the current study are as follows The average values of variables and properties of the mixture are defined based on the volume fraction of each phase at a given location The continuous surface force model (CFS) is used to reformulate surface tension into an equivalent body force 1, ρ2 are the density of phase 1 and 2, respectively. According to the CSF model, the surface curvature k is computed from local gradients in the surface normal to the interface, which is given as: where, ρ is the volume-averaged density computed using Eq. 5, ρwhere n = ∇a is the normal vector. Wall adhesion is included in the model through the contact angle:w, tˆw represents the unit vector normal and tangent to the wall, respectively.where nˆ is the unit vector normal to the surface, A simple equation introduced by Zeng et al. h is the capillary hydraulic diameter, θ is the static contact angle and σla is the liquid air surface tension. In Chen et al. where, Dw) and in the height (with angle of αh) vary equally until the meniscus arc in height direction stops changing. The meniscus arc in the width of capillary continues changing until the liquid bursts into the expanded volume (where αw = π/2- θ-β).where w and h are the width and the height of the capillary channel, β is the expansion angle. The expression is developed based on two stages of fluid advancement in a capillary which can be summarized as: a) prior to the valve, b) at the valve (transition period). At the first stage, pressure in the liquid can be derived from either the Young-Laplace equation or the change of total interfacial energy of the solid–liquid–air system by using the static contact angle. The second stage considers the influence of the changes in advancing contact angle and the changes of the meniscus shape from a concave to a flat and from the flat to a convex meniscus. At this stage it is assumed that the meniscus arcs in the width of the capillary , which is also the center of the disk, for Geo. 1 and Geo. 2 is 31.5 mm and 43.57 mm, respectively. The surface tension of water is set at 0.072 N/m. These parameters in addition to the value of advancing and equilibrium contact angles are the same as those used in the experiments carried out by Chen et al. Two common types of microstructures, Geo. 1 and Geo. 2 see  are used−6.The Pressure Implicit Split Operators (PISO) method, which uses the splitting of operations in the solution of the discretized momentum and pressure equations is used for coupling of pressure and velocity The mesh used was based on quad grids with an element size of 0.01 mm. Grids in the area of the waterfront and the edges adjacent to expansion areas i.e., at the water outlet to the right chamber were refined to smaller element size of 0.005 mm. Mesh dependency tests were carried out for each case and the meshes eventually used were justified by the quality of the results. For instance, increasing the original mesh by 100% to obtain finer grids does not give any significant difference for the burst frequency (<1.3%). The quality of the grids used for Geo. 1 and Geo. 2 at the outlet of the right chamber is shown in Forty-four simulation cases were created from Geo. 1 and Geo. 2 by varying the ratio of height/width (aspect ratio) and the contact angle (θ°). The range of these parameters, as listed in The rectangular microstructures were fabricated using a Computer Numerical Control (CNC) machine . The micro structures were engraved on compact disc-like platforms layer made of a 2 mm thick Polymethyl methacrylate (PMMA) and bonded by Pressure Sensitive Adhesive (PSA) material  to a 2 mm PMMA layer with venting holes cut through. A cutter plotter  was used to cut the microfluidic design in the PSA layers corresponding to the design of the PMMA layer. A custom-made system consists of a digital disk Spin Test System, laser sensor and a high speed camera was used to perform the experiments see .Several experimental studies In the second step, the burst pressure measurements for rectangular microstructures by Gliere et al. The difference of flow motion in a hydrophilic and a super hydrophilic capillary is illustrated in The fluid motion in a super hydrophilic capillary with an assumed contact angle of θ  = 20° (Case 11) is shown in Similar flow motion to The fact that on contrary to the theoretical studies, the numerical data shows the fluid leakage at the low contact angles can be due to several reasons. The theoretical expressions i.e., Eqs.11 and 12 have been verified experimentally for a particular operational parameters e.g., specific geometry of the capillary expansion and specific contact angles i.e., 68°, 70°, 93°. In addition, the theoretical models do not explicitly include the effect of wall adhesion that has a significant influence on fluid interfaces at a low contact angle. However, in the VOF method, the wall adhesion effect is fully considered in the governing equations (see Eq. 9).The results show that for a constant height (300 µm), burst frequencies increase as the capillary width decreases. It shows similar trend to experiments by Chen et al. The computed results show the minima burst frequencies for square capillaries (AR = 1). The minimum burst frequency in square capillaries can be seen in For super hydrophilic centrifugal microfluidic platforms (<40°) numerical results and Eq. 10 both show the increase of burst frequencies with the increase of the contact angle which is in contradiction to Eq. 11. Despite of the high burst frequencies predicted by theoretical model our numerical results show that fluid flows over the capillary expansion walls before it actually bursts. In fact, the fluid leaks into the desired chamber before applying adequate centrifugal pressure that causes the change of the meniscus shape in hydrophilic centrifugal microfluidic platforms . At the Flow in passive capillary microvalves in centrifugal microfluidic platforms for a range of superhydrophilic to less hydrophilic contact angles has been studied using the VOF model within the commercial CFD code of ANSYS Fluent. Our experimental data in addition to experiment results from Chen et al. In common capillary dimensions (>100 µm) for the cases of a low contact angle especially less than 20°, the capillary valve is unable to retain the fluid from leaking and it loses its function.The computed results suggest that the theoretical models cannot be used for super hydrophilic materials since they are unable to predict the fluid leakage. While they predict that high pressure is required for pushing the fluid over capillary valves, the computed results show that fluid flows consistently over the capillary valve into the next chamber at low pressures.In general, computed results show that burst frequencies of wide capillaries (w>h) are always lower than those of narrow capillaries (w<h). Theoretical models predict similar to our computed results for wide and narrow capillaries.The computed results for narrow capillaries (w<h) show a consistent increase of burst frequencies with the increase of the contact angle. However, for wide capillaries (w>h) the computed results predict three divisions of burst frequencies. First, as the contact angle increases, the burst frequency increases to a peak where it starts decreasing at a low burst frequency. After a low burst frequency it slightly increases with the increase of the contact angle.The results show that burst frequencies of square capillaries are lower than those of rectangular shapes. However, the theoretical models used for comparison are not able to predict pressure drops in square capillaries; our experimental data and the theoretical model that we have developed


Clostridium acetobutylicum is useful as a producer of biofuels, particularly butanol. Recent work has concentrated on trying to improve the efficiency of the fermentation method, either through changes in the environmental conditions or by modifying the genome to selectively favour the production of one particular solvent over others. Fermentation of glucose by C. acetobutylicum occurs in two stages: initially the acids acetate and butyrate are produced and excreted and then, as the external pH falls, acetate and butyrate are ingested and further metabolised into the solvents acetone, butanol and ethanol. In order to optimise butanol production, it is important to understand how pH affects the enzyme-controlled reactions in the metabolism process. We adapt an ordinary differential equation model of the metabolic network with regulation at the genetic level for the required enzymes; parametrising the model using experimental data generated from continuous culture, we improve on previous point predictions ) The acetone–butanol (AB) fermentation process in the anaerobic endospore-forming Gram-positive bacterium  Some other members of the Clostridium genus generate useful chemicals in much the same way as C. acetobutylicum, while some are highly pathogenic and cause devastating diseases (e.g. Clostridium difficile and Clostridium botulinum).C. acetobutylicum has two distinct components—acidogenesis and solventogenesis. Initially, acids such as acetate and butyrate are formed from glucose and other hexose sugars, then, as the concentrations of acids increase, the metabolism shifts to produce solvents  from the acids, as well as producing spores. These latter processes are conjectured to be survival mechanisms in order for the organism to survive in more hostile conditions. The metabolic pathways leading to both acid and solvent production have been clearly defined and elucidated The metabolism of The reduced metabolism network introduced elsewhere Reinternalised butyrate and acetate can be converted to butyryl-CoA and acetyl-CoA respectively in Ping-Pong-Bi-Bi reactions, in the process converting acetoacetyl-CoA to acetoacetate. Both reactions are mediated by CoA-transferase, which has two subunits, CtfA/B C. acetobutylicum can prioritise different pathways depending on external conditions, and can produce ethanol independently of acetone and butanol fermentation whilst in continuous culture—ethanol is produced at the same rate in both the acidogenesis and solventogenesis phases of the metabolism C. acetobutylicum: cells primarily produce acids  if the external pH is greater than 5 but produce primarily solvents  if the external pH is less than 5 C. acetobutylicum, unlike some other bacteria, is unable to maintain its internal pH at a constant level, which makes it particularly susceptible to changes in the external pH Solventogenesis is postulated as part of a survival strategy AIC to fall below that of all models considered so far, but to a non-physical situation where less than 30% of the cells metabolise post-shift, which is not seen in the experiments. To improve the fit, reduce the parameter ranges found, and break some of the correlations, we recommend that for further continuous culture experiments, concentrations of the important intermediates acetyl-CoA and butyryl-CoA need to be measured, as well as a spore fraction assay.The study 2•C. acetobutylicum strain ATCC824 is grown under anaerobic conditions at 37 °C, and the precultures are prepared as previously described.•2PO4 and 4% (w/v) glucose in the medium and a dilution rate of The culture is kept phosphate-limted in a chemostat , with 0.5 mM KH•The external pH is kept constant at pH 5.7 and 4.5 by automatic addition of 2 M KOH.•–‘Forward 1’: acidogenesis was maintained for 137 h, then the pH control was stopped, allowing the metabolic shift to occur. 22 h after the removal of the pH control the shift had completed, and final measurement was taken at 215 h.–‘Forward 2’: as in ‘Forward 1’—the pH control was switched off after 137.5 h, the metabolic shift occurred over 33.5 h and steady state was reached at approximately 236 h.–‘Forward 3’: as in the previous two, but the pH control was switched off at 121 h, the metabolic shift lasted 29 h and the final measurement was taken at 215 h.–‘Reverse’: the pH was controlled via the addition of KOH so that the change was in the opposite direction—the pH was initially kept low for 129 h, and then increased over 17 h to the high level.Three forward shift experiments are performed by shifting the culture from pH 5.7 to 4.5, and one reverse shift experiment is performed by shifting the culture from pH 4.5 to 5.7:We use experimental data generated for a previous study The experimental data are given in the 3Ah, which represent intracellular concentrations of the enzymes CtfA/B, Adc and AdhE, respectively):i, and F in that paper never increased beyond one so the omission was of no consequence.) For each experiment, the function Following the previous work 4•As in the previous work •A dataset consisting of the three forward experiments, starting at the relevant initial conditions for each experiment and using all available data from these experiments.•A dataset consisting of all available data from the four experiments .As there are four experimental datasets from which we could estimate the parameters, we have chosen three separate ways of combining the data for comparison of parameter estimates  used in the datasets.5n and n and In this section we will sketch out the method used for parameter fitting. There are 23 parameters over which we wish to optimise , and as each one is restricted to being positive, we optimise using the logarithms (to base 10) of the parameters. This has the side-effect that the logarithms of the previously estimated parameters  are all The main optimisation method is a modified simplex method 6Once a parameter set We can use the OLS score function to compute likelihood ellipsoids, from which we calculate the dependent and independent confidence intervals, and estimated correlation coefficients. The dependent confidence interval assumes that the other parameters are exactly determined, and the independent confidence interval assumes that all parameters are equally uncertain. Geometrically, the dependent confidence interval is the intersection of the ellipsoid with the line given by the other parameters being fixed at their MLEs, and the independent confidence interval is the projection of the ellipsoid onto the coordinate axis.77.1The best fits for the three datasets are compared against each other in adhE2 gene which is antagonistically regulated with adhE1 (with only the latter being included in the model). adhE1 is expressed during solventogenesis and so adhE2 is downregulated here, but adhE2 is upregulated during acidogenesis and could be responsible for ethanol production at the higher pH values.The predictions from the lowest scorers of the three different selections of experimental data in some way validates the approach used in the previous work Boxplots of the fitted parameters for the three separate datasets are shown in 7.2The dependent and independent 95% confidence intervals for six representative parameters are shown in n and Note that in some cases, the independent confidence intervals are very large, with the minimum and maximum outside the r of 0.98, which is unsurprising as the two reactions The average (over the three parameter sets) of the average correlation coefficients of the 25 lowest-scoring parameter values is shown in  diagram , the twotabolism .Of course, it may be that the score function does not have ellipsoidal contours around the minima  and the minima themselves may only be local not global.8•(Model 1a) A step function with fixed height, width and time-at-half-saturation:•k is restricted so that the maximum concentration of spore cells is the same as in the previous non-dynamic model. Three choices for the function –m is restricted to lie in the interval  and (Model 1b) Constant production rate post –(Model 1c) Production related to pH through –(Model 1d) Production related to pH through A production model of the form:It is known that the build up of acids and the change in the external pH triggers spore formation as well as solvent production k are calculated such that it does not exceed the maximum spore production rate (which is in turn directly related to the largest concentration of the sporulation protein Spo0A) used in other work when examining the sporulation cascade in a batch culture All three of the ODE models (i.e. 1b–1d) have the same basic form, namely that the production is stepped up some time after the pH is shifted—however, they differ in their biological interpretation: model 1b assumes that sporulation and solventogenesis are triggered independently of each other through different enzyme pathways; models 1c and 1d assume that the same (presently unknown) enzyme is involved in triggering the solventogenesis and sporulation cascade, both with (1d) and without (1c) a delayed response in sporulation compared to the solventogenesis response see . The resBiologically, the final three production models are more plausible, as they relate the production of spores to some internal state of the cell. We could use different trigger pHs in the final two models than for the shift in metabolism, but if, as suspected, changes in the level of the same enzyme begin both the metabolic shift and sporulation response 9RSS is the score function , N is the number of points to be fitted and m is the number of parameters. As the optimisation method is stochastic, we take the lowest 25 scorers  and compute the average AIC for the no sporulation model , which would suggest that we would need to accept a model in which more than 75% of the cells do not metabolise post-shift. A large spore fraction such as this would be immediately evident in the culture, but as this is not seen  this suggests that the model as it stands is deficient. Further improvements to this metabolism model, including the introduction of two separate cell populations (one with a primarily acidogenic physiology and one with a primarily solventogenic physiology) may be required to improve the fit further and narrow down the parameter ranges.The parameter estimates for the models 1a–1d are shown in 11In summary, the parameters we find here (for the no sporulation model), give no better or no worse fit than those found elsewhere AC, and are hence negatively correlated. There are some exceptions, where there is no linking ODE, for instance r of 0.98, which is reflective of the fact that the reactions The signs of the correlations in general are related to the signs of the relevant reaction terms in the ODE model—We introduce sporulation into the model, triggered post-shift, which reduces the differences between the model predictions and the data post-shift, but does not reduce the average AIC below that of the no-sporulation model. However, the lowest AIC values are reduced. There is therefore some evidence that sporulation could occur, but more data would be required to confirm this, such as spore fraction assays. The three models which link spore production directly to the pH shift perform better than the simple step-function model—which is consistent with the hypothesis that spore production is linked to the solventogenic response, allowing the organism to survive if conditions worsen significantly. If we allow the spore fraction to reach approximately 75%, we can improve the fit still further (as measured by the Akaike Information Criterion), but a spore fraction of this size was not seen in the experiments. Further improvements to this metabolism model, including the introduction of two separate cell populations (one with a primarily acidogenic physiology and one with a primarily solventogenic physiology) may be required to improve the fit further, but this would require more parameters, and more data to determine them, either through more measurements during the culture or through independent experiments testing specific reactions in the metabolism.The parameter ranges being large, the large confidence intervals and the clusters of correlated parameters show the network as it stands cannot be determined completely without more data: we suggest that we would need at the minimum the concentrations of the branch point intermediates acetyl-CoA and butyryl-CoA, as well as a spore assay during both phases of the experiment. At present, there is too much leeway in the data to make definitive conclusions about the metabolism, aside from concluding that there could be some weak evidence for sporulation in the data (as the fit is improved by including a non-metabolising fraction). Nevertheless, we are investigating the use of approximate Bayesian methods (such as approximate Bayesian computation via stratified Monte Carlo


Ankle reconstruction prior to limb lengthening for was performed in 13 patients with fibular hemimelia with complete radiological absence of the fibula (type II). There were different degrees of absence of metatarsal rays. The hindfoot deformity was a heel valgus in 12 patients and equinovarus in 1 patient. The patients’ ages ranged from 9 to 26 months. Excision of the fibular anlage was performed with lateral subtalar and ankle soft tissue releases to restore the ankle and subtalar joint relationships. In all cases, the fibular anlage ended distally in a cartilaginous lateral malleolar remnant that was fused to the talus in two patients. This fibular remnant was advanced distally and fixed to the tibia with 2 Kirschner wires to recreate an ankle mortise. The period of follow-up ranged from 12 to 38 months. All patients had a stable ankle without tendency to valgus deformity or subluxation. The ankle range of movement was a mean of 27.3° plantarflexion (25–30) and 18° dorsiflexion (15–20). Reconstruction of the ankle in type II fibular hemimelia using advancement of the cartilaginous lateral malleolar remnant has produced encouraging results in the short-term but longer follow-up is needed. The fibula transfers weight from the knee to the ankle, estimated at 10–16 % by different authors; this reduces to <1 % after a partial resection . The latFibular hemimelia is a congenital disorder characterized by partial or complete absence of the fibula. It is the most common longitudinal deficiency of long bones and is a spectrum of anomalies ranging from mild fibular shortening to bilateral involvement with associated defects of the femur, tibia, ankle, and foot. There is a leg-length discrepancy usually with an equinovalgus deformity of the foot; sometimes, there are associated anomalies including a flexion contracture of the knee, femoral shortening, instability of the knee and ankle, and a stiff hindfoot with absent lateral rays . Fibula Using the Ilizarov technique for the reconstruction of the leg through lengthening and deformity correction –10, the Thirteen patients with fibular hemimelia type II according to the Achterman–Kalamchi classification were treated by the method described below. There were eight girls and five boys. The youngest patient in this series was 9 months and the eldest 26 months with a mean age of 13 months. The right limb was affected in seven patients (53.8 %) while the left affected in four (30.8 %) and both limbs in two (15.4 %). On the basis of the intraoperative findings, 10 patients had a talocalcaneal coalition. The patients were clinically and radiologically followed up for an average of 18.6 months (12–38 months).The principle of this method is to use the rudimentary fibular anlage for the reconstruction of the ankle by fixing it to the lateral side of the distal tibia and talus with the aim of fusion and therefore prevent a valgus deformity and lateral subluxation of the ankle.A lateral soft tissue release of the ankle and Z-Plasty of peroneal tendons and tendo Achillis is performed through a posterolateral incision Fig. a, bFig. The underlying fibular anlage and malleolar remnant is excised and released from its proximally displaced position Fig. c.A sharp dissection and division of the talocalcaneal coalition, if present, is performed to reestablish the alignment of the talocalcaneal relationship.The ankle is centralized by placing the talus under the tibia and fixation with a transcalcaneal wire Fig. d.The cartilaginous anlage is made to serve as a lateral support to the ankle by fixing it to the distal tibia and talus using two transverse smooth K-wires; it becomes a ‘lateral malleolus’ buttressing the talus Fig. e, 1f.The wound is closed in layers and an above knee plaster cast applied Fig. g.The sutures were removed after 2 weeks and the cast after 6–8 weeks.The period of follow-up ranged from 12 to 38 months. The patients were found to have a stable ankle without tendency to valgus deformity or subluxation of the ankle with a range of ankle movement ranging from a mean of 27.3° (25–30) plantarflexion and 18° (15–20) dorsiflexion  applied for all patients to compensate for shortening until the age of lengthening at 4–5 years. X-ray follow-up showed good alignment in all cases but without any evidence of ossification of the fibular anlage.The absence of the lateral malleolus in fibular hemimelia causes subluxation and valgus deformity of the ankle joint due to lack of the lateral support at the ankle. Excision of the fibular anlage and centralization of the ankle corrects the valgus deformity but does not prevent the gradual subluxation and valgus deformity that usually recurs with growth and with tibial lengthening.Attempts have been described to increase ankle stability in patients with fibula hemimelia. The osteotomy described by Ulrich Exner which altered the convexity of the distal tibia to a concave shape showed promising short-term results . ReconstGrucca described transepiphyseal division of the plafond and a proximal and medial translation of the medial tibial fragment with the talus accompanying it, but many complications, notably damage to the physeal growth and bony bar formation, growth retardation, and ankle arthrosis, were reported , 14.Michael Weber described a lateral malleolus reconstruction in a 5.5-year-old boy with type II fibular aplasia using a triangular iliac crest transplant to the lateral distal tibia with the apophysis as a growing portion and the gluteal fascia as a lateral ligament. He reported a good result after 2.5-year follow-up with ankle stability and a good range of movement .Finally, as a salvage procedure, arthrodesis of the ankle has been recommended by several authors.Using the fibular anlage for the reconstruction of the lateral malleolus solves the problem of instability and prevents subluxation of the ankle and progressive valgus deformity of the ankle. This is illustrated in this series of short-term follow-up. All patients were found to have a stable ankle joint without any subluxation or valgus deformity and a good range of motion. This suggested that the transplanted anlage had not fused and acted only as an extra-articular buttress to the lateral side of the ankle. We did not identify ossification of the cartilaginous anlage in the period of follow-up (12–38 months). In those patients who underwent an MRI scan (6 patients), we were not able to identify the anlage remnant. Longer follow-up is needed to ensure that the stability of the ankle with growth of the child or during leg lengthening is maintained.We can reservedly recommend the anlage remnant of fibula as a potential donor material for lateral malleolar creation in ankle reconstruction for type II fibular hemimelia. It acts as a buttress to the talus and prevents the recurrence of valgus deformity.


Publicly funded trials regularly fail to recruit their target sample size or find a significant positive result. Adaptive clinical trials which may partly mediate against the problems are not often applied. In this paper we investigate the potential of a form of adaption in a clinical trial - a futility analysis - to see if it has potential to improve publicly funded trials.Outcome data from trials funded by two UK bodies, the Health Technology Assessment (HTA) programme and the UK Medical Research Council (MRC), were collected. These data were then used to simulate each trial with a single futility analysis using conditional power, undertaken after 50% to 90% of the patients had been recruited. Thirty-three trials recruiting between 2002 and 2008 met the inclusion criteria. Stopping boundaries of conditional powers of 20%, 30% and 40% were considered and outcomes included the number of trials successfully stopped and number of patients saved.Inclusion of a futility analysis after 75% of the patients had been recruited would have potentially resulted in 10 trials, which went on to have negative results, correctly stopping for futility using a stopping boundary of 30%. A total of 807 patients across all the trials would potentially have been saved using these futility parameters. The proportion of studies successfully recruiting would also have increased from 45% to 64%.A futility assessment has the potential to increase efficiency, save patients and decrease costs in publicly funded trials. While there are logistical issues in undertaking futility assessments we recommend that investigators should aim to include a futility analysis in their trial design wherever possible. Publicly funded trials in the United Kingdom (UK) have poor rates of recruitment ,2, with Trials with poor recruitment may also be more likely to request an extension to attempt to reach their target sample size. Between 2002 and 2008 nearly half of all publicly funded trials received an extension of some kind , with siInnovative trial designs have been applied to avoid this problem in the pharmaceutical industry , where tPublicly funded trials, such as those funded by the UK Medical Research Council (MRC) or National Institute for Health Research (NIHR) Health Technology Assessment programme (HTA), have different practical issues. For example, the trials may be assessing treatments which already are licensed but for which further information is required to get them used in practice. Valuable information is still obtained regardless of the final result . Indeed,Interventions are often complex  in publiMethods to improve the overall efficiency of clinical trials must account for the practical difficulties in undertaking the studies; it is the logistics and not the statistics which most impact a trial design . Methodset al. [This paper will focus on futility assessment during a trial as a method for stopping trials which have low chances of finding a statistically significant result. We have applied existing conditional power methods for futility analysis retrospectively to the database of trials used by Sully et al. . The reset al. [et al.:Analysis was undertaken using the trial database created by Sully et al.  with som1. They were multicentre.2. Recruitment started on or after 1 January 2002.3. Recruitment was originally planned to close on or before 31 December 2008.4. They were not a cluster randomised trial.Additional inclusion and exclusion criteria added were as follows:1. They were a two-armed, parallel group superiority study.2. The primary endpoint was binary or continuous.3. They had evidence of a sample size calculation.4. Data relating to the endpoint used in the sample size calculation were available at follow-up.We restricted the studies to binary and continuous outcomes as in these cases it was always possible to determine the treatment effects in the trials and undertake calculations using summary data.The existing database was sufficient for identifying most trials and further data extraction was undertaken to obtain the planned and observed effect sizes and type of endpoint for each trial. While utmost care went into ensuring that extracted data were correct there were occasional ambiguities in reporting; in these cases trial managers were contacted for clarification. In the event that no response was received, the trial was excluded from the study.The types of data collected, split by endpoint (binary or continuous), are shown in Table Trials were also broken down into those which were ‘successful’ and those that were not. For this paper a successful trial is primarily defined as a trial finding a treatment effect greater than or equal to that specified in the sample size calculation; it was not necessary for this effect to be statistically significant. This is because in publicly funded trials it is not just important to see a statistically significant effect, but to see a worthwhile effect size; there are often many secondary endpoints, so the primary endpoint being non-significant does not mean the trial has failed to be worthwhile . The finAs a secondary analysis the data were re-investigated with the requirement that the primary outcome must be statistically significant for a trial to be considered successful.γ, of either γ = 0.2, γ = 0.6 or γ = 0.4. Thus, for the first boundary value, if the conditional power is less than 0.2, for instance, then the study will stop early for futility.For each trial, a simulation was run using the planned sample size of the trial and observed number of successes  or effect size and standard deviation . Trials were simulated using a binomial distribution or Normal distribution, as appropriate. The conditional power of the trial was then calculated at every 10% of the planned sample size , as well as after 75%, using the methods shown below. Each simulation was repeated 50,000 times. The mean conditional power was calculated at each proportion of the sample size for each trial and used to judge whether a trial should be stopped for futility using a boundary, When calculating the conditional power of a trial an assumption must be made about the distribution of the remainder of the trial data. Three common choices are that it will follow the null hypothesis, the alternative hypothesis or the observed trend so far . In thisΖ is the standard Normal random variable, Ζt is the Ζ-score at interim, τ is the proportion of patients recruited, and α and 1-β are the planned type I and II errors, respectively. Conditional power was calculated on a two-sided basis, so the conditional power was the probability of finding either a significant positive or negative result.where Φ is the cumulative density function of the standard Normal distribution, In our analysis the assumption is that only one assessment of futility is planned in the study.We also make the additional assumption that we have all patients recruited available who have been recruited for the futility assessment. The implications of this assumption will be discussed later.As highlighted, one of the assumptions is that the remainder of the trial will follow the alternative hypothesis. Thus, the estimate of effect will be taken from that used in the sample size calculation. If the effect in this original calculation was overstated then the conditional power will be low and a trial will stop for futility even when a possibly clinically meaningful difference has been observed .The results in this paper are particularly sensitive to this assumption as we define ‘success’ as observing the pre-specified estimate of effect.The seventy-three trials from the previous database  were assP-values of P = 0.050 [Figure  = 0.050 . This waConversely, one trial had very high average conditional power despite finding an effect much smaller than originally planned; this is because the trial undertook a sample size re-estimation and went on to recruit far more than its original planned sample size – based on a new effect size - eventually finding statistically significant evidence but one that was smaller than used in the original sample size calculation .The majority of trials which eventually ‘fail’ appear to have conditional power below 40% by the time 70% of patients have been recruited, and below 30% by the time 80% of patients have been recruited. This is summarised in Table It is important to avoid false negatives – in this case, higher numbers in the top-left cell of the tables – as this indicates a trial has been stopped due to futility, when it would have been successful had it continued. One trial could be considered to have incorrectly stopped for futility in our analysis Table . This stThe analysis was repeated using our secondary definition of a successful trial – that the trial has a ‘statistically significant’ outcome. Ten 30%) of the trials were considered successful using this classification compared to eight 24%) using the previous definition; this is higher because of statistically significant results for effects smaller than planned on the sample size. Table 0% of the4% using It is worth noting that for both Table For our analysis we are concerned with the overall proportion of trials, particularly how a futility assessment will impact on the proportion of trials that successfully recruit to their target sample size. It is worth highlighting, however, that trials that continue after a futility assessment will not necessarily go on to be successful. With a futility assessment after 70%, 80% and 90% of patients, 62% 8 of 13), 64% (7 of 11) and 50% (4 of 8), respectively, would still go on to be unsuccessful according to Table , 64% (7 γ = 0.30 boundary (807 patients) and a γ = 0.40 boundary .Table Numbers and percentages of trials considered to have ‘successfully recruited’ after undertaking a futility analysis are given in Table Undertaking a futility analysis after 50% of patients have been recruited leaves the proportions unchanged: 15 (45%) of the trials are classed as successfully recruiting. This proportion is highest  when the futility analysis is undertaken after 70% or 75% of patients are recruited. It should be noted that because some of the trials failed to ever reach higher proportions of their target sample size (80% or 90%) they would not have had chance to undertake a futility assessment.We recommend that a futility assessment be considered for all trials and in particular for publicly funded trials. With a futility assessment there is the potential to considerably increase the proportion of trials that successfully recruit to the target sample size. There would need to be a small inflation in the target sample size but this would be offset by the savings made from the trials stopping early.Including a futility assessment can increase the chances of a study successfully recruiting. In our analysis, with a futility assessment after 75% of the target sample size has been recruited, the proportion of studies successfully recruiting can be increased from 45% to 64%.It should be noted, however, that a futility assessment will not ensure the trial goes on to be successful. In our analysis with a futility assessment after 75% of patients with conditional power of 30% the majority of trials - 69% (9 of 13) - would still go on to be unsuccessful. Altering the conditional power at the futility assessment will have an impact on the proportion of trials that go on to be successful.In our study, only 24% of trials were successful by our definition which was consistent with the work of Dent and Raftery using slightly different definitions . If the A balance must be struck when undertaking a futility analysis earlier in a trial, which allows for greater opportunity for savings in the sample size compared to undertaking a futility assessment later when a trial is more likely to be correctly designated as futile. In our analysis, we found that a futility assessment after at least 70% of the target sample size has been recruited maximised the chances of stopping for futility with little chance of stopping after 50% of the target sample size. Our results are consistent with those of Lachin . Assuminσ). If the observed effect was zero  then the simple average would be d/2 and the naïve power calculation would be 37%. Thus, half way through the trial, the effect size  can be small and the study may still not stop for futility. Futility assessments are not without their downsides. Introducing an extra chance to stop the trial due to lack of a positive result will increase the Type II error β up to a maximum of [The issue of early futility assessment can be illustrated with simple calculations. Imagine a conventional sample size calculation where the sample size, n, is estimated for a clinically significant difference, d, for 90% power and a two-sided significance level, 5% . Supposeximum of γ = 0.30); however, this maximum is only reached if the trial is continuously monitored for futility [ is very small relative to the total time spent recruiting (T), that is, t/T is small. This assumption means that when assessing futility after n0 evaluable patients have been recruited, outcome data will be available on all of them. However, if this assumption does not hold and t/T is not small there will be those who have been recruited to the trial but do not have evaluable data for the futility assessment. These patients are known as pipeline patients [It should be noted that the methods applied and results obtained in this paper assume that the time between recruitment and primary outcome measurement  by and a successful futility assessment will only savet/T ≥0.25 then there is little benefit to be gained from a futility assessment as the trial will have fully recruited. The results found in this paper are, therefore, indicative of the maximum savings attained if the time between recruitment and outcome is very short relative to the total recruitment time of the trial, such as, for example, trials in emergency medicine [patients. Clearly if medicine .In cases where the follow-up time for the primary outcome is relatively long, alternative methods must be considered. Hampson and Jennison discuss this situation in detail, proposing methods for delayed responses which include analysis of pipeline data .The implication of what we have highlighted is that the potential reductions in the sample size highlighted in Table There are many benefits to including a futility analysis in a trial. Stopping a futile trial early allows patients who would otherwise have been assigned to an inferior treatment to instead receive the best possible care. Additionally, with only a finite capacity for research the patients, no longer being recruited into a stopped trial, could go on to take part in other, more promising studies. Additionally, investigators can focus the time saved on further research rather than completing a negative trial.γ = 0.3.A futility assessment in a clinical trial has the potential to increase efficiency, save patients, and decrease the costs of publicly funded trials. While there are logistical issues in undertaking a futility assessment, whenever possible investigators should aim to include a futility analysis in their trial design, with the results from the 33 trials in the paper suggesting maximum savings found by doing so when 75% of the target sample has been recruited, using a boundary of CP: conditional power; HTA: Health technology assessment; MRC: Medical Research Council; NIHR: National Institute for Health Research.The authors declare that they have no competing interests.BGOS carried out the data extraction, simulations, analysis and report writing. SAJ conceived the study, assisted with interpretation of results and reporting. JN assisted in interpretation and contributed towards report writing. All authors read and approved the final manuscript.


Radiologic determination of pediatric femoral fracture rotation has been debated. Measuring the antetorsion angle of the fractured femur by computed tomography and comparing it with the opposite side has been the method of choice for this purpose. However, no simple method for direct measurement of femoral fracture rotation exists in the literature. In this study, our aim was to test a mathematical method of measuring the axial plane malrotation from direct roentgenograms.A pediatric femoral shaft fracture model was produced. The bone was secured to a wooden frame that allowed the distal part of the fracture to rotate around an axis. Radiographs were taken at known intervals of rotation ranging from the neutral position to 60° external rotation and to 60° internal rotation in 5° increments of rotation. Five independent, blinded observers measured the radiographs and calculated the fracture rotation according to a standard formula. Calculated rotation values were compared with known rotation values.Calculated rotation values were close to actual rotation values throughout the arc of rotation. The mean absolute error of five observers for all measurements of external and internal rotation was 3.97° (±0.83). The correlation coefficient between calculated and actual rotation values was 0.9927. The interobserver intraclass correlation coefficient for calculated rotation was 0.997.Absolute error and correlation coefficient values indicate that this method is accurate and reliable in determining the fracture rotation. Femoral shaft fractures are one of the most common lower extremity fractures in children . TreatmeSeveral direct roentgenograms have to be taken during the traction period or during the follow-up with spica casts to assess the maintenance of the reduction. Angular deformities on coronal and sagittal planes can be determined easily with conventional roentgenograms but rotational deformities on the transverse plane cannot . The amoAlthough it has been reported that up to 25°–30° of rotational malunion seems to be well tolerated , in mostCalculation of rotational deformity on a direct roentgenogram was performed by Henderson et al.  on a pedWe developed a mathematical method to calculate fracture rotation from a direct roentgenogram. We obtained a well-preserved human cadaver femur and reamed its medulla starting from the tip of the greater trochanter down to the intercondylar notch. We performed an oblique midshaft osteotomy to create an oblique femoral fracture model. Then we inserted a wooden rod into the medullary cavity through the length of the proximal and distal fragments to function as the axis of rotation. We distracted the fracture ends to see the rotation more clearly. Then a wooden frame was constructed, and the bone and wooden rod were secured to the frame with metal pins. This system allowed the distal part of the fracture to rotate around the wooden rod while the proximal part was stable. The degree of rotation was measured with a goniometer, which was adapted to the frame Fig. .Fig. 1PeDigitized anteroposterior radiographs of the femur were obtained. Radiographs were taken with the fracture rotation ranging from the neutral position to 60° external rotation and to 60° internal rotation in 5° increments of rotation.Three orthopedic surgeons and two orthopedic residents independently measured the radiographs. The observers were blinded to the protocol used to take the radiographs. They were allowed to use the photo management program of their choice and could modify the images to their liking.d, while its perpendicular distance h to the fracture edge does not change . Figure Our method of calculating the rotation is based upon the displacement of a certain point on the distal fracture fragment on a horizontal line. For easier measurement, we chose this point as the intersection of the fracture line and the vertical midline of the femoral diaphysis (the axis of rotation) Fig. a. When tnge Fig. b. On thef α Fig. c. As theram Fig. , the ang−1 function. Since a calculator may not be available everywhere, a list of ratios and corresponding angles was prepared  do not change. On the other hand, if a fracture is angulated on the lateral plane, the parameter h changes, and its new value can be determined on the lateral radiograph according to the amount of angulation, or the measurement can be performed directly on the lateral radiograph if there is no angulation on the anteroposterior (AP) radiograph. So in the application of this method in the clinical setting, AP and lateral radiographs should be taken. If there is no angulation on the lateral plane, measurement can safely be made on the AP radiograph and vice versa.In our model, the fracture is only rotated and not translated or angulated. But angulation on the anteroposterior plane and translation do not change our measurement, because the parameters in the formula , and measurement is difficult. With our landmark technique, transverse fracture rotations can be measured more easily.The fracture in our model is an oblique osteotomy, and the fracture ends are smooth. In a real fracture, such smoothness does not exist, and there are spikes on the fracture ends. These spikes can be used as landmarks in rotation measurements. In this case, measurement of parameter Fracture comminution may preclude accurate measurement with this method if the comminution is severe. But if there is only a butterfly fragment or there is a segmental fracture, we can still calculate the overall rotation by measuring the rotation of the fragments with reference to each other.A good radiologic technique is important for the highest accuracy using our method. We prefer using digitized roentgenograms and magnification programs. If digitized roentgenograms are not available, scans or digitized photographs of standard roentgenograms can be taken and magnification can be performed on a computer. Magnification quality of the roentgenogram is also very important. Resolution should be sufficient to allow adequate magnification. The more magnification that can be performed, the more precise is the measurement. Fracture ends have to be clearly visible.To the best of our knowledge, this is the first study in the literature that defines a method of calculating fracture malrotation of a long tubular bone from a simple direct roentgenogram. Techniques for quantifying supracondylar humerus fracture malrotation have been described previously , 25. Our


Sex predominates in eukaryotes, despite its short-term disadvantage when compared to asexuality. Myriad models have suggested that short-term advantages of sex may be sufficient to counterbalance its twofold costs. However, despite decades of experimental work seeking such evidence, no evolutionary mechanism has yet achieved broad recognition as explanation for the maintenance of sex. We explore here, through lineage-selection models, the conditions favouring the maintenance of sex. In the first model, we allowed the rate of transition to asexuality to evolve, to determine whether lineage selection favoured species with the strongest constraints preventing the loss of sex. In the second model, we simulated more explicitly the mechanisms underlying the higher extinction rates of asexual lineages than of their sexual counterparts. We linked extinction rates to the ecological and/or genetic features of lineages, thereby providing a formalisation of the only figure included in Darwin's “The origin of species”. Our results reinforce the view that the long-term advantages of sex and lineage selection may provide the most satisfactory explanations for the maintenance of sex in eukaryotes, which is still poorly recognized, and provide figures and a simulation website for training and educational purposes. Short-term benefits may play a role, but it is also essential to take into account the selection of lineages for a thorough understanding of the maintenance of sex. Some traits are considered to be “evolutionary dead ends”, initially developing due to short-term selective advantages, but resulting in lower rates of speciation or high rates of extinction in the long term, resulting in a species level selection Parthenogenesis is probably also an evolutionary dead end The two-fold short-term advantage of asexuality led many researchers to search for counterbalancing two-fold short-term advantages of sex . The fate of lineages is determined exclusively by extinction, speciation and transition rates. Our simulation is thus more illustrative than Nunney's model, as it represents a branching process similar to that acting in nature. Further, we let the transition rate change during the course of evolution. In our model, new lineages have a transition rate slightly different from that of their parents , so lineage selection can act on this trait. We thereby investigated whether lineage selection favoured lower rates of transition to asexuality .The second model attempts to explore the reasons for the higher rates of extinction in asexual than in sexual lineages. It can thus be seen as a submodel of the first one, because it gives a justification for the values of parameters used in the first model. This second model is inspired directly from the single figure included by Darwin in “The origin of species”  or any other taxon. Such process linking “distance” between lineages at a given time and divergence (or extinction) has been formalised and simulated in some models of speciation Here, we address the following questions: (1) Can the transition rate to asexuality evolve towards lower values by lineage-selection? (2) Can the specificities of sexual vs. asexual lineages in ecological competition influence the probability of maintenance of sex by lineage selection? Our aim was also to provide representations of the evolutionary history of sexual and asexual lineages for training and educational purposes.Sp for sexual species and Ap for asexual species) or not speciate . Sexual lineages may thus be transformed into asexual species, with a probability Su, or may remain sexual, with a probability (1−Su). Asexual lineages cannot become sexual again, so the rate of transition from asexuality to sexuality is 0. We assume that sex is too complex to be restored after this function has been lost. Lineages may become extinct with a probability Se for sexual lineages and Ae for asexual lineages, or they may persist, with a probability (1−Se) for sexual lineages and (1−Ae) for asexual lineages.The first model includes a limited number of parameters. Simulations represent a branching process, with a single lineage at the beginning (as opposed to the constant number of species in Nunney's model) and transition rates evolve and are therefore potentially subject to selection. Mutations can change the rate of transition to asexuality at each step, in contrast to Nunney's model. The transition rate to asexuality reflects the cost of sex: it represents the probability that a parthenogenic female appears within the species and rapidly invades the sexual populations. i at each generation N is calculated from the transition rate i (i−1) in the previous generation (N−1) as follows:uσ. There is no direct selection at the individual level on this transition rate.The transition rate Ap, Sp, Ae, Se, uσ and initu, where initu is the initial rate of transition assigned to the first sexual species at the start of the simulation.Simulations were run for 2000 generations, with various values for Some relationships and constraints were assigned to the parameters presented above. The speciation rate of asexual lineages was assumed to be twice that of sexual lineages α indicates the magnitude of the difference between the extinction rates of sexual and asexual lineages. We performed simulations for α = 2 and α = 5.Speciation is assumed to be easier in asexual lineages because they do not experience recombination that is predicted to prevent speciation by breaking the adaptive combination of alleles Sp > Se), to ensure that not all lineages became extinct before the end of the simulation.Finally, we restricted the analysis to situations in which the speciation rates of sexual lineages were higher than their extinction rates , we assigned the same value to its descendants, which were thus also unable to give rise to asexual descendants. This choice was based on the observation that sexual species that appear today to have a zero probability to lose sex due to strong constraints will probably not be able to lose sex on the long term either. Humans for example will very unlikely become ever asexual naturally.This second model was inspired from Darwin's diagram in “The Origin of Species” and is presented in The outcome of this competition depends on the similarity of the lineages in terms of their ecological traits, as indicated by a value on an axis, regardless of the sexual state of the lineages. Basically, if two lineages have very similar values for these traits, they have a higher probability of extinction. This illustrates the classical niche competition hypothesis, according to which, two species with similar niches cannot coexist.The persisting lineages can give rise to descendant lineages at the next generation, with a probability corresponding to their speciation rate. The number of descendant lineages is sampled from a Poisson distribution (see the parameters section).An ecological trait value is assigned to each new lineage, based on both the value of the parental lineage and some random evolution of this trait.Ld and Rd are the distances to the nearest neighbours of the trait considered, to the left and to the right, respectively, on the horizontal axis . If the lineage has no neighbour on the right or left (i.e. if it is the last lineage on the left or on the right of the axis), the corresponding fraction 1/d is removed.Competition between lineages controls the probability of extinction of each lineage, in each generation. It is dependent on the distance between the ecological trait value of the lineage and those of its neighbours to the left and to the right on the horizontal axis. The equation giving the probability of a lineage becoming extinct is:λ, which is different for sexual (Sλ) and asexual (Aλ) lineages. We set Sλ to Aλ/2, to reflect the demographic advantage of asexual lineages, which are therefore more likely to generate larger numbers of daughter lineages, due in particular to the reproductive isolation induced by the absence of sex. A larger number of separate lineages would therefore be expected than for sexual lineages, in which gene flow occurs.The number of lineages descending from each lineage is chosen randomly from a Poisson distribution with parameter x and is calculated as follows for each lineage i at each generation N:Pi represents the parent of lineage i and σε is sampled from a normal distribution with a mean of 0 and a standard deviation of σ. The value of σ differs for sexual (Sσ) and asexual (Aσ) lineages. The long-term advantage of sex is included in the model by assuming that Sσ > Aσ, implying that sexual lineages can evolve more rapidly to new environments.The trait value is 0 at the beginning of the simulation. This trait is called initu is set at the beginning of the simulation for the first  sexual linage. Each time a new lineage is formed, the probability of its becoming asexual is S→Ap  = u and the probability of its remaining sexual is S→Sp  = 1 – u. The probability of a reverse transition A→Sp is set to 0 so that A→Ap  = 1. The transition rate u evolves at each speciation event. It is calculated as follows for each lineage i at each generation N, as long as the lineage remains sexual:Pi represents the parent of lineage i and uσ.An initial transition rate As for the first model, for lineages reaching a transition rate of zero , we assigned the same value to its descendants, which were thus also unable to give rise to asexual descendants.Refer to Sp set to 0.002, 0.003, 0.004 and 0.005 and Se set to 0.001, 0.002, 0.003 and 0.004, applying the constraint that Sp had to be higher than Se. We tested values of initu and uσ between 0 and 10−2 .For the model without competition, we simulated 2000 generations for each set of parameters and carried out this operation 20 times. We performed simulations with Sσ fixed at 100 and Aσ varied between 10 and 100 . We set Sλ to 3 and Aλ to 6 and we tested four different values for initu and uσ: 0, 10−5, 10−3 and 10−1. We carried out 10 simulations for each set of parameters.For the model with competition, the simulations were as follows: 500 generations, For clarity, Sp, x-axis) and extinction  rates, initial transition rate initu and the rate of change of the transition rate uσ are summarised in α = 2 , the success of sexual lineages depends solely on the initial transition rate initu and the extinction and speciation rates for sexual lineages . The effect of the initial transition rate on the success of sexual lineages is clear in conditions in which the difference in extinction rates between sexual and asexual populations is not too large  seems to be associated with a decrease in the proportion of sexual lineages at the end of the simulation. This is expected, as an increase in the initial transition rate implies an increase in the probability of a sexual lineage giving rise to asexual descendants. The effect of the speciation and extinction rates of sexual lineages is also very clear when we consider the situation in which α = 2, uσ  = 0 and initu  = 10−4. When the speciation and extinction rates of sexual lineages are similar , the percentage of sexual lineages at the end of the simulation is higher than in conditions in which the rate of speciation is higher than the rate of extinction. For example, for α = 2, uσ  = 0 and initu  = 10−4, only 25% of the lineages are sexual at the end of the simulation if the speciation rate of sexual lineages is 0.005 and their extinction rate is 0.001, versus 86% if sexual lineages have an extinction rate of 0.004.If the transition rate cannot evolve  and the speciation rate (Sp  = 0.004) of sexual lineages, and for an initial transition rate (initu) of 10−3 and a rate of change in transition rate (uσ) of 10−5. We varied the value of α from 1 to 20, by steps of 0.2. We performed 100 repetitions for each value of α, allowing the simulation to run for 3000 generations. A value of α = 4 indicates that the extinction rate is four times higher for asexual lineages than for sexual lineages. The mean percentage of sexual lineages at the end of simulations was found to be a function of the value of α , sex may be maintained if the initial transition rate is low or if the initial transition rate is high but can change.α = 10 over the 3000 generations. Transition rate seems to decrease over time, with lineages having too high a transition rate being transformed into asexual lineages that eventually become extinct due to their higher extinction rate.We demonstrate the importance of changes in transition rate for the success of sexual lineages in This new model provides access to the same information as the previous one: percentage of sexual and asexual lineages at the end of simulations, and transition rate over the simulations. It also involves the recording and visualisation of additional relevant biological features, such as the timing of the transition events and the number of sexual and asexual forms present in the past. Graphical representations of the trees are also produced, in which each line links an ancestral form to its descendants .initu) on the success of sexual forms with a fixed rate of transition (uσ  = 0) , the probability of transition is high enough for such transitions to occur many times, and the asexual forms predominate after 300 generations.We first looked at the effect of different initial transition rates (σu  = 0) . We set rd tree in uσ) appeared to increase the proportion of sexual lineages. Indeed, the proportion of sexual lineages at the end of simulations increased with increasing uσ. This may be because the sexual lineages lost the ability to become asexual rapidly, such that the asexual lineages did not have time to expand. Furthermore, as asexual lineages had a higher extinction rate , sexual lineages eventually predominated, as such lineages tend to lose their ability to become asexual, whereas the asexual lineages that emerged at some point in the simulation eventually become extinct. As an illustration of this process, we generated a plot similar to those presented in If we begin with a set of parameters favourable to asexual forms , and the rate of change of the transition rate (uσ).initu = 0.001, Aσ  = 60, 30% of the lineages are sexual at the end of simulation), increases in the rate of transition may lead to a final proportion of sexual forms at the end of the simulation of more than 80% , the easier it was for sex to be maintained. This is intuitive but had not been explored before, and corresponds to Darwin' unique figure in “The Origin of Species”. Empirical studies have in fact shown that asexual and sexual closely related populations or species differ in ecological niches, with a wider niche for sexuals ). These studies however remain scarce while our simulations indicate that it could be an important feature to explore.Our study reinforces the view, still too poorly recognized, that the maintenance of sex can be explained by lineage selection, based on long-term advantages. Our model indeed allowed the maintenance of sexual lineages without the need to assume the existence of short-term advantages. We show that lineage selection can favour the gradual evolution of low rates of transition to asexuality and that the specificities of ecological competition between asexual and sexual lineages influence the possibility of maintenance of sex. Indeed, the greater difference in the ability of sexuals versus asexuals to diversify and occupy more diverse ecological niches . Many examples of short-term advantages of sex unrelated to recombination making it difficult for an organism to lose sexual reproductive function have been described, including the formation of dispersal structures, eggs or resistant spores et al.We do not claim that there are never short-term advantages to recombination in particular conditions or lineages, and we agree with West 6. This means that, to be maintained at the same frequency across 20 generations, asexual lineages need to have short term disadvantages by the same factor 106, due for instance to maladaptation to parasites or the accumulation of deleterious mutations. Asexual lineages are recent, but have nonetheless existed for many thousands of generations Three main lines of evidence can been put forward to support the idea that lineage selection is responsible for the maintenance of sex. First, most asexual lineages appear to be recent, as shown by their “twiggy” pattern in phylogenetic trees The hypothesis that lineage selection can account for the maintenance of sex could be tested by investigating whether asexual lineages indeed have higher extinction rates and are younger than sexual lineages. Indeed, if clonal lineages experience higher extinction rates, then there will be an automatic sorting (i.e. a selection) of lineages. If there is variation in the rate of transition to asexuality, this sorting will result in an over-representation of lineages with a low probability of losing sex. It is widely accepted that clonal lineages have higher extinction rates, but this hypothesis has been surprisingly little tested. Some recent studies have suggested that the hypothesis of a “twiggy pattern” of asexual lineages should be tested against a hypothesis of neutral processes The idea that recombination has huge long-term benefits is so widely accepted that ancient asexual lineages, such as the bdelloid rotifers, have been called “evolutionary scandals” th century organismal complexity was built), but rather to allow the diversity of selective processes at the different scales of the tree of life “to sing” (that is to understand how the complexity of the tree of life was built).Several major scientists with strong personalities have contributed to this debate. Stephen J. Gould made the point that the rejection of selection at species level was a firmly established feature in the ideas of most of the leading figures in evolutionary genetics of the late 20We are therefore unlikely to be able to develop a complete understanding of the maintenance of sex in eukaryotes if we do not consider all levels of selection. Indeed, one possible answer to the question “Why do aphids reproduce sexually?” would be “because they cannot produce resistant forms in any other way”. However, the most correct answer to the question “Why do most species reproduce sexually?” may be “because all the other species became extinct”. Both answers may be correct, but these two questions do not address the same level of selection. Ideally, if we wish to obtain a comprehensive understanding of the maintenance of sex in eukaryotes, we need to identify all the short-term causes in each species, together with the long-term forces in action. However, it is becoming clear that we do not necessarily need to know all the proximate causes, whereas ignoring even one final cause would greatly damage our overall understanding of the evolutionary problem.Lineage selection can occur as soon as the different lineages become isolated from each other, such that lineages cannot be invaded by the individuals with short-term demographic advantages that have invaded other lineages. The finding that so many different proximal causes seem to be responsible for the maintenance of sex in different lineages strongly suggests that a higher level of selection is at work, consistent with the widely accepted view that recombination has long-term advantages.http://lineage-selection.cgenomics.org.The C source code for the second model and a browser for exploring and visualizing all the trees generated with this model is available online at


The implementation on a BLDC motor demonstrates the effectiveness of the design approach.The variable sampling rate system is encountered in many applications. When the speed information is derived from the position marks along the trajectory, one would have a speed dependent sampling rate system. The conventional fixed or multisampling rate system theory may not work in these cases because the system dynamics include the uncertainties which resulted from the variable sampling rate. This paper derived a convenient expression for the speed dependent sampling rate system. The varying sampling rate effect is then translated into multiplicative uncertainties to the system. The design then uses the popular  The brushless DC (BLDC) motor applications are getting popular due to their efficient operation and high power density characteristics. Because of its high efficiency and easy maintenance, the application range has been found from large system like electrical vehicles to small systems like computer peripherals. The BLDC motor usually uses hall-effect sensors to determine the commutation timing among the armature windings. The same hall sensor signal is also used to measure motor speed. There are many commercial industrial driver ICs which apply such kind of design.BLDC motors may be standard application, but ones still encountered many difficulties when engineers try to implement in precision speed control. As mentioned before, the typical BLDC motor feedback signal is from the few hall-effect sensor signals which originally are designed for commutation purpose. The time interval between the consecutive feedback signals depends on the motor speed. Notice that BLDC motor is used in many high performance industrial applications like computer hard disk drive and the optical drive servosystems. High performance controller design is necessary. The BLDC motor system design maintains high resolution speed measurement through high frequency clock. The traditional driver design uses current transducer to achieve constant current sampling for the acceleration loop; however, the velocity loop has to be based on the speed dependent sampling, and some sort of robust variable sampling rate controller design is thus desirable.Most previous researches on multirate sampling systems in the literatures have focused on multiple but fixed sampling rate problems –6. If onIn the preceding work , the autμ-synthesis can now provide a necessary and sufficient condition for the controller design. The simulation and the experimental results demonstrate very satisfactory results from this method.In this paper, the authors realized that the variable sampling rate system model derived for the observer can be reduced to describe the variable sampling rate system itself. Instead of treating the system with a prespecified sampling rate, one can look at the system only when the feedback is available and lump the effects of the variable sampling time into system uncertainties. It is now possible to describe the system with the standard linear fractional transformation. The k, and the other one describing the measurement update is denoted by i. Therefore, as mentioned above, the i sampling system has a variable sampling rate. Consider two time scales, the underlying system sampling rate is denoted by t, the system could be described asA = eFΔt, B = ∫0tΔeFτGdτ, F ∈ ℜn×n is the continuous-time system matrix, G ∈ ℜn×1 is the continuous-time input matrix, n is the system order, and C ∈ ℜn1× is the same in the discrete-time system as in the continuous-time system.If the basic sampling time for the system is Δu(k) is known at every instance. This is a valid assumption because the system model can be measured with various system identification processes, and the control input is from the known controller. For the design purpose, assume that the system model is precisely known, and the manipulated input ki” stands for the instance when the ith measurement is available. “mi = ki+1 − ki” stands for the number of fast samples from the instance “i” to the instance “i + 1.” The control law is updated only when the measurement is available; therefore the system description becomes the following equations:Consider that “i, (u(ki + j) = u(i), for ki ≤ k < ki+1 in our case; if one considers the (i + 1)th measurement instance, then the system becomesj=0mi−1Ami−j−1B. One can treat the power of A's as an uncertainty term. Then, the system block diagram representation could be shown in Thus, for every instance i,  could be becomesx(i+1)=Ammi is the number of underlying samples between two consecutive feedback measurements. This is basically an unknown value depending on when the next feedback measurement is available. In the case of BLDC motor driver, the number of samples between measurements depends on the speed of the motor. When the motor is still, mi may be infinite, but the number should be small when the motor is running at high speed.Note again that Ami−1 and ∑j=0mi−1Ami−j−1 are the uncertainty terms influencing A and B, respectively. As discussed before, A is the system matrix and is completely known; however, mi cannot be determined before the next sample measurement is available. Notice that the A matrix at this stage is still arbitrary. The authors have not yet discovered any way to further decouple the power of A. Therefore, for the moment it seems the best way to treat this variation as a lumped uncertainty. The uncertainty setup is still clearly structured; for the terms Ami−1 and ∑j=0mi−1Ami−j−1 each affects different part of the system. The structured singular value control synthesis is therefore the best suited design approach to arrive at a less conservative design.The system in μ-synthesis framework, the system uncertainties are represented in their magnitudes:To facilitate the The control system can be represented with the familiar linear fractional transformation (LFT) as in A and ΔAB affect the constant matrices A and B, respectively. Therefore, there is no need to consider the number of right half plane poles in these cases. The constants βA and βAB are scalars to represent the magnitudes of the singular values A and the power of A, the maximum singular values of Ami−1 still need to be calculated for each mi to determine the worst case singular values. The same is true for ∑j=0mi−1Ami−j−1, and the calculation has to be carried out for every mi. Notice that the structure and the variation of the uncertainties are quite certain in this case. One needs only a step by step multiplication to understand all the powers of A, the possible uncertainty and the worst case uncertainty. WP in The uncertainties ΔThis experiment uses a BLDC motor from Troy Co., Taiwan, and the TMS320F243 from Texas Instrument for the controller implementation. The driver uses a simple protection circuit to read the hall sensor signal and determine the rotor angle and the rotor speed. Another protection circuit is used to drive the six MOSFET switches for the three-phase winding. The controller determines the proper on-off sequence and the controlled PWM duty cycle for the MOSFET to achieve the control purpose. The schematics of the control circuit and the physical setup are shown in The capture unit built in the TMS320F243 is used to detect the hall sensor signals and determine the rotor angle. The proposed driver also uses these signals to determine the proper MOSFET to turn it on. There are shunt resistors for constant rate sampling of the phase currents, but the motor speed measurement still depends on the hall sensor signals. The time interval between two consecutive hall sensor signals is used to determine the rotor speed. The sampling rate of the feedback system is thus speed dependent. In this experiment, the underlying sampling frequency for the TMS320F243 is set at 4 KHz. And the PWM module built in TMS320F243 is set to generate PWM waveform whose carrier is also set at 4 KHz. In an other site, there are 12 hall sensor signal updates per revolution for a 4-pole permanent magnet arrangement. Therefore there would be 66 samples between measurements when the motor runs at 3000 rpm.The filtered signal can serve as the bases for system identification and for the control feedback. The identified system transfer function ismi, between feedback measurements would range from 11 to 66. Now it is necessary to calculate the singular values of the powers of mi = 11~66 and to obtain the expressions for βA and βAB. Again, there are 12 measurement updates per cycle. For the motor operating at 300~3000 rpm, the sampling frequency would have range of 60~600 Hz. The sampling frequency for the underlying control is 4 KHz. Therefore the number of sampling periods, μ-synthesis tool box for the control synthesis. The simulation also uses the bilinear transformation to convert the discrete-time system for continuous-time simulation environment. The performance bound is set to the transfer function in μ value come below 1 before the bound is imposed. The iteration procedure becomes fairly easy when the bound is imposed.The experiment then uses the MATLAB The synthesis resulted in a 6th order controller with the following form:The controller calculates a PWM duty cycle in the actual system. Inherently, there is no negative command from the duty cycle. There is a separate logic for the reversed rotation, but it will not be discussed here. The lack of negative command would result in slower recovery from the overshoot. The MOSFET switches also impose limits on the output current. The TMS320F243 is a 16-bit fixed-point digital signal processor. There are special considerations when implementing high order control algorithm. The compensator is first decomposed into fractional expansion in The first two terms in μ-synthesis procedure is introduced to calculate the controller. The experimental verification is based on the TMS320F243 digital signal processor. The experimental results over different speed settings agree with the original design specifications and have demonstrated the effectiveness of the proposed method. This paper proposed a controller design for systems with variable sampling rate. By variable sampling rate, the authors mean a system with undetermined sampling frequency or speed dependent sampling frequency. The paper first presented the system modeling for the variable sampling rate system. The changing sampling rate was then translated into system uncertainties. The uncertainties appear in different locations in the system with fixed structure; therefore, the popular