Cerebral Palsy (CP) is the most common cause of motor disabilities in children and young adults and it is also often associated with cognitive and physiological challenges. Climbing requires a multifaceted repertoire of movements, participants at all levels of expertise may be challenged functionally and cognitively, making climbing of great potential interest in (re)habilitation settings. However, until now only few research projects have investigated the feasibility of climbing as a potential activity for heightening physical activity in children with CP and the possible beneficial effects of climbing activities in populations with functional and/or cognitive challenges. The aim of this study was therefore to test the feasibility of an intensive 3 weeks indoor-climbing training program in children with CP and typically developing (TD) peers. In addition we evaluated possible functional and cognitive benefits of 3 weeks of intensive climbing training in 11 children with cerebral palsy (CP) aged 11–13 years and six of their TD peers.The study was designed as a feasibility and interventional study. We evaluated the amount of time spent being physically active during the 9 indoor-climbing training sessions, and climbing abilities were measured. The participants were tested in a series of physiological, psychological and cognitive tests: two times prior to and one time following the training in order to explore possible effects of the intervention.p < 0.01), increased rate of force development in the least affected hand during an explosive pinch test and increased muscular-muscular coherence during a pinch precision test (p < 0.05). We found no improvements in maximal hand or finger strength and no changes in cognitive abilities or psychological well-being in any of the groups.The children accomplished the training goal of a total of nine sessions within the 3-week training period. The time of physical activity during a 2:30 h climbing session, was comparably high in the group of children with CP and the TD children. The children with CP were physically active on average for almost 16 h in total during the 3 weeks. Both groups of participants improved their climbing abilities, the children with CP managed to climb a larger proportion of the tested climbing route at the end of training and the TD group climbed faster. For the children with CP this was accompanied by significant improvements in the Sit-to-stand test  contains supplementary material, which is available to authorized users. Cerebral palsy (CP) is the most common cause of motor disabilities in children and young adults and it is also often associated with cognitive and psychological challenges , 2. As aPerforming intensive sports activities may facilitate the motor and cognitive development of the children and ensure a strengthening of their social integration, and should therefore be made available for children with CP.Climbing involves strength, endurance , posturaBecause climbing on artificial climbing walls at the same time can be considered safe in terms of number of injuries pr. 1000 h of activity , 14 and http://www.ep-uk.com/configurations/disabled_climbing.html). The children had to understand the intention of moving against gravity by climbing, and the fact of falling by losing the grips. Fear of high, severe joint deformities, acute inflammatory joint diseases or increased risk for bone fractures needed further evaluation if climbing is an optimal sport for these children.Climbing, can be a sports activity for almost every child, including children with physical disability and cognitive deficit. For children with severe physical disability specific climbing platforms has been developed  and without (N = 6) cerebral palsy were recruited to this non randomized, non-blinded feasibility and intervention study. The children with CP were recruited according to their CP diagnosis regardless of CP type or functional level. The only exclusion criteria was severe functional dysfunction that made climbing impossible. Four of the participants knew each other prior to project . Seven of the children with CP attended mainstream school and 4 attended a school with a special program for children with physical and cognitive disabilities. All children in the TD group attended normal school.Seventeen children from the age of 11 to 13 years with . The study was performed in accordance with the Helsinki Declaration.The participants were recruited through online advertisements. All the participants were unfamiliar with climbing training as an organized sport, a few of the participants had tried an indoor climbing wall before, but none were regular users.Table Neurological examinations were made of all participants by an experienced physiotherapist specialized in child neurology. This examination included test of reflexes, muscle tone, muscle strength, range of motion and gross motor function. History of surgery for in relation to treatments and current antispastic drug use in children with CP were recorded (see under Results).2 of climbing walls with walls and routes up to 12 m of height. The gym was reserved for this particular purpose during the hours that this intervention took place, which meant that the instructors and participants were the only ones present in the gym during the intervention. The gym was formerly the main climbing gym for the Danish Mountain and Climbing Club, but the club had moved to new larger (2000 m2) facilities. This older and smaller gym was mainly used for climbing activities with school classes. The first climbing day was performed in the larger facilities of a commercial climbing gym belonging to the Danish Mountain and Climbing Club, where the participants were provided with climbing shoes, which they used throughout the climbing period. During the first day other climbers were present in the commercial climbing gym. Nine days of climbing were planned within a period of 17 days, week 1 & 2: climbing on Monday, Tuesday and Friday, week 3: Climbing on Monday, Tuesday and Wednesday. All climbing was performed in the afternoon after school hours. Each climbing day consisted of approximately 2.5 h of physical activity. The 2.5 h were each time split into approx. 30 min of warm up exercises specifically focused on climbing. Subsequently the participants were split into two half with an approximately equal number of children with CP and TD participants in each group, where one group started with bouldering exercises, The other half of the participants were engaged in wall climbing with a top rope as safety. In order to avoid long waiting times during the top rope session, 3–4 instructors were allocated for the top rope practice, and one instructor was responsible for the bouldering session. After approximately 1 h, the two groups swapped activity. Each climbing day ended with a wrap up where the participants told what they had learned during the day. The intervention was planned as a sports activity, where the focus for the participants was to improve their climbing performance and experience that they learned new skills. Without focusing on specific needs of the individual participant with disabilities, we hoped that the focus of climbing performance also would give rise to therapeutical benefits, but intentionally without focusing on these specifically. This intentional lack of focus on therapeutical benefits was made, because we believe such a focus would remove the intrinsic enjoyment of climbing when applying external goals, see for instance ) .x and y. Power spectra were constructed from sections of data taken at a fixed offset time with respect to a trigger point in each trial. Estimates of the power spectra were constructed by averaging periodograms across all trials. Fxx(λ) and fyy(λ) represent the Fourier transforms of processes x, and y, at frequency λ. The cross spectrum between x and y is denoted by fxy(λ), and is estimated in a similar manner. Two functions were then used to characterize the signals’ correlation structure: coherence, and cumulant density. Coherence estimates are bounded measures of association defined over the range . The cumulant density provides an unbounded time-domain representation of the EMG-EMG correlation structure analogous to the motor unit cross-correlogram and is defined as the inverse Fourier transform of the cross-spectrum [APB-EMGFDI analysis except that the data was generated during a 2 min static contraction and not a concatenation of several epochs of static contraction.Normalized signals are assumed to be realizations of stationary zero mean time series, denoted by spectrum . For thespectrum . The intspectrum . EMG froForce was measured using a hand held pinch device with a strain gauge providing voltage output corresponding linearly to the applied isometric force. Force onset was calculated using an initial visual inspection of the force measured. A time point prior to visible force onset was identified and a stable force baseline 500 ms prior to that point was calculated. Actual force onset was then calculated as the time point after the visually identified point that exceeded 2 times standard deviation of the 500 ms force baseline.All MVC data was exported from Signal as Matlab files and analyzed using Matlab 2014b. A customized Matlab script provided information regarding the peak values of the participants’ trials to each experimental session . The trial with the highest MVC value for the left and right hand respectively to each of the sessions was used for further analysis. The peak magnitude of the MVC data, originally measured in Volt, was converted into N.RFD data was converted into Matlab files and analyzed in Matlab 2014. A customized matlab script was designed and the RFD performed by the voluntary contraction of the thumb and index finger was calculated at different time points  following onset of contraction . RFD for each participant  was first analyzed by visual inspection and all trials contaminated by activity prior to the actual contraction (resulting in upward deflection of baseline force) were discarded. The RFD for the retained trials was then obtained from the slope of the force-time curve (Δforce/Δtime) between time periods of 0-30 ms, 0-50 ms, 0-100 ms, and 0-200 ms relative to the onset of contraction, which is used to express the rate of rise in contractile force . The tri(RStudio Team (2015) RStudio: Integrated Development for R. RStudio, Inc., Boston, MA URL http://www.rstudio.com/.) for further statistical analysis. A linear mixed model was used to investigate the effect of the climbing intervention on coherence by testing the specific interactions between group and intervention time and the differences between the values for pretest2 vs. the posttest in the CP and TD group. The large variability between participants was accounted for by including the ‘participant’ in the model as random effect, reparametrized interaction between group and time as fixed effects and age, height, gender as fixed effects. For all linear mixed models, Normal Quantile-Quantile plots were made, and analyses were rejected if these did not visually display an approximate linear relationship. In addition, residual plots were inspected visually in order to ensure near normal distributions.Analysed data from the different tests were imported into R-studios In order to test specific hypotheses concerning the effect of the climbing intervention on the two groups we tested differences between pretest 2 and postest in the children with CP and in the TD group, and in order to test for unspecific learning effects due to multiple exposure to the same test, we tested the differences between pretest 1 and 2 in the children with CP and in the TD group.Correlations were tested using a Pearson’s product-moment correlation.o) p < 0.1, as well as statistical significant results *) p < 0.05, **) p < 0.01, ***) p < 0.001 are indicated when tested.Statistical trends Four children with CP had tendon lengthening surgery 2–6 years prior the intervention and no children received antispastic medication.All children with CP were diagnosed with bilateral CP except one participant that was diagnosed as unilateral CP. All children with CP were ambulant and categorized with GMFCS 1 except in one case with GMFCS 2. No walking aids were used by any of the children with CP but in 6 of the children ankle orthoses were used during daytime, but not during the climbing training. No hand or elbow orthoses were used in the group of children with CP.The neurological examination revealed generally reduced ability to produce force bilaterally in dorsal flexors, plantar flexors, elbow flexors and elbow extensors. Spasticity was identified  in the plantar flexors in 7 cases and elbow flexors in 3 cases. Reduced range of motion was identified  in the ankle joint in 6 cases whereas no reduction in elbow was found in any of the cases.The TD children all had normal age related strength, normal range of motion in all joints-, reflex activity and walking ability.The primary outcome of this study is that children with CP perform an equal amount of physical activity as their TD peers without having to perform training interventions that are specialized for children with physical disabilities.Table Nine of the 11 children with CP took part in all 9 climbing days, one child with CP missed 1 day due to a logistic problem, another child with CP cancelled one climbing day due to a school arrangement. Three of the TD participants took part in all climbing days and three took part in 6 of the 9 climbing days. Cancellations in the TD group were due to illness, logistics or other arrangements, not lack of interest in the training.Using the individual’s own time spent doing physical activity, which on average was 1:45 h. We calculated the total time of physical activity performed during the climbing sessions. The children with CP performed on average 16:05 h of physical activity during the climbing specific training and the TD group performed on average 13:01 h of physical activity taking into consideration how often each individual was present at the training. The IPAQ questionnaire disclosed that all children except two (one child with CP and one TD) made the climbing training as addition to their normal activities.The measures of the performance during the climbing intervention test climbing route is summarized in Table During a warm up play on one of the last days of climbing one of the TD participants accidentally sprained the wrist in a collision with one of the other participants.p < 0.01). In order to investigate whether the improvement in Sit-to-Stand was related to climbing performance, we performed correlation analyses of the change in Sit-to-Stand with the change in climbing performance (Climbing speed and fraction of route climbed) from pretest 2 to posttest in the children with CP. Neither of the two correlation analyses revealed significant relationships. No significant improvements were observed in the Romberg 30 s tests. Further details can be found in the Additional file Significant improvement was seen after the climbing intervention in number of times the children with CP were able to perform a sit-to-stand movement , 0-100 ms (p < 0.01) and 0-200 ms (p < 0.05). No significant improvements were found for whole hand-and pinch strength (MVC) for any of the groups. Further details can be found in the Additional file We found significant improvements in the children with CP for the pinch grip RFD calculations of the least affected hand as a result of the climbing intervention in the time intervals 0-50 ms  increases in the children with CP, when tested in a mixed linear model.Results of the EMG-EMG coherence analysis for m. FDI and m. APB are summarized in Fig. 2 difference confirms a slight, above the confidence interval, increase in EMG-EMG coherence for the children with CP, especially around the 20 Hz bin. To further investigate the changes in coherence from pretest2 to posttest we calculated the log normalized area under the coherence curve and tested it in a mixed linear model in the 15–30 Hz beta band. This confirmed the trend  of coherence increases in the children with CP, but did not obtain significant difference.Results of the EMG-EMG coherence analysis during the 2 min of static ankle dorsi flexion are summarized in Fig. p < 0.05) for the children with CP but not for the TD group. No significant differences were observed in ankle joint stiffness for any of the groups. Further details can be found in the Additional file Unfortunately the PSAD device did not work properly on Pretest2 and we therefore only compared the measurements of ankle joint stiffness and ankle joint range of motion from Pretest1 with the Posttest. These measurements showed a significant improvement in range of motion (ROM) , in a 1-back working memory task, in a continuous paired association learning task and in a maze learning task.The linear mixed model of the combined detection score did not reveal a significant difference between the pretest 2 and post test in the cardbased CogState computerized tests  or the spatial location based tasks  in any of the groups. Further details can be found in the Additional file The results of the self-reported questionnaire “Sådan er jeg”, where the participant should self-report their experience of themselves in 72 questions in 5 different domains did not show significant differences in any of the groups when comparing pre2 and post. Further details can be found in the Additional file We have shown that children with CP can engage in climbing training on equal terms with their TD peers. The children with CP performed equally many hours of physical activity as their TD peers, which amount to more than 5 h per weeks of moderate or vigorous physical activity.Furthermore, four children with CP and three of the TD children and their parents expressed a wish to continue the climbing training after the 3-weeks ended. On a side note it is also worth mentioning, that three of the children after the 3-week period continued climbing as an organized sport in a local climbing club.In the present study we have shown that children in the age range of 11 to 13 with and without cerebral palsy can improve climbing abilities with less than 3 weeks of intense climbing training. Furthermore, we have demonstrated that some aspects of physiological measures as well as functional measures can be improved in children with CP with less than 3-weeks of climbing.In summary, children with CP improved the fraction of the test route climbed, but at the expense of producing more hand placement mistakes while climbing. The children in the TD group did not improve the fraction of route climbed but increased their climbing speed.With respect to functional improvements, children with CP improved their Sit-to-stand score, a measure used as a general measure of motor abilities.The physiological tests revealed a significant increase in EMG-EMG coherence in the pinch task for the children with CP after the climbing intervention. Furthermore, the children with CP increased RFD in their least affected hand after the climbing intervention and had increased range of motion in the ankle joint.None of the cognitive computer tests or the psychological questionnaire revealed any significant changes in either of the two groups when comparing before and after climbing training.This study extends upon previous studies of biomechanical and physiological effects of sport climbing , 46–50. We have shown that a short intense intervention period of climbing training can improve climbing performance in children with CP GMFCS level I and II. We have also shown that a very basic test of general functional performance reveals improvements after the climbing intervention. Furthermore we have shown that fine motor control of the index and thumb is changed as reflected by an increase in coherence in the most affected hand of the children with CP. In the least affected hand the children with CP improved RFD from 0 to 100 ms. Finally, we have shown that climbing training increase the ankle joint ROM in the children with CP measured on the most affected lower limb. These findings suggest that climbing training have physiological impact on both upper and lower limb physiology and that climbing training can improve aspects of general functionality as measured by the Sit-to-stand test.The psychological as well as cognitive tests did not reveal any significant improvements. All though climbing training has mental aspects related to management of fear and anxiety and cognitive aspects related to decision making and planning we were not able to see any effects on the employed measurements. This may be due to several reasons. The number of participants was rather small compared to the groups that the psychological test (Sådan er jeg) has been evaluated on.Climbing training seems to be a way to engage children with CP in many weekly hours of physical activity in a fun and motivating way that may help improve climbing skills, muscle strength, balance, as well as mental and social skills. Therefore, we propose climbing as an efficient clinical training method with several benefits for children with CP.This is also what we experienced during the intervention. Although the children had very diverse physical capabilities, they all found routes that suited their different skill-sets. With guidance from the instructors the participants could progress on these routes, often in collaboration with their peers from both the CP and the TD group. The mixture of top rope and bouldering provided a good foundation for the participants to try many different aspects of climbing, and for the instructors it increased the possibilities for setting up playful climbing games where the participants could interact independently of their physical foundation.The participants self-reliance in relation to their climbing abilities may have improved during the 3 weeks and that the immediate feedback that the nature of climbing provided (e.g. overcoming specific obstacles on the route), may have fueled the children’s pursue of their inherent enjoyment rather than external reward or influence. Some of the children even personalized the routes, as one of the children with CP expressed it: He wanted to overcome his ‘arch enemy’ . The participants had surprisingly few complaints about the level of their physical exertion, even though the sessions were relatively long and challenged the participants both physically and mentally. This may reflect that the climbing environment and the instructors encouraged the participants to develop the right mindset and self-determination for pushing their limits. In that context, it was enjoyable to see that the children with CP met the challenge with the same enthusiasm as the healthy controls, even though the training was not specifically adapted to the children with CP. Future studies could have even more focus on customizing the climbing training in order to accommodate the specific needs of participants with disabilities. Especially it could be of interest to investigate if climbing training can be used as an effective tool to treat people with even more severe CP. It should also be noted that the clinical significance of climbing has started to be recognized in other patient groups as well (e.g. in people with multiple sclerosis and geriatric patients) –56.Only very few studies have been conducted looking at effect of (therapeutic) climbing training in children and adolescents –23. We dWith respect to the intervention intensity, previous studies have used climbing once or twice a week for 8 weeks  for chilChildren with CP are often confronted with the challenge that they are not disabled enough to participate in adaptive sports, but fall short in traditional competitive sports, which can lead to a loss of motivation and self-confidence . ClimbinOne can hope that the climbing environment in the future can be an arena for children with CP where they can develop social, mental and motor skills alongside their healthy peers, without the fear of falling short of certain competitive standards. The use of climbing as a tool in physical therapy is still a largely uninvestigated area and therefore much more research should be done on the possible physical, psychological and social benefits of climbing.This study was designed as an exploratory study in order to investigate whether climbing training has the potential to be used as a possible therapeutic intervention, which combines physical and psychological challenges. In order to take into account possible exposure bias to the employed tests, we decided to compare two pre-intervention tests against the pre-post intervention comparison. We tested the differences between the pre1 and pre2 test and only found a significant improvement of the chase test in the CogState test battery for the TD group. All other tests did not show any significant differences between the pre1 and pre2 test. We therefore believe possible learning effect are minimal, and therefore differences between pre2 and post are more likely due to an interventional effect rather than an unspecific learning effect. Some of the employed tests, such as the questionnaire “Sådan er jeg” and the CogState tests may be very general, and not specific enough to test effects of the intervention. In particular we believe the “Sådan er jeg” questionnaire, after having employed it, is not suited for these short term intervention studies. With respect to the CogState tests we hoped to see effects on the MAZE-LEARNING test because the skills that the test resembles are similar to what you learn when trying to memorize a climbing route. The remaining tests in the CogState battery were less climbing-skill related.The study was designed with the purpose of exploring whether children with CP can engage in climbing training together with their peers and explore possible physiological, psychological and social gains from climbing training. To explore these parameters in the most reliable way, practical issues from the surrounding activities, such as getting to the gym, organizing climbing coaches, etc. was taken care of by the experimenters and not the participant’s and their parents to the same degree as “normal” climbing training would require. Therefore, the surrounding circumstances were somewhat easier to cope with for the participants and their parents. We employed a relatively high instructor-to-participant ratio with 3–5 instructors to 17 children. This was done because we wanted to focus on both bouldering and top rope climbing, where the latter requires someone who can belay (hold the rope through a breaking device) the climber. Under normal circumstances, part of a longer climbing training course, belaying would be something that was taught the participants in the beginning or taken care of by parents. But because we wanted to focus on climbing as a physical activity (with it’s related aspects) we did not teach belaying techniques during the 3-week session. Furthermore, the activities took place in the early afternoon after school, where the parents of the children were still at work. It has to be considered that the instructors were highly motivated towards the intervention and already prepared that special precautions had to be made in order to properly adjust the climbing training to the children with CP. We therefore acknowledge that this fact potentially could bias the effect of the intervention to favor the children with CP, but when an instructor deals with a participant it would be impossible not to individualize the instruction without paying attention to the individual participant’s needs. At the same time the setting of the climbing intervention took place in a safe and inclusive environment, where the instructors facilitated interation between all participants. Therefore, in order to implement children with CP in real life climbing settings, a challenge could be the recruitment of dedicated instructors that are willing to make an extra effort, preferably with some prior knowledge of CP. However, a specialized team has been established for 6 children with CP where they climb once a week in a public climbing gym. Further, the disability level of the children with CP in this study has to be considered. The Children in our study was (with one exception) level 1 on GMFCS scale and therefore had a reasonable level of functional capability. Therefore, another consideration is how well a climbing intervention like ours could be implemented for more severely challenged children with CP.Despite the very small group of participants included in this study, we were able to show some changes in the functional and physiological tests. However, given the large number of tests we performed none of the tests would survive strict Bonferroni correction for multiple comparisons. We have therefore decided to show all tests uncorrected in order to give a better impression of the raw results from the individual tests. We also decided not to correct for multiple comparisons because many of the tests cannot be considered completely independent. As an example, we found high correlations between performance in the computer based chase test and most of the other computer based cognitive tests. Also, tests like the four different RFD measurements are non-independent because they include data from the same time-interval.These findings show that it is possible to use climbing as a setting for peer socialization (climbing in a mixed group of children with CP and their TD peers) as a motivating and effective training alternative in children with CP. The improved motor abilities obtained through the training is likely reflected by increased synchronization between cortex and muscles, which results in a more efficient motor unit recruitment system that may be transferred to daily functional abilities. It was not possible with this short-lasting intervention to demonstrate larger muscle strength or cognitive or psychological effects of the training.


A network motif is a sub-network that occurs frequently in a given network. Detection of such motifs is important since they uncover functions and local properties of the given biological network. Finding motifs is however a computationally challenging task as it requires solving the costly subgraph isomorphism problem. Moreover, the topology of biological networks change over time. These changing networks are called dynamic biological networks. As the network evolves, frequency of each motif in the network also changes. Computing the frequency of a given motif from scratch in a dynamic network as the network topology evolves is infeasible, particularly for large and fast evolving networks.In this article, we design and develop a scalable method for counting the number of motifs in a dynamic biological network. Our method incrementally updates the frequency of each motif as the underlying network’s topology evolves. Our experiments demonstrate that our method can update the frequency of each motif in orders of magnitude faster than counting the motif embeddings every time the network changes. If the network evolves more frequently, the margin with which our method outperforms the existing static methods, increases.We evaluated our method extensively using synthetic and real datasets, and show that our method is highly accurate(≥ 96%) and that it can be scaled to large dense networks. The results on real data demonstrate the utility of our method in revealing interesting insights on the evolution of biological processes. Biological networks capture complex interactions among molecules  which perform various cellular functions . These nnetwork motif is a small subnetwork that occurs frequently in a given network  is a path that is shown in Fig. length of a path as the number of edges on that path. We define the distance between two nodes in a graph as the length of the shortest path which connects them.A u and v of graph G are connected if there is a path between u and v in G. Thus, a graph G= is connected if all node pairs u,v∈V in G are connected. Figure We next define the concept of connectedness of nodes and graphs. Nodes G=, we define the k-neighborhood of an edge e= as a subgraph G′=, where E′⊆E contains all edges which have at least one incident node whose distance to either u or v is at most k. The set of nodes, V′⊆V contains all nodes connected by E′.In a given graph G= and H= are isomorphic if there is a bijection function π:VG→VH such that edge ∈EG⇔(π(u),π(v))∈EH. Let us denote the graph in Fig. G. Figure P. There are four subgraphs in G which are isomorphic to P. These subgraphs consist of the following sets of edges {,,}, {,,}, {,,}, and {,,}. We denote these subgraphs with S1, S2, S3 and S4 respectively. We denote the set of all subgraphs in G which are isomorphic to P with We say that two graphs P represents the equivalence class Isomorphism is a transitive relationship. This indicates that all subgraphs in overlap if they share at least one edge. Otherwise, we call them disjoint. For example, in Fig. S1 and S2 overlap since they share the edge . Subgraphs S1 and S4 on the other hand are disjoint.We say that two subgraphs of a given graph frequency of a pattern P in a graph G is a function of the set of subgraphs of G which are isomorphic to P. We call these subgraphs as the embeddings of P in G. The fact that these subgraphs may overlap with each other has led to alternative definitions to the frequency function. The classic definition of frequency ignores overlaps between isomorphic subgraphs. It thus reports the frequency of a pattern as the cardinality of its equivalence class. This frequency function is denoted as the F1 count of a pattern [P in G. Therefore, the F1 count of pattern P in G is four. An alternative measure counts only the disjoint embeddings of a pattern. This frequency is denoted as the F2 count. For example, in Fig. F2 count of P in G is two . A more restrictive version of frequency definition of a pattern does not allow two embeddings to have a common node. This is denoted as the F3 count of a pattern. For example, F3 count of pattern P in G is one since node a is common to all embeddings in The  pattern . For exaG= such thatBiological networks are inherently dynamic structures. Their topologies change with time through network edit operations. There are four possible network edit operations that perturb the topology of a network, namely (i) node insertion, (ii) node deletion, (iii) edge insertion, and (iv) edge deletion. It is easy to represent node insertion (deletion) using edge insertion (deletion). For example, a node insertion can be represented as a set of edge insertions between existing nodes and the new node. Similarly, a node deletion is equivalent to deleting all edges connected to the deleted node. Formally we define a dynamic network as follows: a sequence of edge deletions and additions Gi represents the topology of the network after the ith edge insertion or deletion and G0=Gwhere We classify existing methods that are related to our work in two categories: methods for static networks and methods for dynamic networks. The methods on static networks can be broadly divided into two categories: (1) those that count motifs in a single network, and (2) those that count motifs in a set of networks. SUBDUE  and GREWThe meaning of frequency changes when we consider methods finding frequent subgraphs in a set of networks. If a subgraph exists in a network, the frequency of that subgraph increases by one, irrespective of the number of embedding found in that particular network. FSG , gSpan  and SPINF1 count of the patterns . Oth. OthF1 cs  with |S|. Recall that the F2 count of pattern P is the cardinality of the maximal set of embedding where two embeddings do not share edges. We denote such set with F2 count of P, we introduce the concept of an overlap graph, which is unique to P and G. Let us denote the overlap graph with Vo corresponds to an embedding of P listed in S. Let us denote the relationship between the nodes in Vo and the embeddings in S with a bijection function ϕ:Vo→S. Each edge ∈Eo indicates that the two embeddings ϕ(u) and ϕ(v) share at least one edge.Assume that we are given a motif topology denoted with u∈Vo with the smallest degree. If there are multiple nodes with the same smallest degree, we randomly select one of them. We insert the corresponding embedding ϕ(u) into u from Vo along with all the nodes v∈Vo, such that ∈Eo. We repeat this process to populate Vo becomes the empty set.We use the overlap graph to generate the maximal, non-overlapping embedding set G=. Also, let us denote the topology of the network after the ith edge insertion or deletion with Gi=. Thus, we have G0=G and ∀i≥0,|Ei−Ei−1|=1. Given a motif topology denoted with pattern M, we compute the F1 and F2 counts of M in the initial network G0 by using the method described in “G evolves , the count of F1 and F2 of M can change. Next, we will show an algorithm for efficiently updating the F1 and F2 counts as the network evolves from Gi to Gi+1∀i≥0. By repeatedly applying our algorithm, after each network edit operation, the motif count is updated for arbitrarily large sequence of network updates.Let us denote the given network with F1 of M as Gi evolves into Gi+1. We assume that F1 for Gi is known. Our algorithm for updating F1 relies on initially constructing and maintaining an auxiliary data structure that allows for the embeddings containing an edge to be efficiently queried. Thus, at the beginning of our algorithm, we find all embeddings of a given motif M in the initial network G0. After finding these embeddings, we create a list of embeddings for each edge e∈E, denoted as De, which stores all embeddings that contain e. That is, for a motif M, let m be an embedding in a given network. Then m∈De if e∈m. This data structure, which we refer to as the edge compressed bitmap, is updated each time an edge is either added or deleted. The F1 is then updated based on the edge compressed bitmap.We now describe our method for updating the count of Gi evolves to Gi+1 the e∈Ei is deleted. This reduces the F1 count of motif M, if the deleted edge is a part of embeddings of M. From the edge compressed bitmap, we find the set of embeddings of M which contain e. We remove this set (De) from the edge compressed bitmap and reduce the F1 count of M by the cardinality of this set.Suppose that as the network e∉Ei is added to Gi. Unlike the edge deletion, prior to this update, we do not know whether e is a part of an embedding of M in Gi+1. We locate such embeddings of M in Gi+1 as follows. Let us denote the diameter of M with k. We search the k-neighborhood of e in Gi+1. The set of embeddings of M which contains e can be formed with its neighboring edges. We add this set to the edge compressed bitmap and increase the F1 count of M by the cardinality of the set of new embeddings.Next, assume that an edge F1 count, we proceed to update the F2 count. Updating the F2 count is more challenging than updating F1 because computing the count of F2 is NP-complete [F2 count we compute even for a single static network may deviate from the optimal result. We would like to minimize the additional errors introduced by dynamic updates.After updating the complete  and the F1 and F2 counts of the given motif M in Gi and the F1 count of M in Gi+1. Next, we describe how we update the F2 count for Gi+1. There are following two possible scenarios: (1) an edge has been deleted from Gi, and (2) and edge has been added to Gi. In the first scenario, the removal of an edge e from Gi will cause the F2 count to either remain the same or decrease by one. The former case occurs when none of the embeddings in the set De contribute to the F2 count in Gi. The latter case occurs when one of the embeddings from the set De, contributes to the F2 count of Gi. Let us denote that embedding with X (X∈De). After removing e, the embedding X does not exist in Gi+1. This reduces the F2 count of M by one. However, it is possible that there is another embedding (say Y), which can be included in the F2 count for Gi+1 to replace X. For this to happen, Y must satisfy two conditions: (i) Y overlaps with X, and (ii) Y does not overlap with any other embedding included in the F2 count of M in Gi. If such an embedding Y exists, we include it in the F2 set. Thus the F2 count remains unaltered. Otherwise the F2 count decreases by one.First, we assume that we have already computed the Y that satisfies the two condition above, we explore the neighbors of X in the overlap graph. Recall that the neighbors of an embedding in the overlap graph are those embeddings of M which share at least one common edge with that embedding. If say, X consists of edges e1, e2 and e3 then the neighbors of X will be the union of sets In order to identify any embedding neighbors of X, we consider each embedding and check if they can be included in the updated F2 count. If an embedding Y in that set, has all of its edges free then we include it in the F2 set for Gi+1. Therefore, if such an embedding Y exists, the F2 count remains unaltered as the inclusion of Y compensates for the deletion of X. Otherwise we decrease the F2 count by one.From the set of e, where Ei+1−Ei={e} is added Gi.This addition will either increase the F2 count of M in Gi+1 by one or has no influence. The new edge can form new embeddings of M in Gi+1. We explain how we obtain such new embeddings in “F2 count. To do this, we consider each new embedding, and check if all of its edges are uninvolved in the F2 count (they could be involved in the F2 count with other embeddings). If such an embedding exists, we include it in the F2 set and increase the F2 count by one.Assume that an edge size and degree respectively. We generate undirected random networks using three generative models, namely (i) Erdös-Rényi (ER) [We use synthetic and real datasets to evaluate our method. First, we describe our synthetic dataset. We use synthetic networks to observe the performance of our method by varying a broad suite of parameters, namely the number of nodes, the average number of edges per node in the network and the topology model. We call the number of nodes in the networks and average number of edges per node as network nyi (ER) , ii) Watnyi (ER)  and (iiinyi (ER) . We expePost stem cell uses all the cell lines except the stem cells. (ii) Pre-pancreatic cell includes all cell lines but the pancreas cells. (iii) Pre-liver cell contains all cell lines except liver cells. We use the datasets (i) and (ii) above to build a pathway of two networks for pancreas cell development. We use (i) and (iii) to build a second pathway for liver development. For the first pathway above we construct two networks as follows. We compute the Pearson’s correlation between the transcriptions of all gene pairs in the two datasets ((i) and (ii)). We use two correlation thresholds; stringent cutoff set to 0.95, and a relaxed cutoff set to 0.75. Assume that the post stem cell dataset is the primary and the Pre-pancreatic cell dataset is the secondary dataset. Given a pair of genes u and v we include an edge between them in the primary dataset network if one of the two conditions hold: (1) correlation between u and v within the primary dataset is above the stringent cutoff, or (2) correlation between u and v within the secondary dataset is above the stringent cutoff while that within the primary dataset is above the relaxed cutoff. We then construct the network for the Pre-pancreatic dataset similarly by assuming it to be the primary and the post stem cell dataset as the secondary one. We repeat the same process to construct another sequence of two networks for the liver pathway. The details of the three simulated networks and two pathways are summarized in Fig. Next, we describe our secondary dataset. We generate gene co-expression network using the transcription data from Rivera-Mulia et al. . This datoff, or  correlata,b) and , such that they share no nodes. It then replaces these edges with  and . Thus, one edge shuffling introduces two edge insertions and two edge deletions in a random order. Such edge shuffling ensures that all the nodes in the network preserves their degrees.We simulate dynamic changes in the synthetic networks as follows. For each network, we perform a sequence of 1000 edit operations . We conduct the topological perturbations using the degree preserving edge shuffling method . Each shbasic motifs. Having said that, our method is generic and can be employed to count any motif topology. Given a sequence of network edit operations, we first compute the number of embeddings for each motif in Fig. F2 frequency of motifs in a static network. We then apply each edit operation in the order it was given to update the network topology. After each edit operation, we dynamically update the count of each motif using our method.In our experiments, we use a small set of motifs that have been used frequently in literature. Figure F2 count of the four basic motifs. We evaluate our method in terms of its accuracy and running time. All experiments were performed on a server with 3 GB RAM and AMD Opteron dual core processors (2.2 GHz) running Linux. Notice that unlike our method, Elhesha et al. computes the F2 count from scratch when the network topology changes. Thus, at any point in the evolution, we consider their count as the ground truth. We report the accuracy of our method as the ratio of the F2 count of our method to the F2 count returned by their method at that instance of the network. We measure the running time of the static method for the initial network topology and denote it with static time. We also measure the running time of our method after each edit operation and denote it with dynamic time.We compare our method against Elhesha et al.  as it alWe first investigate the effects of different network models on the running time of our algorithm. We set the network size to 5000 nodes and the average node degree to 15. We perform 1000 random edit operations and dynamically update the motif counts for all motif topologies in Fig. F2 count from scratch every time the network topology changes. To illustrate this, we plot the cost of running the static algorithm twice, i.e., one for initial topology and the other for an arbitrary intermediate topology as the network evolves. We observe that, despite the additional cost our method incurs for initialization, the total cost of our method for 1000 dynamic topologies is less than that of the static method for only two topologies. More specifically our method updates the motif count for 1000 evolving topologies more than four times faster than the static method on a single network topology.We observe that the running time of our method for all three models are significantly less than that of the static method. The static method needs to recompute the We observe that both static and dynamic methods incur the largest running time for the BA model. This is because the degree distribution in the BA model forces motif counting algorithms to evaluate many possibilities not encountered in other models, hence the increase in the running time. We observe much faster running time for the other two models.In this experiment, we fix the average degree of the networks to 15, and vary the number of nodes from 1000 to 4000 with increment of 1000. We perform the experiments for each motif and random network model combination. We process 1000 random edit operations and dynamically update the motif counts. We compute the total update time as the processing time starting from the first edit operation until the end of the last edit operation. We compute the average update time as the total update time divided by the number of edit operations. Figure We observe that the average update time of our method is very small even for the largest network size. This shows the effectiveness of using dynamic update in the motif counting problem.In this experiment, we fix the network size to 4000 nodes and set the average node degree to 10, 15 and 20 respectively. We perform experiments for each motif and random network model. We process 1000 random edit operations and dynamically update the motif counts. We report the average update time for the edit operations. Figure We observe that the average update time of our method is very small for all degree values. Average update time increases as the node degrees increase. Among the three models, we observe that the BA model takes longer time to update and the other two models show similar performance.These experiments support our conjecture that dynamic update time per edit operation is very small for all parameter combinations. It scales well to networks with large sizes and node degrees. The potency of our method is important in evolving biological networks where running static method after each topological change is not practical.F2 count as compared to the static method. Faster running time is desirable only if our method produces accurate results. Here, we evaluate the accuracy of our method as compared to the static one. We run our experiments with network size and degree set to 5000 and 15 respectively. To compare with static method, we also run static method [In the previous section, we show that our method dynamically updates the motif count very fast in all parameter combinations. Recall however that it is possible that our dynamic method may yield fewer c method  after evF2 count reported by our method is very close to that of the static method. As the number of edit operation increases, the gap between the two F2 counts gradually grow. This is expected since the static method computes the number of embeddings from scratch, whereas our method dynamically updates the count from the initial computation. That said, the gap remains to be negligible even after applying 1000 edit operations.We observe that our method is highly accurate for all motifs and random network models. The In this experiment, we fix the average degree of the networks to 15, and vary the network sizes from 2000 to 5000 with increment of 1000. In this experiment, we dynamically compute the motif counts after each of the 1000 edit operations. In the final topology, we also compute the results using the static method. We report the accuracy of our method for each motif and random model combinations. Figure Our results demonstrate that our method is highly accurate for all network sizes, motifs, and random network models. The accuracy of our method is over 95% in all of our experiments. This shows the robustness of our method.In this experiment, we fix the network size to 5000 and vary the degrees to 10, 15 and 20. We dynamically compute the motif counts after each of the 1000 edit operations. In the final topology, we also compute the results using the static method. Here, we report the accuracy of our method in different random models. Figure Our method is highly accurate for all degree values. For instance, accuracy values of motifs one, two and four are over 97% for each network size. For all degree and random network combinations, our method counts the motifs with more than 95% accuracy.In summary, our method yields highly accurate results for all network sizes, density values, motif topologies, and random network models. These results suggest that our method is robust to a broad spectrum of parameter settings.starting network and the second one as the final network in the dynamic evolution. The difference in the set of edges between the starting and the final networks is the set of edit operations that the starting network goes through to evolve into the final network. As the order in which these edit operations take place are not known, we rank them randomly. We then use the static algorithm on the starting network to find the motif embeddings for the same motifs used in “We perform experiments on two pathways using the gene co-expression dataset described earlier in “F2 counts for over thousands of edit operations as compared to running the static algorithm once on two network instances only . This is a dramatic improvement as our method computes the motif count for not only those two networks but also thousands of network topologies in between these two states of networks. In other words our algorithm can identify motif counts for thousands of networks while the static algorithm fails to do that without running it from scratch. We observe that while the static algorithm takes around 60 s per network to count motifs, our dynamic algorithm updates the motif count in only a few milliseconds per intermediate network.Figure F2 motif count until we reach to the final network. Let us denote this count with cdynamic. We then compute the same count using the static algorithm when it is applied on the final network only. Let us denote this count with cstatic. We report the accuracy as the percentage of the motif count our method finds over the static one (i,e., accuracy = 100×cdynamic/cstatic. Figure The performance improvement is desirable only if the algorithm remains accurate. Here, we evaluate whether our method correctly computes motif counts. We focus on the final network of each pathway for this purpose. We run our algorithm to dynamically update the Similar to the synthetic datasets, we observe that our dynamic algorithm is highly accurate on the real datasets as well. For all the motifs tested, the accuracy is over 97% for both pathways. We observe higher accuracy for the second pathway. This is possibly because the number of network edit operations for the second patway is lesser than that of the first one see Fig. . The resF2 count. This is because it initializes the relevant data structures which are needed to quickly update the F2 count as the network evolves. Once the data structures are set up, however, the cost of dynamic update becomes negligible. This demonstrates the key advantage of our dynamic method over the static one. The average update time grows very slowly as the network size increases. This suggests that our method scales to large networks and very large number of network edit operations. Among the three models, we observe that relative to the other two models, the BA model takes longer time to update. This is because the networks in BA model yields significantly more motif instances. The potency of our method is important in evolving biological networks where running static method after each topological change is not practical.From the results on the simulated dataset we observe that the running time of our method for all three models are dramatically less than that of the static method. For the initial network topology, the dynamic method spends a little more time than the static method to compute the We also observe that our method is highly accurate for all motifs and random network models. The accuracy of our method remains stable with increasing network size while the accuracy values for BA model increase as the network size grows. Since the BA model represents the biological networks well, these results promise the applicability of our method in real networks. We also observe that our method is stable for growing node degree values.Our experiments with the real dataset shows that our methods have similar efficacy on real data as well. Therefore, it can be used on biological dynamic networks to uncover interesting observations on the evolution of biological processes.Several approaches exist to compute the number of embeddings for a given motif in static network; however, no such method exists for dynamic networks. In this article, we address this problem and describe a method that incrementally updates the motif count as the network changes its topology. We evaluated our method extensively using synthetic and real datasets, and show that our method is highly accurate and that it can be scaled to large dense networks. The results on real data demonstrate the utility of our method in revealing interesting insights on the evolution of biological processes.


The resolution of super-resolution microscopy based on single molecule localization is in part determined by the accuracy of the localization algorithm. In most published approaches to date this localization is done by fitting an analytical function that approximates the point spread function (PSF) of the microscope. However, particularly for localization in 3D, analytical functions such as a Gaussian, which are computationally inexpensive, may not accurately capture the PSF shape leading to reduced fitting accuracy. On the other hand, analytical functions that can accurately capture the PSF shape, such as those based on pupil functions, can be computationally expensive. Here we investigate the use of cubic splines as an alternative fitting approach. We demonstrate that cubic splines can capture the shape of any PSF with high accuracy and that they can be used for fitting the PSF with only a 2–3x increase in computation time as compared to Gaussian fitting. We provide an open-source software package that measures the PSF of any microscope and uses the measured PSF to perform 3D single molecule localization microscopy analysis with reasonable accuracy and speed. Approaches to accurately modeling the PSF for the purpose of localizing single molecules include PSF correlation7 and pupil function fitting8. In the PSF correlation approach, the PSF of the microscope is measured by scanning the objective (or the sample) in Z to capture the shape of the PSF at multiple focal planes. The resulting z stack of PSFs can then be cross-correlated with the image of a single molecule to determine its position. This is computationally expensive as it requires multiple 2D Fourier transforms, followed by further fitting if sub-pixel position information is desired. Alternatively, the molecule’s position can be determined directly by providing an initial estimate for the molecule’s position and computing the fitting error between the molecule’s image and the PSF, then appropriately updating the position estimate. This can be done using a simplex optimization algorithm such as Nelder-Mead9. However this class of algorithms, which are used when the derivatives of the error function are not known or are hard to calculate, converge slowly on the optimal solution. An additional complication of this latter approach is the need to interpolate between the measured PSF values in x, y and z.The accuracy of the fitting algorithm is an important factor in the final resolution of single molecule localization based super-resolution microscopy. This accuracy is in turn determined in part by how well the fitting algorithm models the PSF of the microscope. This is particularly important for 3D super-resolution imaging, where the PSF shapes are often not a simple Gaussian function11, as interpolation is not necessary and the derivatives of the error function can be calculated, enabling the use of more rapidly converging algorithms such as those based on the Newton-Raphson algorithm. In the pupil function approach, the PSF is again measured by scanning the sample containing point emitters (such as beads) in Z, passing through the focal plane of the objective. The measured PSF is then fit to determine the pupil function of the microscope. As the pupil function captures the PSF of the microscope with high accuracy, fitting the images of single emitters to the pupil function allows the positions of these emitters to be determined with high accuracy. However calculating the PSF and its derivatives with respect to the fitting parameters at a particular x, y and z location from the pupil function is computationally expensive, and could be slow unless GPU acceleration is used8.These problems are largely avoided with a pupil function fitting approach12, or single molecules13, the latter using the DAOSTORM algorithm, the adaptation of DAOPHOT for single molecule localization microscopy. B-splines, of which cubic splines are a sub-set, were used previously in localization microscopy to estimate the maximal localization accuracy in x, y and z of an arbitrarily shaped 3D point spread function (PSF)14. B-splines were also used in a general method for 3D localization fitting similar to what is presented here15.Another approach to accurately model the PSF for the purpose of localizing single molecules is to use splines. The most commonly used splines approximate the function of interest using one or more polynomials. This class of splines can be calculated rapidly, and it is also easy to compute their derivatives. 2D cubic splines were used in the DAOPHOT astronomy package to provide higher order corrections to a 2D Gaussian for the purpose of fitting the locations and magnitudes of starsThis current work builds on previous work by further exploring the use of splines for rapid and accurate analysis of 3D super-resolution microscopy data based on single molecule localization. In particular, we provide an open-source maximum likelihood estimation (MLE) based cubic spline analysis package, side-by-side comparisons of its performance with elliptical Gaussian fitting, and a demonstration of its broad application by fitting simulated 3D super-resolution microscopy data derived from several different PSFs.rd order polynomials of the form:fi is valid on small equal length intervals . Any function with a continuous first derivative can be approximated with arbitrary accuracy by choosing an appropriate interval size. Here we follow DAOPHOT/DAOSTORM and choose an interval size that is ½ of the image pixel size as a good compromise between accurately modeling the PSF and minimizing the total number of intervals and coefficients.A cubic spline approximates a one dimensional function using piecewise 3rd order spline interpolation  and their first derivatives fi(x)/dx are continuous at the red points, the interval boundaries. We chose to use the additional constraint that the second derivative of the spline is zero at the start of the first interval and at the end of the last interval. We use the approach described in ref. In Fig. 17 with 3rd order spline interpolation. Next, the up-sampled images of these emitters were aligned to their centroid positions using the ndimage.interpolation.shift function and averaged to obtain a single image for each z position. Finally, images covering 50 nm z ranges were further averaged together to generate the PSF for the z position that is at the center of the 50 nm range and the average PSFs determined from a series 50 nm intervals were used to create a 3D array that represents the 3D PSF. To measure the 3D PSF in real experiments, small fluorescent beads can be attached to a coverslip. Then z scan movies can be taken with a piezo positioner. In this situation the true emitter locations are not known, but they can be estimated with sufficient precision for this purpose by using another fitting algorithm such as 3D-DAOSTORM18 because the fluorescent beads typical emit a very large number of photons.We used the following procedure to measure the PSF in three dimensions. We placed 4x up-sampled PSFs with high photon counts on a xy grid. Then we scanned the z values of the PSFs to create z scan image stacks. Finally, we downsampled the image stacks by 4x in x and y to create the final PSF measurement image stack. Then, using the known emitter locations, images of isolated emitters were cropped out of these image stacks and up-sampled by 2x in xy using the ndimage.interpolation.zoom function from the Scipy Python packagefi,j,k is valid on ½ pixel length intervals  in x, uk, uk+1] intervals in z. We start with a 3D PSF that is determined by either simulation or experiments, and then up-sampled such that it has an equal number of elements in x, y and z. This is not strictly necessary but having an equal number of elements for each axis simplifies later computations. We then use a series of 1D cubic spline interpolations in x, y and z to further upsample the PSF, by 4 fold in each dimension, to obtain 64 values of the PSF within each element. These 64 values then allow us to uniquely specify the 3D cubic spline fi,j,k and determine it’s 64 coefficients ai,j,k,m,n,o in the intervals that correspond to each element of the PSF.The extension of the cubic spline construction to 3 dimensions is described below. In three dimensions, the cubic spline is:fi,j,k at a single point will require 128 operations, as compared to 10 operations to evaluate the elliptical Gaussian that is often used for 3D PSF fitting:xmynzo terms as for fitting the evaluation is always done at the same point in each of the spline intervals. Similarly, we have ignored the calculation of the wx(z) and the wy(z) terms in the Gaussian as these are also the same for each point in the PSF. We wrote a simple C program to time how long it took to perform these calculations, and found that the cubic spline calculation is approximately 2.3x slower. However, we note that this is a best case estimate as the cubic spline calculation is more memory intensive and thus is likely to be somewhat slower in general. However, this rough estimate suggests that the computational expense of the cubic spline fitting will be similar to that of Gaussian fitting.We then use this cubic spline approximation of the 3D PSF to fit the image of individual emitters and determine their 3D coordinates. Based on the 3D cubic spline formula it can be seen that the evaluation of The highest accuracy estimation of the parameters of the localization of an emitter is typically achieved using non-linear least squares or maximum likelihood estimation (MLE) based fitting. This can be done efficiently if both the error function and its derivatives are easily calculated for any point in the PSF. As described in ref. xc, yc, zc (the 3D coordinates of the emitter), h (the height of the PSF), and b (the baseline of the PSF) are easy to calculate, and they are:The cubic spline PSF and its derivatives with respect to the fitting parameters 20. Given a reasonable starting point the optimal values for xc, yc, zc, h and b can be found with high accuracy in just a few (~10) iterative updates for isolated emitters. The algorithm executes the following pseudo-code on each frame of a single-molecule imaging movie until either no new peaks are found, or the maximum number of iterations is reached.We minimize Note that this algorithm ignores cross-terms for overlapping localizations. Each localization’s fitting parameters are updated in a context where all the other localizations are fixed. This increases the simplicity of the algorithm implementation at the price of a reduced convergence speed for overlapping localizations.18, differing only at the peak finding and peak fitting steps. In order to do peak finding the image is convolved with the spline representation of the PSF at one or more z values, then local maxima in the convolved images are used as the initial locations for the subsequent peak fitting step. We found this approach to be more robust, particularly with PSFs that do not have a single local maxima, such as the double helix PSF2, than using local maxima in the original image. When we performed convolutions at multiple z values each of the convolved images was treated as a single plane in a z stack. Then local maxima were found in the z stack and peak fitting was started at the x, y and z values of the local maxima. In our experience the use of multiple z values was only helpful for analyzing data taken with the double helix PSF2, probably because among the PSFs that we tested, the double helix PSF is the only one that is not degenerate in z. By this we mean the PSF shape at one z value cannot be approximately recreated by a linear sum of PSF shapes at other z values. This is in contrast with other PSFs such as the astigmatic PSF whose shape at large z offset values can be approximated quite well by a sum of multiple z = 0 PSFs with differing xy displacements.This approach is very similar to that used by 3D-DAOSTORM11 to generate 3D PSFs for a purely astigmatic PSF (Zernike mode z22 = 1.3) and for the saddle-point PSF described in ref. 3. For each condition we generated noise free images of the PSF at z positions ranging from −1 um to 1 um in 10 nm steps. Then we determined the coefficients of the cubic spline for this PSF as described above. We found that the maximum pixel wise amplitude difference between the astigmatic PSF and the cubic spline was 3.3% (at z = 0 nm) over the entire 2 um z range  and wy(z) calibration curve for elliptical Gaussian z fitting. Then, to measure the fitting performance, a test movie was created using the same approach but with 4000 photons per localization and a constant background of 100 photons. In the test movie, we chose the z position of the emitters to range from −300 nm to 300 nm, as it has been noted previously that the localization precision of the astigmatism approach depends on the z position of the emitter and that relatively high localization precisions have been achieved experimentally in the z range of −300 to 300 nm1. These movies were created using a Poisson noise model, with no camera read noise, 600 nm emission wavelength and a 160 nm pixel size. The test movie was analyzed by fitting the single-emitter images to the cubic spline representation and elliptical Gaussian representation of the astigmatic PSF . The z localization accuracy of cubic spline fitting was noticeable better, with an average improvement of ~15% within ±300 nm of the focal plane. Spline fitting is able to come very close to the CRLB for the x and y locations of an emitter, however at the largest z values tested it is ~20% worse than the CRLB.22. Since the cubic spline representation can capture these experimental PSFs with arbitrarily high accuracy given a sufficiently small interval size, we expect cubic spline fitting to provide a greater improvement in the localization accuracy in experiments.We note that in real experiments, the PSF often deviates from the ideal PSF derived from the pupil function due to non-ideal optics and optical alignment errors, which can lead to systematic errors in the determination of the x, y and z positions of the emitters when a simple elliptical Gaussian function is used to fit the PSFThe average amount of time that it takes to analyze a single image with the cubic spline approach as a function of emitter density is shown in Fig. It is worth noting that the fitting time will depend on the number of pixels that the PSF covers. In the above test, we used a spline that covers 14 × 14 pixels in x and y. This is a reasonable size for most PSFs, but a larger size may be necessary for PSFs with a larger spatial extent or for microscope setups with a smaller pixel size. We found a size of between 12 and 16 pixels (in each dimension) was sufficiently large to work well for the PSFs that are typically used in single-molecule imaging and our pixel size of 160 nm.2 and the saddle-point PSF3. We generated two kinds of simulated movies using these PSFs, a high photon count movie for measuring the PSF and determining the spline and a low photon count movie similar to typical single-molecule localization conditions. The saddle-point PSFs were generated using the pupil function approach11, and we rotated the saddle-point PSF by 45 degrees relative to the orientation reported in ref. 23 and spline interpolation was used to position it at arbitrary x, y and z values. In the movies simulating single molecules, we used 4000 photons per localization, a constant background of 100 photons, and a camera pixel size of 160 nm. The x, y and z errors were again determined as the average distance in x, y and z between the localizations determined by the fitting and their nearest ground-truth emitter positions and shown in Fig. An important advantage of the cubic spline approach is that it can be used to fit arbitrarily shaped PSFs, which makes it widely applicable to many different 3D localization methods using various engineered PSF shapes. Here, we demonstrate the versatility of the cubic spline approach by using it to analyze simulated data from two additional types of PSFs that have been used for super-resolution imaging, the double helix PSFWe note that the localization errors reported in this work are all determined by fitting the relatively ideal, simulated images of single molecules with the same photon numbers for each PSF. In real experiments, the photon loss associated with generating relatively complex PSFs, for example by using spatial light modulators or deformable mirrors, also needs to be taken into account when estimating the final localization error.24.In this work, we demonstrate that a cubic spline approach provides a 5% improvement in x and y localization accuracy and up to a 15% improvement in z localization accuracy relative to elliptical Gaussian fitting for a purely astigmatic PSF. Although the improvement in localization accuracy is moderate, the cubic spline approach has the additional advantage of being highly versatile, as it is easily adapted to fit any experimental realizable PSF. It is also a practical choice as it is only ~2–3x slower than a Gaussian fitting approach at typical emitter densities encountered in super-resolution imaging experiments. To make it easier for others to explore this approach to single-molecule localization analysis, our implementation of this algorithm, Docker images, instructions and an example of its use are available on GithubPython code and Data


For more than a century, Dutch scientists and clinicians have contributed importantly to the understanding of cardiac electrophysiology and to the diagnosis and treatment of cardiac rhythm disturbances. The first example supporting this statement that comes to mind is the development of a string galvanometer and the registration of the human electrocardiogram by Willem Einthoven in 1901 . SimilarThis issue of the Netherlands Heart Journal is focused on electrophysiology and aims at answering that very question. Having left the pioneering era behind us, the field of clinical electrophysiology has grown over the past years, both worldwide and in the Netherlands. In 2016, our country saw ≈ 10,000 pacemaker implantations, more than 5000 ICD implantations and 8750 catheter ablations, of which 4000 for atrial fibrillation (AF).With the increase in the number of AF ablations, a shift in the characteristics of patients undergoing these procedures is to be expected. Teunissen et al. describe how patient and procedure characteristics of pulmonary vein isolation for AF changed in 975 procedures performed in a single centre between 2005 and 2015 . Ninety Device therapy can be life-saving, but is also associated with complications related to lead failure and pulse generator exchanges, which in turn may cause infection. Battery longevity therefore forms an important determinant of device safety. De Vries et al. investigated the temporal trends in service time of pacemakers between 1984 and 2006, making use of a national device database . Almost The five original contributions to this issue of the Netherlands Heart Journal show that clinical electrophysiology is being practised and studied at a high level in the Netherlands. All studies were generally well designed and carried out with care. Importantly, relevant and complete follow-up of patient outcomes is emphasised by the investigators. This will truly improve understanding of the potential and limitations of different therapies, also beyond the electrophysiological community. With these contributions, being representative for contemporary Dutch electrophysiology, the future can be looked forward to with confidence.


CT images but is well visualized in MRI images. However, implants used for spinal stabilization can produce artifacts on the MRI images which can interfere with identification of the cord. CT myelography in conjunction with CT simulation helps to clearly delineate the cord.The spinal cord is poorly visualized on CT simulation was done in a patient with vertebral hemangioma. Pre‐ and post myelography images were obtained. Two plans were generated on pre and post myelography CT images using Eclipse™ treatment planning system (TPS) version 10.0, Varian Medical Systems, USA. The prescribed total dose to PTV was 40 Gy in 20 fractions.CT images. The titanium implants used for spinal cord stabilization produced artifacts in the MRI images. Figure The cord was poorly visualized in the premyelogram PTV in the premyelogram images. This volume was reduced to 0 cc in the post myelogram images. There was an overlap of 5.4 cc volume of the PRV of spinal cord with the PTV in the premyelogram images which was reduced to 1 cc in the post myelogram images. The overlap region between the PTV and spinal cord received around 71% of the prescribed dose in premyelogram CT could be reduced to 0% in the post myelogram CT. The mean dose received by the overlap PRV spinal cord and PTV could be increased from 70% in the premyelogram to 92% in the post myelogram plans.There was an overlap of 0.75 cc volume of the spinal cord with the CT myelogram in conjunction with CT simulation is particularly useful in cases where the tumor margin is very close to the cord and spinal implants are causing distortion of magnetic resonance images. For tumors in and around the spinal cord it is imperative that we delineate the cord precisely.The spinal cord is poorly visualized on CT images. Traditionally, it is the spinal canal which is often contoured as the spinal cord in CT scans. Spinal cord can be accurately contoured on the registered MR images. However, if implants are used for spinal stabilization it can produce artifacts on the MRI images which can interfere with identification of the cord accurately.It is of paramount importance to identify the cord accurately and limit the dose to the cord, since excessive dose to the spinal cord is associated with irreversible myelopathy.2The patient was immobilized using a Vac‐Lok [manufacturer ‐Klarity medicals] in the supine position. CT simulation was taken using CT Somatom Spirit a dedicated CT machine for simulation. CT was acquired from T6 to L5. Our CT scanner does not have a metal artifact reduction algorithm.Images acquired from CT had a slice thickness of 3 mm and the reconstructed image from the Eclipse registration software was also 3 mm. After premedication a lumbar puncture was done, dye was injected into the spinal canal and the patient was immobilized using the same Vac‐Lok. The neurosurgeon performed the lumbar puncture and injection. No additional staff was required for the simulation. The normal CT simulation consent form was modified to inform the patient about any potential risks of CT myelography.Premedications for myelogram and post myelogram instructions were given. However, the procedure of CT simulation remained the same.A second set of simulation images were taken using the same parameters. Two volumes were contoured – Gross tumor volume and Planning target volumes. Spinal cord, kidneys, lungs, stomach and intestines were the organs at risk. Contouring was done separately for each CT image. The tumor in both the instances was drawn based on the registered MRI images. However in Plain CT images, the area of the tumor drawn was an approximation since the cord was not seen due to the susceptibility artifacts in the fused MR images. However in the post myelogram CT images, the tumor edges could be drawn clearly where it was seen distinct from areas occupied by the contrast in the spinal canal and the cord. This led to a reduction in the PTV volume by almost 20 cc.The spinal cord is suspended in the CSF and is tethered by roots and the dentate ligament. Hence it is unlikely to move significantly in the spinal canal on a day to day basis. However, small movements were accounted for by giving a small margin around the spinal cord of 3 mm. We gave a margin of 3 mm around the cord to create the PRV cord [planning organ at risk volume].Plans were generated on pre‐ and post myelography CT images using Eclipse™ treatment planning system (TPS) version 10.0, Varian Medical Systems, USA. The TPS used analytical anisotropic algorithm (AAA) algorithm for calculation of monitor units and progressive resolution optimizer (PRO) version 10 for VMAT plans. Since the contouring was done separately for each CT image and there was a difference in the PTV volume between the two images, separate plans had to be created for the pre and post myelogram images.During planning, the contrast around the cord had a Hounsfield unit (HU) of 200–350 HU. We have assigned HU manually for the contrast. Since the spinal canal contains CSF and during the actual treatment it is the CSF which fills the spinal canal, we assigned tissue equivalent value for the contrast which is zero HU.The prescribed total dose to PTV was 40 Gy in 20 fractions and the optimization constraint was that 95% isodose line encompasses 95% of PTV. The optimization constraint given for the spinal cord was that the D‐Max dose to any part of the spinal cord was kept below 40 Gy and both the plans met the specified objectives.The Varian Clinac iX with 6 MV photon energy was used for delivering the treatment.3The cord was poorly visualized in the premyelogram CT images. The titanium implants used for spinal cord stabilization produced artifacts in the MRI images. Figure Contouring was done separately for each CT image. There was a difference in the PTV volume between the two images. The volume of PTV CT myelogram was 128.38 cc and PTV CT plain was 140.42 cc.There was an overlap of 0.75 cc volume of the spinal cord with the PTV in the premyelogram images, which was reduced to 0 cc in the post myelogram images. There was an overlap of 5.4 cc volume of the PRV of spinal cord with the PTV in the premyelogram images which was reduced to 1 cc in the post myelogram images.The dosimetric parameters in percentage and absolute value in Gy are outlined in Table The D‐max received by the cord in the pre‐ and post myelogram images are 88.4% and 84%, respectively. The D‐mean received by the cord reduced by half from 52% in the premyelogram image to 27% in the post myelogram image.The D‐max received by the PRV cord in the pre‐ and post myelogram images are 108% and 101.8%, respectively. The D‐mean received by the PRV cord reduced by half from 69% to 33.7%.The overlap region between the PTV and spinal cord received around 71% of the prescribed dose. This was reduced to 0% in the post myelogram CT. However, the mean dose received by the overlap PRV‐spinal cord and PTV could be increased from 70% the premyelogram CT to 92% in the post myelogram plans.We have also applied the same plan done on the premyelogram CT to the post myelogram CT.The only complication which was reported was headache which developed 24 hr after the procedure. Headache lasted for 1 day and subsided with adequate hydration, paracetamol and foot end elevation.4Radiation damage to the spinal cord is one of the most dreaded complications in the treatment of cancer.When the tumor is very close to the cord often the dose to the tumor is compromised to spare the cord. This will inevitably result in recurrence of the tumor. Hence, it is imperative that the cord is identified as accurately as possible in such cases and not use the spinal canal volume as a surrogate for the cord. Registration with MRI and contouring the cord on the registered MRI is often the solution. However, this has its own errors especially due to the artifacts caused by the implants used for spinal stabilization. Titanium implants can cause distortion of the image or even obscure surrounding anatomy around them due to susceptibility artifacts. Figure In our patient, the cord was poorly visualized in the plain CT images. We could not reliably contour the cord using the registered MR images because of the severe distortion caused by the implants. Figure We wanted to see if there was an impact on the planning and dosimetric parameters. Identifying the cord clearly helped to decrease the overlap between the PTV and the cord and PRV cord. In the premyelogram images this overlap between PTV and cord was 0.75 cc and could be reduced to zero in the post myelogram CT images.Similarly, the PRV cord volume also decreased from 5.4 to 1 cc. The mean dose received by the PTV and PRV cord overlap increased from 70.8% to 92%.It is clear from our data that the dose to the overlap region was more in the post myelogram CT which indicates that this area which can potentially harbor tumor tissue could receive a higher dose. Thus, CT myelogram helped us to increase the dose to the PTV and decrease the dose to the spinal cord.Thariat et al. studied the value of CT myelography in the delineation of the spinal cord and found that CT myelography had superior resolution to MRI when Cyber knife stereotactic radiotherapy was used.As far as we know this is the first description of the use of CT myelogram to achieve dose optimization in radiation therapy of vertebral hemangioma in cases where spinal implants have been used.We wanted to describe a simple method which will identify the cord accurately which is applicable in cases such as this where titanium implants obscured the normal anatomy. Since we could identify the cord clearly we could reduce the radiation dose well below the recommended tolerance dose of 40 Gy while keeping an optimal PTV dose. Even the recommended tolerance dose of 40–44 Gy to spinal cord has a 5 yr probability of complications of 5%. By staying below this dose, we reduce even this small probability of complication and such a strategy may reduce the risks of myelopathy in case the patient requires re‐irradiation in the future. An important finding in our study was that dose to the overlap region was more in the post myelogram CT, which indicates that this area which can potentially harbor tumor tissue could receive a higher dose. Thus, CT myelogram not only helped us to decrease the dose to the spinal cord but also to increase the dose to the PTV.However, myelogram is contraindicated if there is a spinal block or if the cord is low lying. One should also be absolutely certain that only non‐ionic contrast is used since inadvertent intrathecal injection of ionic contrast media is fatal.5CT myelogram in conjunction with CT simulation helps clearly visualize the cord. The technique we have described is particularly useful in cases, where the tumor margin is very close to the cord and spinal implants are causing distortion or obscuration of magnetic resonance images. Precise identification of the cord enabled us to deliver more dose to the tumor.The authors declare no conflict of interest.


Editorial on the Research TopicVestibular MigraineVestibular migraine (VM) is now the accepted name for vestibular symptoms that are causally related to migraine. The International Headache Society and the International Bárány Society for Neurootology together have developed a consensus document with diagnostic criteria for VM , but theFor those that acknowledge the existence, VM is considered the most common cause of recurrent spontaneous vertigo attacks. It has a lifetime prevalence of about 1% and a one-year prevalence of 0.9% in the general population  and accoThe author confirms being the sole contributor of this work and approved it for publication.The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


Aims. Predicting the risk of severity at an early stage in an individual patient will be invaluable in preventing morbidity and mortality caused by dengue. We hypothesized that such predictions are possible by analyzing multiple parameters using mathematical modeling. Methodology. Data from 11 adult patients with dengue fever (DF) and 25 patients with dengue hemorrhagic fever (DHF) were analyzed. Multivariate statistical analysis was performed to study the characteristics and interactions of parameters using dengue NS1 antigen levels, dengue IgG antibody levels, platelet counts, and lymphocyte counts. Fuzzy logic fundamentals were used to map the risk of developing severe forms of dengue. The cumulative effects of the parameters were incorporated using the Hamacher and the OWA operators. Results. The operator classified the patients according to the severity level during the time period of 96 hours to 120 hours after the onset of fever. The accuracy ranged from 53% to 89%. Conclusion. The results show a robust mathematical model that explains the evolution from dengue to its serious forms in individual patients. The model allows prediction of severe cases of dengue which could be useful for optimal management of patients during a dengue outbreak. Further analysis of the model may also deepen our understanding of the pathways towards severe illness. Sri Lanka has experienced a marked increase in dengue from 2004 to 2014. The increase from the year 2013 to 2014 was 47.4%  . The fuzzy membership function determines the degree to which an element belongs to a defined set.In a wider sense the fuzzy logic principles are based on the fuzzy set theory . In fuzzf : n → . For this study we use the Hamacher operator along with the Ordered Weighted Aggregation (OWA) operator.A fuzzy operator can be defined as a function μA(x), μB(y) are membership functions defined on sets A and B.Hamacher operator is defined as follows:wi ∈  and wi is the ith element in the weights vector where weight values are arranged in the descending order. xi denotes the ith largest membership value.The Ordered Weighted Aggregation operator (OWA operator)  is defina, b, r ∈ . Hence the weights of the OWA operator are defined as in The weights vector is defined according to the following conditions and constraints.ed by Xu . The linMembership Function for Platelets3 [3 and the severity level continuously increases as the platelet count drops.Membership Function for NS1 Panbiotifiable   and UB(x) denote the corresponding degree of favorability to develop DHF.(1)A and B are favorable to develop DHF, then If (2)UA(x) < UB(x) < 1. Then the effect of an increase in the favorability of A ) on UA∩B(x) may depend on UB(x).Suppose (3)UA(x) and UB(x) < 1. The effect of a decrease of UA(x) on UA∩B(x) can be negated by an increase in UB(x), that is, an increase in the unfavorability to develop DHF through B.Suppose Accordingly we have chosen the Hamacher operator to be applied to each of the clusters of parameters as it proves to be a suitable function [Our aim is to capture the combined effect of the parameters. Hence the operator should satisfy several conditions. Let function , 19.H1, H2) for each of the groups is as follows:The modified Hamacher product (μP is membership value of the platelet count, γP is concentration value for platelets, μN is membership value of the NS1 Panbio, and γN is dilution value for NS1 Panbio.μI is membership value of the IgG Panbio units, γI is concentration value for IgG Panbio, μL is membership value of the lymphocyte count, and γL is dilution value for lymphocyte count.Concentration and the dilution of the membership values are carried out according to the relative importance that each of these parameters plays as an early indicator of severity. Platelet count and the IgG Panbio level play critical roles in the identification of the severity; thus their effects are concentrated. But NS1 Panbio level and lymphocyte counts are not early predictors of severity. Hence their effects are diluted. The values are assigned in an intuitive manner according to the information gained through medical expertise.γP, γN, γI, and γL are 1.2, 0.2, 1.1, and 0.2, respectively.Accordingly the values assigned to a = 0.1, b = 0.6 to the linguistic quantifier (ρ = 0.8.The Hamacher operator value resulting from platelet count and the NS1 Panbio levels  and the antifier  and then1OWA=ρ∗maxAccordingly the orness  of the OThe ambiguous region is determined using the ambiguous parameter levels which are mentioned earlier. The method we use to identify the ambiguous operator value range is as follows.The range of the ambiguous parameter values of each individual parameter is applied to the modified Hamacher operators of the relevant groups.The range of the values given by the modified Hamacher operators is calculated.The range of values given by the modified Hamacher product is applied to the OWA operator.The range of the values given by the OWA operator is taken as the ambiguous region of the operator.The resulting partitions of the operator are illustrated in the The values from the data set are applied to the model and In order to evaluate the performance of the model further we have to determine how many severe patients and nonsevere patients are classified correctly. Accuracy of classifying, classification as ambiguous, and misclassification are defined as follows:Accuracy  patientsClassificThe objective of carrying out a sensitivity analysis is to identify whether classification of the patients according to the severity changes significantly in case of a shift in the degree of fuzziness.δ > 0 value and we observe the change in the boundary values for severe and nonsevere regions. The degree of change in the boundary value determines the robustness of the model and the relevant results are indicated in δ values. According to Each of the membership functions has been shifted by a certain The concentrations and dilutions that are applied to each of the parameters are also shifted in order to identify the sensitivity of the model. The relevant results are illustrated through Since the boundary values do not change significantly for either a shift in the membership functions or concentrations it can be inferred that the accuracy levels of the model also remain relatively constant. Hence the model is not affected by changes in the degree of fuzziness which leads to the conclusion that the model is indeed robust.Dengue is currently a significant issue not only in Sri Lanka but also in the South Asian region. Although the mortality rate of this disease is low, it can be further controlled by the developing a model which can predict the severity level of the patients.Most of the research that is carried out so far focuses on predicting severity based on the information derived by a single parameter. In this research, the cumulative effects of the platelet count, NS1 Panbio levels, IgG Panbio levels, and lymphocyte count of the patient are considered. The statistical analysis shows that the behaviour of the platelet count and NS1 antigen levels are similar; furthermore the behaviour of the IgG antibody levels and lymphocyte count also indicates a similarity. The fuzzy logic principles involving fuzzy set theory and operator theory are used in order to construct the mathematical model which can predict the severity of the patient.The model can be further improved by the inclusion of more parameters into the model. Cytokines and liver enzyme levels might be vital in increasing the accuracy of the model.