CK) is a unique system with respect to radiation protection shielding and the variety and number of non‐coplanar beams are two key components regarding this aspect. In this work, a framework to assess the direction distribution and modulation factor (MF) of clinically applied treatment beams of a CyberKnife M6 is developed. Database filtering options allow studying the influence of different parameters such as collimator types, treatment sites or different bunker sizes. A distribution of monitor units (MU) is generated by projecting treatment beams onto the walls, floor and ceiling of the CyberKnife bunker. This distribution is found to be highly heterogeneous and depending, among other parameters, on the bunker size. For our bunker design, 10%–13% of the MUs are delivered to the right and left wall, each. The floor receives more than 64% of the applied MUs, while the wall behind the patient's head is not hit by primary treatment beams. Between 0% and 5% of the total MUs are delivered to the wall at the patient's feet. This number highly depends on the treatment site, e.g., for extracranial patients no beams hit that wall. Collimator choice was found to have minor influence on the distribution of MUs. On the other hand, the MF depends on the collimator type as well as on the treatment site. The MFs (delivered MU/prescribed dose) for all treatments, all MLC treatments, cranial and extracranial treatments are 8.3, 6.4, 7.7, and 9.9 MU/cGy, respectively. The developed framework allows assessing and monitoring important parameters regarding radiation protection of a CK‐M6 using the actually applied treatment beams. Furthermore, it enables evaluating different clinical and constructional situations using the filtering options.Compared to a conventional linear accelerator, the Cyberknife ( This leads to the typical task of a medical physicist to optimize the design of bunkers such that radiation protection issues are managed, while keeping the corresponding costs and resources of the bunker construction as low as possible. This is a challenging task for dedicated delivery systems such as the Cyberknife (CK) system . In general, the dose given to a person can be expressed as follows:D˙i = dose rate due to exposure iti = time duration of exposure iti. Another possibility is to place attenuating material between the source of radiation and the person, e.g., build a bunker, which reduces the dose rate. This bunker shielding problem can be separated into primary and secondary barriers.Based on eq. It is important for the motivation of this work that, a few years ago, a new CK model  has been released, which differs from the previous versions22.AFor this study, clinical treatment plans delivered by a CK‐M6 at our center were analyzed. This robot‐based stereotactic radiation therapy system was initially equipped with a fixed field size interchangeable cone collimators (Fix) and an Iris (Iris) collimator systemFor this study, we retrospectively analyzed a database of 364 CK treatments performed at our center. Altogether, these patients received 1115 treatment fractions, which lead to a total number of 166'125 applied beams. An overview of treated tumor sites and used collimator types is provided in Table 2. For reasons of simplicity, the entrance barrier was neglected in this study and dimensions in left and right directions were adapted to be symmetrical. Figure The primarily used bunker design in this work is illustrated by Fig. 2.BThe CK data management system stores information about all applied beams into a database. This information can be extracted as a log file in xml format. In a first step, the newly developed framework reads for each patient the prescribed dose and the log files from all delivered patient treatments and creates a file containing the following information for each delivered beam: robot position as well as beam direction, number of delivered monitor units (MUs), collimator system used, applied tracking mode, and field size at SAD 800 mm. Note, that for the MLC the field size is defined as the area that is not covered by the leaves. All those parameters, together with the treatment site, which is mapped from a separate database, are recorded and stored into the treatment list. This treatment list is then purged from all sensitive information and serves as a completely anonymized repository of the clinically applied treatments. In a second step, the CK bunker is defined as a rectangular geometry with freely selectable dimensions.www.python.org), version 3.5. For all routines, data may be filtered according to collimator type or treatment site. These routines allow an analysis of the mentioned parameters as well as projecting the beams onto the inner surfaces of the considered bunker design by simply ray‐tracing the beam on its central beam axis.In order to extract and combine the available information, several routines are implemented in Python  and the distribution of the applied MUs are evaluated. This includes the MU distribution for each single barrier as well as a top view of the bunker, resulting from integrating all the applied MUs of the walls along the z‐axis , as defined by Purwar et al32 bunker. From this figure, it can be concluded that the applied MUs are neither distributed homogenously between the different barriers nor is the distribution homogenous on a single barrier itself. A more quantitative analysis is presented in Table The resulting MU distribution for all primary radiation beams in the treatment list is visualized in Fig. As there are only few irradiations using the Fix collimator for extracranial treatments and until now only few cases using the MLC for cranial treatments, the following situations are compared: Fix collimator vs. Iris collimator for cranial treatments and Iris collimator vs. MLC for extracranial treatments. The largest difference between collimators is, that using the MLC, about 5% more MUs are delivered to the left or right wall instead of to the floor.The ‘top‐views’ in Fig. The dependency of the MU distribution on the bunker size is presented in Table The analysis of the applied field sizes, represented by histograms of the field diameters in Fig. Finally, MFs of the treatment plans for the different collimation devices and treatment sites are compared. While the mean MF for the Fix collimator is the highest, the MLC shows the smallest MF Fig. . Althoug4A framework to evaluate important parameters regarding the radiation protection considerations of a CK‐M6 was developed. The resulting MU‐weighted direction distributions represent the current situation at our Institute and are very heterogeneous for the different barriers of the bunker. Whereas the left and right wall receive between 10% and 13% of the MUs irrespective of the collimator system or treatment site choice, the situation is different for the wall at the patient's feet. Cranial treatments result in a MU fraction of about 5% delivered to that wall, while extracranial treatments do not deliver direct beam there. This is explained analyzing the allowed beam directions for those two treatment types. As the CK‐M6 system just allows beams with an elevation up to 22° from the horizontal direction, the low number of MUs for ceiling is expected. However, for cranial treatments some beams hit the corners of the ceiling. The floor and the wall behind the patient's head represent the extreme values in this study, receiving 65%–79% and 0% of the MUs, respectively. Even if the orientation of the CK‐M6 and the location of the adjacent rooms are already fixed, the varying MU distribution over the barriers allows identifying suitable places to create e.g., a cable duct.The enhanced symmetry in the node distribution for the CK‐M6 compared to the CK‐G4 explains the observed differences in the right and left wall MU distributions between the work of Yang and FengGenerally, the different collimator systems used had a minor effect on the direction distribution, compared to the influence of different treatment sites.While enlarging the area that has to be shielded, larger bunkers naturally lead to a smaller number of MU/pixel. Furthermore, bunkers with a nonsquared footprint lead to substantial changes in the MU ratio between the left/right wall vs. the wall at the patient's feet and floor.Regarding the secondary radiation, the mean MF for the Fix and Iris collimators (8.6 and 8.3 MU/cGy) are larger than the mean MF for the MLC of 6.4 MU/cGy. This is due to the segmented manner of delivery as well as the larger field sizes, which are offered by the MLC and also used during the treatments Fig. . In addiSo far, all clinically delivered beams at our institution are included in the database. By always incorporating the most recent treatments, it is possible to monitor the radiation protection issues in almost real‐time. Furthermore, the whole framework was developed in a two‐step approach, in which the creation of the treatment list removes any sensitive information. This allows easily comparing data for different centers in future works.5The developed framework allows analyzing and monitoring radiation protection parameters for the present situation as well as filtering for collimators or treatment sites and exploring different bunker sizes.The authors declare that they have no conflicts of interest.


Continuum neural field equations model the large-scale spatio-temporal dynamics of interacting neurons on a cortical surface. They have been extensively studied, both analytically and numerically, on bounded as well as unbounded domains. Neural field models do not require the specification of boundary conditions. Relatively little attention has been paid to the imposition of neural activity on the boundary, or to its role in inducing patterned states. Here we redress this imbalance by studying neural field models of Amari type  with Dirichlet boundary conditions. The Amari model has a Heaviside nonlinearity that allows for a description of localised solutions of the neural field with an interface dynamics. We show how to generalise this reduced but exact description by deriving a normal velocity rule for an interface that encapsulates boundary effects. The linear stability analysis of localised states in the interface dynamics is used to understand how spatially extended patterns may develop in the absence and presence of boundary conditions. Theoretical results for pattern formation are shown to be in excellent agreement with simulations of the full neural field model. Furthermore, a numerical scheme for the interface dynamics is introduced and used to probe the way in which a Dirichlet boundary condition can limit the growth of labyrinthine structures. Importantly it gives a reduced description of solutions to a neural field model without any approximation. On the down side the approach cannot be generalised to treat smooth firing rate functions, though simulations by many authors have shown that the behaviour of the Amari model is consistent with neural field models utilising a steep sigmoidal function. Here we show how the interface dynamics approach can be generalised to include the effects of Dirichlet boundary conditions for arbitrary choices of synaptic connectivity.Neural field models are now widely recognised as a natural starting point for modelling the dynamics of cortical tissue. Since their initial inception in the 1970s by Wilson and Cowan , Amari , 4, and et al. , which aet al. , synaptiet al. , and coret al. , as wellnditions , and newnditions , 16. Neunditions , who pronditions  or even nditions . Amari’s [et al. , albeit In Sect. et al. [clamps neural activity at the boundary to a specific value. Of course other choices are possible, though this one is a natural way to enforce a functional separation between cortical areas.Although single neuron models are able to predict dynamical activity of real neurons that have a wide variety of spiking behaviour , 23, thecription . As wellcription , the undcription , visual cription , 27, andcription , 29, thecription , 31. In cription . This wa [et al.  on dynamΩ is a planar domain u represents synaptic activity and the kernel w represents anatomical connectivity. For simplicity we shall only consider the case that this depends on Euclidean distance. The nonlinear function H represents the firing rate of the tissue and will be taken to be a Heaviside so that the parameter κ is interpreted as a firing threshold. We assume that a suitable initial condition is specified for  on infinite domains, as well as the construction of these on finite domains without a boundary condition. Our key observation that allows the extension of the Amari approach to handle the boundary condition  is to exmentclass2pt{minimWe shall now consider Eqs.  and 4)  4) as thPrior to describing the analysis for a two-dimensional Amari neural field model with a Dirichlet boundary condition, we first consider the more tractable one-dimensional case. This illustrates the main components of our mathematical approach, as well as delivers new results about stable boundary induced bumps.edges of the bump interface, as they naturally separate regions of high and low firing activity. The differentiation of the level-set condition   4) on thψ given by (bump width is given by x (rather than just at the bump edges). Here we pursue the latter approach, though it is straightforward to check that the former approach gives the same answer.Since it is well known that the Amari model supports a stationary bump solution when the synaptic connectivity has a Mexican hat shape we now revisit this scenario and choosegiven by  will alsiewed in , which cz is given by L to x and use To determine the linear stability of a bump we write xpanding  to firstFrom  we may gκ for a neural field posed on a finite domain Fig. κ, and two of these bumps are stable. In other words, the Dirichlet boundary condition induces a new stable bump, whose active region occupies a large portion of the domain. In Fig. mentclass2pt{minimBefore discussing the extension of Sect. loped in , albeit mentclasspt{minimaΩ by writing We consider the integro-differential equation given by  with \doentclass1pt{minimanator in , using vs is a parametrisation for points on the contour Thus the denominator in  can be erator in  in termsentclass1pt{minimarator in  can be wity rule  can be eAs an example of the approach above let us consider a difference of Gaussians with mentclasspt{minimaentclass1pt{minimaψ given by , by . Using (+(t), by  the levez and ψ from Sect. Γ defining this integral. Each of the paths Γ connects a point x in the interior of the domain to a point on the boundary, and we set Hence, using the formulae for Note that we do not have to numerically integrate along this path , and that we need only to determine the values of Figure cs using  and 24)\documentD with a spot (circularly symmetric) solution of radius R. In this case t, and ψ by Given the large amount of historical interest in spot solutions of neural field models on infinite domains, and those on finite domains without incorporating the role of boundary conditions –39, it ited from .An implicit equation for the radius of the bump is obtained after setting the normal velocity to zero. Using  and 33)33) this For the specific choice of a difference of Gaussians given by  we may wet al. [0 and radius r and radius σ subject to the constraint Let us first consider a top-hat connectivity defined bytly from  as38\dor a spot  takes thUsing the results from Appendix Another natural piece-wise constant choice is the piece-wise constant Mexican hat shape given byThe stability of spots without boundary conditions has been treated by several authors, and see  for a rez is given by λ in the form To determine the linear stability of a spot we write xpanding  to firstgiven by  it is siκ. Moreover, the narrow spots are always unstable , cells respond preferentially to lines and edges of a particular orientation. A standard neural field model that links points at r and θ  readily confirm that boundary induced patterns can also be seen in models with a smooth sigmoidal firing rate. Thus it would also be of interest to extend the elegant functional analytic treatment of localised states in bounded domains by Faugeras et al.  to incoret al. , or subset al. . If thiset al. , or itseet al. , then thet al. . Anothering rate , and is  [et al. , and theg brains .One obvious caveat to all of the above is that the interface approach is restricted to Amari style models with a Heaviside firing rate. Nonetheless the qualitative similarities between Amari models and those with a steep sigmoidal firing rate are well known. In summary the treatment of neural fields with boundary conditions is a relatively unexplored area of mathematical neuroscience whose further study should pay dividends for the understanding of neuroimaging data, and in particular waves of activity in functionally identified and folded cortices.


The major therapy for ischemic stroke is thrombolytic treatment, but severe consequences occur when this method is used to treat hemorrhagic stroke. Currently, computed tomography and magnetic resonance imaging are used to differentiate between two types of stroke, but these two methods are ineffective for pre-hospital care.We developed a new brain diagnostic device for rabbits based on electromagnetic induction to non-invasively differentiate two types of stroke. The device includes two coils and a phase difference measurement system that detects the magnetic inductive phase shift (MIPS) value to reflect the tissue’s condition. The hemorrhage model was established through the injection of autologous blood into the internal capsule of a rabbit’s brain. Ischemia was induced in the brain of a rabbit by bilateral carotid artery occlusion. Two types of animal models were measured with our device.The MIPS value gradually decreased with increasing injected blood and increased with ischemia time. The MIPS changes induced by the two types of strokes were exact opposites, and the absolute values of MIPS variation in the hemorrhagic and the ischemic groups were significantly larger than those of the normal control group (P < 0.05).The tested technique can differentiate ischemic stroke from hemorrhagic stroke on rabbit brain in a non-invasive, continuous, and bulk monitoring manner by using a simple and inexpensive apparatus. Stroke can be classified into two kinds, namely, hemorrhagic and ischemic. The incidence rate of this medical condition continuously increases with the aging of the global population and the intensification of social pressure. The latest statistics issued by the Ministry of Health of the People’s Republic of China reveal that stroke is the first cause of death among all diseases in China, and its standardized mortality rate in the country ranks first in the world. Besides, such a mortality rate continues to increase at a yearly rate of 9%. Approximately 2 million people suffer a stroke each year. Among this population, nearly half die, and roughly 3/4 of the survivors experience varying degrees of disability. Stroke has brought severe economic burdens to China and its people, and this situation emphasizes the urgency for the prevention and control of such a medical condition . IschemiThe early thrombolytic treatment of ischemic stroke is an established procedure today, but it could be disastrous when performed on patients with hemorrhagic stroke –5. At prSeveral technologies for distinguishing ischemic from hemorrhagic stroke have been developed over the years. Doppler ultrasonography can accurately identify the states of the cerebral artery , but it cannot exclude hemorrhagic transformation from the resulting ischemic regions . ElectriThe MIPS technique is based on the principle of electromagnetic induction and measures the phase perturbation of the induced magnetic field (IMF) to the excitation magnetic field (EMF). The phase perturbation is proportional to the conductivity of the measured object. Changes in conductivity can be detected by the MIPS technique . HemorrhOur group has long been engaged in MIPS detection research on cerebral hemorrhage, cerebral ischemia and brain edema, and carried out a large number of animal experiments. The experimental results show that MIPS would gradually reduce as the amount of bleeding increased –21. In tV and ΔV, respectively) can be represented by the phasor diagram shown in Fig. V is related to V as follows:ω is the signal frequency; σ, εr, and μr are the electrical conductivity, relative permittivity, and relative permeability of the sample, respectively. ε0 and μ0 are the permittivity and permeability of free space, respectively. Q and R are geometrical constants. Thus, the total signal (V + ΔV) detected by the receiving coil lags the primary signal by an angle φ, which is approximately proportional to ω and σ.In MIPS technique, the measured object is always placed between a transmitting and receiving coil, and the current flow in the transmitting coil induces a primary magnetic field. This primary magnetic field then incurs eddy currents in the object that in turn produce a secondary magnetic field. Both the primary and secondary fields are detected by the receiving coil. The primary and secondary signals  inserted into the slot of a PC. The output signal from the receiving coil was connected to the other input port of the PCI5124. The rabbits were placed on a polyvinyl chloride plate platform, which could move up–down, left–right, and back–forward. Before measurements, the height and horizontal position of the platform were readjusted, placing the rabbit brain exactly between the two coils. The geometrical position of all subjects was carefully maintained as similar as possible for all measurements. A phase difference measurement program based on a Fast Fourier Transform (FFT) algorithm was compiled on LabVIEW software. This program was used to measure the phase difference between the two input signals to the PCI5124 acquisition card. The sampling rate was set at 100 M/S, and the number of sampling points was 1 million. The changes in the phase were recorded as a function of the ischemia time or injection volume.The experimental setup is shown in Fig. All experimental protocols were approved by the Animal Experiments and Ethics Committee of the Third Military Medical University, and the care of the animals was performed in accordance with the Declaration of Helsinki and the guidelines issued by the International Association for the Study of Pain , 24.A total of 20 New Zealand rabbits (2.0–2.5 kg in weight) were randomly selected and divided into two groups, i.e., hemorrhage and ischemia groups (each n = 10). To lessen killing and keep consistency, four animals were measured two hours earlier before blood transfusion or ligation in two groups. The data before transfusion or ligation were regarded as control group data. Eight rabbits’ data before transfusion or ligation were randomly selected and regarded as a control group data. All animals were purchased from the Animal Center of Chongqing Daping Hospital and were well attended to before and after the experiment.Considering that ICH mainly occurs in the internal capsule, we established the internal capsule hemorrhage model by injecting self-body blood into the hindlimb of the right capsule. The operation procedure in this study is similar to those in previous research –21. The The bilateral carotid artery permanent ligation method was used to establish the ischemia model. The rabbits were first anesthetized by injecting their ear vein with urethane . Under anesthesia, the necks of the rabbits were depilated and disinfected. Subsequently, the middle neck skin was incised and bluntly dissected to expose and separate the bilateral common carotid artery, which was knotted using two 1 mm-thick nylon lines, but not ligated Fig. . After tData of the absolute value of MIPS measured in the three groups were presented as mean ± SD. The Wilcox rank sum test was applied to test the difference in the MIPS between the stroke groups and the normal control group in SPSS . A value of P < 0.05 was considered statistically significant.The breathing and heart beat signals in each animal data were filtered with wavelet decomposition. For the result of each animal in ischemic and control group, the MIPS data during 2-h were equally separated into 42 successive parts. Then the mean of each part was determined. This processing method is essentially equivalent to obtaining one MIPS data at every 2.85 min. Then the all group animal data were averaged, and standard deviation (SD) was determined. For the result of each animal in hemorrhage group, the MIPS data during 1-h injection were equally separated into 21 successive parts. Then the mean of each part was determined. This processing is essentially equivalent to obtaining one MIPS data at every 2.85 min or every 0.047 ml of injection. Then the all group animal data were averaged. Figure A gradual decrease in the MIPS is observed as a function of the injection volume in Fig. A gradual increase in the MIPS is observed as a function of the ischemic time in Fig. The variation of MIPS in control group is little with the time. The average MIPS drift (maximum–minimum) in 2 h is 0.333 ± 0.059°, which is significantly less than those in the other two groups (P < 0.05).Despite the MIPS changing trends for two kinds of stroke were opposite, the amplitudes of variation were small (0.942 ± 0.307° vs 1.448 ± 0.467°), indicating the low sensitivity of the coil system. Therefore, it is necessary to change the coil structure for improving the MIPS sensitivity. Second, all group data show poor consistency. The standard deviation for the ischemic group is particularly large. The reasons are as follows: First, there are large individual differences for all animals. The weight and age have a great impact on the results. Second, it is hard to ensure that each animal was given a sufficient depth of anesthesia. Some rabbits were restless in experiment for poorly anesthetized. Third, it is difficult to maintain that all animal heads were at the same position between the two coils. The head position is of importance, because MIPS sensitivity varies from the different positions between the two coils, which directly affect the measurement results. Fourth, it is also hard to keep the consistency on the injection position and injection depth for all animals. In general, the experimental method should be improved. The positioning device of higher accuracy is required.In addition to the above problems, animal ICH and ischemia model also should be improved. Although the used auto blood injection model is perfect for our experiment, because its bleeding amount and bleeding rate are controllable for the convenience of comparative studies, its bleeding mechanism differs from the clinical ICH. Another common bleeding model is collagenase-induced bleeding. The bleeding of this model is indeed caused by rupture of blood vessels, but the bleeding amount and rate are uncontrollable. After the initial trial is completed in the future, we will attempt to adopt this model for MIPS experiment. The ischemia model is often created by middle cerebral artery occlusion (MCAO) which conforms to the actuality. However, this model is mostly used for rat and mice. The rabbit has different arterial structure, and the operation is very difficult, so this model is rarely conducted on rabbit for low success rate. Therefore, we used the bilateral carotid artery permanent ligation method to establish the ischemia model on rabbit. Many papers indicate that bilateral carotid artery permanent ligation may only cause mild cerebral ischemia, which may partly explain the low variation of MIPS in our experiment. We will perform this ischemia model test on rat in the future.Ischemia in the brain is an important clinical problem that causes serious consequences when treated as hemorrhage. In this study, a novel coil structure based on MIPS technology was tested to distinguish between ischemic stroke and hemorrhagic stroke. Two stroke animal models were used, and the measurements showed that MIPS gradually decreased with the increase of bleeding amount, but it increased with the prolonging of ischemic time. The changing trends of MIPS because of the two types of stroke were completely opposite and the absolute values of variations in MIPS of the hemorrhagic group and the ischemic group were significantly larger than the normal control group. The mechanisms of MIPS changes due to these two types of stroke are different. The analyses given in the previous Sections indicate that the reduction of MIPS at early hemorrhage is caused by the regulating effect of CSF, and the MIPS changes are mainly caused by the volume changes of tissues at early hemorrhage. The ischemia-induced MIPS rise is attributed to the changes of tissue components. At high frequencies, the increase of MIPS is primarily caused by the change of the total ion content in the tissue. With the prolonging of ischemic time, the accumulation of metabolic products in the tissue is increased, which leads to small impedance, but large MIPS. The consistency among all animal results for each group is low mainly because the placement of the rabbit head between the two coils cannot be guaranteed with full consistency. The individual differences  among the samples used may be another reason for the low consistency. In general, this is just a pilot study to validate the feasibility of this approach to discriminate two types of stroke, and more work need to be performed. Based on the experimental results, the measurement sensitivity for two kinds of stroke is low; the coil system should be improved. Second, the experimental setup also needs to be improved for higher precision. An animal positioning device should be made to maintain the consistency for all animals in the experiment. Besides, work should be done to establish new ICH and ischemia models which are more consistent with clinic.


One of the test images in each set was per design free of noise and, therefore, genuinely 2D periodic so that it adhered perfectly to the constraints of a Bravais lattice type, Laue class, and plane symmetry group. Gaussian noise with a mean of zero and standard deviations of 10 and 50% of the maximal pixel intensity was added to the individual pixels of the noise-free images individually to create two more images and thereby complete the sets. The added noise broke the strict translation and site/point symmetries of the noise-free images of the four test sets so that all symmetries that existed per design turned into pseudo-symmetries of the second kind. Moreover, motif and translation-based pseudo-symmetries of the first kind, a.k.a. genuine pseudo-symmetries, and a metric specialization were present per design in the majority of the noise-free test images already. With the extraction of the lattice parameters from the images of the synthetic test sets, we assessed the robustness of the algorithms’ performances in the presence of both Gaussian noise and pre-designed pseudo-symmetries. By applying three different computer programs to the same image sets, we also tested the reliability of the programs with respect to subsequent geometric inferences such as Bravais lattice type assignments. Partly due to per design existing pseudo-symmetries of the first kind, the lattice parameters that the utilized computer programs extracted in their default settings disagreed for some of the test images even in the absence of noise, i.e., in the absence of pseudo-symmetries of the second kind, for any reasonable error estimates. For the noisy images, the disagreement of the lattice parameter extraction results from the algorithms was typically more pronounced. Non-default settings and re-interpretations/re-calculations on the basis of program outputs allowed for a reduction (but not a complete elimination) of the differences in the geometric feature extraction results of the three tested algorithms. Our lattice parameter extraction results are, thus, an illustration of Kenichi Kanatani’s dictum that no extraction algorithm for geometric features from images leads to  Direct space imaging techniques such as scanning tunneling microscopy (STM) and (scanning (S) electron probe and high-resolution (HR) parallel illumination) transmission electron microscopy (TEM) provide nowadays atomic resolution in detected images on a routine basis –4. STEM The information theory approach to the analysis of more or less 2D-periodic images is quantitative and considers microscopes as channels through which human beings obtain structural information about solids at the atomic level. The images that the microscopes deliver are “data planes”  from whi3(CH2)7CH=CH(CH2)7COOH, could possibly be imaged in the future with a low electron dose in 3D-atomic resolution by electron exit wave reconstructions from HRTEM through-focus series  for which the individual images were recorded with parallel illumination either in a single projection  and [01] vectors of this lattice needs to be within an error bar that contains the 90° value.The same example can also be discussed on the basis of the conventional  unit cell parameters of the mp → oc → hp, see Fig. mp, oc, and hp), the noisy data, and the set of corresponding maximal likelihood lattice parameter extraction results. Future data with a lower noise level could result in one of the other two models in the set being the most likely translation symmetry. We will expand on this elsewhere.Reducing the widths of the error bars on the extracted lattice parameters of this example sufficiently  so that the approximately equal (≈) sign between the lattice vector magnitudes can be safely ruled out and/or the 60° value is excluded from the extracted lattice angle interval would be one way to deal with this ambiguity. Utilizing a G-AIC and Akaike weights  for the When error bars are zero, i.e., in a mathematically strict and abstract sense, the conceptual basis for the above discussed translational pseudo-symmetries disappears. Correspondingly, approximately equal (≈) signs in the relations between the lattice parameters would no longer be allowed.Limiting cases for lattice parameters that lead in the abstract mathematical sense from a lower symmetric Bravais lattice type to its higher symmetric counterpart(s) are listed in column 3 of Table mp, one for op and two for oc. It is also clear from the fourth column of Table oc Bravais lattice setting (which encompasses two lattice points). This is because the oc limiting case of the oblique (mp) Bravais lattice type possesses twice the area of the oblique unit cell. Correspondingly, the hp limiting case of the (oc) rectangular centered Bravais lattice type possesses only one half of the rectangular centered unit cell area. Alternative translation vectors need, therefore, to be chosen when one “moves up” in the mp → oc → hp hierarchy branch, see middle sketches in Fig. The number of limiting cases that lead to higher symmetric types of Bravais lattices in Table γ = 120° to γ* = 60° .As is very well known, Fourier transforms relate corresponding pieces of information in direct and reciprocal space to each other per mathematically defined relations. The intertwined symmetry and metric properties of a Bravais lattice type are, as a consequence, independent of the space in which one chooses to work , 18. It  cm or c2mm, one needs to label the ‘diffraction peaks’ in a way that index sums are always even because one considers all odd integer sum ‘reflections’ as being systematically absent  oriented transmission electron diffraction pattern of a very thin cubic body-centered crystal (such as tungsten) are of the {110} type, i.e., This is all very different from systematic absences that are due to glide lines in 2D (as well as glide planes and screw axes in 3D), where certain “odd reflections” obtain zero amplitude by destructive wave interference in single scattering experiments or correspondingly by mathematical superpositions of complex-number valued FCs . These r2, 2mm, 4, 4mm, 6 and 6mm. Four of these point symmetries, i.e., 2, 2mm, 4mm, and 6mm, are holohedries and, therefore, in 2D responsible for the one to one correspondences between the lattice systems and crystal systems.Since the point/space symmetry and metric properties of the Bravais lattice types are intertwined, programs such as CrysTBox and CRISP that display dFT amplitude maps of more or less 2D-periodic images from which they extract lattice parameters allow the user to assess the 2D-Laue symmetry class of the images visually. This kind of symmetry is based on the amplitudes of the FCs around the central (00) peak, rather than their positions in this map. The Laue classes in 2D are the six point symmetry groups References , 26 provAll of the 12 synthetic 2D-periodic images used for our lattice parameter extraction review are presented in Fig. The noise-free images are at the top of each of these columns in the first row of Fig. The second row of this figure consists of images where independent Gaussian noise of mean zero and a standard deviation of 10% of the maximal pixel intensity was added to the individual pixels. The third row in Fig. One may consider the 10% amount of added noise to be “small to moderate” relative to the signal in the images because the latter is highly redundant due to its 2D-periodic nature. The 50% added noise may then be considered as “moderate to excessive” with respect to the signal in the images for the same reason.Of all of the test images, the first two in the second column, i.e., images #4 and # 5, should present the least challenge to any lattice parameter extraction algorithm because there is a clear difference in the magnitudes of the two lattice vectors and a 90° lattice angle per design while additional noise is either non-existent or small to moderate. Also these two images are composed of approximately 175 “sub-images” of individual unit cells so that an effective averaging can take place by suitable algorithms to reduce the effective noise level of the average unit cell.There are also no genuine pseudo-symmetries per design in the noise-free image #4. Note, however, that all symmetries in image #5 are only pseudo-symmetries of the second kind because all originally existing symmetries were unavoidably broken by the addition of 10% Gaussian noise. .calculated images in Vasco Ronchi’s sense rather than experimentally detected images  due to adding Gaussian noise to a synthetic (noise-free) image with metric specialization (see footnote 3) suffice.  close to 90° or 120° (within error estimates) and the magnitudes of the unit cell vectors are close to being equal (within error estimates), many researchers would not consider the possible existence of a rectangular centered Bravais lattice where the tp) or hexagonal (hp) Bravais lattice types, if the error bars allow for these conclusions. There is, however, no objective basis to rule out lower symmetric Bravais lattice types to justify these conclusions either by a human being or by a currently existing computer program. The Committee on Statistical Descriptors of the International Union of Crystallography was well aware of this fact when it stated that “Thoughtless use of established procedures in widely distributed software may be as harmful as the natural tendency of most people to prefer results in agreement with preconceived ideas.” * and [01]* labels to the two shortest lattice vectors in the amplitude map of a dFT regardless of their intensity when she or he intents to extract the direct space lattice parameters for subsequent Bravais lattice type assignments. As already mentioned in the “Since we saw from the amplitude map of the dFT of an image which reciprocal lattice spots had been selected as the reciprocal basis vectors by the CrysTBox program, we made re-interpretations and re-calculations of the corresponding direct space lattice parameters and the derived unit cell areas. The latter was particularly easy as we only needed to count the number of reciprocal lattice nodes that correspond to one CrysTBox determined reciprocal unit cell and multiply the derived direct space unit cell areas with the corresponding factor.For images #1 to #3, there are four extra reciprocal lattice nodes that are completely included within the four lattice nodes that outline one reciprocal unit cell that CrysTBox has assigned. This means the four extra reciprocal lattice nodes count full because they are completely included within the algorithm assigned reciprocal unit cell. The four reciprocal lattice nodes that outline the assigned reciprocal unit cell itself count, on the other hand, just for one quarter of a full node each, because they are each shared with three other reciprocal unit cells. Four times one quarter plus 4 sums to 5 as the factor by which the direct space unit cell areas of these images as listed in Table The corresponding entries for these three images as extracted by the CrysTBox program in Table Analogously for the other images for which there are red ink entries in Tables a* axis. The direct space lattice vector magnitude along the a axis is, therefore, to be doubled so that the listed b/a ratio for this image in Table In the case of image #6, there is one extra reciprocal lattice node completely included within the reciprocal lattice cell that the CrysTBox algorithm assigned and this node is located parallel to the a* axis. The entries for the b/a ratios of these three images in Table For images #7 to #9, there are two extra reciprocal lattice nodes completely included within the reciprocal lattice cells that the CrysTBox algorithmic implementation came up with and they are both located parallel to the b/a ratios for the entries for these three images for the CrysTBox program in Table For images #1 to #3 there is per design no crystallographic convention for the origin of the unit cells (as mentioned above). There are, therefore, no unique sets of direct space lattice vectors so that there are consequently no unique Probably due to the motif-based pseudo-symmetry of the first kind that apparently “fixes” the origin to the positions of the pseudo-twofold rotation points in these three images, both the PUCE and the CRISP program extracted lattice parameters in good agreement with each other as listed in Table As anticipated above in the “For image #8, a lattice of the rectangular (primitive) reciprocal Bravais lattice type was clearly visible in the amplitude map of the dFT of the image when processed with CRISP. This program would however in its (default) automatic reciprocal lattice assignment setting ignore the first two weak peaks in the amplitude map of the dFT of image #8 and extract parameters of a pseudo-square Bravais lattice instead regardless of whether or not the maximal circular or square area of the image was selected for the processing, see corresponding entries in Tables When the reciprocal basis vector assignment was, on the other hand, made in the alternative manual setting, rectangular vectors of uneven magnitudes were handed over to the rest of the algorithms of the CRISP program so that qualitatively correct results were obtained, see Table For the noisiest image of this test image set, image #9, the first two weak peaks in the dFT amplitude map of the CRISP program were almost indiscernible to us due to the very noisy background although one could still make them out if one “knew” that they must be present. The results for this image are, therefore, analogous to that of image #8 as we also utilized the alternative (non-default) manual reciprocal lattice assignment feature of the CRISP program in order to compile the entries for image #9 in Tables It is remarkable that the noise-free image of this test set, image #7, follows the same pattern in as far as the CRISP program in its default setting is concerned, see Tables The extraction results from the images that possess per design the rectangular centered Bravais lattice, i.e., #10 to #12, require a separate discussion as they possess either a metric specialization or extreme cases of translational pseudo-symmetry of the special kind. The “detached lattice symmetry” (see footnote 3) or very pronounced pseudo-symmetry  was easy to detect in the amplitude map of the dFT as provided by the CrysTBox and CRISP programs. For a thorough elucidation of different types of pseudo-symmetries, one needs to determine the most likely Laue classes and plane symmetries of the test image set in addition . We willWhile the lattice parameters needed to be re-calculated from the outputs of the CrysTBox program for images #10 to #12, the CRISP program possesses  an alternative setting to test for the existence of a rectangular centered Bravais lattice type whenever the ratio of the magnitudes of the shortest reciprocal lattice vectors is approximately unity. We utilized this feature under both the default and non-default settings of the CRISP program to arrive at qualitatively correct results in both cases, see Tables We re-calculated the entries for the rectangular centered Bravais lattices for listings as entries for images #10 to #12 for the CrysTBox program in Tables We have to note in passing that the results of the PUCE program depended sensitively on image format conversion processes that were performed prior to the lattice parameter extractions. Some of the utilized image format conversion programs changed the nature of the noise inadvertently so that it was no longer Gaussian.Surely, any image format conversion software should not do this kind of thing because it is equivalent to the inadvertent introduction of systematic errors into the synthetic test images. The results of both the CrysTBox and the CRISP program are, on the other hand, quite insensitive to image format conversions that were done to their inputs for all of the cases we studied in this review. This is probably due to the built-in “noise-filtering feature” of lattice parameter extraction algorithms that work in reciprocal space.Note finally that the results from the moderately noisy image #5 are within reasonable error bars in qualitative agreement across all three programs/algorithms in their default settings, see Tables b/a ratio for this image is 0.601 ± 0.011, the average lattice angle is 89.93° ± 0.08°, and the average unit cell area is 373.3 ± 1.0 square pixels as obtained from the combination of the results of the three programs in their default settings. For comparison, the design parameters for image #5 are: b/a ratio = 0.60, lattice angle = 90.0°, and unit cell area = 375 square pixels. The agreements between the averaged extraction results and the design parameters are for this particular image pretty good, but our initial error estimates that took the outputs of the CRISP program as significant numbers were, as Tables For image #5, one would be justified to average the lattice parameter extraction results and what has been derived from them over the three different algorithms and to obtain a higher accuracy and precision. Indeed the average As a matter of fact, the initial error estimates were throughout the whole review far too optimistic. Ten to twenty times larger error estimates than the initially assumed significant number outputs of the CRISP program are obviously more realistic given the totality of the results discussed in this section. Typically extracted lattice parameter magnitudes are, therefore, at least for the kinds of noisy images that are shown in Fig. Three different algorithms (as implemented in three different computer programs) were put to the task of extracting lattice parameters from four sets of synthetic test images that were 2D periodic per design but also contained images that were noisy so that all site and translation symmetries were broken. While one of the images in each of these sets was free of noise , independent Gaussian noise of mean zero and a standard deviation of 10 or 50% of the maximal pixel intensity was added to the individual pixels of that image in order to create two noisy images for each set of test images. While the added noise obscures the translation and site symmetries in these images, it obviously cannot change them in a systematic way. The presence of noise is supposed to present a greater challenge for any computer program to extract accurate lattice parameters with a high precision.Our sets of calculated test images can be considered to be equivalent to images that were detected with different signal-to-noise ratios by an instrument that is free of systematic imaging errors. The signal in the images of one test set is then a combination of individual pixel intensities that obey designed restrictions that are set by the combination of the chosen plane symmetry group, i.e., a combination of translation and site symmetries, with the metric properties of the unit cell.A (non-existing) ideal algorithm for the extraction of lattice parameter information applied to any one of these test images would have quantified the magnitudes of the two basis lattice vectors and the angle between these vectors accurately to the values that were put into the images’ designs. The translation symmetry/Bravais lattice type that is to be inferred from these parameters would also be the one that was put into the images’ designs.definitive algorithm for the extraction of geometric-structural features from noisy images for the task at hand.Moreover, an ideal algorithm would not have introduced systematic errors into the geometric extraction results and delivered the same results for the three individual images of each test set. An increase of the error estimates on the lattice parameters would be expected with an increasing amount of noise in the images as the ideal algorithm would propagate random errors faithfully. This kind of an ideal algorithm could be considered as the (non-existing) A minor complication arose in our review by the fact that none of the three tested programs provides explicit statements on error bars. We were, therefore, initially forced to take the numerical outputs of the computer program that provides the fewest number of digits for the extracted lattice parameters as significant numbers. This was the CRISP program, which had also the best overall test performance. The resulting error estimates were then also used for the outputs of the other program that extract lattice parameters in reciprocal space . The error bars on the unit cell areas and other derived results were for these two programs obtained by standard error propagation calculations. As a result of this review, we have to conclude that our initial error estimates were way too optimistic by a factor of ten to twenty for the noisy images at least.For the program that allows for lattice parameter extractions in direct space , we made a reasonable assumption for the precision with which the Cartesian coordinates of the start and end points of lattice vectors could be extracted from the images. For quantities that were derived from the extracted coordinates, e.g., the ratio of the magnitudes of the lattice vectors, the lattice angle, and the area of the unit cell, the error estimates of the coordinates were propagated to the derived end results.Contrary to our expectations, the sets of lattice parameters that the three programs extracted in their default settings from the same image disagreed in the vast majority of cases within both the originally anticipated and more reasonable error estimates. On the one hand, this fact reflects positively on the nature of the test images in the sense that they present tough challenges to lattice parameter extraction algorithms because many of them possess pseudo-symmetries of the first or special kind per design. This fact, however, also reflects somewhat negatively on the tested computer programs/algorithms because most researchers (including the programmers) would probably expect them to perform much better for the tasks at hand.definitive extraction algorithm for geometric-structural features in all real-world applications. Nevertheless, a very brief summary of the test performances of the three computer programs is in order in this final section of this review.The main thrust of this paper was, however, not at all a ranking of the relative performance of the three tested programs. The test performances of the corresponding three different algorithm implementations are supposed to serve collectively as an illustration of the fact that there is simply no Of the three tested programs, the one that has been around for more than a quarter of a century as a windows executable, i.e., CRISP, performed best. The CRISP program is also the only one of the three tested programs that allows for a direct route to the extraction of the parameters of rectangular centered Bravais lattices. Plane symmetries and Laue classes of more or less 2D-periodic images can also be determined with this program (in a somewhat subjective manner) so that it is the only one of the three tested programs that offers a systematic route towards elucidating pseudo-symmetries of the first and special kind.The application of the CRISP program in non-default settings resulted in extracted lattice parameters that were entirely consistent with the designs for all test images. This could be due to both its classical 2D-crystallography approachNaturally, lattice parameter extraction from noisy images is more difficult in direct space so that the PUCE program is at a disadvantage in this task in comparison to the other two programs. For three of the four noise-free images, the PUCE program extracted lattice parameters in very good agreement with the results from the CRISP program and the a priori known Bravais lattice types. In the fourth case, where there was a rectangular centered Bravais lattice type per design, the PUCE program extracted the parameters of its primitive sublattice (because it is designed to extract primitive lattice parameters only).The lattice parameters that the CrysTBox program extracted in its default setting had to be re-interpreted/re-calculated in 10 out of 12 cases. The reason for this was in seven cases the implemented strategy of this program to assign a highly precise reciprocal lattice in reciprocal space that does not need to outline the smallest reciprocal lattice unit cell. This meant ignoring the shortest reciprocal lattice vectors when they had small amplitudes in the corresponding map of the dFTs of these images. After re-interpretation/re-calculation, all results from the CrysTBox were qualitatively correct. Also the CrysTBox program is optimized for work with electron diffraction patterns rather than more or less 2D-periodic images. This program possesses, in addition, many non-default settings that we did not test as part of this review.For the detection of pseudo-symmetries of the first and special kind, the user benefits greatly when a computer program displays the amplitude map of the discrete Fourier transform of a more or less 2D-periodic image. This is because the dFT amplitude map displays the Laue class that corresponds to the underlying plane symmetry group. When there is no pseudo-symmetry of the first or special kind , there is also no obvious mismatch between (or detachment of) the visible point symmetry around the center of this map and the translation symmetry that is governed by the metrical properties of the extracted lattice parameters. While the CRISP and CrysTBox programs both possess such a feature, it is absent in the PUCE program as the latter works in direct space exclusively.For dealing with pseudo-symmetries of all kinds, it is also helpful when a computer program allows the user to overwrite an automatic (default) assignment of the reciprocal basis vectors in the amplitude map of a dFT of a more or less 2D-periodic image. The CRISP program possesses this feature as well. The only negative thing that could be said about this program is that, typically, error bars on the extracted unit cell angles are at least one order of magnitude larger than one would expect based on the assumption that the CRISP program output are significant numbers. (The other two programs provide even more non-significant numbers as outputs).no definitive algorithms/computer programs for the extraction of geometric-structural features from detected images. As there is much more hierarchical geometric-structural information beyond Bravais lattice types that could be extracted from noisy images , we hope to have brought the implications of Kanatani’s no definitive geometric feature extraction algorithm/results in all real-world applications dictum and his comments on the vast majority of computerized attempts to extract symmetries and other hierarchical geometric features from noisy images to the attention of the wider scientific community. This will hopefully lead to a more thoughtful “use of established procedures in widely distributed software” and a disengagement from “the natural tendency of most people to prefer results in agreement with preconceived ideas” as encouraged by the Committee on Statistical Descriptors of the International Union of Crystallography.We conclude finally that our testing of the lattice parameter extraction capabilities of three different algorithms (as implemented in three different computer programs) was useful because not all readers might have been aware that there are simply There are many more geometric-structural feature extraction programs to be written and thoroughly benchmarked with respect to each other on sophisticated test image sets in order to make progress collectively as a community towards the shared goals of model-based imaging, materials informatics, and the knowledge-based designs of new materials. Kanatani’s geometric AICs could become very useful in the pursuit of these overarching goals because they provide model parameter spaces of comparatively small dimensionalities, noise-level dependent rankings that are free of arbitrarily set thresholds, and are applicable whenever very small systematic imaging errors are negligible with respect to small random imaging errors with an approximately Gaussian distribution. Note that in the case of nested models, the noise level of an image does not even need to be estimated when one wants to find out which crystallographic model minimizes the unavoidable information loss (that is associated with the model’s usage as representation of the image data).We also hope to have achieved the secondary goal of this paper so that misconceptions surrounding Bravais lattices in 2D and plane symmetry groups that existed in the wider scientific community are now cleared up. Addressing these shall hopefully foster the widespread application of G-AICs by microscopists, computer scientists, and applied crystallographers in the future.


Caulobacter crescentus, a scaffolding protein, PopZ, localizes to both poles and aids the differential patterning of proteins between mother and daughter cells during division. Polar patterning of misfolded proteins in Escherechia coli has also been shown, and likely plays an important role in cellular ageing. Recent experiments on both of the above systems suggest that the presence of chromosome free regions along with protein multimerization may be a mechanism for driving the polar localization of proteins. We have developed a simple physical model for protein localization using only these two driving mechanisms. Our model reproduces all the observed patterns of PopZ and misfolded protein localization - from diffuse, unipolar, and bipolar patterns and can also account for the observed patterns in a variety of mutants. The model also suggests new experiments to further test the role of the chromosome in driving protein patterning, and whether such a mechanism is responsible for helping to drive the differentiation of the cell poles.The spatial patterning of proteins in bacteria plays an important role in many processes, from cell division to chemotaxis. In the asymmetrically dividing bacteria  C. crescentus and misfolded protein aggregation in E. coli and provides predictions about how this mechanism could spontaneously lead to asymmetric patterning of the poles.A key process in biology is the self-assembly of biomolecules into highly organized structures. This spontaneous assembly can give rise to complex spatial patterns that help give spatial order to the cellular environment. In many bacteria, the patterning of proteins to the cell poles allows the bacteria to differentiate one end of the cell from the other. What mechanisms can lead to the spontaneous organization of proteins to the cell poles? Prior work has shown that such patterning can emerge from interactions between proteins and the cell membrane. In this paper we use computational modeling to show that a novel patterning mechanism involving only the presence of the bacterial chromosome and a self-associating protein is sufficient to generate polar patterning in bacteria. This model explains recent experiments on polar patterning in  Caulobacter Crescentus show that the presence of the chromosome may also play a key organizing role in positioning protein scaffolds at the poles independent of interactions with the membrane Escherechia coli is also preferentially localized to the poles, in particular to the older mother cell's pole A variety of molecular mechanisms have been identified for localizing proteins in bacteria cells. The emergence of spontaneous patterns from instabilities arising from the reactions of diffusing proteins C. crescentus the scaffolding protein PopZ forms domains that occupy the cytoplasmic space at the two poles C. crescentus , PopZ exists at only one pole, inherited from the previous division. After division it begins to assemble and form at the other pole, leading to a bipolar pattern in the dividing cell. However, PopZ can also display a variety of subcellular localizations, not just bipolar, depending on both the amount of PopZ in the cell and on the cellular shape. Amazingly, such patterning can also be reproduced by expressing PopZ in Escherichia coli that possesses no popZ homologue.In E. coli, from unipolar to bipolar localization, with the occasional localization midcell Similar patterning was observed for misfolded proteins in E. coli. These experiments have led to the hypothesis that the presence of chromosome free regions along with protein self-association could be a potential mechanism driving localization What mechanisms could lead to the diversity of observed patterns of aggregating proteins with bacterial cells, such as PopZ or misfolded protein? Cell curvature has been suggested as a mechanism for sorting proteins to the poles via attraction to lipid domains that prefer negative curvature. Experiments on spherical cells showed that PopZ can show diffuse C. crescentus and misfolded protein in E. coli and suggest new experiments that would help to test the model further.In this article we provide a model along with simulations that show that protein multimerization in chromosome free regions can be a sufficient mechanism for polar localization. To model the localization of aggregating proteins in bacterial cells we consider a cellular environment that contains the circular bacterial chromosome and a diffusing protein that is able to self-associate. In bacteria the DNA compacts into a structure called the nucleoid that occupies a smaller volume than that of the cell, and it is estimated to have a volume fraction between With respect to the aggregating protein, assays on PopZ's structure have revealed that it exists as a multimer and grows as a complex, highly branched three dimensional lattice With respect to interactions between the protein and DNA, we assume that there is no interaction other than self-avoidance. Both the DNA polymer and protein are confined within the cell by the cellular membrane that is modeled as a barrier. For further details of the model and the various interactions between the constituents see the Materials and Methods section. We now present results for the patterning of protein in the presence of DNA in different cellular geometries and variation in model parameters.E. coli from Here we consider the patterning of aggregating proteins in cells possessing a cylindrical geometry with length In  one end  since thpole see . The locE. coli and C. crescentus due to the fact that their genomes are roughly the same size yet C. crescentus has a smaller cellular volume. To separate the effect of DNA volume fraction from cell size, we decided to increase the volume fraction of DNA within the cell from C. crescentus, favors bipolar patterning of PopZ, whereas the unipolar pattern occurs more frequently in E. coli. These results would also predict a potential change from unipolar patterning to bipolar patterning under osmotic shock that would cause the cell size to shrink, thereby increasing the DNA volume fraction The volume fraction of DNA within the cell affects the types of patterns that can be generated and at what protein concentrations they occur. This effect is particularly relevant since there is a difference between the volume fractions of DNA between To further test the effect of DNA and its polymer structure in the model, we also consider the possibility of breaking the DNA into smaller fragments. In We have also modeled the nucleoid using just a self-avoiding DNA polymer confined within a smaller cylindrical volume and find that the similar progression of patterns is found. We find that we need to confine the chromosome to a cylinder whose diameter is Next we examined the effect of scaling cell size for the situation where unipolar patterns were favored. From the simulations discussed above, using a DNA volume fraction of erns see . Only a E. coli when the properly folded protein did not localize whereas it did upon unfolding. For strong protein interactions, In our model, the interaction between protein subunits is governed by a single energy, cell see . In Fig.We next consider the situation where the aspect ratio of the cell is changed. First we consider the situation of a growing cell, where through the replication of DNA, the production of protein and by dilution as the cell grows, the volume fractions of both protein and DNA remain fixed. The DNA volume fraction is C. crescentus that already has a PopZ domain at one of its poles, in a newly divided cell where the PopZ concentration is likely at levels to satisfy bipolar domain formation, we predict that instead of the preexisting domain continuing to grow, a 2nd polar domain of PopZ will form. The same would also be true for misfolded protein, where if the concentration of misfolded protein is large enough, another aggregate will begin forming at the new pole. Adding more protein to a preexisting bipolar pattern caused the polar domains to grow further, similar to what was seen in PopZ overexpression experiments.The above simulations were performed with fixed protein concentrations, allowing the system to come to equilibrium from an initially random spatial distribution of protein within the cell interior. We explored the effect of initial conditions by allowing the system to come to equilibrium and then we changed the amount of protein in the cell. In particular this allows us to address whether a cell starting with a protein domain at one pole will continue to grow only at that pole or will a bipolar pattern ultimately emerge as more protein is added? We see the system transition from diffuse to unipolar and then to bipolar localization as the protein concentration is increased from one initial condition to the next. For the diffuse to unipolar transition, unsurprisingly the unipolar pattern emerged with the same frequency as found above using random initial conditions. For the unipolar to bipolar transition, we found that the bipolar pattern was favored although at slightly less frequency than the situation when the initial protein distribution was random  cylindrical geometry with a cylindrical region of length With respect to energetic interactions, for the beads making up the chromosome, they are tethered together using the following potential between neighboring beads, We use a Lennard-Jones potential to model the interactions between the various types of beads in the system, given byThe system of DNA and aggregating protein is simulated using Metropolis Monte Carlo (MC). Each simulation starts with the circular DNA in a stretched out configuration in a long cell and is allowed to equilibrate as the cell contracts to its final volume. After the chromosome has equilibrated within the cell's final volume, protein beads are added to the cell randomly. Both protein and the DNA have a fixed amplitude step size set to We use a heuristic procedure to define protein patterns. At the end of each simulation, the final configuration of beads is used to calculate the density of protein along the length of the cell. To define the pattern, the cell was divided into equal sized bins of width equal to the diameter of the cell. If the protein density anywhere within a bin exceeded a predefined cutoff , then a protein domain was deemed to exist in that region. Patterns were then classified as either gas, unipolar-left/right/middle, bipolar or multidomain if more than two domains exist.Figure S1C using a protein volume fraction of fP = 1.0% that gives diffuse behaviour, leading to well mixed protein throughout the cellular volume. In (a) the density of DNA beads as a function of radius is shown for a nucleoid model that has attractive Lennard-Jones interactions (black) or is a cut-off, purely self-avoiding Lennard-Jones potential (red). In (b), the radial protein density is shown for the above two nucleoid models and for the case of smaller protein diameter size, in this case σP = 0.35 σC. In all cases, there is a significant protein density in the nucleoid interior with a slight increase nearer to the membrane.Radial probability densities for (a) DNA and (b) protein. Cell geometry was as in (0.02 MB EPS)Click here for additional data file.Figure S2C = 10% and protein is fP = 1.5% (a) Changing cell radius, keeping an aspect ratio of three. (b) Increasing the protein-protein interaction strength from εPP = 1.1 to εPP = 1.3. (c) Affect of making the protein diameter smaller from σP = 0.5 σC to 0.35 σC. (d) Affect of making protein diameter larger from σP = 0.5 σC to 0.7 σC.The frequencies of protein spatial patterns as a function of the model's parameters. In this case ‘OTHER’ represents patterns that do not fall into either unipolar or bipolar cases. In all simulations the volume fraction of DNA is f(0.02 MB EPS)Click here for additional data file.


Functional imaging based on tomographic X-ray imaging relies on the reconstruction of a temporal sequence of images which accurately reproduces the time attenuation curves of the tissue. The main constraints of these techniques are temporal resolution and dose. Using current techniques the data acquisition has to be performed fast so that the dynamic attenuation values can be regarded as static during the scan. Due to the relatively high number of repeated scans the dose per single scan has to be low yielding a poor signal-to-noise ratio (SNR) in the reconstructed images. In a previous publication a temporal interpolation scheme in the projection data space was relaxing the temporal resolution constraint. The aim of this contribution is the improvement of the SNR. A temporal smoothing term is introduced in the temporal interpolation scheme such that only the physiologic relevant bandwidth is considered. A significant increase of the SNR is achieved. The obtained level of noise only depends on the total dose applied andis independent of the number of scans and the SNR of a single reconstructed image. Theapproach might be the first step towards using slowly rotating CT systems for perfusion imaging like C-arm or small animal CT scanners. Hence, the temporal evolution of the gray value of a voxel is proportional to the temporal evolution of the average concentration of contrast agent within the voxel. TACs are used as input for algorithms based on a tracer kinetic model which compute functional parameter maps. Several approaches have been presented in the literature for this purpose. We refer the reader to  taken every Ts. Let us now consider the problem of the estimation of the continuous projection value p(t) is an accumulation of time attenuation curves along the direction of the x-ray beam. In order to find an appropriate method to estimate p(t) it is convenient first to characterize TACs in order to be able to extract the maximum information from the measurements. The model, however, should be as general as possible in order for it to be valid both in pathologic and nonpathologic cases. The morphology of perfusion signals depends on the patient, on the tissue, and on the injection parameters  so that it is very variable. The task to find a model that represents their variety is a very challenging one. The model we propose is motivated by the linear systems approach for the modeling of tracer kinetics  − νmax, νmax the B-spline coefficients. The polynomial B-splines are uniquely determined by its order n so that, for a given order n, we only need to find the coefficients cλ[k]. For this purpose we minimize the following functional: We denote with rem (see ), sλn, it can be shown  that thF{cλ,y}=∑cλ[k], y[k], and bn[k], respectively. Substituting the spline coefficients C^λ(ei2πν2πν′) of  we obtaiy[k] with a continuous polynomial spline low-pass filter ηλn(t′). When λ = 0, η0n(t′) is the polynomial spline interpolator of order n and it verifies lim n→∞η0n(t′) = sinc(t′) . The values are all between 0.914 and 0.95. We would like to have a simple rule of thumb to estimate the value of the variance of a sequence filtered with a low-pass spline filter. For this purpose, we propose to use the average value of the curve in In this section we discuss the modification of the TIA-TFDK algorithm to exploit the ideas presented in the previous sections. For the sake of simplicity we initially assume that the source is always switched on during acquisition time, that is, it operates in continuous acquisition mode.  The TIA-TFDK algorithm can be seen as a dynamic extension of the T-FDK algorithm for cone-beam reconstruction on circular trajectories described in . We firsw is a weighting function which only depends on the detector coordinates and , v′) are the coordinates of the detector pixel where the ray passing through x intersects the detector for the projection at projection angle β  as in . While in the original curve the enhancement cannot really be perceived, the TAC of the low-noise sequence clearly shows an increase with time in the concentration of contrast agent. On the other hand, the TAC of the low-noise sequence contains low frequency oscillations which are not of physiological nature. These are due to low-frequency noise in the frequency band [−t = 19 s. Sequence 4 has samples at t = 16 s and t = 20 s so that the frame at t = 19 s has been computed at the filtering step. The image quality of the frame estimated from sequence 4 appears to be equivalent to the original frame although four times fewer input data were used for the computation of the sequence. Sequence 1, with strong temporal smoothing, presents substantially reduced noise while visually preserving spatial resolution. The reconstructed sequences were used as input for the Perfusion CT software  that computes the functional parameter maps. This software first segments vessels and bone, performs a strong spatial smoothing, and subsequently computes the functional parameters. The segmented regions are excluded from the functional maps and represented in black. We would like now to verify if the reduction of the noise caused by the temporal smoothing is in accordance with the predictions of our model. Noise is measured as the variance of pixel values in a certain region of a homogeneous object. The variance might be due not only to quantum noise but also to pixels in the region of interest that correspond to different tissues and have therefore a different temporal behavior. Since our model concerns the temporal behavior of the statistical fluctuations caused by quantum noise only, we must segment a region within a tissue where all points have the same temporal behavior. This is not practicable in clinical images since we do not know a priori the exact temporal behavior in any region, except for in those regions where the attenuation is constant because contrast agent does not reach them. For this reason we used a segmented ROI within the ventricular system to estimate the variance. This delivered a set of 2300 pixels with the same temporal behavior. We then used the whole 2300 × 40 pixels to estimate the variance in each reconstructed sequence. The ratio of the variance of the original sequence to the variance of the postprocessed sequences is shown in the second column of νmax = 0.15 Hz from data acquired by scanning with any rotation time up to 5.33 s. In a simulation study similar temporal resolution and noise level are achieved by acquiring with slow and fast rotation time. We have illustrated the effect of the low-pass filtering using a clinical data set, and the results obtained were in accordance with the predictions obtained from a model for the temporal behavior of noise in dynamic X-ray imaging presented. The proposed approach opens the possibility to use other tomographic X-ray techniques such as small animal imaging CT scanners for perfusion studies. It might even be a starting point for the development of dynamic reconstruction algorithms for C-arm systems by adapting the estimation step to the irregular sampling scheme caused by the back and forth movement. However, the assumption of no motion limits the possible clinical applications to fixed patients.In this paper we have presented the dynamic reconstruction algorithm TSA-TFDK, which is a further development of the previously published TIA-TFDK algorithm . A tempo


Haplotype Pattern Mining (HPM) method is a machine learning approach to do exactly this.Identifying the genetic components of common diseases has long been an important area of research. Recently, genotyping technology has reached the level where it is cost effective to genotype single nucleotide polymorphism (SNP) markers covering the entire genome, in thousands of individuals, and analyse such data for markers associated with a diseases. The statistical power to detect association, however, is limited when markers are analysed one at a time. This can be alleviated by considering multiple markers simultaneously. The We present a new, faster algorithm for the HPM method. The new approach use patterns of haplotype diversity in the genome: locally in the genome, the number of observed haplotypes is much smaller than the total number of possible haplotypes. We show that the new approach speeds up the HPM method with a factor of 2 on a genome-wide dataset with 5009 individuals typed in 491208 markers using default parameters and more if the pattern length is increased.The new algorithm speeds up the HPM method and we show that it is feasible to apply HPM to whole genome association mapping with thousands of individuals and hundreds of thousands of markers. Identifying the genetic causes of common diseases has long been an important research area in genetics. Where early studies were limited to studying few genes at a time, due to economical and technological constraints, development in genotyping technology has revolutionised the field. It is now cost effective to obtain hundreds of thousands of genotype markers in thousands of individuals for a single study. This makes it possible to scan the entire genome for disease associated markers in a single analysis and such genome-wide association studies have recently lead to a virtual flood of newly discovered disease genes -7.χ2-test. However, a marker by marker approach is limited in statistical power due to the indirect testing for association, where so called "tag SNPs" are used as proxies for unobserved markers, but by using multiple markers, this problem can be alleviated [Most studies search for disease association through a marker-by-marker approach where each marker in turn is tested for association to the disease phenotype, e.g. using a simple Fisher's exact test or a leviated ,9. A traet al. [The Haplotype Pattern Mining method is a multi-marker approach introduced in 2000 by Toivonen et al. ,11. It iet al. -17 HPM iet al. -20 capabThe goal of association mapping is to find disease-predisposing regions of the genome. This can be done by looking for differences in the frequency of genetic variants between cases and controls. Since genome sequencing is expensive the whole genomes of the case and control individuals in a case-control study are usually not sequenced. Instead only single base pairs that are known to frequently differ between humans, called SNP markers, are sequenced.k SNP markers are typed then we can represent a chromosome by a haplotype vector H of length k, where H =  and hi ∈ alleles(i) for all i, 1 ≤ i ≤ k; alleles(i) is the domain of the ith marker. The input to an association mapping method then consists of a set A = {A1,..., Ap} of disease-associated haplotypes and a set C = {C1 ... Cq} of control haplotypes.If k markers is a vector (p1 ... pk), where pi ∈ alleles(i) ∪ {*} for all i, 1 ≤ i ≤ k, where * is the "dont't care" symbol. The haplotype pattern occurs in a given haplotype vector (chromosome) H =  if either pi = hi or pi = * for all i, 1 ≤ i ≤ k. The length of a pattern is defined as the maximum distance between two non-"*" characters in the pattern. Gaps are subsequences of "don't care" symbols in a pattern that are surrounded by non-"*" characters on both sides. Since long patterns are not likely to exist we only want look at a subset of the possible patterns. We call the patterns that we want to consider for legal patterns. A pattern is legal if the pattern length is less than the parameter l, it contains fewer than g gaps, and no gaps are longer than s.A haplotype pattern P over χ2 measure ± χ2(P) of a haplotype pattern P is the standard χ2 measure where the sign is positive if the relative frequency of P is higher in cases than in controls, and negative otherwise. Given a positive association threshold x, we say that P is strongly associated with the disease if ± χ2(P) ≥ x.The signed A = {A1,..., Ap} and control haplotypes C = {C1 ... Cq} the goal of the HPM algorithm is to find all strongly associated patterns that are legal.Given a set of case haplotypes Haplotype patterns close to a susceptibility locus are likely to be more associated with the disease than patterns further away. If we have found all strongly associated patterns we can score each marker by counting the number of times that it is contained in a strongly associated pattern. The HPM point prediction is then the marker that is most frequently contained in the strongly associated patterns. Fig. The algorithm presented in  recursivdepthFirst will take time O(n·l) where n is the number of individuals and l is the length of the pattern. If we remember which individuals match the pattern at a given time then we only need to look through these when a new non-"*" symbol is inserted in the pattern. Pseudo code for the algorithm with this improvement is shown in fig. Some simple improvements can be made to this algorithm. As presented in the paper counting the number of affected and unaffected individuals that match the pattern in each call to n possible haplotypes if you look at n neighboring SNPs. Instead of looking at all different haplotype patterns spanning a region we look at all combinations of haplotypes over the region. We search these haplotype sets in a depth-first-search but stop examining a branch if there is no legal haplotype pattern that could occur in all of the haplotypes in the current set.The idea of the new algorithm is to exploit that LD structure means that you usually only see a handful of the 2h1 ... hk the induced pattern of the set is the haplotype pattern that occurs in all of the haplotypes and contains fewest possible "*"("don't care") symbols.Given a set of haplotypes s > 2. This happens if a pattern is illegal because it contains too many gaps but would become legal if two gaps were joined into one. If for example l = 5, g = 1 and s = 3 then "0 * 1 * 0" is an illegal pattern because it contains more than g gaps. The pattern can however be made legal by substituting the "1" for a "*" yielding the pattern "0 * * * 0".An induced pattern over a set of haplotypes that is not legal can sometimes be made legal by inserting extra "*" symbols if S of haplotypes is said to be valid with regard to S, if the pattern occurs in all of the haplotypes in S but not in any of the other haplotypes found in the input data.An induced pattern over a set A haplotype pattern will split the set of individuals into those that match the patterns and those that do not. We say that two patterns are equivalent if they result in the same bipartitions of the set of individuals.The new algorithm Fig.  looks atSNPfile library [Beagle [We have implemented both the old and the new algorithm in Python using the  library  to read  library . This da [Beagle  program First we tested the running time as a function of the number of individuals. From the WTCCC data we created test data by picking subsets of individuals, keeping the affected/unaffected ratio constant, and we then ran both algorithms on chromosome 22. Figure l. Figure l, when analysing the full chromosome 22 from the WTCCC data set. The running time of both algorithms clearly grows super-linear, but with the time for the new algorithm clearly growing slower.An important parameter for the running time is the maximal allowed pattern length, l = 11) and the number of haplotypes along chromosome 22 of the WTCCC data: The plot on the left shows both the running time per marker  together with the number of distinct haplotypes starting in a given marker. Figure Since the time usage of the new algorithm depends on the number of different haplotypes over a region we expect it to use less time in regions with few distinct haplotypes and more time in regions with many distinct haplotypes. Figure The same dependency on haplotype diversity is not seen for the old algorithm (results not shown), nor is it expected to be as the old algorithm does not depend on the number of distinct haplotypes seen in the data. Instead, the running time could depend on the maximal score we see when scoring a marker, since this is the threshold used in the branch and bounds heuristic. From the data, however, we do not see a significant effect here.As the final comparison of the old and the new HPM algorithm, we compare the running time on the full Crohn's disease data set. Table We have developed a new algorithm for the haplotype pattern mining method and shown that it outperforms the original algorithm on genome wide association data. As a function of the number of individuals or the maximal pattern length, both the new and old algorithm appears to have the same asymptotic running time, with the new algorithm having a significantly smaller time overhead.l = 11) where for the new algorithm the mean running time per marker is 2.5 ± 9.6 seconds. It might therefore be worthwhile to use a hybrid algorithm where the new algorithm is used in areas with lower haplotype diversity and the old algorithm is used in areas with high haplotype diversity. If this would reduce the time usage on markers now taking more than 4 seconds to only 3, the hybrid algorithm would spend 1.4 ± 0.63 seconds per marker.The new algorithm is very sensitive to the haplotype diversity. The same is not the case for the old algorithm, but here the mean running time per marker is 8.8 ± 0.57 seconds (with pattern length The authors declare that they have no competing interests.SB developed the algorithm and implemented the software. All authors designed the experiments. TM and SB draftet the manuscript. All authors read and approved the manuscript.


The Working Group on Peer Review of the Advisory Committee to the Director of NIH has recommended that at least 4 reviewers should be used to assess each grant application. A sample size analysis of the number of reviewers needed to evaluate grant applications reveals that a substantially larger number of evaluators are required to provide the level of precision that is currently mandated. NIH should adjust their peer review system to account for the number of reviewers needed to provide adequate precision in their evaluations. On February 21, 2008 the recommendations of the Working Group on Peer Review of the Advisory Committee to the Director of the National Institutes of Health (NIH) were posted on the internet Thus, the Advisory Committee has left the actual number of reviewers to evaluate each grant application ambiguous. No guidelines were provided to determine the number of reviewers that would be needed. Consequently, we have conducted a statistical analysis to provide guidance in arriving at appropriate numbers. Our analysis shows an inherent statistical inconsistency in the NIH peer review recommendations concerning the number of reviewers. We also demonstrate how crucial this number is and how it influences the precision of the eventual score.For each grant proposal reviewers from the relevant scientific community are asked to report their evaluations within a pre-defined scale. The average grade obtained through this process is considered a valid estimate of the “true” value of the proposal.α/2 is the upper percentile of the standard normal distribution. For a 95% confidence interval and an alpha  of .05, Zα/2 is equal to 1.96. The parameter σ represents the underlying standard deviation. Finally, L indicates the desired half-width of the interval between two consecutive evaluations or the precision of the evaluation.The survey sample size is a crucial parameter in determining whether we can rely on these mean estimates. Elementary sampling techniques give us the minimum number of respondents that are needed for the evaluation procedure to deliver reliable estimates:There are two important implications of this equation. First, the inverse correlation between n and L indicates that more reviewers are needed to obtain a more fine-grained or precise evaluation. Moreover, this relation is exponential so that greater precision comes with an increasingly greater number of reviewers.Second, typically the standard deviation σ of a population is not observed and needs to be estimated. Since the data necessary to estimate σ for the review of biomedical research proposals have not been collected in a statistically robust sampling system, we have relied on a model system of peer review with short movie proposals reviewed on a scale from 1 to 5 by undergraduate students . We used short movie proposals in order to increase the potential sample size since all undergraduate students could be considered expert enough to grade the proposals. In this study 10 proposals were scored by an average of 48 reviewers. The average standard deviation was approximately 1.0 with a standard deviation considerably less than 0.1. Therefore, we estimate σ to be equal to 1. Obviously, a more accurate estimate of the standard deviation can eventually be obtained for each form of application requested by NIH, although it should be clear that a large number of independent evaluators is required to make any estimate of σ reliable.Using equation (1), we can assess the effect of having 4 reviewers for each proposal. With four reviewers and a standard deviation of 1, the review would be expected to distinguish applications at the level of the unit interval:Yet, in the evaluation of grant proposals NIH currently uses a 41-grade scale with a range of scores from 1.0 to 5.0 In The disconnect between the needed precision in order to allocate funds in a fair way and the number of reviewers required for this level of precision demonstrates a major inconsistency that underlies NIH peer review. With only four reviewers used for the evaluation of applications, an allocation system that requires a precision level in the range of 0.01 to 0.1 is not statistically meaningful and consequently not reliable. Moreover, the 4 reviewers NIH proposes are not independent which degrades the precision that could be obtained otherwise.Consequently, NIH faces a major challenge. On the one hand, a fine-grained evaluation is mandated by their review process. On the other hand, for such criterion to be consistent and meaningful, an unrealistically high number of evaluators, independent of each other, need to be involved for each and every proposal.5, a small number of evaluators would lead to unreliable mean estimates.Further insights can be derived from the analysis of expression (1). The value of σ is a measure of the underlying variability in the ratings. The minimum number of reviewers for any given degree of ratings precision decreases with decreasing standard deviations. The standard deviation across ratings is also an indicator of the degree of agreement among different reviewers. If the standard deviation is small, for instance equal to 0.01 instead of our previous working estimate of 1.0, there is essentially consensus among the referees. If σ = 0.01, then the following relation holds:Our estimate of σ is not based on an analysis of biomedical research experts judging research projects close to their area of specialty. Scoring standard deviations for large numbers of experts obtained in a statistically acceptable sampling system have not been collected. Instead, as described above, we have used a model system that has allowed us to readily collect opinion data about proposals with undefined potential. Although we believe our estimate is reasonable, it is informative to visualize how the sample size estimate varies with different values of standard deviation for a level of precision of 0.1 . It is eThe importance of scoring accuracy ultimately relates to the rank ordering of proposals. In our model system there were 5 movie proposals with mean scores ranging from 3.46 to 3.64. We have analyzed how the rank ordering of these 5 proposals varied as reviewers were randomly included in the analysis from 1 to 40 reviewers . What isIt is clear from our analysis that NIH needs to adjust their peer review system to account for low precision evaluations. Additionally, it would be valuable to determine the standard deviations of scores given by independent reviewers. This information could be used to obtain more appropriate estimates of σ and consequently would be invaluable in designing and implementing a statistically rational system of social choice for NIH.Our data demonstrate that funding decisions will vary widely with the number of reviewers in considering proposals that are closely scored. Making choices between applications that vary by less than 1 will require larger numbers of reviewers than NIH has been contemplating. Recognition of the statistical inconsistencies of NIH peer review will allow for the implementation of new policies that take into consideration the accepted relationship between the number of reviewers, the precision of scoring needed, and the standard deviation of the scores given.2. Obviously, the length of the application impacts the number of reviewers that could possibly be used for scoring. More reviewers can be used for shorter applications.The Working Group also recommended shortening the length of the application although no specific suggestions were includedIt is commonly accepted that NIH will not fund clinical trials that do not include a cogent sample size determination. It is ironic that NIH insists on this analysis for clinical studies but has not recognized its value in evaluating its own system of peer review. We posit that this analysis should be considered in the revisions of NIH scientific review.The NIH peer review structure has not been based in rigorous applications of statistical principles involving sampling