All combinations in the region within the curves (containing the origin) are valid, while all combinations outside the curves will result in a collision. The data for the charts are collected from measurements of the gantry angle that just clears each couch angle. The patient presence was modeled by placing a stereotactic body frame on the top of the couch. Separate charts were created for couch angles between 0° and 90° and between 360° and 270° over all gantry angles. The graphs are easy to create, implement, and use in the clinic and help reduce the time, complications, and uncertainties of planning with non‐coplanar fields.The use of non‐coplanar radiation fields can potentially lead to collisions between the gantry and the couch or patient. The collisions are often not realized until the plan is finished and the fields are checked on the machine, or even later when the patient is already on the table. This paper presents an easy method of gauging if a collision is likely between the gantry and couch or patient during treatment planning. The method involves creating a chart of allowable gantry and couch combinations. The charts contain curves on a polar graph of the gantry and couch angle “plane”. The curves display the limits of collisions for each gantry and couch combination for vertical couch positions 10, 15 and 20 cm below isocenter and for couch lateral positions of PACS numbers: 87.55.‐x; 87.56.‐v The purpose of this paper is to present a method of creating a series of collision charts that can be easily printed out or reproduced and used in the planning process without the need to create, install, and use separate software. This paper will also present ready‐made charts for the Varian Clinac system. These charts cover a range of common couch heights and lateral offsets for the full range of couch and gantry angles. The charts are to be used as a guide in the planning process. The planner can quickly tell whether the gantry‐couch combination will clear or results in a collision, or whether it needs to be validated on the machine if it is close. As couch height and lateral offset determine what gantry‐couch combinations are allowable, separate charts were created for each lateral offset that includes multiple couch vertical positions. These couch positions cover the majority of treatments. The chart was created for a Varian 21EX linac with Millennium multileaf collimator (MLC) and Clinac exact couch.The treatment couch position may sometimes be required to deviate from its neutral position during treatment in order to improve a planned dose distribution as measured by coverage of the target and sparing of surrounding normal structures. Treatment planning for several anatomical sites requires couch rotations while the gantry is also rotated in order to achieve optimal dose distributions. The use of couch rotations allows the dose to be spread out longitudinally, reduces hot spots in the body, and often improves conformality. However, if one is not careful, the use of couch rotations may also result in couch‐gantry and patient‐gantry collisions. A few authors have reported on computer programs for virtual collision detection using geometric calculations.–8 ThMeasurements were performed on a Varian 21EX – Exact Couch combination with a Millenium MLC . The convention for gantry and couch angles are: a) gantry: 0° when beam is pointing vertically down, increasing angles in the clockwise (facing the gantry) direction, and b) couch: 0° is the nonrotated, neutral position with 90° being to the left and 270° to the right, when facing the machine (clockwise increase when viewed from above). Couch positions between 90° and 180° or 180° and 270° are not reachable.± 10cm were chosen as representative offsets. The positive direction is to the right when looking at the gantry (patient's left when lying supine on the table).Patient plans were reviewed to determine the most common vertical couch positions and lateral offsets, and their respective ranges utilized in cases that require couch rotations such as SBRT. Couch vertical positions of 10, 15, and 20 cm were selected since they encompassed the vast majority of the cases. That is, the couch top moves below the isocenter by 10, 15 or 20 cm. Couch lateral offsets of 0 and In order to determine the maximum couch angle that is achievable for each gantry position, the gantry was rotated at 5° intervals and the couch was moved until it was within 1° of colliding with the gantry. Measurements were taken at higher sampling rate near collision zones. Measurements were taken without the accessory tray.The patient's presence was simulated by placing an Elekta stereotactic body frame (23 cm high and 48.5 cm wide) on the table . Any near collisions with the frame were assumed to be patient collisions also.+10, and −10, respectively. The first figure of each pair (1(a), 2(a), and 3(a)) are of couch angles between 0° and 90°, and the second figure of each pair (1(b), 2(b), and 3(b)) are of couch angles between 360° and 270°. Overlaid on each polar style graph are the results for the three couch vertical positions discussed, 10 cm, 15 cm or 20 cm below the isocenter. The graphs may be read by finding the gantry angle on the outside of the polar plot and then moving in and determining at which radii, couch angle, one of the curves is intercepted. The innermost intersection is the most conservative estimate of a collision‐free combination of gantry and couch angles. The couch angle is read from the radial scale. It can be seen from the figures that the most conservative estimate for safe gantry‐couch angle combination is not always associated with the same couch height. This reflects consideration for anterior versus posterior oblique gantry angles. Every combination on the “inside” of the curves (the region containing the origin) is valid and free of collisions. In addition, there are alternate renderings of the graphs that can be employed. For example, some of our planners prefer to have only one couch height per graph and have more pages. Shading the regions of the graph that are collision zones makes it easier for them to see what combination leads to a collision.The measurements for couch and gantry clearance were collected and graphed with MATLAB . and 1b),,1b), 2. These positions were chosen because they cover the vast majority of couch positions used at our clinic. However, there are a few setups that might require a couch vertical position of greater than 20 cm, or lateral positions greater than 10 cm away from the central position and, in these cases, the gantry‐couch combinations need to be checked on the machine.While this chart is easy to navigate, it does have limitations: 1) accuracy of the measurements, 2) determining lateral offset from planning CT, 3) collisions with the patient using anterior fields, and 4) variability in couch longitudinal. The accuracy of the chart is determined by the number of angles measured. However the more accurate (finer resolution) the chart, the more time it takes to perform all the measurements and create the chart. One way to deal with the large amount of measurements is to use a variable measurement increment. A smaller measurement increment can be used where the changes occur rapidly in the couch gantry combinations and larger increments where they change slower. Any combination near the curve should be checked before the plan is approved and the patient is on the table. The range of useable couch parameters will also determine how many measurements are taken. We chose three couch heights  and three lateral couch offsets  than the “true” collision chart. Having slightly smaller angles available to use is preferable to discovering a potential collision with a patient in the treatment room.There are multiple ways to present the data, from all curves on one graph to every curve having its own graph. We felt that when using more than three couch positions per graph and two pages of graphs, it becomes cumbersome to keep track of what is being presented. The purpose of the graphs is to supply a quick reference that can be an easy aid in planning. With too many pages or lines on the graph, the system will be too cumbersome to use. At that point, a program such as the ones referenced earlier would be easier to use. These programs use either extensive measurements or complex modeling and calculations to determine the gantry couch combinations that cause collisions. These programs can be built to cover a much larger range of couch positions but at the cost of being more difficult to use and create. These charts, unlike custom computer programs, are literally at the fingertip of the planner, and the data for the charts can be easily obtained in a few hours of work.This simple chart can be printed out or easily recreated for a particular linear accelerator. It allows for quick determination if a gantry‐couch combination is going to clear, or whether it may require special investigation at the machine. As a result, the use of these charts greatly reduces the number of angles that need to be verified at the machine during the planning process. Perhaps more importantly, it should reduce the number of times that a plan needs to be modified due to angles not clearing during the patient setup. Our clinical experience has shown that planners have an easier time referring to the sheet of paper than trying to manipulate another program on an already crowded computer screen.I would like to thank Keith DeWyngaert, Gabor Jozsef, Allison McCarthy, and Christine Hitchen for their insightful input and detailed editing of this paper.


The procedure is well established for linear tandem applicators. However, with the advent of ring applicators, this has become more critical and challenging. This work describes a new approach to determine positional inaccuracies for ring applicators in which the dummy markers are imaged just once and their dwell positions characterized with respect to an applicator‐defined axis. The radiograph serves as a reference for dummy markers for comparison with all subsequent measurements in which the active sources are autoradiographed at different offsets – thus obviating the back‐and‐forth transferring of setup between afterloader and simulator. The method has been used specifically to characterize the Varian GammaMed 60° ring applicator, but it may be adapted to any other applicator. The results show that an offset of 1–2 mm minimizes the overall inaccuracy to within PACS numbers: 87.53.Jw, 87.56.Fc, 87.56.B, 87.56.bg, 87.56.‐v The American Association of Physicists in Medicine Task Group No. 56 recommends this correspondence be within ±2 mm referenced to the applicator system.Intracavitary brachytherapy in combination with external beam radiotherapy is a standard treatment for cancer of the cervix. For brachytherapy, the use of high dose rate (HDR) remote afterloader, with its dosimetric challenges due to high source activity and short treatment distance, requires vigilant quality assurance.–5 On have reported positional inaccuracies with their ring applicators that do not always meet the ±2 mm criterion. Their test data show a 2–4 mm proximal offset from the distal‐most dwell position. They recommend that each applicator be individually characterized prior to use to determine the offset correction and verified at each source change. Waid et al. have reported differences between the planned and treatment positions of up to 5 mm. Similarly, Stern et al. found the differences in the range of 2–6 mm, and used a global shift of 3 mm to correct all but the most proximal dwell positions to within ±2 mm.With the advent of ring applicators, the verification of dwell position accuracy has become even more critical and challenging, since the trajectory of the source wire along the circular lumen of the applicator with an inner diameter larger than source wire diameter results in so‐called “snaking” within the lumen. Varian Medical Systems6 have to verify dwell position accuracy uses kilovoltage radiography to image the dummy markers followed by autoradiography with active sources on a single film with the applicator in the same geometry. To determine the optimum offset, this requires the procedure to be repeated for a number of different offsets, ensuring reproducibility of setup between radiography and autoradiography. This work describes a new approach in which the dummy markers are imaged just once and their dwell positions characterized with respect to an applicator‐defined axis. The radiograph serves as a reference for the dummy markers for comparison with all subsequent measurements in which the active sources are autoradiographed at different offsets – thus obviating the back‐and‐forth transferring of the setup between afterloader and simulator. The method has been used specifically to characterize the Varian GammaMed 60° ring applicator, but it may be adapted to any other applicator. The results show that an offset of 1–2 mm minimizes the overall inaccuracy to within ±2 mm. For comparison, measurements were also done using the conventional method; the two sets of results are comparable within experimental error.A conventional method4 to vIn the Varian GammaMed high dose rate remote afterloader unit , when the active source is transported to a tandem applicator it hits the distal end of the applicator and is retracted by 1 mm, so that its position coincides with that of the distal‐most dummy source. It is important to verify this correspondence with a tandem applicator before proceeding to any measurements with ring applicators. Indeed, this is the accepted QA procedure for tandem applicators.,4KVp and 200 mAs for radiography. It was also found that the film quality can be enhanced significantly by placing a high‐Z material shield in the central cavity of the ring to block the cross‐exposures from dwell positions across the ring. The available eyeshield was used for this purpose.To determine the optimum film quality, test films were exposed with different combinations of exposure parameters. The best combination in terms of contrast and brightness was a dwell time of 0.3–0.5 seconds (corresponding to a 10 Ci source) and a step size of 1.0 cm for autoradiography, and 125 For the actual measurements with the Varian 60° GammaMed ring applicator GM11000760), a custom‐fabricated Perspex jig was used to immobilize the applicator for positional reproducibility, as shown in Fig 1000760, With the HDR afterloader, the treatment is delivered with the active source at a number of dwell positions along the applicator. Because of this, an optimum offset needs to be determined to correct all dwell positions for a particular applicator. With the ring applicator in the jig, a simulator radiograph with the X‐ray marker in place is acquired. This serves as the reference radiograph and provides dwell positions for the dummy markers for comparison with all subsequent measurements with active sources Fig. . Next, a workstation V1.6. The angles with respect to the axis were measured for each of the nine source dwell positions on the autoradiograph , and α= the angle measured from the reference horizontal axis of the ring. At each offset, the average deviation between the two sets of path lengths was determined.To analyze and determine the inaccuracies in the dwell positions, the films were first digitized with VIDAR scanner  and then imported into a K‐PACS9 workaph Fig.  and comp an autoradiograph was acquired with the nine dwell source positions, ensuring that the applicator axis remained aligned between the exposures. The setup was then moved to the simulator to acquire a radiograph with the X‐ray marker in place, as shown in Fig. To assess our method against the conventional one,4 an aThe data analysis for the offsets of 0, 1, 2, 3 and 4 mm for the two methods are shown in Tables ±2 mm.The data show that the average deviation in the dwell positions decreases, at first, as the offset increases; reaches a minimum of ~1 mm at an offset of 1–2 mm, and increases again. The optimum offset lies between 1 and 2 mm. It is noted that, except for the most distal positions, the dwell position error in this offset range is within ±0.2 mm. It also depends on the uncertainty in measuring the angular spacing of sources with respect to the reference axis. This can be determined within ±1° using the software, which translates to ±0.3 mm in path length. Hence, the estimated error in the deviation between the X‐ray markers and the active dwell positions is ±0.5 mm.The experimental error depends on the precision with which the applicator reference axis can be marked on the film by aligning the pinpricks with the groove on the Perspex jig. This is estimated to be ±2 mm. Our results agree with the previously reported findings with ring applicators, both in terms of the range and the direction of the correctional offset.(–) However, the magnitude of optimum offset determined in this work differs with the suggested offset of 3 to 4 mm. Based on this, we agree with the Varian recommendation that each applicator be individually characterized prior to its use to determine the offset correction and verified at each source change.The positional inaccuracy measured for the ring applicator without an offset varies from 1.8 mm proximally (towards afterloader) to 5.0 mm distally (away from afterloader). The optimum corrective offset of 1–2 mm minimizes the dwell position error to within the l offset.–8 HoOur results, using the “autoradiograph alone” approach, agree within experimental error with those obtained using the conventional method. However, the method reported in this work is simpler and less laborious and is cost effective. The reference radiograph with the dummy markers is obtained just once, and can be used for comparison in subsequent characterizations. The multiple autoradiographs are acquired on a single film; hence, alleviating the back‐and‐forth transferring of the setup between the afterloader and the simulator and the need for vigilance for reproducibility between the transfers. that the correspondence between the planned and the actual dwell positions be within ±2 mm with reference to the applicator system.Finally, in this work all measurements have been referenced to the axis of the ring. This is in compliance with TG56 recommendation2 that±2 mm. The results are in agreement, within experimental error, with those obtained with the conventional method.The goal of this paper was to measure the dwell position inaccuracy and quantify the corrective offset in HDR ring applicators using a new approach based on autoradiography alone. We have carried out the measurements with the Varian GammaMed 60° ring applicator to demonstrate the method and its applicability to other ring applicators. The results show that an offset of 1–2 mm minimizes the overall inaccuracy to within  recommended tolerance of ±2 mm in the dwell positions is not met with the Varian 60° GammaMed ring applicator (GM11000760). Based on our findings and those reported elsewhere in literature, we conclude that each ring applicator should be individually characterized prior to its use to determine the corrective offset and verified at each source change. Such characterization should be referenced to applicator geometry in compliance with the TG56 recommendation.This work shows that the TG562 recoFinally, it is suggested that the clinical significance of positional inaccuracies in dwell positions using ring applicators should be explored to quantify the resultant change in dose distribution in HDR brachytherapy.The authors would like to thank Zubair Usman, Information Systems Department at the Aga Khan University Hospital, Karachi, Pakistan, for his help in digitizing and processing the films.


Such a diffusion is called type II diffusion in order to distinguish it from the more usual “Fickian” diffusion proceeding without such a constant concentration front and yielding, at least in the beginning, a weight gain proportional to the square root of time. It turns out that the conventional diffusion equation without any special new term but with a diffusion coefficient rapidly increasing with concentration has a series of solutions representing exactly such type II diffusion with v as a completely free parameter which determines the steepness of concentration front. With the usual boundary conditions and infinite medium the diffusion coefficient has to become infinite at the highest penetrant concentration. This case can be considered as an extreme limit which is approached to a high degree in an actual experiment. The finite sample thickness, however, requires only a very large but not an infinite diffusion coefficient. Hence type II diffusion is only a special case of possible diffusion processes compatible with the conventional diffusion equation without any need for new terms if only the diffusion coefficient increases sufficiently fast with penetrant concentration.Very often a non-solvent diffuses into a glassy polymer with a steep concentration profile proceeding at an almost constant rate  As a consequence the specific weight gain per unit cross section of the diffusing front, W increases almost linearly with time as expected from the almost constant velocity, v, of progress of the swelling boundary between the low and high concentration of the penetrant. An effect of minor importance for the diffusion process itself is the partial destruction of polymer with craze and crack formation in the swollen region of the polymer.In recent years a large amount of experimental –12 and tv. Type II diffusion is replaced by a more or less conventional Fickian diffusion. In this case the polymer has both a glassy shell and glassy core. The swelling by the penetrant is not sufficient for the reduction of the glass transition point of the swollen material below or to the temperature of the experiment. According to some analyses of the low molecular weight paraffin diffusion into polystyrene, the effect occurs at relatively high penetrant activity [W with time most clearly at very low activity [As a glassy polymer is cooled the speed of advancing front falls off and a critical point is reached with zero velocity activity , betweenactivity .S) and diffusion (D) coefficient are strongly dependent on concentration c of diffusant or its chemical potential μ.Since the advancing of a sharp boundary between low and high solvent concentration at an almost constant velocity is a most unexpected feature of a diffusion process, a new name, type II diffusion was created  and the c0 or maximum chemical potential of the diffusant. This condition is a natural consequence of the geometrical boundaries of the sample extending from −∞ to +∞. A constant current j from the boundary at −∞ can be only sustained if D increases to infinity while (dc/∂x)−∞ goes to zero. The more realistic consideration of the finite dimensions of the sample removes the need for an infinitely large diffusion coefficient but also makes the mathematics a little more complicated than attempted in this paper.In this paper it will be demonstrated that the general diffusion equation in an infinite medium can have a steady state solution with a time independent concentration profile progressing uniformly through the polymer at a constant velocity. The sole condition for such a solution is that the mobility becomes infinite at the maximum concentration x-axis direction. The specific weight gain of the sample per unit time, dW/dt, is constant as in the case of type II diffusion if one neglects the transients at the beginning and the end of the experiment. The transients are of course completely neglected in the consideration of the steady state boundary propagation through an infinitely thick sample extending from −∞ to +∞.For the sake of simplicity only the linear case will be treated. The swelling boundary has a constant cross-section and proceeds with constant velocity in the D = 10−12 cm2/s in a glass to 10−6 cm2/s in a gel. The above mentioned steady state solution in the infinite medium is therefore the asymptotic approximation to the actual diffusion process if the mobility increases by a few orders of magnitude while the sorbent concentration increases from c = 0 to c0. This is indeed the case with a diffusant which transforms the glassy polymer into a rubbery gel. The gel must be so much lower in polymer content that the sorbate flows through it almost freely and thus easily supplies the amount of liquid requested for the swelling at the propagating concentration front.In the actual membrane the diffusant mobility may increase quite drastically with glass to rubber transformation but still remains finite. Yet the changes are quite substantial, from t = 0 with c = 0 throughout the sample to the steady state solution is a combination with the usual type I diffusion with the initial weight gain proportional to the square root of time. During this transition time the steady state of the concentration tail gets established in front of the almost constant concentration profile. The shape of the profile depends on the propagation velocity [t1/2 and one proportional to t [n- pentane into glassy polystyrene just before the diffusion process is completed. Such an effect finds a simple explanation in the superposition of the concentration tails in front of the concentration discontinuities as they approach the center plane of the film. Hence the concentration at each point between the advancing fronts increases faster than formerly when the fronts were farther apart. As a consequence the critical concentration for the transformation from glass to gel is reached earlier. This shows up in an acceleration of specific weight gain at the end of type II diffusion.Transient from the start at velocity . The totnal to t . The fornal to t , 9, the v and hence do not determine the velocity of profile propagation. The velocity must be connected with some independent material property. It seems to be a good suggestion that the increase of concentration of the penetrant produces by the ensuing membrane swelling a sufficiently high stress on the polymer network for the rupture or partial disentanglement of most strained chains. But since the chain rupture or the pulling out of chains is not an instantaneous process an effect of this type sufficiently large to permit a substantial swelling is only achieved if the stress persists for a sufficiently long time. This condition determines the propagation velocity of the concentration profile. If the condition cannot be met the transport of permeant proceeds in the usual way as type I diffusion, i.e., without a discontinuous concentration front and with a weight increase which is for a long while proportional to the square root of time.The diffusion equation and the boundary conditions do not impose any limitation on c* to c0, moving through the sample at constant velocity v. In this area the Fickian diffusion equation reads [All the theoretical work up till now is purely descriptive without any serious attempt of explanation. Peterlin , 14 baseon reads ∂∂x′ and hence the current j′ in the x′,t frame equal to zero. In the laboratory system x,t the diffusion current j is by vc larger than j′. Hence it turns out to be −D∂c/∂x in front of discontinuity and vc0 behind it. Exactly speaking the very large value of D in the swollen region permits even a higher value of current than vc0 in order to supply the liquid needed for the gradual establishment of the steady state concentration tail in front of discontinuity.Behind the discontinuity the sorbate concentration is practically constant 1/2 increases and the total equilibrium sorption c*D/v in that region decreases with increasing v. The constant rate progression of the concentration discontinuity yielding the linear weight gain according to v per second in spite of the steadily increasing length of the supply route from the outer surface of the film to the steadily progressing discontinuity. In all practical cases that requires that the diffusion coefficient of the gel (D2) is some orders of magnitude higher than in the not swollen glass (D1). In the ideal case of an infinite extension of the film in the direction perpendicular to the front this condition amounts to an infinite value of D2.There is no limitation about t = 4Da2/v2 with a → 0) with the weight gain proportional to t1/2 may be over in such a short time that it is practically unobservable. After that a linear increase of weight gain with time is established. In a sample with a very large distance between the border where the liquid enters and the concentration discontinuity the weight gain must show some incipient drop as soon as the influence of diffusion time through the swollen region becomes preceptible on the solvent supply at the concentration discontinuity. This does not occur in the above-quoted experiments by Hopfenberg et al. [μm thickness the diffusion from both surfaces of the film was completed between 12 min and 400 hours, dependent on temperature, activity of n-pentane vapor, and on polymer orientation. The velocity of concentration front propagation in the same cases varied between 1.3 × 10−9  and 2.65 × 10−6 cm.s−1 . Since the maximum liquid concentration was about 13 g per 100 g of polymer one needs indeed such an extremely small gradient for steady supply of so small an amount of liquid increase per unit time that its decrease would be hard to detect.One concludes that the type II diffusion can be sufficiently well approximated on the basis of the classical diffusion equation by the limiting case of constant rate propagation of a concentration discontinuity. The very beginning , or at the point where the concentration jumps from c* to c0 (model of Peterlin). Hence the experimental data of Kwei and Zupko [None of the two concepts yields a front propagation containing linear and square root terms in time. Both yield a constant or almost constant velocity of propagation of concentration front if this is taken as the point of fastest concentration increase, i.e., at nd Zupko  yieldingxxS which at completed sorption increases linearly from 0 at the first boundary, x = 0, to sc0d at the other boundary, x = d. The gradient xx∂S/∂x would simply continue to pump the liquid through the film at the same rate v and in the same direction as during the preceding type II diffusion. There is no end to the process. The way out of this impass is the simultaneous consideration of the identical type II diffusion pumping the liquid at the same rate v and in opposite direction from the other boundary as a consequence of xxS(d) starting with value 0 at x = d and increasing to sc0d at x = 0. At the end of sorption the sum of xxS(0) and xxS(d) is constant, sc0d, throughout the sample. Its gradient disappears and so does the current corresponding to type II diffusion. This happy end effect has however the disadvantage that it yields the same negative result for all previous and later times. The sum of both xxS is a constant as far as x is concerned although it increases with time during sorption and decreases during desorption as does the total uptake of penetrant. Hence it cannot generate any steady flow as observed in type II diffusion. The modification of xxS in A further difficulty for the model of Frisch et al. shows up in desorption experiments , 8 whichB yields the velocity of concentration front propagation seems to be a constant of the penetrant-polymer system and hence independent of c. All the concentration dependence is in the factor B which affects equally the type I (the first term) and type II (the second term) diffusion. Hence v is proportional to the conventional diffusion coefficient. The proportionality factor s depends on the penetrant-polymer combination but not on concentration. That leads to the peculiar consequence that the normal type of diffusion is not the classical “Fickian” diffusion but the type II diffusion.The formulation according to v with decreasing activity a and the complete cessation of such diffusion below a limiting activity a* ~0.3. Hence s cannot be a constant but must be a function of a-a* vanishing at a* and for any a below a*. Although this dependence is the crucial point of explanation of type II diffusion it was never attempted to be derived from material properties.In contrast to that the experiments by Hopfenberg et al. , 8 on thsB in diffusion equation completely changes the concentration increase in time ahead of the concentration front as soon as the local concentration c surpasses the value c* corresponding to the limiting activity a*. This consequence may be less disturbing if one assumes that ahead of concentration front c is always smaller than c* and behind it larger than or equal to c*. But it still may cause some problems in the front itself if the concentration does not jump discontinuously from c1= c* ahead of the front to c2 behind the front. The variation of propagation velocity with c must create a special type of front profile which was never yet analyzed from this point of view. Without any more detailed analysis one can only guess that any concentration increase in the front will become steeper until it will be almost discontinuous.Moreover the presence of the term S and D dependent on chemical potential one can easily formulate the conditions for a constant rate propagation of a concentration profileOn the basis of the general diffusion equation with x′ moves to the right with the same velocity v as the concentration profile. Hence in this system the concentration C is constant at each x′ and the current density j′ = 0. With these assumptions the solution of the diffusion problem is truly conventional without a need for any additional more or less arbitrary term in the diffusion equation.The coordinate system x is proportional to the chemical potential gradient of the diffusantf is the frictional resistance yielding the diffusion coefficient D = RT/f. The chemical potential μ can be expressed as function of vapor pressure p in equilibrium with the sorbate at position x at time tAccording to the thermodynamics of irreversible processes the diffusion current density in the laboratory fixed system μ0 is still a function of T and of any other parameter independent of p and x. The concentration c is a product of sorption S(p) and pressure. The conservation of mass yields the concentration variation with timeThe constant S, the diffusion coefficient D and permeability P are functions of p or concentration c and not constants as assumed in the ideal Fickian case. But they are independent of time and location. The membrane is homogenous and does not change although at each point x its swelling and diffusivity vary drastically with time. There is also no temperature effect considered.In this well-known classical one dimensional diffusion equation the sorption The drastic change of sorption in the transition from glass to gel implies quite a substantial swelling of the polymer causing eventually the formation of macroscopic cracks as observed in the earlier experiments , 3. For vS(p) is independent of x and l. Hence the value p can be expressed as functions of a single variable x′ = x − vt. That means that the partial derivatives in x′. In such a case the differentiation on time is equivalent to differentiation on locationv transforms The thermodynamic equilibrium correlation between concentration and vapor pressure permits to express the concentration profile according to x′ travels with the same velocity v as the concentration or vapor pressure profile. This yieldsThis formulation actually means that there is neither a current nor a change of concentration since the coordinate system q = p(+ ∞) = 0 at x′ = + ∞ is met automatically with a finite, not vanishing D at q = 0. The finite q = p0 at x′ = − ∞ demands an infinite value of D(p0). D beyond those mentioned in connection with the boundary condition, i.e., at x′ = − ∞. Note that the value of S and its dependence on p or c do not enter The boundary condition v if only the diffusion coefficient goes to infinity at the maximum pressure p0 applied at the infinitely distant negative boundary x′ = − ∞ of the medium. The profile q(x′) depends on D(p) and v as shown in D on pD0= A + B at p = 0 and D = ∞ at p = p0, i.e., at the maximum pressure of the sorbate at the infinitely distant film boundary, x′ = − ∞, one obtains the profilevx′/D0= y for different values of B/D0. If the constant is equaled to zero one only displaces the q(x′)/p0 curves horizontally. This does not affect their shape which is our primary interest. The larger v and the smaller D0 the steeper the true profile if plotted against x′  which exhibits a large increase with p approaching p0 if one wishes to describe the transition from a glass at p = 0  to a highly swollen gel with c = c0 at p = p0 (S large). Hence the concentration profile is expected to vary substantially faster than the pressure profile.One sees that in an infinite medium the diffusion ainst x′ . The conformerly  for a coD and S on p. In particular, one can choose constant D and S in the glass and rubber with a discontinuity at p* thus producing a change of slope of P and a step-like concentration profile at p* /D(o) ~ ∞ is a very good asymptotic approximation of the actual material transport. But the finite maximum value of D imposes a reduction of the permeant supply with increasing transport length trl because the pressure gradient at the outer boundary of the sample, (dp/dx)x=0 ~ (p0 − p(trℓ))/tℓ, decreases with this length. That means a slow but steady decrease of propagation velocity of the concentration profile with increasing distance of the profile from the outer boundary of the sample. In a very thick sample the initially constant velocity propagation of the profile and hence the linear increase of weight gain with time are expected to show an observable decrease. But since the most precise measurements were made on extremely thin films with total liquid path less than 0.1 mm one was never faced with this limitation.The diffusion coefficient D2 in the highly swollen material can be derived from the very schematic D2, independent of concentration which varies from c*2 at the concentration front to the maximum c0 in thermodynamic equilibrium with the penetrant pressure p0 at the outer boundary of the sample. In order to simplify the matter one assumes in that which follows that the profile shown in t = 0, at the outer boundary, x = 0, so that the initial transient effects can be completely neglected. At t = 0 one hence has x′ = x − vt = 0.A very instructive general picture of the effects caused by the finite although very large diffusion coefficient c(0) at the outer boundary has the initial value c*2 which steadily increases with time up to the maximum value c0. If one assumes that during this time the concentration front moves with a constant velocity v one deduces a slight increase of current density and a more than linear increase of weight gain with timeOne first notices that the concentration v2t/D2 remains small during the whole experiment. Its maximum value is reached at the moment when the front hits the opposite boundary of the sample or meets the front proceeding from this boundary. In the latter case one derives from film thickness d and tmax= d/2v the maximum value of this parameterThe parameter n-pentane into cast annealed PS film [μm = 3.8 × 10−3 cm, tmax = 60 hours = 2.16 × 105s for p0 = 550 mmHg at 30 °C. From these data one obtains maximum values for the parameter between 1.7 × 10−4 and 1.7 × 10−2 if D2 varies between 10−7 and 10−9 cm2s−1. In biaxially oriented film at penetrant activity 1 and T = 35 °C the values are up to 300 times higher so that a sufficiently small value of the parameter is only obtainable with the higher diffusion coefficient which is indeed more probable than the lower limit. One can be rather certain that in most cases the parameter is so small that the current j is practically constant, the weight increase W almost linear with time, and a very small difference c(0) − c*2 required for a constant supply of liquid to the progressing concentration front. The small value of the parameter also tells that the ideal case, In the diffusion of  PS film  the filmD and of any other parameter of the diffusion equation.But the finite increase in time of penetrant concentration at the sample boundary in contact with the liquid or gas seems to be an important feature of type II diffusion. It is a consequence of the fact that the polymer glass simply cannot expand instantaneously to such an extent that the equilibrium concentration of penetrant could be accommodated. Since the polymer is quite inhomogeneous on molecular scale it exhibits a wide variation of penetrability in very small regions. The more penetrable areas expand substantially more and faster than the less penetrable sections. The forces exerted by the penetrant on taut molecules connecting the latter sections across the former regions will either rupture these chains or pull them out slowly. Both effects require a finite time. As a consequence of such enforced polymer expansion the sample crazes and even cracks. One gets convinced that just this relaxation of the specimen under the osmotic pressure of the penetrant determines the velocity of propagation of the concentration front which is independent of c*2, let us say c*1. This means that at low activity with c(0) < c*1 the sorption and diffusion are conventional without any concentration front propagating at constant velocity through the glass. If, however, the activity is so high that c(0) > c*1 a finite time is needed for proper polymer expansion thus creating the circumstances observed with type II diffusion. The higher the driving force c(0) − c*1 as compared with c*1 and the higher the volume change with the linear expansion of the polymer the more rapidly the glass adjusts to the space requirements of the penetrant, i.e., the higher the velocity of concentration front propagation in perfect agreement with observations.In a more realistic approach the inability of the polymer to expand sufficiently for the accommodation of the equilibrium amount of penetrant most likely starts at a substantially smaller concentration than On the other hand, the steady state profile cannot be established instantaneously. Even if a constant propagation rate is imposed it takes a certain time before the pressure or concentration distribution in the frontal tail assumes time independent values. This was explicitly demonstrated for a concentration discontinuity moving with a constant velocity . The rest1/2 and the final stage with the weight gain less than proportional to t because at fixed maximum values of D and P the sorbate has to be transported to the concentration front through a steadily increasing depth of the fully swollen film. The large ratio between Dmax and Dmin and the small maximum linear extension trℓ, max of films through which the material transport takes place may make very short the duration and rather difficult the observation of the initial and final stages. Therefore, as a rule, the scene is dominated by the pseudo-stationary type II diffusion as formulated in The steady state solution of the simple diffusion equation with the constant rate of propagation of the fixed concentration profile according to P, i.e., of S and D with sorbate activity. The increase in S is only needed for a rapid increase of penetrant mobility. There is no need for introduction of a new mechanism although a new name may have some practical use. The old Fickian formulation in concentration terms is of course rather inadequate for the description of such less conventional effects because it is applicable only to perfectly ideal material with activity independent sorption. But this limitation was already so thoroughly demonstrated [But one also sees that type II diffusion is nothing exceptional requesting any change or modification of diffusion equation. It is a simple consequence of a very rapid change of nstrated  that it v of propagation of concentration discontinuity is not in the slightest manner determined or limited by the diffusion equation and the dependence of D on concentration. Experiments, however, very clearly show a drastic increase of v with activity of the permeant and temperature of experiment. There is also a substantial dependence of v on thermodynamic properties of penetrant and polymer and on mechanical and thermal history of the polymer. Hence one will have to consider the molecular effects connected with swelling much more thoroughly than it was done up till now in order to be able to find the connection between the velocity and the mechanical and thermodynamic properties of the penetrantpolymer system. A next paper will try to discuss some possible approaches to such a molecular theory of unconventional diffusion.An important result of this investigation is also the confirmation of the very early finding  that the


The demand for metal nanoparticles is increasing strongly. Transferred arc synthesis is a promising process in this respect, as it shows high production rates, good quality particles and the ability of up-scaling. The influence of several process parameters on the performance of the process in terms of production rate and particle size is investigated. These parameters are the electrode design and adjustment, the gas flow rate and power input. A novel feeding mechanism allows process operation over an extended time period. It is shown that the process is capable of producing pure metal nanoparticles with variable primary particle sizes and comparatively high production rates. Optimal process conditions for a single transferred arc electrode pair are found, which allow further scale-up by numbering up. Metal nanoparticles are used in numerous applications and products  and the other one as a quench gas flow (Q2). The arc is ignited by a commercial TIG welder (2) between a tungsten rod cathode and a graphite crucible anode, which carries the material to be evaporated by the arc. The crucible serving as anode is a crucial part of the transferred arc synthesis. Multiple crucible designs and electrode arrangements are therefore tested. All crucibles are made of a massive graphite rod. A cylindrical recess with a diameter of 20 mm and a depth of 10 mm is used to fill in the feedstock material to be evaporated. On the other side, a recess with a diameter of 21 mm allows to place it on top of a tungsten rod serving as crucible carrier. The crucible is not cooled in order to minimize the energy losses and to keep the reactor chamber as simple as possible.The reactor chamber is evacuated by a vacuum pump (4) prior synthesis and refilled by two gas flows, adjusted by mass flow controllers (3). The process pressure is atmospheric. During processing, one gas flow is used as carrier gas flow  in which the particles are separated from the gas and collected as a powder. This powder can be used for BET measurement in order to determine the primary particle size of the produced particles. Before the filter, a measurement line is installed, which allows the adaption of several measurement devices to characterize further particles attributes. As measurement equipment, gravimetric measurements (8), an Electrostatic Precipitator (ESP) for microscope measurements (9), and a combination of several online measurements are arranged. A Tapered Element Oscillating Microbalance  is used to determine the aerosol mass load. A combination of Scanning Mobility Particle Sizer  and Electrical Low Pressure Impactor  is used to obtain information about agglomerate size and structure and also the primary particle size. The online measurement procedure is discussed in detail elsewhere  is installed, which replenishes the material loss from the crucible. By keeping the crucible filled and therefore the distance between the electrodes constant, a stable production over time can be realized. The feeder is designed to refill the crucible with bulk metal with minimal vacuum leakage. Therefore, it is integrated into standard ISO-KF 40 flanges, which guarantee full vacuum capability at minimal costs. An electric motor drives a cylinder, with a small drill inside Fig. . While tThe online measurement devices are used throughout this work in order to get information about particle and agglomerate size, production rate and specific electricity consumption in dependency of different process parameters. The goal is therefore to increase the production rate, while keeping the particle size on the nanoscale. The investigated parameters and their effect on the production are the electrode arrangement and crucible design, the effect of carrier gas composition, gas flow and arc power. Furthermore, the long-term stability of the process is investigated using the dedicated feeding mechanism.This section describes the development of the crucible design for optimal conditions of particle synthesis via transferred arc. The reactor chamber allows due to its design multiple arrangement possibilities, allowing to test different crucible designs. The electrode arrangement allows changing the position of the two electrodes relative to each other, as well its position relative to the direction of the gas flow.Q1, which enters the reactor chamber at the bottom flange streaming along the crucible or a cross flow Q2, coming from the side streaming along the cathode directed on the arc. The aerosol outlet is the top flange of the reactor housing, unless stated otherwise.Figure Q1), cross flow (Q2) or both gas flows combined (Q1 + Q2). The total gas flow  and the applied power (35A) are kept constant.Figure Q2) being much smaller. This can be explained by the fact that a gas flowing in the same direction as the buoyancy leads to less turbulences and therefore to smaller particle losses. Another advantage of the axial setup is a much more stable arc, which is especially for lower electric currents useful.Crucible 1 is the simplest to manufacture and leads to a production rate of about 700 mg/h when using axial flow, the production rate in the cross flow setup , but the production rate is much smaller than the axial flow mode of crucible design 3.Further crucible designs also have a reduced effective diameter to minimize heat losses. The main goal of optimization is to reduce the particle losses inside the reactor chamber, as the water-cooled walls of the reactor are a potential thermophoretic trap for the particles formed. In crucible design 4, a graphite tube guides the flow Q1). By adding a cross flow Q2 to the main flow Q1, the production rate is increased further, while the particle size is reduced. The combination of cross and axial flow appears to be an optimum solution to increase the production rate up to 1.5 g/h, while keeping the particle size in the desired nanoscale range. With this setup, almost no particle deposition inside the chamber is visible after synthesis. However, clogging at the inside of the crucible tube occurs after some time  only leads to particle deposition closer to the melt. A further problem of this design is the thermal conductivity of the tube which results in high temperatures at the outlet flange of the reactor housing damaging the rubber sealing. Additional cooling of the flanges would require a new design of the housing and might lead to increased thermophoretic losses. Crucible design 6 is therefore only suited for low-production rate processes where the clogging does not present a problem.The axial flow mode has been used in further crucible designs, as this configuration showed in the preceding investigations to be more productive. In these designs, the aerosol flow is guided more directly out of the reactor, avoiding flow recirculation within the reactor chamber. This allows also gravity-based feeding of feedstock material through the outlet into the crucible, which is discussed in a following section. Crucible design 5 has a graphite tube with four elongated gas inlets placed above the integrated crucible which should guide the aerosol directly out of the chamber. However, the arc plasma tends to leave the graphite tube through one of the holes opposite to the cathode at higher currents. This design shows a significant decrease in production rate in comparison to the axial mode of crucible design 3 due to particles losses inside the chamber. To overcome the problem of the arc leaving the tube, the tube of crucible design 6 has only one inlet. It shows an increased production rate (1300 mg/h) at the axial flow mode , it deflects the arc away from the centre  flow Q2 is also beneficial in terms of production rate. By adding 3 l/min from the side, the production rate can be increased about 1 g/h (Q1 = 40 l/min), to a value of approximately 5.5 g/h. If the cross flow Q2 increased further (dotted line), the production rate decreases significantly. On the one hand, the arc gets unstable at a cross flow = 5 l/min so that the production process is quite discontinuous. On the other hand, much smaller particles are formed at this flow. The particle size decreases significantly with the cross flow Q2, as it directly influences the residence time in the nucleation and coagulation zone. The primary particle size is the result of coagulation between particles  and sintering within agglomerates if the local temperature is below the melting point. Higher gas flow rates will lead to shorter residence times, reducing not only the time molten particles that might grow due to Brownian collisions, but also reducing the time primary particles that can increase their size due to solid-state diffusion or other sinter mechanisms. The increased gas flow might also decrease the total arc temperature and hence the effective zone in which the primary particle can increase their size due to the aforementioned mechanisms. Principally, one can adjust the primary particles size of the copper nanoparticles in the range from 60 to 130 nm by varying the nitrogen gas flow. In order to reach the maximal production rate of metal nanoparticles, a main carrier gas flow Q1 as high as possible and a cross flow Q2 at about 3 l/min appear to be optimal.Further optimization of production rate in the nanoscale size range can be realized by the amount and direction of the gas flow, using CD 7. Figure Q2 is advantageous for the synthesis with CD 7. At a value of 3 l/min, the arc remains stable, while significantly smaller particles can be produced. The particle size of argon produced particles is much smaller, so that an additional decrease of particle size is not required. As discussed before, the applied electric current impacts the production rate and hence particle size strongly, which means that the highest current is not automatically the optimal process condition. The evaporation rates of copper and zinc are very high even at lower currents. An increase in current might lead to a marginal increase in evaporation and production rate, but it is not beneficial in terms of electricity consumption and particle size. Due to the heat development at higher currents, current values above 60 A have not been used.Table Table In order to ensure that the process is indeed energy- and cost-efficient, the system has to run as long as possible with the minimal possible number of interruptions and operator actions. This section reports on the long-term stability of the process, using a dedicated crucible feeding mechanism, which fills the evaporated amount of metal back into the crucible. The production rate of the different metals at different electric currents, determined by gravimetric measurement, using CD 7 is plotted over a time span of 4 h as shown in Fig. The production rate and particle size in transferred arc synthesis are optimized by a variation of reactor design and process conditions in comparison to a previous setup. By optimizing the crucible design, an increase of production rate by a factor of 4 as compared to the earlier reported results (Stein et al.


Increasingly, the advent of multislice CT scanners, volume CT scanners, and total body spiral acquisition modes has led to the use of Multi Planar Reconstruction and 3D datasets. In considering 3D resolution properties of a CT system it is important to note that both the in‐plane  and z‐axis (slice thickness) influence the visualization and detection of objects within the scanned volume. This study investigates ways to consider both the in‐plane resolution and the z‐axis resolution in a single phantom wherein analytic or visualized analysis can yield information on these combined effects. A new phantom called the “Wave Phantom” is developed that can be used to sample the 3D resolution properties of a CT image, including in–plane  and z‐axis information. The key development in this Wave Phantom is the incorporation of a z‐axis aspect of a more traditional step (bar) resolution gauge phantom. The phantom can be examined visually wherein a cutoff level may be seen; and/or the analytic analysis of the various characteristics of the waveform profile by including amplitude, frequency, and slope (rate of climb) of the peaks, can be extracted from the Wave Pattern using mathematical analysis such as the Fourier transform. The combined effect of changes in in‐plane resolution and z‐axis (thickness), are shown, as well as the effect of changes in either in‐plane resolution, or z‐axis thickness. Examples of visual images of the Wave pattern as well as the analytic characteristics of the various harmonics of a periodic Wave pattern resulting from changes in resolution filter and/or slice thickness, and position in the field of view are shown. The Wave Phantom offers a promising way to investigate 3D resolution results from combined effect of in‐plane (x‐y) and z‐axis resolution as contrasted to the use of simple 2D resolution gauges that need to be used with separate measures of z‐axis dependency, such as angled ramps. It offers both a visual pattern as well as a pattern amenable to analytic analysis using Fourier Transform methods, and is believed to offer an image quality test closer to the diagnostic task where the 2D image has the hidden third (z) axis effects.PACS number(s): 87.57.Q‐ Classical measures of CT resolution generally involve separate measures of in‐plane  resolution and slice thickness (z‐axis). In fact, the actual clinical image is produced by the separate measures acting together in tandem.The ability of the tomographic imaging device to produce accurate images, including 3‐dimensional renditions of objects, may be vital in applications involving diagnosis of disease, volume measurements, and three‐dimensional planning of invasive medical procedures.The aim of this work is to use a newly patented phantom that can be used to simultaneously sample the x, y  and third‐dimension (z‐axis) extent of a tomographic image rather than making separate measurements of in–plane  or z‐axis (thickness) information.The Wave Phantom is illustrated in Our approach uses a phantom as illustrated in When the test object is imaged, the axial CT image of the test object will show a waveform profile that comprises a pattern of repeating on and off signal values of, as seen in A perfect image with near zero loss of resolution in the x‐, y‐, z‐planes of a tomographic slice of the test object would produce a consistent pattern in the square wave waveform profile taken across a set of angled ramps in the image, due to the consistent parallel spacing of the angled ramps, as well as the consistent thickness of each angled ramp. However, due to the finite resolution in the spatial performance of a tomographic imaging device, the CT waveform profile across the set of angled ramps will not yield the perfect waveform profile. The actual profile will be influenced by spatial resolution limitations caused by the finite z‐axis slice thickness and x‐y in‐plane resolution, as well as stochastic (noise) limitations and any other sources of nonuniformity such as beam hardening, faulty calibration of the CT scanner, and geometric mismatch (such as angulation) of the object with the scan plane.Various characteristics of the waveform profile including, but not limited to, amplitude, frequency and slope (rate of climb) of the peaks, as well as associated mathematical analyses of the waveform profile, such as the Fourier transform, can be analyzed to determine and evaluate imaging performance of the tomographic imaging device. Spatial performance includes the device's ability to accurately image an object, including in‐plane resolution characteristics, slice thickness, angular orientation of the slice plane, and uniformity of response across the scan field. If, for instance, there is a variation in slice thickness throughout the slice or from one side of the slice to another, or if the mean of the profile is changing due to nonuniformity of the scanner, these variations will be reflected in the image of the test object and the waveform profiles taken across the angled ramps in the image will encode these properties.The waveform profiles from the tomographic images may be visually examined which in some circumstance may be sufficient to provide a general evaluation of the performance of the imaging device. Alternatively, or additionally, the waveform profiles may be mathematically analyzed — for instance, using automatic processing software.These concepts will now be examined theoretically as a function of slice thickness and in‐plane resolution before examining experimental results in the following section.To further illustrate some of the concepts of how a waveform profile can encode characteristics of a tomographic imaging device reference is made to Likewise, v is the spatial frequency and the weighing factors on the contributing sine waves constitute the harmonics of the function and decreases as: 1/1; 1/3; 1/5; … etc.It is well known that an (infinite) square wave pattern is characterized by the well‐known Fourier series and resulting harmonics, as provided from However, a waveform plot extracted from images obtained by a tomographic imaging device will exhibit a rounding and/or blurring characteristic of the profile resulting in a repetitive more sine wave‐like pattern. As mentioned previously, the two major reasons are (i) the influence of finite z‐slice thickness and (ii) in‐plane (x‐y) point‐spread function (psf), or blur resolution limitations in the x‐y plane of the imaging device.The effects of changes of in‐plane (x‐y) resolution will show similar changes in wave form plots of To try to illustrate some of the attributes of the Wave Phantom, and particularly differences from existing methods of measuring resolution, such as two‐dimensional (2D) resolution gauges, In this case, a medium–high resolution kernel was used to scan the phantom on a Toshiba Aquilion One Scanner  at slice thicknesses of 0.5 mm, 1.0 mm, and 2 mm. Of course, since the reconstruction filter is constant, the only difference in the resolution gauges should be the increased statistical noise from the lower slice thicknesses. In contradistinction, the Wave Pattern with the added variation in the third (z‐axis) dimension shows differences in the periodic pattern variation. This is the key qualitative difference in a resolution pattern that has 3D variation as opposed to the more classical 2D resolution patterns.As seen in the scans in 15lp/cm section of the 2D resolution gauges vs. the 3D Wave Pattern for three different slice thicknesses  all taken with the same resolution kernel (medium‐high).Thus, for example in To illustrate the further analytic use of the Wave Phantom, Thus, by visual inspection of these Fourier Transform harmonics, because the same in‐plane resolution is used in both cases, one can gauge relative slice thickness effects.The Fourier Transform of the wave pattern encodes information about the in‐plane spatial resolution of the imaging device as described for example by the MTF. As is well known, a convolution (or blurring) due to resolution capabilities of the imaging device becomes a multiplicative filter  in the spatial frequency domain. Thus, As the paper has tried to point out harmonic amplitudes of the waveform will reflect the combined effects of (x‐y) in‐plane resolution and z‐axis resolution  which combine to produce the 3D point spread function, psf .Vx,Vy) and the modulation and amplitude from the z‐axis MTF . Moreover, the overall MTF can be calculated from analysis of the relative amplitude of the Wave harmonics, which will be reduced by the product of the in‐plane MTF and z‐axis MTF, appropriately scaled for ramp angle, at the corresponding harmonic frequency. This concept is illustrated in As a result, the amplitude of the harmonics in the Fourier Transform are influenced by the in‐plane modulation of Fourier amplitude as is characterized by the classical in‐plane MTF  regarding the overall MTF, in‐plane MTF, and z‐axis MTF are illustrated in As seen from Thus, A series of experiments were conducted to study the effect (if any) of the addition of a 30 cm body annulus to the Wave Phantom and to position the Wave Phantom both centered and off‐centered in the body annulus , 12.In this way, variation of response in resolution resulting from BOTH x,y resolution and z‐axis (slice thickness) resolution can be studied. As a figure of merit for the combined effect, the amplitude for the third harmonic from the Fast Fourier Transform (FFT) of the Wave pattern is chosen. This choice results from the observation that the third harmonic is close to often perceived cutoff value of about 10% to 20% on a classical high‐resolution test pattern. The exact choice of harmonic would not significantly affect the trends reported in the experimental data shown below.The results of Thus, it appears that measures extracted from the Wave analysis, such as the third harmonic, may provide an interesting way to study spatial frequency response that results from combined in‐plane  and z‐axis resolution effects.The type of analysis suggested in the Materials and Methods section E on variation within the field of view will be extended to include the z‐axis location within the scanned volume, to investigate whether z‐axis variation due to divergence in multislice scanners and/or volume scanners can be usefully described with figures of merit, such as third harmonic amplitude.The current results are based on 30° angled ramps that are used to augment z‐axis effects because of trigonometric magnification. However, ongoing work using 45° ramps avoids correction for 30° angulation and equally weights in‐plane and z‐axis effects.The use of the Wave pattern will be further investigated as a tool for offering visual and analytic measures of 3D tomographic imaging devices, and may also offer a way to study nonuniformity effects throughout the CT plane (volume). The ideas of the Wave Phantom will be investigated for use with other modalities such as MRI, PET/CT, and SPECT/CT.The wave phantom offers a promising way to investigate 3D resolution as results from combined effect of in‐plane (x‐y) and z‐axis resolution, as contrasted to the use of simple 2D resolution gauges that need to be used with separate measures of z‐axis dependency, such as angled ramps. It offers both a visual pattern, as well as a pattern amenable to analytic analysis using Fourier Transform methods, and is closer to the diagnostic task where the 2D image has the hidden third (z) axis effects.This work is licensed under a Creative Commons Attribution 3.0 Unported License.


Norm inequalities for several classical operators of harmonic analysis have been widely studied in the context of Orlicz spaces. It is well known that many of such operators fail to have continuity properties when they act between certain Lebesgue spaces and, in some situations, the Orlicz spaces appear as adequate substitutes.M and the Riesz potential operator r centered at x and The Hardy-Littlewood maximal operator b and the operator The commutators generated by a suitable function b the operators et al. gave the sufficient conditions for the boundedness of the commutator Given a measurable function mentclass2pt{minimby Nakai , 4, and by Nakai , Fu et ab belongs to the BMO and Lipschitz spaces, respectively.The main purpose of this paper is to give characterizations for the strong and weak boundedness of the Riesz potential on Orlicz spaces. Our characterizations for the boundedness of the operator mentclass2pt{minimC independent of appropriate quantities. If A and B are equivalent.We use the notation We recall the definition of Young functions.A function From the convexity and The set of Young functions such that For a Young function Φ and It is well known that A Young function Φ is said to satisfy the Orlicz spacef such that For a Young function Φ, the set f and For a measurable set The weak Orlicz space We note that The following analogue of the Hölder inequality is well known .Letbe a measurable set and functionsfandgmeasurable on Ω. For a Young function Φ and its complementary function Φ̃, the following inequality is valid: By elementary calculations we have the following property.Let Φ be a Young function andBbe a set inwith finite Lebesgue measure. ThenBy Theorem For a Young function Φ andthe following inequality is valid: In the next section, where we prove our main estimates, we use the following theorem.Let Φ be a Young function. (i)The operatorMis bounded fromtoand the inequalityholds with constantindependent off.(ii)The operatorMis bounded onand the inequalityholds with constantindependent offif and only ifIn this section we find necessary and sufficient conditions for the strong/weak boundedness of the Riesz potential operator on Orlicz spaces.c such that We recall that, for functions Φ and Ψ from In the theorem below we also use the notation In , CianchiLetLet Φ and Ψ Young functions and letandbe the Young functions defined as in (and (respectively. Then(i)is bounded fromtoif and only if(ii)is bounded fromtoif and only ifdominatesglobally anddominates Ψ globally.ed as in  and  For an arbitrary ball ick, see , we obtaThus, by  we obtaii.e.Let be as in . Then byi.e.Let be as in . Since \(2) We shall now prove the second part. Let (3) The third statement of the theorem follows from the first and second parts of the theorem. □From Theorems Letand the regularity condition  ondition  holds ifondition  holds.Moreover, ifthen condition (holds if and only if (holds.(2) ondition  holds if only if  holds.The following result is due to Nakai .LetandAssume that the conditions ,  becomes mentclasspt{minimaBy Theorems LetandThen the conditionis necessary and sufficient for the boundedness ofonand Φ be a Young function. The conditionis necessary for the boundedness ofonSuppose that t}Rn, by  \documenBy Theorems Let Φ be a Young function withThen the conditionis necessary and sufficient for the boundedness ofonIn this section we find necessary and sufficient conditions for the boundedness of the commutators of Riesz potential on Orlicz spaces with the help of the previous section.et al. found the sufficient conditions for the boundedness of the commutator In , Fu et aLetandLet Φ be a Young function and Ψ defined, via its inverse, by setting, for allIfthenis bounded fromtoThe following lemma is the analogue of the Hedberg trick for Ifandthen for allandwe getWe have Ifandthenfor everywhereIf The following theorem gives necessary and sufficient conditions for the boundedness of the operator LetandIfandthen the conditionfor allwheredoes not depend onr, is sufficient for the boundedness offromtoIfthen the condition  For arbitrary For Let us estimate Applying Hölder’s inequality, by , 4.1), , 4.1),  and LemmA geometric observation shows Consequently, we have i.e.B in  We shall now prove the second part. Let (3) The third statement of the theorem follows from the first and second parts of the theorem. □Theorems The following theorem is valid.LetandIfand the condition  The first statement of the theorem follows from the first part of Theorem (2) We shall now prove the second part. Choose any ball (3) The third statement of the theorem follows from the first and second parts of the theorem. □b belongs to the Lipschitz space, by which some new characterizations of the Lipschitz spaces are given. Such a characterization was given in [In this section, as an application of Theorem given in  as an apb belongs to the Lipschitz space C such that, for all C is called the b and is denoted by Let To prove the theorems, we need auxiliary results. The first one is the following characterization of Lipschitz space, which is due to DeVore and Sharply .Letwe havewhereLetandthen the following pointwise estimate holds: If The following theorem is valid.LetIfand the conditionshold for allwheredoes not depend ont, then the conditionis sufficient for the boundedness offromtoIf the conditionholds for allwheredoes not depend ont, then the conditionis necessary for the boundedness offromtoIfcondition (holds andthen the conditionis necessary and sufficient for the boundedness offromtoondition  holds an(1) The first statement of the theorem follows from Theorem B in (2) We shall now prove the second part. Suppose that (3) The third statement of the theorem follows from the first and second parts of the theorem. □The following theorem is valid.LetIf the conditions  The first statement of the theorem follows from Theorem (2) For any fixed ball Let Set (3) The third statement of the theorem follows from the first and second parts of the theorem. □We have obtained necessary and sufficient conditions for the boundedness of the Riesz potential and its commutators on Orlicz spaces. We have also compared our results with the existing results. Lastly, we conclude this paper by remarking that some new characterizations of the Lipschitz spaces have been given as an application of the above-mentioned results.


In fact, special numbers and polynomials play a significantly important role in the progress of several fields of mathematics, physics, and engineering. They have many algebraic operations. That is, because of their finite evaluation schemes and closure under addition, multiplication, differentiation, integration, and composition, they are richly utilized in computational models of scientific and engineering problems. For more information related to special numbers and polynomials, see –11 and tnormal function.By this motivation, we find a number of poles of generating functions of Bernoulli numbers and associated Euler numbers, denoted by Let cf. [In the case when ion (cf. ):\documHere we first consider the generating function of Bernoulli numbers It means The functionis not a meromorphic function over complex plane includingWe now modify the generating function of Bernoulli numbers as follows:The functionis a meromorphic function at everywhere.Let us now consider The function D are as follows: If If A number of poles over disk Then we have the following corollary.A number of poles of the functionover diskwhere the notationdenotes the largest integer less than or equal tox.Now we give the following theorem.The functionholds forover diskDSince mentclass2pt{minimThe mean value of the functiononis thatwheremeans bigOnotation; for information about this notation, see [ion, see .Setting The Nevanlinna characteristic function of the functionis thatSince it follows from Definition The Euler polynomials In the case when The functionis a meromorphic function at everywhere.Let us now consider The function D are as follows: πi: that is, If πi, If A number of poles over disk Then we have the following corollary.A number of poles of the functionover diskNow we give the following theorem.The functionholds forover diskDSince from Eq.  that\doBecause of Theorem The mean value of the functiononis thatAs has been used in Theorem The Nevanlinna characteristic function of the functionis thatSince it follows from Definition f be a meromorphic function in a domain f is a normal function if there exists a positive number K such thatspherical derivative of f, cf. [Let f f, cf. .We now find spherical derivative of the function n by Eq. . Since\Here we consider By the triangle inequality It is easy to see thatWe deriveBecause of the inequality r is a real number greater than 1, we can write Since Thus we get the following theorem.The functionis a normal function.Although Nevanlinna theory was studied intensively in complex analysis by many mathematicians, it was not considered to apply the tools of Nevanlinna theory to generating functions of special numbers. This work was first done in this issue. We dealt mainly with the generating functions on Nevanlinna theory. It seemed interesting with the following properties:In fact, the identities of special numbers have been studied in great detail. From some of relations, it is possible to obtain further properties on Nevanlinna theory by making use of some of the identities. For example,ring Eq.  with Eq.By using this relation, one may derive easily the Nevanlinna characteristic function of the generating function of Genocchi numbers the same as the Nevanlinna characteristic function of the generating function of associated Euler numbers.


This article provides supporting data for the research article “A simple automated system for appetitive conditioning of zebrafish in their home tanks”  [1]. In that article, we described overall movements of zebrafish toward a food source as a response to auditory or visual cues as conditioned stimuli in a novel learning paradigm. Here, we describe separate analyses of the vertical and horizontal components of the learned response. These data provide evidence that the conditioning might result from both classical conditioning of an innate response of zebrafish to move to the surface in response to food cues and secondary conditioning of the fish to associate a food presentation with a specific location in the tank. Movement data from the twenty trial acquisition period and probe trials from 2–32 days post conditioning are included. Specifications TableValue of the data•Contributes data to compare to those obtained from other zebrafish conditioning paradigms.•Establishes a behavioural baseline for zebrafish movement in a home tank environment.•Provides data for comparison in zebrafish studies where learning behaviour or memory retention is affected.1We examined the vertical and horizontal components of zebrafish movement in response to a sound  and 2A oIn addition to the data collected during acquisition of conditioning, we also examined the vertical and horizontal movements of the zebrafish in probe trials when the conditioned stimuli were presented alone some days after training. Retention was examined separately in both individual and groups of adult zebrafish presented with the auditory  and 2B a22.1Training consisted of 10 pairings of the auditory or visual stimulus with food on each of two consecutive days. Conditioning was performed by either playing FM tone sweeps or illuminating green LEDs for a 20-second period. The conditioned stimulus was immediately followed by the presentation of the food reward. In trials with control fish, the unconditioned stimulus (food) did not immediately follow the conditioned stimulus, but was instead administered at variable times after the conditioned stimulus.2.2Probe trials to test memory retention were conducted at various times after training. Fish were either tested in the groups in which they were trained or tested individually. For testing single fish, one animal at a time was removed from the group tank and transferred to a new tank one day before testing. On the day of testing, fish were exposed to the stimulus to which they were conditioned for 20 s without the food reward to test the memory of the association. Each group or individual fish was given only a single probe trial at 2, 4, 8, 16 or 32 days after training.2.3Trials were recorded for the 20 s immediately before exposure to the auditory or visual conditioned stimulus and the 20 s period during presentation of the conditioned stimulus. Videograms 3t-tests were conducted for the probe trials in adults and juveniles, respectively  and Welch two sample


The effect of the ceiling-mounted radiation shielding on the amount of the scatter radiation was assessed under conditions simulating obese patients for clinically relevant exposure parameters. Measurements were performed in different projections and with different positions of the ceiling-mounted shielding: without shielding; shielding closest to the patient; and shielding closest to the physician performing the procedure. The protection provided by the shielding was assessed for cardiology when the femoral access is used and for radiology when the physician performs the procedure in the abdominal area. The results show that the use of the ceiling-mounted shielding can decrease the dose from the scatter radiation by 95% at the position of the performing physician. In cardiology, the impact is more pronounced when the left oblique projection is used. In radiology, a large decrease was observed for right oblique projections, compared to cardiology. The ceiling-mounted shielding should be placed as close to the physician as possible. The idea of creating the largest radiation shadow by placing the radiation shielding as close to the patient as possible does not provide as effective radiation protection of the operator as it might be thought. It is widely accepted that low dose exposures can increase cancer risk, and as a consequence the scatter radiation produced by patients during fluoroscopy guided procedures may present a risk for performing physicians . PhysiciIn some situations, shielding placed correctly can significantly improve the radiation protection of the staff; Fetterly et al.  reviewedIn this study, the influence of the ceiling-mounted shielding and its positioning on the occupational doses of performing physicians due to the scatter radiation from different projections under conditions simulating the exposures during interventional procedures was investigated.Two angiography systems, each equipped with a ceiling-mounted radiation shielding, were chosen for this study. The first, Artis Q  has a 25 cm flat panel detector and is used for cardiac procedures. The cine mode Coro_CARE (dose saving mode) was chosen for measurement, each cine acquisition running for 4 s with the frame rate 15 fr/s, as used clinically. The exposure parameters were set automatically by the use of the automatic dose rate control (ADRC), which is used on both angiography systems. The exposure parameters were as follows: 86–96 kVp, 340–780 mA, pulse width 5.2–9.0 ms, 0.0 mm Cu additional filtration, and small and large focus.The second system, Artis Zee , with a flat panel detector of 48 cm diagonally is used for interventional radiology. On Artis Zee, the mode for abdominal procedures  was chosen for measurement, with each cine acquisition running for 4 s with the frame rate 7.5 fr/s . The exposure parameters chosen by the ADRC were as follows: 81–115 kVp, 478–668 mA, pulse width 36–39 ms, 0.0 mm Cu additional filtration, and large focus.The attenuation of the patient was simulated by the anthropomorphic Alderson male adult RANDO phantom, the torso with head, but without arms and legs. Due to the relatively small size of the RANDO phantom , higher attenuation was simulated by adding the PMMA slabs to the RANDO phantom. For the cardiology system, 4 cm of the PMMA was placed above the phantom and 2 cm under the phantom. The RANDO phantom with added PMMA slabs simulates a patient weighting between 85 and 90 kg, which corresponds more closely to our patient group. For the radiology system, an additional 2 cm of the PMMA was placed both right and left side of the RANDO phantom to simulate larger patients. Larger patients produce more scatter radiation, because they require more radiation to be used (a patient with the PA diameter of 29 cm requires about 200% more radiation than a patient with the diameter of 24 cm ) and hen3). Correction for energy dependence [The amount of scatter radiation was measured by the Radcal 9095 system  with the cylindrical 1800 cc ionization chamber 10X6-1800  together with a panel curtain 0.5 mm Pb equivalent  which provides a patient contour cutout. At all the measurements, a table-mounted (undertable) shield for the lower body with equivalent of 0.5 mm Pb was also used, but only in one position.For the cardiology system, the heart of the RANDO phantom was irradiated. The ionization chamber was placed 75 cm from the centre of the X-ray field, which corresponds to the chest of the physician when femoral access is used With the shielding placed closest to the physician (the dose measured just behind the shielding).The amount of the scatter radiation was measured for the position of the physician standing at the right hand side of the patient at the physician's left chest side for three positions of the ceiling-mounted shielding:All the three positions are illustrated in For cardiology, the doses from the scatter radiation per 10-frame acquisition were measured for 22 different projections that are routinely used in cardiology. Each measurement was performed twice. The results are given in For radiology, the doses from the scatter radiation were measured for 9 different projections that are routinely used. Doses are included in PKA values that were gained during the measurement from the KAP-meter (with uncertainty 5%) and are shown in The dose to the phantom for each projection was driven by the ADRC. It is known that the dose for steep projections is higher –17. TherPKA values and are shown in In the same way, as for cardiology, the doses from the scatter radiation for each projection were normalized to the μGy per 10 fr when the ceiling-mounted shielding is not used , the dose from the scatter radiation without the use of the shielding could not be measured due to the high dose rate.In radiology, the doses to the physician from the scatter radiation are also the highest from LAO projections, when the X-ray tube is on the right side of the patient. In LAO projections, the dose can be as high as almost 140 The dependence of the dose on the shielding positioning is smaller for some steeper projections in radiology, for example, LAO 30°, CD/CR 20°, because the freedom in the positioning is limited by the small space between the X-ray tube and the physician or the detector and the physician.The doses from scatter radiation are approximately 35–180 times higher for radiology than for cardiology, depending on the projection. The physician-interventional radiologist usually stands much closer to the exposed area of the patient , so his dose from scatter radiation is much higher. As the doses from scatter radiation are much higher for radiologists, the eye lens of radiologists may be at higher risk of cataracts making the wearing of glasses with Pb equivalent particularly important for radiologists.The use of the ceiling-mounted shielding is crucial at LAO projections when the performing physician is standing at the right hand side of the patient. The amount of the scatter radiation for the physician in LAO projections is more than 10 times higher than in the RAO projections.PKA value used in the projection, as mentioned above, so the dose from scatter radiation in the steeper projection is higher. The doses from scatter radiation normalized to PKA were determined and are included in Tables The physician's dose depends on the PKA are higher for radiology than for cardiology. There may be a number of reasons why. In the case of abdomen imaging, which was simulated here, the physician stands closer to the source of scatter radiation. Other causes might be the larger X-ray fields that are used (10 cm for cardiology versus 18 cm for radiology) and also as more attenuating areas are investigated , more scatter radiation may be produced.The values in Tables PKA were of the same order for their and our study, for example, LAO/RAO 0°, CR/CD 0° 0.023 versus 0.022 μGy/μGy ∗ m2, RAO 90°, CD/CR 0° 0.017 versus 0.011 μGy/μGy ∗ m2, LAO 90°, CR/CD 0° 0.062 versus 0.086 μGy/μGy ∗ m2, and LAO 60°, CD 10–20° 0.072 versus 0.081 μGy/μGy ∗ m2.The results were compared with results from the study of Kuon et al.  on invasThe results correspond with results from the study of Fetterly et al. , where tIf the ceiling-mounted shielding is placed closer to the patient, a larger solid angle is shielded but with lower efficiency. On the other hand, if the shielding is placed close to the operator, a smaller solid angle is shielded but with higher efficiency. This should be taken into account when more people are present in the catheterization laboratory.This study has its limitations. The measurements were performed for one main operator with changed positions of the upper body shielding, but one position of the lower body shielding. The situation would be different when there are two or more operators or other staff in the catheterization laboratory.Another limitation is that the operator's position may be different for procedures performed from different access, for example, the patient's left side in cardiology, typical in electrophysiology, when pacemakers are implanted. Similarly in radiology, different approaches will be used when the procedure is performed from the jugular vein access or for procedures outside of the abdominal area, for example, limbs.The measurements were performed for one size of the phantom and one size of the X-ray field. The results would be different for patients of different sizes and also for different sizes of X-ray field. For smaller patients and smaller fields, less scatter would be expected.The use of radiation shielding has a significant impact on the amount of scatter radiation at the position of the performing physician, so the use of radiation shielding should be recommended in all projections.The dose of the physician performing interventional procedures may be reduced significantly, when the ceiling-mounted shielding is used correctly during the procedures. The influence of the position of the ceiling-mounted radiation shielding on the physician's dose is more pronounced for LAO projections and also for LAO projections tilted in the CD/CR direction. For cardiology, the position of the ceiling-mounted shielding in RAO projections does not affect the dose from the scatter radiation as significantly as it does for LAO projections. However, this is not the case for radiology, where the position of the shielding affects the dose from the scatter radiation in all projections performed on the abdominal area.In summary, the ceiling-mounted shielding should be used wherever possible, placed as close to the physician as possible. This approach should be applied only to situations, when there is only one operator in the catheterization laboratory where upper and lower body shielding is used. The situation would be different, when there are more people in a catheterization room. The idea of creating the largest radiation shadow by placing the radiation shielding as close to the patient as possible might be used in these situations but we were not dealing with this aspect in our study.