The physics of wound and unwound strings on the electric guitar are presented here, and the pitch intervals produced by the movements of a Fender Stratocaster tremolo unit are explained. Predicted changes in pitch sensitivity of different strings are given, and experimentally verified, for changes in saddle height, the distance of string free to move behind the nut and ratio of diameters/masses of the core and windings of wound strings. Also, it is shown that changes to the gauge of strings (assuming the string tension is sufficient for linear behaviour and in absence of changes to other construction details) don’t alter the pitch intervals produced by a given angle of tremolo arm use assuming the instrument is set up with the same sounding pitches and starting bridge angle. It is demonstrated that it not possible to equalise the relative sensitivity of unwound steel stings on a Fender Stratocaster type tremolo unit through string construction techniques. The ratio of core to winding mass in the string, on the other hand, was found to be a very powerful design parameter for choosing the sensitivity of the string to tremolo arm use and standard pitch bends. For instance, the pitch intervals produced by operation of tremolo arm for wound strings may be made to approximately match that for one of the unwound strings if they share very similar core gauges (assuming the winding masses are chosen to give approximately the same tension at their sounding pitches). Such a design, only available currently by custom order, also delivers the optimum equalisation in sensitivity of strings for standard string bends . The physics of string bending techniques have been covered recently by Grimes . When frAnother significant method of creating vibrato effects on the guitar involves the use of what has become variously known as a “tremolo arm”, “vibrato arm” or “whammy bar”. The player applies force to the tremolo arm in order to move the bridge and adjust the tension of the strings and thus the sounding pitches. Since modulating the force on the arm produces vibrato (modulations in frequency) the term “vibrato arm” is technically appropriate and the first commercially successful device was named the Bigsby Vibrato Tailpiece (patent filed by Paul Bigsby in 1952) [Since the term “tremolo” is usually reserved for modulations in amplitude (and this is something that is often achieved using electronic effect units), the term “tremolo arm” is a misnomer for mechanical pitch adjustment systems, but this nonetheless became standard terminology through Leo Fender’s introduction of the Fender Stratocaster and its “synchronized tremolo” in 1954 . This deThis particular source of tuning problems was largely solved with the invention of the Floyd Rose Locking Tremolo (patent filed in 1977 ) in whicThe pitch of different strings deviate by different relative intervals during tremolo arm operation in the designs of Fender and Floyd Rose, meaning that in-tune chords become discordant when significant tremolo arm usage is applied. Ralph Jones developed a moveable tailpiece with strings mounted at adjustable distances with respect to the pivot in order to achieve chords that could be adjusted in tune (patent filed in 1965 ) and theThe properties of tremolo arm systems, how they should be set up, the relative intervals with tremolo arm use and problems and solutions for tuning stability are popular topics of discussion with electric guitarists today. Tremolo arm use is dealt with fleeting by Grimes  without While the topic has been explored by various patents, to the author’s knowledge, the only treatment of the physics of tremolo arm usage that accounts for relative tuning of different strings on tremolo arm usage is to be found in a student project by Jeremy Ozer  at CarneIt is worth examining the physics of the tremolo arm systems in order to provide information for players, technicians, instrument manufacturers, string manufacturers in addition to musical acousticians and those interested in accurate physical modelling synthesis of this type of system. Naturally this also involves a clear treatment of how the behaviour of wound strings relates to the fundamental properties of the material construction and the relative masses in the core and windings. The subject of how string slippage at the nut alters the relative pitch intervals is also an important part of this study.θ, around the pivot for the saddles, bridge plate and tremolo block. The initial position of the moving components are shown in grey while their position after applying a force to move the tremolo arm tip away from the body are shown in black. After tuning and before the player applies force to the tremolo arm, the sounding length of the string is l0 + Δl. This portion of string is defined as having had a length of l0 before being tuned up to pitch using the tuner . After using the tremolo arm, the sounding length of the string is further increased by a distance δ, further increasing the tension and therefore sounding pitch. The sections of string not contributing to the sounding length (to the left of the nut and to the right of the saddle) are denoted by dash-dot lines. Note that pressing the tremolo arm tip towards the body produces a rotation in the opposite direction  resulting in a decrease in pitch. In a double locking Floyd Rose system, the string is fixed at the nut and saddle meaning that the geometric value of δ corresponds exactly to the change in length of the portion of string responsible for the sounding length , and after, , the change using polar coordinates). In the older Fender design the string slips over the nut and saddles to (within the limits of friction) equalise tension meaning that the lengths of string either side of the nut and saddle must be taken into account.A typical tremolo system in an electric guitar, based on the design of Leo Fender , is showWhile it is sometimes stated that “string tension on the top equals spring tension on the underside of the guitar” , this isThe design of Leo Fender involves six strings mounted under similar tension and this tension applies a torque (twisting force) around the pivot (which is defined as the line connecting the points of contact between knife edges in the bridge plate and the screws mounted in the body). The support posts for vintage Fender Stratocaster designs feature six screws while the current American Standard Fender Stratocaster features two (in order to give fewer points of friction). Assuming that the tremolo arm has no additional force applied to it and that the instrument has been set up for floating operation (meaning that the only point of contact between the bridge plate/tremolo block and the body is at the mounting posts) then the torque generated by the springs should be equal and opposite to the torque generated by the strings. There are usually between two and five springs mounted in such a tremolo system in order to balance the the torques.θ and δ). Setting the instrument for floating bridge operation .Adjustment of the rest position of the bridge plate may be achieved by tightening or loosening the screws attaching the tremolo claw to the body. If the screws are tightened until the spring torque overcomes the string torque sufficiently for the bridge plate to rest against the body then it is only possible for the player to adjust the playing pitch downwards  involveA0 is the cross-sectional area of the unstretched string, l0 is the rest length of the portion of the string that has a sounding length of lf = l0 + Δl after being tuned up to normal playing tension, Tf, and E is the Young’s modulus. Note that this assumes that the Young’s modulus measured for nominal stress (constant cross-section) is valid. In reality the linear regime may be exceeded to some extent and this may occur due to the string cross-section contracting in diameter as the string is lengthened (due to the Poisson effect) and due to nonlinearity in the stress-strain graph for the particular material used.In order to explain the behaviour of tremolo systems in more detail it is worth first reviewing the relationship between string elongation and tensions. The tension in a plain or unwound string is given by the equation:lf = l0 + Δl as a function of the tension is [μ is the mass per unit length of the string.The fundamental frequency of a stretched string of sounding length nsion is :νf=12lfμ0, the mass per unit length after stretching will be:ρ0 is the density and A0 is the original cross-sectional area of the string when unstretched. The sounding frequency is then given byl ≪ lf gives:The mass per unit length does not remain precisely constant when the string is stretched . Defininlfνf)2E).The binoνf)2E)3-…Note EqsThe D, A and low E (lower pitch) strings on the electric guitar are usually wound. We will consider the most common case, known as round wound strings, which consist of a core  wrapped with cylindrical wire. This is done in order that the degree of stretching involved in getting them up to pitch is large enough to give the desired approximately linear behaviour (where the pitch is close to being largely independent of the amplitude of vibration) and to prevent excessive resistance to flexing (which would lead to inharmonicity) . The G sThe tension in wound strings is, to the first approximation, produced by the stretching of the core, as the  winding is angled and doesn’t stretch along its length as significantly. It worth proving the limited extent to which the winding contributes to the overall axial tension of the string.Na active coils of wire diameter dw, with a mean diameter of Dw with respect to the axis of the core, and modulus of rigidity G, the spring constant will be [l0 = Nadw and the spring force on stretching this portion of the spring/windings to increase the length by Δl will beAw = π(dw/2)2 is the cross-sectional area of the wire used in the winding. The most common choice of material in electric guitar strings is steel, and the winding is often nickel plated steel. In order to approximate typical values we will assume steel music wire construction [E = 207 GPa and the modulus of rigidity is G = 79.3 GPa. Remembering that Dw is the mean diameter of the winding around the axis, we may define the core diameter as dcore = Dw − dw. In practice the core diameter is larger than the diameter of the wire in the winding, so that dw < dcore and thus dw/Dw < 1/2, impling thatAssuming the winding behaves like a helical spring consisting of  will be :k=dw4G8truction  such thaIt is possible to construct a spring that is pretensioned (for instance through rotating the wire during cooling) and theoretically this could give the winding a stronger contribution to the overall tension force but doing so would reduce the benefits of using wound strings.μcw, this will equal the sum of the mass per unit lengths of the core, μcore and the mass per unit length of the windings, μw, giving:τ is the mass per unit length of the string divided by the mass per unit length of the core. Expressing the mass per unit length of the core after stretching asρcore is the density of the core when un-stretched and the cross-sectional area of the core when un-stretched isdcore is the maximum core width and γcore is a factor which depends on the cross-sectional shape of the core. For a cylindrical core, dcore is the core diameter and for hexagonal cross-section cores, dcore is defined as the distance between opposite points on the hexagon. The factor γcore is given by:Dw is the diameter of the centre of the winding with respect to the axis of the core, the mass per unit length of the spring/windings (after stretching), μw, is derived using the mass of a helical spring [l0 ≈ Nadw and again Aw = π(dw/2)2 is the cross-sectional area of the wire used in the winding, allowing the overall mass per unit length of the wound string to be given as:dcore, and using Dw = dcore + dw gives:τ is given byτ > 1 for wound strings. The above equations are also valid for unwound strings of circular cross-section by substituting dw = 0 and γcore = π as then τ = 1. Hexagonal cores wound with a circular cross-section winding have τ for a string destructively, based on cutting a length of string, unwrapping the winding from the core and measuring the ratio:mcore is the mass of the core and mcore + mw is the total mass of the same length of string.Defining the total mass per unit length of wound strings as corel0lf,where ρcl ≪ lf gives:Assuming that the tension supported by the winding is negligible, we may return to coreΔll0.1Now usising Eqs  gives:νlf = l0 + Δl and fundamental frequency νf when no pressure is applied to the tremolo arm. Suppose that the sounding length the string is then changed by a distance δ using the tremolo arm to give a new sounding length of lt = l0 + Δl + δ. It is important to note that, in a standard (non-locking) nut design, the string will slip across the nut to approximately equalise the tension either side of the nut. This is clear from the fact that pressing the string behind the nut to increase the tension there also produces a corresponding pitch increase in the main portion of the string . This means that, when the sounding length changes by δ, a fraction of that length change is taken up by the portion of string behind the nut. The portion of the string responsible for the original sounding length  actually slides to occupy a different length lt = l0 + Δl + δc where:lnut being the length of string behind the nut. The resulting fundamental frequency, labelled νt, will change  and  are the initial and final positions of the point of contact between the string and the saddle from the pivot in polar coordinates. Due to slippage at the nut, the length change responsible for changes in tension and mass per unit length is then:Returning to l, is larger for the higher pitched strings and hence they are less sensitive to the similar values of string length change δ in A common setup for a Stratocaster style tremolo system is to “float the bridge”  strings were assumed to be constructed of plain steel. The wound  strings featured in the initial experiments and calculations (D’Addario EXL120) feature hexagonal cross-section steel cores.An American made Fender Stratocaster was used for the measurements presented here (Oiled Ash 10 for 15 limited edition). This instrument has the American standard two post Fender Stratocaster tremolo system with a 9.5 inch radius neck (as on the American Standard Stratocaster) and was modified with Graph Tech String Saver Classics saddles and Schaller F-Series Locking Tuners. Adjustment of the saddle positions and tremolo claw screws was performed to obtain typical string heights, good intonation and 300 ± 6 cents of pitch increase for a full pull up. The measured (and where necessary assumed) parameters for the open string playing frequencies in standard tuning and values for the relevant distances are given in lf + lnut). The winding was then pulled from the core by hand. While pulling the winding, the core rotated rapidly around the winding, exposing a section of core, typically around 10 cm in length on each pull. The exposed section of core was then cut and retained for weighing and the process repeated until all the core sections and windings were separated for weighing. The masses of the cores and windings were then measured using laboratory scales (Unimatic CL41) and τ for each string.After all tests were complete, the strings were removed from the guitar and cut at the positions were they showed permanent bends from meeting the tuners and saddles  so that the  bridge plate contacted the body and plucking the string lightly, noting the pitch in cents  on the Logic tuner before releasing the tremolo arm.Plucking the strings lightly and noting the tuning of the strings after the tremolo arm is released.If the final pitch of the string is more than 2 cents out of tune then the tuning and measurement process is repeated from step 1.In practice it takes a small number of repeated pull ups and re-tunings before stable operation with accurately reproducible results is achieved on instruments with a standard (non-locking) Stratocaster style bridges.Low dynamic level plucks were performed by the author to check the tuning of notes in cents in order to avoid downward pitch drifts of over 10 cents observed during the sustain of some louder notes, a nonlinearity resulting from increased displacements causing increased tensions. The neck was supported at the heel by the left hand (at the connection between the neck and the body). It should be noted that supporting, pressing or pulling the neck with the left hand can lead to additional sources of pitch changes. All strings other than the one being plucked were damped lightly with the left hand fingers in order to prevent sympathetic resonance causing instability in the tuner output.4) string in comparison to the error on the G string cannot be explained by this.It is clear that there is reasonable agreement between experiment and theory for the pitch interval produced by pulling the tremolo arm fully up. The pitch intervals for the unwound strings are reasonable close to the 100 cents, 200 cents and 300 cent values both in the theoretical predictions and in the experimental measurements. Errors in determining the angle of tremolo arm movement would give a similar error for different strings so the small under-prediction in the number of cents on pull up for the high E (E4) has the largest engineering strain of almost 0.7%. Such strains ensures that new strings exceed the linear elastic region of the stress-strain graph for steel (see [It is worth noting from eel (see ) and areeel (see  (and theEcoreΔl/lf ≈ 1.41 GPa meaning that this string is operating at around 60 ± 10% of the Ultimate Tensile Strength (even before any tremolo arm movement). With the tremolo arm movement increasing the pitch as described, the engineering strain increases to approximately (Δl + δc)/lf giving a stress of E(Δl + δc)/lf or 70 ± 10% of the UTS for the high E string (E4). Considering the low E string (with the lowest stress on the core) the resulting stress at rest is 0.43 GPa meaning that this string is operating at around 19% of the UTS (and operates at around 28 ± 3% for a full pull up). It is worth noting that it has been stated that the best tone available for stress corresponding to around 40% to 70% of the Ultimate Tensile Strength [The Ultimate Tensile Strength (maximum stress before breakage) of the type Ia or IIa steel used in strings is typically around 2.1—2.5 GPa  and the θ are shown in 3. The maximum θ value plotted corresponds to a full pull up of θ = 0.085 rads = 4.9° to 2 significant figures. Negative values of θ show decreasing sounding pitches for the strings with the 6th string ) going completely slack first (due to its large core diameter) as θ becomes more negative. Again the dimensions and variables are taken from n is the number of the string ). Note that the relative tunings for the 1st, 2nd, 4th, 5th and 6th strings with respect to the 3rd string correspond to the standard tuning intervals of +900, +400, -500, -1000 and -1500 cents respectively. Dashed grey lines on the figure show the tuning of the 3rd string transposed through these numbers of cents so that the relative tuning of the strings is clear. It should be noted that the 3rd and 5th strings are closest to retaining an in tune relationship for this set of strings while the pitch deviations of the 1st and 2nd strings are much smaller than that required for remaining in tune with tremolo arm use (as dictated by their unwound construction).Theoretical pitch deviations for a wider range of values of The theoretical sounding frequencies implied by the equations above may be used to assess the effect of various adjustments in the parameters which could be typically observed in design and setting up electric guitars and their strings. Ra, by a distance of 0.5mm (as would happen if the height of the saddle on screws was adjusted by around Ra/sin ϕ ≈ 0.6 mm) produces a slight increase in the sensitivity  of between 3.6% and 5.0%. Such height differences are typically observed between instruments with a different fretboard radius.An adjustment of the distance between the point of string contact with the saddle and the pivot, 4) string by 0.6 mm (and retuning the strings to standard pitch). The result was that the interval produced by a full pull up changed from 105 ± 6 cents to 111 ± 6 cents, validating the theoretical prediction from Experimental verification of the effect of saddle height was conducted by increasing the saddle height on for the Fender Stratocaster’s high E  string instead of the high E (E4) string, produces a relatively large reduction in the number of cents of change with tremolo arm use (of roughly 14%) for the low E string.Using a reverse neck  guitar, where the longest distance behind the nut, 4) being adjusted by the greatest fraction (roughly 23%). Experimental verification of the effect of a locking nut was conducted using a Floyd Rose locking nut equipped guitar (Washburn KC-40V). All three bolts that clamp the strings at the nut were first removed  and the guitar tuned to standard tuning. A block was then inserted into the tremolo cavity to give 300 ± 6 cents increase in pitch for the G string. The high E (E4) string was observed to sharpen by 113 ± 6 cents due to the insertion of this block. With the process repeated with the high E string clamped at the nut using one of the locking nut bolts and leaving the G string unclamped. The same block was then inserted to give 300 ± 6 cents for the unclamped G string, and the resulting interval for the high E (E4) string was 137 ± 6 cents, an increase of 21 ± 11%. While this is a different instrument to that used for the theoretical predictions from Ra for instance), the effect is clearly measurable and of the correct order.Using a locking nut raises the pitch sensitivity of all strings, with the high E string (Elnut) for such designs introduce adjustments to the relative intervals obtained under tremolo arm use and these adjustments are similar in magnitude to the difference between the Fender and Locking Nut designs previously discussed. Most alternative bridge/tailpiece designs such as the Bigsby vibrato tailpiece are expected to have similar relative pitch intervals to the Fender design because they operate by simultaneously introducing similar length changes to the different strings . A full treatment of this topic is beyond the scope of the current paper.Various alternative headstock designs are also popular, including the traditional design featuring three tuners on each side of the headstock as featured on most guitars by the manufacturer Gibson. The altered lengths of string behind the nut . It was found that by adjustment of the tremolo claw screws together with retuning the strings to standard pitch enabled the same pitch intervals to be obtained from the “Experimental” column of In deriving the equations used in this model, it was noted that the engineering strain required to tune up a string from In practice setting up an instrument for a new gauge (including truss rod adjustment to counteract string tension changes) may lead to small, indirect changes in relative tuning with tremolo arm use in addition to the expected changes in force required to produce a given distance of tremolo arm movement.2 . The change in cents with tremolo arm usage increases from 348 cents to 419 cents. This illustrates the fact that detuned (flattened) strings are oversensitive to player’s control changes in general.Changing to an alternative tuning will have a big impact on the pitch changes obtained by tremolo arm use because the engineering strain is roughly proportional to the sounding frequency squared for a given string construction from . Table 3dw/dcore by 40% from 0.609 to 0.853 results in a 28% reduction in the interval produced by tremolo arm movement for the low E (E2) string for instance. In practice this can be achieved by using a narrow core and wider winding to obtain a string with a similar tension at pitch. Conversely, increasing the diameter of the core and reducing the diameter of the winding in the D3 string helps to equalise the sensitivity with the G3 string. This is simulated in the table by showing the theoretical effect of decreasing dw/dcore by 40%, increasing the interval for a full pull up by 35%.The most dramatic result from the theoretical analysis is that changing the ratio of the diameters of the winding and core results in large change in sensitivity to tremolo arm use. As an example, increasing the ratio 2) and G strings but also reducing the sensitivity of the D string. It should be noted that strings designed with pure nickel windings will often feature different diameters for core and windings and the sensitivity of the strings is heavily dependent on this parameter (as discussed below).Using pure nickel windings (instead of nickel plated steel) on steel cores increases the mass in the windings and this reduces the sensitivity of the wound strings, again helping to equalise the low E (Eτ) and hence a larger engineering stain at pitch and therefore a lower sensitivity to length changes with tremolo use.Experimental verification of the effect of different core to winding mass ratios was conducted by replacing the strings with D’Addario EPN120 “Pure Nickel” Super Light 9-41 strings (round wound with pure nickel). While labelled as “Pure Nickel”, the unwound strings and the cores of all the wound strings in these sets  are made of steel due to the low Ultimate Tensile Strength of nickel  string again completed 300 cents of pitch increase on a full pull up (again giving θ = 0.085 radians for a full pull up). While the intervals produced by the unwound strings were unchanged, those of the wound strings changed to those given in τ, for the strings, accurate to three significant figures. This enabled the theoretical calculations for a full pull up with θ = 0.085 radians to be added to With the “Pure Nickel” strings fitted, it was necessary to tighten the tremolo claw screws a fraction of a turn (and retune) to counteract a slightly larger tension of the strings at pitch so that the 3rd (G3) string was significantly wider and the winding significantly narrower for the EPN120 “Pure Nickel” set and this lead to a noticeably lower value of τ and hence a larger pitch deviation on tremolo arm use in comparison to the standard EXL120 set for this string. Experimental data differs from the theoretical prediction for this string by only 8 cents, validating the theoretical prediction of the intervals produced by tremolo arm movement. In this case, the increase in core diameter and decrease in winding diameter for the D3 string reverses the result which would expected if only the volumetric density of the winding had changed  is likely to be principally due to the effect of the finite thickness of the strings.The cores of the 5th and 6th strings of the EPN120 set were close to identical to those of the EXL120 set while the corresponding diameter for the windings was slightly reduced in comparison to the EXL120 set, compensating for the increased density and leading to a similar values for Tf ≈ EcoreAcoreΔl/l0 from l/l0 is the same for the different strings and this implies that they should share approximately the same cross-sectional area for the core, Acore. This also implies that the strings would then have close to the same value for the stress Tf/Acore meaning that the stress will be approximately the same fraction of the Ultimate Tensile Strength. A benefit of this is that the low E string may be prevented from having a stress below the range recommended in Jahnel [Tensions for different strings in the same set tend to be within about 50% of one another in order to prevent warping of the neck and for a consistent playing experience. Different wound strings in the same set will therefore tend to have similar values for n Jahnel  for optim) of diameter dcore, and a wound string (string number n) of core diameter also equal to dcore but with windings constructed using wire of diameter τn)(π2/γcore) and it should be noted that τm) in order to utilise available gauges of wire and to compensate for various approximations made during the derivation of lnut, and the different positions of the saddles with respect to the pivot). The result is a set of strings with a significantly lower fraction of the mass in the windings  than usual on the D3 string, and a significantly greater fraction of mass in the windings (τ approximately 30% greater) than usual on the E2 string.A set of strings  was therefore designed that show close to equal pitch change intervals for the GThe theoretical pitch deviations for this set of strings is shown in τ using 3, D3, A2 and E2 strings. A comparison of the theoretical and experimental pitch deviations on full pull up for the custom string set and for the D’Addario EXL120 Super Light 9-42 strings are shown in The strings were tested for their pitch deviations and then destructively tested to obtain the mass ratio 3, A2 and E2 strings respectively if the winding thickness given in the specifications is simply added in the distance to the pivot, Ra).It is interesting to note that the low E string shows the biggest discrepancy between the theory and experiment in two of the three measured string sets. In calculating the theoretical results it was assumed that the distance of string stretch was based on the point of contact between the string and saddle. In doing so we ignore any effects due to the finite thickness of the cores and windings of strings, the slightly rounded shape of the saddles and the slightly rounded profile of the string nearby and any slipping of the strings at the saddles. Adding the thickness of windings into the calculation of the string stretch (giving larger string stretches for the thicker wound strings) would improve the agreement between experiment and theory . After a change in temperature of dt Celsius, the section of string formerly of length l0 when not under any tension would therefore have a rest length of l0 + δa when not under any tension, wherea being the thermal coefficient of expansion of steel. We will neglect the very small decrease in Young’s modulus with temperature.The vibration frequencies of metal strings on musical instruments are known to decrease with increasing temperature, necessitating retuning. In the case of steel (or steel cored) guitar strings the thermal coefficient of expansion of the steel in the strings  is signil to Δl − δa. Following a similar derivation to νa and νf are the frequencies after and before the change in temperature respectively.The extension in the string  therefore changes from Δdt degrees Celsius thus becomes:Using 2) on the standard D’Addario descends in pitch to the greatest extent and the low E (E2) in the custom set is improved in this regard, with the strings with close to equal engineering strain staying close to being in good relative tuning to one another with temperature changes.Theoretical and experimental results are displayed in 3 string described above (featuring 4 thousandths of an inch winding on a 16 thousandths of an inch core) required the string to be displaced along the fret by a distance of 12.5 ± 0.5 mm for a whole tone bend while the D3 string from the standard D’Addario EXL120 set required a larger displacement of 14.7 ± 0.5 mm. In comparison, the unwound G3 strings made by D’Addario and by Newtone Strings both had a core of 16 thousands of an inch and both required a displacement of 12.5 ± 0.5 mm. Since there is only around 9 mm between the strings when at rest, the small but measurable difference in displacement amounts to a noticeable improvement, matching the displacement required for conventional pitch bending when strings with approximately matched engineering strains are used.Conventional string bends involve pressing the string just behind a fret and sliding the finger parallel to the fret to support the string in a triangular shape with increased total length (and forcing other strings sideways to make room as necessary), producing an increase in tension and thus pitch . StringsThe theoretical pitch variation with tremolo arm usage for electric guitar strings has been derived and experiment matches theory with a mean error in cent values of less than 3% and a maximum error of 20 cents for changes of order 300 cents (or less than 7%), thus agreeing to within the experimental error inherent in precise measurement of the point of contact between the string and saddle with respect to the pivot. Setting out the engineering strains involved in tensioning wound and unwound strings was also an important aspect of this work, proving the assumption that the windings support negligible tension at pitch. Saddle height, the ratio of diameters/masses for core and winding and the distance of string behind the nut all have a measurable impact on relative tuning of strings under tremolo arm usage in agreement with the theory developed here.In summary, unwound strings of higher sounding frequency will detune by a much smaller amount with a given control change assuming they are constructed of the same material . The relative pitch interval through which wound strings detune due to tremolo arm movements or conventional string bends, on the other hand, is heavily dependent on the ratio of winding diameter to core diameter and this is a very effective control parameter in designing string sets that detune more evenly. It should be noted that while string sets are currently commercially available that have balanced tensions across all strings at rest, none of these are currently designed to have close to equal engineering strains or equal pitch deviations for control changes on standard hardware.2 string in standard string sets, are also improved by using close to equal engineering strains. Other similar arrangements are possible, for instance giving matched sensitivities for wound strings and the B3 string for instance by approximately matching cores to the gauge of that string. While the paper focuses on the electric guitar, the analysis presented here could have applications to strings for other instruments including the bass guitar.In a standard Fender Stratocaster style tremolo (whether a locking nut is fitted or otherwise), the 3rd, 4th, 5th and 6th strings may be given close to equal pitch deviations with tremolo arm use and conventional pitch bends through the use of similar core cross-sectional areas for those strings . Temperature change related tuning problems, which introduce the most problematic tuning for the low EUltimately, the choice of core and winding dimensions and materials are a matter of taste, with the relative tone quality, output level, tuning stability, durability and similarity to the strings used by famous performers all being of great importance to typical players and these topics are not the subject of the current paper. Nonetheless, the sensitivity of the strings to pitch bends (whether through tremolo arm use or standard pitch bends) is a crucial factor in the feel and musicality of the instrument and this paper sets out the conditions for understanding and potentially controlling this factor.The University of St Andrews’ intellectual property rights contained in this publication are covered by a UK patent application.


Quanty4RIXS program is presented with instructions for setup and usage.Documentation of the  Quanty4RIXS program written in MATLAB® are provided. The program assists in the calculation of 1s 2p RIXS and 1s 2p RIXS–MCD spectra using Quanty. Furthermore, 1s XAS and 2p 3d RIXS calculations in different symmetries can also be performed. It includes the Hartree–Fock values for the Slater integrals and spin–orbit interactions for several 3d transition metal ions that are required to create the .lua scripts containing all necessary parameters and quantum mechanical definitions for the calculations. The program can be used free of charge and is designed to allow for further adjustments of the scripts.Some initial instructions for the  Quanty4RIXS is related to the CTM4XAS program  or X-ray emission spectroscopy (XES) spectra based on the Cowan approach  simplifying the creation of the Quanty. The Quanty containing all parameters, quantum mechanical operators and definitions used for the calculation. (Examples of the different elements can be found at http://quanty.org/documentation/start.) The created Quanty4RIXS to initiate the calculation, or the Quanty4RIXS is written in MATLAB® and the source code is published ‘as is’ to make it available to the scientific community. It can be altered and adjusted to meet the user’s own requirements as long as the original author  is mentioned along subsequent versions.2.® version 2016a or newer must be used. In older versions some GUI elements are not present, leading to errors. To enable the program to start the calculations locally, Quanty must be installed on a local machine. To obtain the latest version and further information on the usage of Quanty in the http://www.quanty.org.In order to run the program, MATLABQuanty4RIXS needs several files, which are stored in the repository directory as shown in Fig. 1Quanty binary and all settings, and therefore the folder’s permissions must be set to allow write access.Quanty binary is not found in the default path the program will ask on first startup for the paths, which can then be selected accordingly.If the repository folder or the 3.Quanty for the computation of RIXS spectra upon calling the Quanty is an exact diagonalization program designed to take advantage of the sparse operators and wavefunctions present in many correlated electron problems.In this section we describe the approach taken in the underlying program Quanty4RIXS, Quanty calculates the RIXS intensity I for a given initial state, i) and outgoing (o) photons with polarizations Of relevance for T one computes the total intensity as a weighted sum over the contributing states asZ is the partition function. In the following, however, we describe only the case for To calculate the RIXS intensity for a given temperature H, one can describe the RIXS intensity byWith This then describes the entire two-step RIXS process where a core hole is created by a transition operator Quanty first calculates the initial state Quanty calculates H in a small Krylov basis [usually dimension Next, This enables the wavefunction to be computed, Quanty computes H to obtain the RIXS spectral function for the state Next, For a finite temperature 4.Quanty input files for XAS, RIXS and RIXS–MCD calculations with circular polarized light. In other words, it creates a Quanty. The job can be executed on a local machine or, for more time-consuming jobs, the script can be saved and then copied to any other computer that has Quanty installed.The program can currently create Quanty4RIXS as it looks after start-up is shown in Fig. 2d transition metal and its ionization level is selected the three corresponding RIXS electron configurations for the ground state (GS), the intermediate state (IS) and the final state (FS) are displayed.In the following we first give a short overview and some instructions on how the GUI can be used. The main panel of s XAS and 1s 2p RIXS for hard X-rays, and 2p 3d RIXS and 2p XAS calculations for soft X-rays are implemented. Additionally the expectation values (Observables) and an energy-level diagram (ELDiagram), also called the Tanabe–Sugano diagram, can be calculated. The Options tick box can be used to adjust the number of microstates to be computed, counted from the state lowest in energy.Currently 1via the settings menu  for the selected 3The atomic, crystal field and broadening parameters are discussed in the following.4.1.d transitions metal ions are stored in the e.g. 1s 3d for the 1s 2p RIXS intermediate state. The contents of the RCN file have been calculated in advance with the Cowan program which is a part of CTM4XAS . The spin–orbit interaction ζ is not screened and thus a scaling of 1.0 corresponds to 100% of the atomic value  as described by de Groot. . Once the folder is chosen, the internal function The resulting The structure of the final e.g. Slater, SOC, CF parameters, all for GS, IS, FS; and the path and filename used for the calculated maps (the latter is convenient when the (i) The header contains the ion information and the corresponding parameters, (ii) Definitions of the XAS- and XES-transition operators, the Hamiltonians for the three states  and polarizations; the Hamiltonians and the transition operators  are hereby defining the entire two-step photon-in photon-out RIXS process.Quanty function (iii) Dynamic section performing the actual calculations of the spectra, ELDs or Observables. The Quanty. If cancelled, the files are just saved allowing the script to be transferred to another computer or changes to the script made.Using the final confirmation window the 4.5.Quanty using Quanty4RIXS. Clicking the Run Quanty button then triggers either of the following two methods:There are two ways to start a calculation with (1) GUI mode selected in the main panel creates the script and starts the calculation.Quanty.(2) LUA mode selected asks for any prepared Quanty. This can be used to run a Quanty that was not created with the GUI. The selected file is then used as an input file for Quanty. (Note: this can cause the plotting procedure discussed in the following to no longer work.)Method (1) was already mentioned in the previous section, where it directly starts the calculation based on the selections made in the GUI. Method (2) is meant as a simple click-and-run approach for any 4.6.Here we introduce the Plotting panel, which is shown in Fig. 7Quanty. One can see that in the raw data each individual RIXS map, for each microstate, is merged into one large map showing all microstates at once .Fig. 8The XAS projection onto the incident energy axis . This will equally add all RIXS maps up to the given microstate. With Split Multimap selected, it can also be used to split the raw data and plot only one specific microstate. In spite of the various plots possible, all plots are rather rudimentary, because its main purpose it to quickly plot the calculated RIXS and RIXS–MCD maps for review.s 2p RIXS map using the join function is shown in Fig. 94+ ion.An example of one individual 1s 2p RIXS–MCD map together with the projections onto its axes is shown in Fig. 10T > 0 K) it is appropriate to use the Boltzmann distribution to produce a reasonable result [equation (1)Furthermore, the magnetic circular dichroism (MCD) can also be calculated. Owing to symmetry considerations the calculations are performed in spherical coordinates for which the LCP and RCP can each be directly represented by a single quadrupole operator, each corresponding formally to one spherical harmonic operator Here one can use the Observables option to compute the expectation values of the Hamiltonian 5.Quanty4RIXS routines as well.We hope that the above descriptions are helpful for those who want to use the software. We want to explicitly point out that adding more or changing the existing Particularly for RIXS–MCD spectra the results can dramatically change even for very small parameter changes. To get started we recommend to begin with the atomic case. From there one has to evaluate each parameter individually to approach the specific system under study. Here, additional information, for example from density functional theory calculations, can be advantageous to estimate good values.Quanty4RIXS, the repository folder and a folder with all dependency graphs (as PDF) illustrating the connection of all functions has been published at https://git.science.uu.nl/P.Zimmermann/Quanty4RIXS/. For full access, a guest account can be created (via contacting the authors).The complete MATLAB source code for The code has been thoroughly tested on Apple OS X 10.11 and in part also on Windows 7. In principle it should work on any operating system as long as the setup requirements described in §2All sources are made available ‘as is’ and it can be altered to the user’s liking as long as a reference to the original author  is given. For subsequent scientific work we appreciate citation of this paper.


The kinetics of the demetallation of chlorophyll in the fused microdroplets  was recorded by controlling the traveling distance of the fused microdroplets between the fusion region and the inlet of a mass spectrometer. The rate of acid-induced chlorophyll demetallation was about 960 ± 120 times faster in the charged microdroplets compared with that reported in bulk solution. If no voltage was applied to the sprayed microdroplets, then the acceleration factor was about 580 ± 90, suggesting that the applied voltage is not a major factor determining the acceleration. Chlorophyll a was more rapidly demetallated than chlorophyll b by a factor of ~26 in bulk solution and ~5 in charged microdroplets. The demetallation kinetics was second order in the H+ concentration, but the acceleration factor of microdroplets compared with bulk solution appeared to be unchanged in going from pH = 1·3 to 7·0. The water:methanol ratio of the fused microdroplets was varied from 7:3 to 3:7 causing an increase in the reaction rate of chlorophyll a demetallation by 20%. This observation demonstrates that the solvent composition, which has different evaporation rates, does not significantly affect the acceleration. We believe that a major portion of the acceleration can be attributed to confinement effects involving surface reactions rather than either to evaporation of solvents or to the introduction of charges to the microdroplets.Kinetics of acid-induced chlorophyll demetallation was recorded in microdroplets by fusing a stream of microdroplets containing 40 μM chlorophyll  Microdroplet fusion mass spectrometry was employed to study chlorophyll demetallation kinetics. We have examined the effect of applied voltage in the generation of electrosprayed charged microdroplets on the chlorophyll demetallation rate. We investigated chlorophyll a and b were purchased from Sigma . HPLC grade methanol, hydrochloric acid and water were purchased from Fisher Scientific .Chlorophyll a and b were prepared at 1 mM in methanol. Working solutions of chlorophylls at 40 μM were prepared by diluting the stock solution in methanol. HCl solution prepared at 70 mM concentration in water was mixed with the chlorophyll solutions at 1:1 (v/v) ratio with the final HCl concentration of 35 mM. The mixed solutions were gently stirred and incubated at room temperature for 20 min. Mass spectrometric analyses of the mixed solutions were carried out to determine the identities of the species present using an Orbitrap mass spectrometer .Stock solutions of chlorophyll 2 gas at 120 psi was used to generate a stream of microdroplets. The voltage applied to the syringes of the two droplet sources was varied between 0 and +5 kV to examine the effect of applied voltage on the kinetics of chlorophyll demetallation. The two microdroplet sources are equipped with an X–Y–Z micropositioning linear and angular stage for accurate alignment of the two microdroplet streams. This alignment is important for ensuring fusion of most of the incident droplets and to maintain a linear trajectory toward the mass spectrometer inlet. The best alignment was acquired with the angle between two crossed microdroplet streams at 78°, which showed the highest probability of droplet fusion and straight trajectories of the fused microdroplets to the inlet of the mass spectrometer. Two solutions of analytes (chlorophyll a or b at 40 μM and HCl at different concentrations ranging between 0 and 100 mM) were injected from the two microdroplet sources with a syringe pump  at a flow rate of 30 μl min−1. The heated capillary temperature was maintained at approximately 275 °C. The capillary and tube lens voltages were set as 13 and 190 V.The kinetics analysis using microdroplet fusion mass spectrometry was conducted as previously described  at m/z 893·53 with a sodiated peak ([M + Na+]+) at m/z 915·52. The chlorophyll a with potassium adduct ([M + K+]+) was observed at m/z 931·50. An oxidized species at m/z 925·53 was detected in a relatively low intensity. Chlorophyll a after treating for 20 min with 35 mM HCl exhibited a protonated phaeophytin mass peak ([M−Mg2+ + 2H + +H+]+) at m/z 871·57. Negligible molecular species with sodium and potassium adducts were detected in the demetallated chlorophyll a with HCl treatment. The high hydrogen ion concentration drove a major portion of formed ions to the protonated species. It is to be noted that oxidized peaks observed in intact chlorophyll a also disappeared in demetallated chlorophyll a, suggesting that acid-induced demetallation interferes with oxidation.a demetallation was recorded using microdroplet fusion mass spectrometry  decreased while the signal intensity at m/z 871·57 corresponding to phaeophytin (Phe) increased as the distance x increased. The reaction time was calculated by integrating the speed of fused droplet over distance x as provided in our previous work  suggests that evaporation was minimal and played a negligible role in determining the kinetics.In other words, the normalized signal intensity of chlorophyll velocity . The sigk of chlorophyll demetallation measured in microdroplets was 46 ± 6 mM−2 s−1. The reported rate constant measured in bulk solution is 0·048 ± 0·002 mM−2 s−1 , as expected. The approximately thousand-fold acceleration was found to be constant regardless of the H+ concentration in the microdroplets. It is noted that there was a sizable rate of demetalation in microdroplets, even when no HCl was added. This behavior is discussed below in terms of enhanced H+ concentration on the surface of the microdroplets.The rate constant mM−2 s−1 . Thus, tSeveral candidates are thought to be responsible for the reaction rate acceleration in microdroplets. These include charge, localization of reactants at the interface, charge separation, micro-confinement, electric field and evaporation of solvent (We then studied the effect of different solvent composition on the reaction acceleration. We measured demetallation rate constant in microdroplet with a different solvent mixing composition between water and methanol. The vapor pressure of water and methanol at 20 °C is about 17·5 and 98 torr. Therefore, the liquid in the microdroplets containing 70% methanol are expected to evaporate more rapidly than the microdroplets containing 30% methanol. The faster evaporation should lead to the increase of concentration of reactants, consequently, the increase of measured rate constant. The reaction rates increased by only about 20% as the percentage of methanol increased from 30 to 70% . This bea and b to assess the effect of different chemical structures on reaction acceleration in microdroplets. a and b. A methyl group on the chlorin ring in chlorophyll a is replaced by a CHO group in chlorophyll b. Different side chains in chlorophylls exhibit different properties including absorption, redox potential, fluorescence, demetallation kinetics, etc. (b is more stable than chlorophyll a against proton attack (b is known to be slower than chlorophyll a.We compared the demetallation kinetics between chlorophyll cs, etc. . Chloropa and b in microdroplets were measured using microdroplet fusion mass spectrometry under the identical conditions of 40 μM chlorophyll concentration, 35 mM HCl concentration, and an applied voltage of 5 kV. a and b. The measured rate constant of chlorophyll b demetallation was lower than chlorophyll a, which is consistent with the literatures (b as a function of the H+ concentration was greater than chlorophyll a, suggesting that in the microdroplet environment chlorophyll b might be more susceptible to H+ attack.The kinetics of chlorophyll We have studied the kinetics of chlorophyll demetallation occurring in microdroplets, and we found that the demetallation reaction rate is significantly faster in microdroplets that in bulk solution. We characterized the reaction rate acceleration in microdroplets by examining several factors influencing the acceleration. We investigated the effect of charge on the reaction rate acceleration by measuring the kinetics of chlorophyll demetallation under different voltages. Although the increased voltage from 0 to 5 kV increased the reaction acceleration factor, the major portion of the reaction acceleration was attributed to the other factors possibly, including the confinement effect and electric field at the microdroplet–air interface rather than the net charge introduced into the microdroplets. It should be noted that a significant reaction acceleration by approximately 600 times occurred with no voltage. It has been reported that neutral water exhibits a separation of the charge between hydronium ion and hydroxide ions distributing non-uniformly between the surface and inside of the water microdroplet , it needa demetallation only by 20%. This observation demonstrates that the solvent composition does not significantly affect the acceleration. Given about four times higher evaporation rate of methanol compared to water at room temperature, the reaction rates in the microdroplets containing a higher portion of methanol are expected to be accelerated compared with water. However, the reaction rates differ between the microdroplets containing 30 and 70% methanol by only 20%. This implies that evaporation does not dominate in causing acceleration of this reaction in microdroplets.We also examined the influence of different solvent composition by varying the mixing ratio between water and methanol. The different solvent composition affected the reaction rate of chlorophyll a and chlorophyll b differed in the microdroplet compared with bulk solution. Several factors might contribute to the behavior, such as different surface adsorption and different effects of electric field at the air-liquid interface. Chlorophyll b possessing a –CHO group is more hydrophilic than chlorophyll a possessing a –CH3 group. We speculate that the differences in the hydrophilicity between chlorophyll a and b might cause the differences in the acceleration factor by affecting the surface adsorption. The surface catalytic effect of liquid microdroplet surface can be also influenced by the different polarity of the different groups between chlorophyll a and b. The electric field may also play a role in accounting for the difference as the electric field can influence the orientation of reactant molecules during reactions.It might be wondered why the acid-induce demetallation of chlorophyll In the present study, the accelerated kinetics of chlorophyll demetallation suggests the reaction acceleration was mainly attributed to the micro-confinement effect rather than the introduced charges, solvent composition, and solvent evaporation effect in microdroplet. The behavior of accelerated reaction rate in biological reactions in the present work illustrates that reactions of biomolecules in a confined environment similar to the cell may behave differently than what has been reported through assays using bulk solutions.


The aim of this study was to establish the suitability of terahertz (THz) transmission measurements to accurately measure and predict the critical quality attributes of disintegration time and the amount of active pharmaceutical ingredient (API) dissolved after 15, 20 and 25 min for commercial tablets processed at production scale.Samples of 18 batches of biconvex tablets from a production-scale design of experiments study into exploring the design space of a commercial tablet manufacturing process were used. The tablet production involved the process steps of high-shear wet granulation, fluid-bed drying and subsequent compaction. The 18 batches were produced using a 4 factor split plot design to study the effects of process changes on the disintegration time. Non-destructive and contactless terahertz transmission measurements of the whole tablets without prior sample preparation were performed to measure the effective refractive index and absorption coefficient of 6 tablets per batch.R2 = 0.86) and API dissolved after 15 min (R2 = 0.96) linearly correlates with the effective refractive index, neff, measured at terahertz frequencies. In contrast, no such correlation could be established from conventional hardness measurements. The magnitude of neff represents the optical density of the sample and thus it reflects both changes in tablet porosity as well as granule density. For the absorption coefficient, αeff, we observed a better correlation with dissolution after 20 min (R2 = 0.96) and a weaker correlation with disintegration (R2 = 0.83) compared to neff.The disintegration time (neff and αeff provide promising predictors for the disintegration and dissolution time of tablets. The high penetration power of terahertz radiation makes it possible to sample a significant volume proportion of a tablet without any prior sample preparation. Together with the short measurement time (seconds), the potential to measure content uniformity and the fact that the method requires no chemometric models this technology shows clear promise to be established as a process analyser to non-destructively predict critical quality attributes of tablets.The measurements of The online version of this article (doi:10.1007/s11095-017-2108-4) contains supplementary material, which is available to authorized users. The strength of these bonds and the arrangement of the particles in the tablet are affected by the material properties  and the process configuration . Besides the formation of interparticulate bonds the compaction of powder results in a complex pore network within a tablet, which is of central importance for the disintegration and dissolution performance. The pore structure defines the capillary and viscous forces which govern the ingress of gastrointestinal fluids into the tablet upon administration and initiate the disruption of the particle bonds. Such pore structures are typically characterised by the relative void space in the tablet, i.e. the porosity, which is the fraction of the volume of voids over the total volume of the tablet. The porosity of a tablet is therefore one of the most important contributors to tablet disintegration and it is directly influenced by the process configuration, such as compaction force and compression speed [Processing of active pharmaceutical ingredients (APIs) into convenient dosage forms for effective drug administration places great demands on excipients, the formulation and the processes used in pharmaceutical production. A variety of excipients, such as diluents, disintegrants, surfactants and lubricants, are mixed to form uniform blends of suitable flowability and compactibility that do not stick to the surfaces of feeder walls, tablet punches or other processing equipment and do not demix during processing . Tablet on speed , 3.Moreover, unfavourable API particle properties, such as their shape or electrostatic properties, may result in the requirement to granulate the API and excipients prior to compaction resulting in denser, larger particles which are more amenable to processing. Granulation thus affects the properties of the particles/granules which are subsequently used for the tablet compaction and ultimately it impacts on the disintegration performance of the final tablet. In particular, the granule density was shown to have a significant effect on the compactibility as well as the performance of the final dosage form . Anotheri.e., porosity and density of individual granules) of complex powder compacts which are required for the accurate prediction of the disintegration performance. Even though the disintegration performance can be mostly inferred from the combination of weight, thickness and hardness measurements, pharmaceutical companies as well as regulatory bodies would prefer a non-destructive direct measure of a large number of tablets providing statistically meaningful insights into the quality of an entire production-scale batch. Such information can only be generated by means of a non-destructive, fast and, most importantly, sensitive method. The low absorption and insignificant scattering losses of terahertz radiation for pharmaceutical formulations result in high penetration power that enable transmission measurements through entire tablets. Together with the non-ionising nature of terahertz radiation as well as the short measurement time (milliseconds) this technology could provide a potential process analytical technology (PAT) tool to non-destructively characterise pharmaceutical tablets, e.g., coating structures, porosity, hardness and density [It is common practice to use weight, thickness and hardness (breaking force) measurements of tablets following the compaction step as a manufacturing in-process control tool to predict the disintegration performance. Since the tablet breaking force is typically called hardness in the pharmaceutical literature, we will henceforth use both terms interchangeably when referring to the tablet breaking force. Therefore, in the context of this study the tablet hardness is not the resistance of a surface to indentation by a small probe as it is defined in other scientific communities such as materials science. However, weight, thickness and hardness are only indirect descriptors of the critical quality attributes  study were measured by terahertz time-domain spectroscopy (THz-TDS). An average effective refractive index per batch was then related to its respective disintegration time.H) varied depending on the targeted tablet breaking force and the tableting speed was 140,000 tablets per hour.Details of the formulation and process used to manufacture tablet cores are provided in . BrieflyThe tablet batches were generated from a split-plot DoE across three granulation factors  and one compression factor (tablet breaking force). Only a subset of the produced batches for the split-plot DoE was selected for this study as indicated in Table The tablet weight, thickness and breaking force were measured using a Pharmatron 10× automated tablet system . Disintegration and dissolution were determined following the pharmacopeial methods (the US and European pharmacopeias). For disintegration testing an Erweka ZT3  was used and the disintegration time was defined as the time at which the last tablet disintegrated out of a set of 6 tablets. The disintegration test was performed once per batch and hence no standard deviation was measured. Dissolution testing was carried out using the Sotax AT7 Smart dissolution bath  together with the Waters Alliance 2695D separation module HPLC system with sample transfer module and 2487 absorbance detector .Terahertz time-domain measurements of 6 tablets per batch  were acquired using a TeraPulse 4000  in combination with a transmission module at an instrument resolution of 0.01 ps over a total range of 150 ps for each time-domain waveform. No sample preparation was required and the measurements represent the transmitted terahertz waveforms acquired at the centre of each respective tablet in axial direction  the pulse delay time Δsee Fig. .t of a terahertz pulse propagating through a sample depends on the effective refractive index, neff, as well as the tablet thickness. The value of neff of a tablet can be readily calculated from the pulse delay time as follows:In general, ΔH is the thickness of the tablet and c is the speed of light in vacuum. Therefore, by measuring the time delay difference of different samples . Variations in neff of different tablets with the same formulation are soly related to changes in the tablet microstructure, i.e. their porosity and density.where Besides the straightforward time-domain analysis the terahertz data can also be analysed in the frequency-domain. Since the electric field of the transmitted terahertz pulse is measured directly, we can extract both the frequency-dependent absorption coefficient and effective refractive index of a tablet from a simple terahertz transmission measurement without any mathematical processing by means of the Kramers-Kronig relationship. Applying a discrete Fourier transform on the time-domain waveforms (reference and sample) yields their respective spectra. The sample spectrum is then normalised by the reference spectrum resulting in the complex-valued transmission spectrum. By assuming the tablet as a homogeneous medium, the frequency-dependent effective refractive index and the effective absorption coefficient can be determined by combining the transmission spectrum, Beer-Lambert’s law and the measured tablet thickness. More details about the extraction of optical constants from terahertz measurements is provided by Zeitler .The frequency-dependent analysis of the terahertz waveforms provides the effective absorption coefficient Fig.  and the In order to analyse the relation of the effective refractive index and the disintegration time in more detail, we chose the effective refractive index at a frequency of 1 THz Fig. . The smaFigure R2 = 0.86). The breaking force primarily reflects changes in the compaction process related to the bonding structure and bonding mechanisms of the tablets [neff accounts for porosity as well as density changes . We additionally observe a linear correlation between neff and the amount of API dissolved after 15 min , 20 and 25 min . The dissolution results from 20 min (R2 = 0.88) and 25 min (R2 = 0.79) are presented in the R2, which decreases with increasing dissolution time. Comparing Figs. neff vs disintegration time) and c (neff vs API dissolved after 15 min) reveals that whilst some batches are outside of the confidence interval  in Fig. i.e., B03, B09, B10 and B17). This apparent discrepancy in the predictive power of the method can easily be explained by the rather limited accuracy of the disintegration testing approach where measurements of multiple samples are aggregated and the endpoint determination of the test is rather ill defined. The magnitude of the error in the amount of API dissolved shown in Fig. A proper comparison of tablets with different shapes would require to calculate tensile strength as well as to normalise the disintegration time by the surface area of each tablet. The results of tensile strength and normalised disintegration time are presented in the supporting information of this manuscript. As pointed out in the supporting information, the relationship is very similar to that of the breaking force and the disintegration time Fig. . It is a tablets , 15 as w tablets , whereasnges see  and thusnair = 1) compared to the API and excipients. A less dense medium can also be caused by a lower density of the particles or granules. Since the density and the porosity of the powder compact are closely related to the mechanical strength of the tablet, the terahertz technique is particularly powerful in providing a measure for the disintegration performance. This is conceptually different to hardness measurements by means of near-infrared [It is well-known that the effective refractive index can be used as a measure of porosity  and densinfrared –20 or Rainfrared  spectrosneff is a combination of the changes in density of the granules and in the pore structure of the tablet. An increase of the porosity of the tablet and/or a decrease of the granule density thus causes a decrease in neff. It was previously shown that a lower granule density is related to a lower moisture content in the granules [In general, the effect of the different process configurations on granules , 22.The amount of water applied during the granulation is listed in Table neff for predicting the dissolution behaviour, which is not surprising given that for this set of samples the thickness and the refractive index correlate almost perfectly  compared to that of the thickness (0.16 ± 0.08%). It will be critical to examine in future whether this variability reflects a better sensitivity to the disintegration process by correlating both the thickness and neff against the dissolution results measured on the exact sample specimens rather than an average batch quantity.In addition to the terahertz measurements we also observe a good correlation between the disintegration time and dissolved API after 15 min with the tablet thickness  and air (n = 1) and that the domain size of the microstructure approaches length-scales of the order of the wavelength (the bandwidth of 0.2 to 1.2 THz corresponds to wavelengths of 1.5 to 0.25 mm) scattering losses will occur upon the propagation of the terahertz pulse through the tablet matrix. The variation in the effective absorption data shown in Fig. Besides the analysis of the refractive index, Shen, Taday and Pepper previously showed that the scattering losses in absorbing media can be described empirically using a power law :2\documeαeff(ν) is the effective absorption coefficient (with the frequency v) and parameters A and B are fitting parameters, with A being constant and B expected to increase with the particle (or here domain) size as discussed by Shen, Taday and Pepper [A and B of Eq. αeff(ν) in the frequency range of 0.65 < v < 1.2 THz.where d Pepper . TherefoB and the amount of dissolved API after 15 and 20 min . The correlation of B with the disintegration time is slightly weaker . In contrast, the ‘domain size’ that leads to the scattering losses reflects the initial size of (agglomerated) particles that form immediately following the disintegration of the dosage form. It is well established that smaller particles have a higher intrinsic dissolution rate and hence it is not surprising to see a relatively strong correlation between the scattering parameter and the dissolution data in the early phase of dissolution of this API (BCS class II), in particular given that all tablets are fully disintegrated at 15 min. The correlation with the dissolution data at 25 min is starting to tail off as for most batches the API is fully dissolved at this point. It is important to highlight that there was relatively little variation in granule size prior to compaction (100 − 130 μm) but that the scattering domain size reflects the structure that forms within the powder compact, and which we assume, is more representative to the size of the particles of the disintegrated tablet. The process configurations of the granulation and the compaction thus clearly impacted the domain size of the tablets causing this variation of the parameter B as well as in the dissolution behaviour between the different batches.The data show an excellent correlation between parameter min Fig. . The besker Fig. . A pooreThis study demonstrates the potential of terahertz spectroscopy as an important process monitoring or quality control tool for pharmaceutical tablets prepared by wet granulation and compaction. The method has the advantage that it is non-destructive and contactless. Due to the fact that the effective refractive index reflects density and porosity changes of the granules as well as of the entire tablet, it provides an excellent predictor for the tablet disintegration performance. Furthermore, we found that the effective absorption coefficient, that is derived from the same measurement, is an independent metric reflecting the internal structural domain size which affects the immediate dissolution rate.Given the potential for automated non-destructive measurements, terahertz technology might be especially advantageous where occupational health categorisation 4 (OHC4) or OHC5 compounds that require high containment are being used in order to prevent the handling of the compounds by the workers. Moreover, the number of measured tablets could be drastically increased, when using this technology as an at-line or on-line tool, to demonstrate uniformity of the product. Further development of the terahertz technology and the processing procedures are, however, necessary in order to transfer the presented concept to sufficiently robust at-line or on-line control of the disintegration/dissolution performance.ESM 1(PDF 1051 kb)Below is the link to the electronic supplementary material.


This study aims to develop and evaluate a manually controlled steerable needle that is compatible with and visible on MRI to facilitate full intra-procedural control and accurate navigation in percutaneous interventions. The steerable needle has a working channel that provides a lumen to a cutting stylet or a therapeutic instrument. A steering mechanism based on cable-operated compliant elements is integrated in the working channel. The needle can be steered by adjusting the orientation of the needle tip through manipulation of the handle. The steering mechanism is evaluated by recording needle deflection at constant steering angles. A steering angle of 20.3° results in a deflection of 9.1–13.3 mm in gelatin and 4.6–18.9 mm in porcine liver tissue at an insertion depth of 60 mm. Additionally, the possibility to control the needle path under MRI guidance is evaluated in a gelatin phantom. The needle can be steered to targets at different locations while starting from the same initial position and orientation under MRI guidance with generally available sequences. The steerable needle offers flexibility to the physician in control and choice of the needle path when navigating the needle toward the target position, which allows for optimization of individual treatment and may increase target accuracy. Liver cancer is the fifth most commonly diagnosed cancer and the second most deadly form of cancer in men worldwide . In addiThe success of HDR brachytherapy and other percutaneous needle interventions is highly dependent on accurate positioning of the needle tip. However, this can be complicated by unpredictable needle deflection and target movement due to tissue deformation and breathing . Source This research aims at the development of a manually controlled steerable needle that is compatible with and visible on MRI to provide active control of the needle trajectory during insertion for accurate tip positioning. Multiple steering strategies have been proposed previously, which include base manipulation , 8, teleThe manually controlled steerable needle consists of three main parts Fig. a: cuttinThe cutting stylet with a diameter of 1.9 mm consists of three parts: a flexible polyether ether ketone (PEEK) tube; a highly flexible part made out of Teflon; and a titanium conical tip connected to each other by means of press fittings. The titanium tip generates a larger artifact than other parts, which enhances its visibility. The working channel is a PEEK tube with an outer and inner diameter of 3.2 and 2.0 mm, respectively. Four grooves are milled along the working channel to provide space for the Dyneema steering cables . In addition, three pairs of radial grooves are milled in the distal part of the working channel to create compliant elements in the steerable tip with a length of 22 mm. The three pairs are positioned with a spacing of 5.0 mm and oriented in 120° with respect to each other to allow for bending of the tip in all directions. The working channel is covered with a biocompatible polyethylene terephthalate (PET) shrinking tube . The polyoxymethylene (POM) and PEEK handle consists of four main parts  were applied to the needle tip in x direction. After this, the gel phantom was replaced by a tissue phantom that consisted of a porcine liver embedded in gel to gain insight in the steering behavior in inhomogeneous tissue. The optical camera was replaced by a transducer  and an ultrasound imaging system  that registered the needle tip at a rate of 25 frames per second with a calibrated accuracy of 0.375 × 0.333 mm. Each experiment was repeated six times per steering angle to gain insight in the variation in deflection while preventing the needle from following the same trajectory twice.The relationship between the steering angle of the tip and the resulting path curvature was evaluated in an in vitro setup Fig. a. The neThe aim of the in vitro MRI-guided intervention was to show how the needle trajectory can be controlled using real-time feedback obtained by MRI. The needle was inserted in a gel phantom (15 % gelatin in 5 L water in a 15 × 17 × 21 cm transparent box) to reach targets at different locations while starting from the same entry point and initial orientation. The targets (olives) were positioned at a depth of 120 mm. The first target was positioned 31 mm to the right and 2 mm below the point of insertion. The second target was positioned 4 mm to the left and 24 mm above the point of insertion. The phantom was positioned in the middle of the bore of the 1.5 T whole-body MRI scanner , and the open side of the box was aligned with the opening of the bore. A coil setup consisting of two elliptical elements of 14 × 17 cm was applied for signal reception. One of the researchers was positioned next to the scanner and inserted the needle in the phantom to subsequently reach two targets starting at the same point of insertion. Figure A volume scan was made for planning. Based on the volume scan, angulated coronal and sagittal planes were selected in such a way that both the needle tip and the target were enclosed in the interception of the planes. These planes visualized the deflections of the needle to the left and to the right and deflections upward and downward. The fast dual-plane dynamic scans of these planes were shown on a display in the MRI-room to provide insight in the required steering motion to the researcher during the procedure with an update frequency of approximately 0.5 Hz. The scan parameters of the volume scan and the intra-procedural dual-plane dynamic scans are provided in Table The 3D volume scans were post-processed according to a reconstruction method known as center-out radial sampling with off-resonance (coRASOR) reconstruction. This reconstruction aims at selective depiction of the needle with high positive contrast. CoRASOR reconstructions were performed in MATLAB  using an off-resonance value of 5000 Hz. A detailed description of both the principles and the workflow of the coRASOR reconstruction method can be found elsewhere , 3, 21. Figure First the needle was inserted and a volume scan was made to determine the location of the needle tip and the target. Based on this scan, the two imaging planes to be displayed intra-procedurally were selected. An example of the dual-plane dynamic scans is shown in Fig. Figure After the intervention, the MRI scans were subjected to post-processing. CoRASOR-reconstructed 3D volume scans enabled positive contrast visualization of the needle trajectories in 3D, as shown in Fig. This work aimed at the development and evaluation of a manually controlled steerable needle to provide active control in MRI-guided interventions that require accurate targeting, such as HDR brachytherapy. The orientation of the tip of the proposed needle could be adjusted by manipulating the handle thereby steering the needle in the required direction. The shaft of the needle passively followed. The needle was designed to be compatible with MRI and mainly consists of plastics. The titanium tip was the only exception and generated an artifact that improved its visibility on MRI.In vitro experiments in gel showed that the steering angle applied to the tip of the needle determined the degree of deviation from a straight trajectory during insertion, which indicated that the needle trajectory could be adjusted by manipulating the tip orientation. Variations in deflection resulting from the same steering angle at the tip may have been due to variable friction and other mechanical losses in the needle. This did not necessarily affect the target accuracy since the trajectory could be adjusted at any time during insertion. The degree of deflection may be sufficient for the clinical application, but the maximum steering angle can be increased if necessary by adjusting the handle design.A second set of in vitro experiments was performed to further evaluate the needle in an MRI-guided procedure. In these experiments, the needle was successfully navigated to two targets at different locations starting from the same entry point with the same orientation, illustrating several important findings: (1) the developed needle can be visualized using a generally available dynamic MR imaging technique using a clinically relevant and realistic slice thickness of 10 mm; (2) the needle does not induce large image distortions or signal voids, enabling the simultaneous depiction of both the needle and surrounding soft tissue; (3) the steering abilities of the needle facilitates accurate navigation, enabling MR-guided targeting of predefined locations; (4) the paramagnetic properties of the titanium tip facilitate selective positive contrast visualization when increased specificity is necessary.Interventional radiologists indicated that the update frequency of 0.5 Hz is sufficient, because needle insertion during the intervention usually takes several minutes. However, a higher update frequency provides more feedback, which may improve the positioning accuracy. Recently we have demonstrated that the update frequency can be increased up to four times by using sliding window reconstruction approaches with compressed sensing reconstruction, leading to update frequencies of 2 Hz for a dual-plane scan technique .A steerable needle offers active control to the physician, so that the trajectory of the needle can be adjusted intra-procedurally. This reduces the need for retraction and reinsertion when the planned trajectory is not resulting in the correct position of the needle tip due to unforeseen needle deflection or tissue deformation. The steerability of the needle also introduces the possibility to reach multiple targets from a single point of insertion, which could be of added value when therapy needs to be applied to multiple locations . In addition, curved paths can be obtained so that penetration of critical and vulnerable structures  can be avoided. The proposed needle could therefore facilitate individual treatment optimization and contribute to the accuracy of needle placement in image-guided interventions such as HDR brachytherapy in the liver.The steering mechanism presented in this work has a number of advantages with respect to previously described steerable needles. In needle steering by means of base manipulation , 8, momeAlthough the needle provides control of the trajectory, some demerits need to be taken into account. The experiment in the MRI showed that steering becomes more difficult when the needle is inserted deeper. This suggests that timely steering is preferred over last-minute adjustments. In addition, the freedom of movement of the physician is limited by the confined space in the bore of the scanner, which may affect the control options. Finally, applicability may be limited when it is expected that avoiding critical structures will most likely lead to inaccurate targeting.The current study evaluated the steering capabilities of the needle in a gel phantom and in a tissue phantom. Neither of the phantoms contained critical structures that need to be avoided. This simplifies navigation thoroughly compared to an in vivo situation. The controllability of the needle is affected by the heterogeneity of tissue. The large heterogeneity of real tissue causes needles to deflect more, which stresses the need for active control of the needle trajectory. Besides this, the needle was controlled by one of the researchers, while one can expect more dexterity from an experienced physician, who could simultaneously insert the needle, control the tip direction, and verify the resulting trajectory on the continuously updated MR images. Extensive testing is required to gain insight in these factors. In such experiments, target accuracy, the number of attempts, and duration of the intervention should be measured to quantify the added value of this technology.An important challenge for implementation in practice will be the selection of the imaging planes or volumes. In some cases, the imaging strategy as described here may be sufficient. In other cases, more sophisticated selection of imaging planes may be necessary, for example when the trajectory is expected or planned to be curved. Automatic selection of imaging plane could be guided by real-time tracking of the needle. This can be accomplished through shape sensing by means of fiber Bragg gratings that are integrated in the instrument, which is work in progress , 10. In A manually controlled steerable needle has been developed successfully. This needle provides active control of the needle trajectory and may increase the accuracy of needle positioning. Steering is initiated by adjusting the orientation of the needle tip. The needle was demonstrated to be compatible with MRI. The image quality in the direct vicinity of the needle was not affected, while the needle and its tip could be visualized clearly.


The accessibility of medical care facilities in sparsely populated rural regions is relevant especially for elderly people which often represent a large segment of the population in such regions. Elderly people have higher morbidity risks and a higher demand for medical care. Although travelling with private cars is the dominating traffic mode in rural regions, accessibility by public transport is increasingly important especially because of limited mobility of elderly people. The aim of this study was to determine accessibility both by car and public transport to general practitioners (GP) and selected specialist physicians for a whole region and to detect areas with poor to no access in the county Vorpommern-Greifswald, which is a rural and sparsely populated region in the very northeast of Germany.Accessibility of medical care facilities by car was calculated on the basis of a network analysis within a geographic information system (GIS) with routable street data. Accessibility by public transport was calculated using GIS and a network analysis based on the implementation of Dijkstra’s algorithm.The travelling time to general practitioners (GP) by car in the study region ranges from 0.1 to 22.9 min. This is a significant difference compared to other physician groups. Traveling times to specialist physicians are 0.4 to 42.9 min. A minority of 80 % of the inhabitants reach the specialist physicians within 20 min. The accessibility of specialist physicians by public transport is poor. The travel time (round trip) to GPs averages 99.3 min, to internists 143.0, to ophthalmologists 129.3 and to urologists 159.9 min. These differences were significant. Assumed was a one hour appointment on a Tuesday at 11 am. 8,973 inhabitants (3.8 %) have no connection to a GP by public transport. 15,455 inhabitants (6.5 %) have no connection to specialist internists.Good accessibility by public transport is not a question of distance but of transport connections. GIS analyses can detect areas with imminent or manifest deficits in the accessibility of health care providers. Accessibility analyses should be established instruments in planning issues. Rural areas with a low population density are characterized by sparsely distributed health care providers and facilities. Catchment areas of physician practices and hospitals are large with long travel distances for patients in many cases. Accessibility is regarded as the ease to reach destinations or activity sites . Equity 2, part of the federal state of Mecklenburg-Western Pomerania in the northeast of Germany). It is a rural and sparsely populated county with an average of 69 inhabitants/km2 [The study region is the county Vorpommern-Greifswald  ArcGIS 10.0  provides a special toolbox to perform network analysis, the Network Analyst. The tools in this toolbox work with an implementation of Dijkstra’s algorithm.The county Vorpommern-Greifswald has an external border to the Baltic Sea in the North and to Poland in the West, internal borders to other counties in the federal state of Mecklenburg-Western Pomerania and a border to the federal state of Brandenburg in the south. People rarely cross the external border to visit a doctor, however, they do cross internal borders; both to other counties and to other federal states. To take this into account, a 15 km buffer zone was defined around the study region to cover adjacent regions within the federal state of Mecklenburg-Western Pomerania. All relevant medical providers within the buffer were considered in the analyses. It was not possible to take into account a similar buffer for the federal state Brandenburg, because public transport data were not available.th September 2012). These data were used to determine the affected population. The population data were merged to the district level (N = 593 districts). Population data were available for 465 districts; some smaller districts had to be aggregated.Population data at the municipality level were retrieved from the Central Information Register (ZIR) that records the population data of Mecklenburg-Western Pomerania at a daily basis , turn restrictions but also house numbers and postal codes for each side of the street.For this network analysis, we used NavStreet Data from NAVTEQ and the geographical data of the Federal Agency for Cartography and Geodesy as at 2012 .The center points of the districts were used as origins for the trips to the medical providers. To determine zones of equal travel times to the physician practices (in 5 min distance-categories) the Network Analyst tool “New Service Area” was used to determine travel time to the practice in minutes. The exact distance values (in minutes as well as in meters) from the center points of the districts to the physician practices were calculated with the Network Analyst tool “OD cost matrix”. These values were used to calculate average distance, standard deviation and maximum distance.To determine the accessibility of physician practices by public transport, bus and train time tables  were used. The geographical coordinates of bus and train stops were actively collected by study staff using a GPS device.Calculation of the footpath distances from the district center point to the three nearest bus and/or train stops;Calculation of the footpath distances from the origin bus/train stop to the destination bus/train stop for vehicle changes;Calculation of the footpath distances from the destination bus/train stop to the physician practice using the ArcGIS OD cost matrix tool.The travel times from each district center point to the physician practices were calculated in three steps:All three calculations were made with the ArcGIS OD cost matrix tool. The footpath distances and the bus and train timetables are essential for the calculation of the overall public transport travel time that is made with a self-developed software based on Dijkstra’s algorithm.The results of the calculation were visualized on a map with ArcGIS using Thiessen polygons. A Thiessen polygon defines an area of influence around its sample point, so that any location inside the polygon is closer to that point than any of the other sample points .Up to 1,000 m from the district center point to the physician practice ;Up to 1,000 m from the district center point to the three nearest bus and train stops;Up to 500 m from the destination bus/train stop to the physician;Up to 250 m between bus/train stops for bus/train changes.The calculation is based on some assumptions: the appointment in the physician practice is on a Tuesday at 11 am out of school holidays (the public transport is mainly organized according to school time requirements ). The paIn the calculation, the three nearest bus and train stops were considered because the nearest bus or train stop must not necessarily provide the fastest connection. The return trip is defined to start latest at 12 am.For both calculations the duration of the doctor’s appointment is not included in the travel time. Also not considered were traffic jams, rush hours, construction sites or the search for parking lots.The differences between the accessibility of the single physician groups were analyzed using the Kruskal-Wallis test followed by a Tukey correction to examine which means differ significantly from each other. The calculations were made with SAS 9.3 © 2002–2010 .Figure 38.6 % of the districts  are located in the zone under 5 min travel time, Table .Table 1TFigure Table p < .0001) as well as for travel distance (p < .0001). The subsequent Tukey correction showed that all four groups differ significantly.The analysis of variance using Kruskal-Wallis showed that the differences in mean of the accessibility of the four physician groups by car are significant for travel time . The Tukey correction showed here that all groups differ significantly with respect to each other except for specialist internists and ophthalmologists.The differences in mean travel time by public transport between the physician groups are significant ) , 29. In We examined the accessibility of different groups of physicians in a rural area with a low population density, both by car and by public transport. On average, travel times to GPs are lower than to any of the medical specialists. Travel times in the public transport system are high for a substantial part of the population and some villages have no connection to a physician practice. About 8,000 to 15,000 people have no access to the examined physicians with public transport (3 to 6 % of the population in the study region).The low population density in rural regions limits the cost-efficient operation of public transport. Concepts that consider the planning of medical care and public transport simultaneously are needed. Adequate medical care should be evaluated not only based on availability but also by accessibility of medical facilities .