People make systematic errors when matching locations of an unseen index finger with the index finger of the other hand, or with a visual target. In this study, we present two experiments that test the consistency of such matching errors across different combinations of matching methods. In the first experiment, subjects had to move their unseen index fingers to visually presented targets. We examined the consistency between matching errors for the two hands and for different postures (hand above a board or below it). We found very little consistency: The matching error depends on the posture and differs between the hands. In the second experiment, we designed sets of tasks that involved the same matching configurations. For example, we compared matching errors when moving with the unseen index finger to a visual target, with errors when moving a visual target to the unseen index finger. We found that matching errors are not invertible. Furthermore, moving both index fingers to the same visual target results in a different mismatch between the hands than directly matching the two index fingers. We conclude that the errors that we make when matching locations cannot only arise from systematic mismatches between sensory representations of the positions of the fingers and of visually perceived space. We discuss how these results can be interpreted in terms of sensory transformations that depend on the movement that needs to be made. Theories about the control of goal-directed movements try to explain things such as how one moves one’s hand to a visual target. A simple model for such a movement is the servo control: One keeps activating muscles of the arm until the hand is judged to be at the same location as the target. If the hand is not visible, the target must be judged visually and the hand haptically . If movements are controlled in this manner, and given sufficient time to move, systematic errors in reaching the target (which we will refer to as matching errors) logically imply that there is some kind of mismatch between the senses. Such matching errors are frequently reported, for instance when moving the finger of one hand to the position of the finger of the other hand  possible origins. A first possibility is that the mismatches are due to systematic, modality-specific misjudgments of positions in space, perhaps arising from systematic differences between the ways in which the positions of the fingers and visual positions are represented. If so, we can consider them to be mismatches between sensory maps of our surrounding. We use the term “map” to refer to any representation of space, irrespective of the reference frame within which positions on the map are described. The maps could simply be shifted relative to each other, but they could also differ in scale or orientation, or have modality-specific complex deformations such as might arise from scaling binocular disparities to help obtain a visual map. All these cases would lead to similar systematic mismatches across trials, regardless of how the match is achieved (both in terms of posture and in terms of the actions involved). Deviations across trials could arise from sensory or motor variability, but the systematic component is the result of the way in which positions in space are coded for the different modalities and possibly effectors.A second possibility is that mismatches arise from systematic errors in sensing joint angles. If so, matching errors will not only depend on the position of the tip of the finger, but on the posture of the arm as well, because systematic errors in judging joint angles will give rise to different errors when the posture of the arm is different, even if the target position for the finger is not different. Consequently, it will be impossible to describe matching errors as aligning sensory maps of the surrounding.A third possibility is that systematic mismatches do not only arise from modality-specific misjudgments of positions in space and systematic errors in sensing joint angles, but also on the way in which the final position and posture is achieved, perhaps because it depends on the specific reference frame transformation that is required for the task  judgments about one’s position relative to the surrounding space. This could be due to errors in the haptic judgment of the position of the hands, but it could also be due to misjudging visually perceived positions. If one were to ask subjects to align an index finger with a visual target, instead of to align it with the other index finger, the proprioception of the arm in question would contribute to the visuo-haptic matching error in the same way as it does when aligning the two index fingers. The contribution of proprioception of the other arm to the matching error would be replaced by errors in visual localization of the target . According to the map-based explanation, we would expect to see a consistent difference between the two arms irrespective of their posture. According to the posture-based explanation, we might expect similar (perhaps mirror-symmetric) visuo-haptic matching errors for both hands when the posture of both arms is similar, and different visuo-haptic matching errors for a single arm when its posture is varied. In the second experiment, we compared the matching errors across sets of tasks that ended in the same configuration , but that involved different actions to reach that configuration. For example, we compared bringing the index finger to a visual target dot with bringing a visual dot to the index finger.In “Ten self-reported right-handed subjects  participated in the experiment, including two of the authors (MG and EB). Except for the authors, all subjects were naive about the purpose of the experiment. All subjects participated voluntarily, had (corrected-to-) normal vision, and gave their written informed consent. The experiment was part of an ongoing research program that has been approved by the ethics committee of the Faculty of Human Movement Sciences of Vrije Universiteit Amsterdam.We used the same setup as we used previously to measure visuo-haptic matching errors  at a sampling rate of 200 Hz. To do so, an infrared-emitting diode was attached to the nail of the index finger. Before each block of trials, we turned on the light below the semi-silvered mirror so that both the finger and target were visible and asked subjects to move to four dots in the four corners of the set up. This calibration allowed us to align the Optotrak coordinates of the index finger with the coordinates of the visual presentations.Subjects moved their unseen index finger to visually displayed targets. This task was done in four different ways: all combinations of the right and left hand above and below the board. Four more ways of performing the task (involving passive matching) were presented in the same session, but not analyzed for this paper. An illustration of the tasks can be seen in Fig. Each matching task was presented as a separate block of 60 trials (ten for each of the six targets). The blocks were presented to the subjects in counterbalanced order. The order of the trials was semi-random: The six target positions were presented once, in a random order, before a next set of six targets was presented in a new random order. Since subjects moved from target to target, we ensured that the first target of a set was never identical to the last one of the previous set. Subjects did not receive any feedback during the experiment other than from their own proprioception. Previous studies have shown that there is little drift or accumulation of errors in this paradigm  in the indicated posture (above or below the board) to the visually presented dot in one continuous movement. Then, the next dot appeared and the subject reached for it in the same way. This continued until the whole block was completed.The end of a movement was detected online and used to stop the trial and start the next one. We started looking for the end after the movement had reached a velocity of 50 cm/s. The end point was then defined as the position of the marker at the first frame after it had moved slower than a threshold of 3.5 cm/s for eight frames.i) we calculated the matching error (ME), which is the vector between the target position in that trial (Ti) and the end point of the finger movement toward this target (Xi).For each subject, block and trial  were calculated for each subject, block and target; we refer to this measure as matching variability. The matching errors and matching variability of an example subject are shown in Fig. 1 and ME2 being vectors as defined in Eq.  were used to determine how the three consistency measures related to each other  for the two postures and the two hands, respectively.Since we are interested in the consistency between the postures of the two hands and the two hands in both postures, we calculate CONSp’s > 0.25), and there were no significant interactions  showed no significant main effects . The interactions were also not significant , except for a significant interaction between hand and posture . The matching variability was larger for the right hand when it was below the board and for the left hand when it was above the board  with no effect of hand or interaction (both p’s > .34). Post hoc pairwise t tests with Bonferroni corrections show that CONSindividual and CONSothers are significantly smaller than CONSvar , whereas CONSindividual does not differ significantly from CONSothers (p = 1.0). Thus, there appears to be very little consistency between the matching errors when moving above and below the board.The 3 × 2 repeated measures ANOVA on the consistency between moving above and below the board , with no effect of posture or interaction (both p’s > .22). Post hoc comparisons with Bonferroni corrections show that CONSindividual and CONSothers are significantly smaller than CONSvar . The tendency for CONSindividual to be larger than CONSothers was not significant (p = .078). Thus, there also appears to be very little consistency in the matching errors between the hands.The 3 × 2 repeated measures ANOVA on the consistency between the left and right hand  and much lower than we found for repeating the same task in our previous study , but that would be likely to involve different sensory transformations. According to any model that assumes that matching errors are due to sensory biases , the waBesides comparing reversed matches, we also compared combinations of matching errors that involved the same (or at least very similar) movements. In such cases, even if the systematic errors depend on the movements that are made, it should be possible to combine the matching errors of two tasks  to predict the errors that will be found in a third task in which the common sensory information (in this example the visible dot) is eliminated (for instance by moving the left and right index fingers simultaneously to the same position). Therefore, we added haptic–haptic matching tasks to the visuo-haptic matching tasks that were used in “We used the same setup as in “In addition to these four visuo-haptic matching tasks, we performed three haptic–haptic matching tasks. The same guidance of the target finger was used for two of these tasks as for the DR and DL tasks described above. Subjects then moved the index finger of their right hand above the board to match the position of the index finger of their left hand below the board (RL), or moved their left index finger to match the position of the right index finger (LR). The third haptic–haptic matching task was matching both index fingers at the same time (B). In this task, a large target area (diameter of 15 cm) was flashed (centered on one of the six target positions), and participants moved their two index fingers simultaneously to some position within that target area, with the fingers aligned on opposite sides of the board.The seven tasks were performed in a single session. Their order was systematically varied across participants. Each task took approximately 5–7 min. After each task, there was a short break. Within each task, there were ten movements to each of the six targets, as in “The end points of finger movements  were determined in the same way as in “For the four visuo-haptic matching tasks, the matching errors were calculated by comparing the position of the index finger with the position of the dot. For the haptic tasks, the matching errors were the differences between the positions of the active index finger with respect to the passive one. In the B task, in which both hands moved, we report the matching error as if the right index finger was the target position.For the visuo-haptic matching tasks, two 2 × 2 × 6 repeated measures ANOVAs (finger × action × position) were used to determine whether the magnitudes of the matching errors and matching variability depended on the task (moving the index finger to the target dot or moving the cursor dot to the index finger) and on which index finger was used (left or right). For the haptic tasks, two 3 × 6 repeated measures ANOVAs (task × position) were used to determine whether the task  influenced the magnitude of the matching errors or their matching variability. If sphericity was violated, Greenhouse–Geisser corrections were used.others) and an expected value based on the subjects’ trial-by-trial variability for each target within a task (CONSvar). We calculated the consistency values between the tasks with the same matching configuration to test for the reversibility of sensory matching errors both within and between modalities; single comparisons  was used to evaluate the same for all three possible pairs of matches.As in “var) for all comparisons. A 3 × 6 repeated measures ANOVA (Consistency type × combination) was used to determine how the individual consistency values related to CONSothers and CONSvar.To test whether sensory matching errors of two tasks can be combined to predict the outcome of a third task, we made several double comparisons : The matching errors were larger when moving the dot to the finger than when moving the finger to the dot. There was no significant effect of hand used . There was a significant effect of position : As reported in the literature, the matching errors for positions closer to the body were smaller than those for positions further away , and between hand used and target position . The RM ANOVA on the magnitudes of the matching errors in the three haptic tasks showed a significant effect of task : The errors were larger for LR compared to RL and B. There was also a significant effect of target position .The magnitudes of the matching errors ranged from 0.1 to 10.1 cm across subjects, target positions and tasks. The mean errors for all tasks are shown in Fig. 2). The RM ANOVA on the matching variability for the four visuo-haptic matching tasks showed a significant main effect of task : Moving the finger to a dot was more precise than moving the dot to the finger. There was also a main effect of finger used : The matching variability for the right index finger was slightly larger than that for the left index finger. Interaction effects of task and finger used  and finger used and target position  were also found, which could be explained by there being exceptionally large matching variability in DR for some of the target positions. The RM ANOVA on the matching variability in the haptic tasks showed a significant effect of task . Post hoc comparisons with Bonferroni corrections show that B was more variable than RL (p = .001).The matching variability , and no significant effect of hand or interaction effect (both p’s > .82). Post hoc comparisons with Bonferroni correction show that CONSindividual and CONSothers are significantly smaller than CONSvar , and CONSindividual was significantly larger than CONSothers (p = .023).The aim of our analysis was to test whether we can explain the matching errors in terms of sensory biases. If so, we would expect that the errors in the two tasks that involved the same matching configuration would be consistent with each other. The consistency is expected to be about CONSF1.2, 10.4 = 23.6, p < .001), and no significant effect of task or interaction effect (both p’s > .34). Post hoc comparisons with Bonferroni correction show that CONSothers was significant smaller than CONSvar (p < .001) and CONSindividual was significantly larger than CONSothers (p = .007), but there was no significant difference between CONSindividual and CONSvar (p = .956).For the single comparisons of the haptic–haptic tasks , and no significant effect of task or interaction effect (both p’s > .17). Post hoc comparisons with Bonferroni correction show that CONSindividual and CONSothers are significant smaller than CONSvar , and CONSindividual was significantly larger than CONSothers (p = .006).For the double comparisons Fig. b, the 3 What is the cause of the lack of consistency that we find? Can it be explained by asymmetries between the tasks? It might for instance be that subjects prefer to minimize muscle activation, and therefore position an active finger closer to the body. To test whether the lack of consistency is due to such systematic differences between tasks, we compared the mean end points in similar tasks that differed in whether it was a certain index finger that was moving to the target during the matching process (we refer to such an index finger as the active finger). We compared the average bias for the active finger tasks (RD and LD) with that in the corresponding passive finger tasks (−DR and −DL). We did not find any systematic differences between the matching errors across subjects in any of the comparisons (not shown). This suggests that the difference between the errors that are made when moving the finger to the dot and moving the dot to the finger is idiosyncratic rather than being consistent across subjects. We also compared the average −RL, LR and B biases, because in LR the right hand could rest a bit on the table, while in the other tasks this hand was actively moving. Again, we did not find a systematic difference in matching errors across subjects, suggesting that the lack of consistency between moving the right to the left index finger and moving the left to the right finger is also due to idiosyncratic errors rather than ones that were consistent across subjects.others), but for all comparisons that involved matching haptics and vision, the consistency was less than we estimated on the basis of the trial-by-trial variability. If the biases had been completely consistent in the complementary tasks , we would have expected the consistency value to be higher than that of CONSvar, because the consistency value is based on the mean error, while CONSvar is based on the error on individual trials, so the former is less susceptible to random variability across trials. Based on the single comparisons of visuo-haptic matching tasks, we can conclude that moving the index finger to a dot does not give rise to the same mismatch as moving the dot toward the index finger. Based on the double comparisons  task is higher than that in the other visuo-haptic matching tasks Fig. . This miIn this paper, we described two experiments that examined errors in aligning the positions of the hands to each other and to visually perceived positions. In the first experiment, we showed that the posture of the arm influences the matched position. The matching errors were also different for the two hands. These results are not in line with a map-based explanation of the matching errors. In the second experiment, we showed that even matching tasks that involve the same configurations of the arm and the visual target do not always lead to comparable errors when performed differently. Moving the index finger to a dot does not lead to the same mismatch as moving the dot toward the index finger. Neither does moving the right index finger to a visual dot and the left index finger to the same visual dot lead to the same mismatch as directly matching the positions of the two fingers. This means that the direction of the transformation of the information between modalities influences the errors that are made, presumably because the different transformations are conducted differently. Only moving the two index fingers to each other with different orders of the movements gives rise to similar mismatches.Simani et al.  tried toOur results are thus not consistent with the hypothesis that the matching errors are simply combined sensory biases (irrespective of whether map-based or posture-based). This finding has important implications for our understanding of the principles by which the brain solves sensorimotor tasks. Apparently, the sensory errors are partly the result of the process of transforming information and sensory representation that depend on the planned action of getting to the target, instead of only being based on evaluating the desired configuration.One could question whether the results are an artifact of a possible distinction between active and passive haptic position sense. It has been suggested that there is a drift in the perceived position of the hand when holding it at the same physical position for a long period of time (Wann and Ibrahim To conclude, we found that sensory matching errors are not simply based on summing sensory biases. Transformations between modalities seem to play an important role, and the transformation seems to depend on the movement that needs to be made.


We demonstrate that for ultracold, optically trapped Cs atoms the photoassociation (PA) can be manipulated by using external uniform magnetic fields due to the alteration of the scattering wavefunction in the region of the free–bound optical transition. We present PA–induced atom loss measurements with the same intensity for PA laser but different external magnetic fields, and analyze main contributions of the PA to the variation of the number of atoms in the trap. The PA rate exhibits a strong dependence on the changing uniform magnetic field. The experimental data are simulated within the model of a single–channel one–well rectangular potential, whose depth is adjusted so as to assure the predicted variation of the scattering length with the magnetic field. The computational and experimental results are in a reasonable agreement to each other. The same model is used to illustrate some general properties of the two–body quantum system in the near–threshold state. At ultralow temperature, extremely precise control over molecular coherent dynamics can be exerted, and even the reactivities of such molecules can be steered at vanishing entropy2. Their rich internal structure makes them sensitive probes in precision measurements of fundamental physical constants up to a check of the hypothesis of the time variation of world constants6. Possibly strong dipolar interactions of heteronuclear dimers facilitate their application in quantum simulation of the strongly interacting regime and quantum computations8, and also enable proxy investigations of exotic condensed–matter phases9.Rapid progress has been witnessed in the formation and manipulation of ultracold molecules, and this is closely related to their wide applications over recent years11. To effectively control over the atom-molecule conversion, an increasing number of theoretical and experimental researches provide new approaches to obtain an enhanced PA rate by increasing the density of atomic pairs in the short–range region17. A much–used method, investigated experimentally, is the Feshbach–optimized PA20, where PA had been also used to observe Feshbach resonances22. The near–Feshbach resonance wavefunction consists of strongly coupled bound and free hyperfine components of both single and triplet symmetry, resulting in greatly increased PA rates23. A similar mechanism is developed theoretically to obtain an efficiently direct stimulated Raman adiabatic passage of ultracold atoms in the continuum state to ground state molecules near a Feshbach resonance24. The efficient formation of ultracold molecules via the effective manipulation of PA enables the multi-purpose applications of ultracold molecules.Laser–induced photoassociation (PA) and magnetic field–controlled Feshbach resonance are two typical ways in which ultracold atoms can be converted to the molecular bound states25. In this paper, we present a feasable study on the efficient manipulation of the PA of the colliding pairs of ultracold Cs atoms by means of an external uniform magnetic field. Compared to the previous investigation18, our Cs atomic sample has a low temperature and is trapped in an optical dipole trap(ODT). Beside, we describe a theoretical model with the depth–adjustable square potential to match the magnetic field-induced variation of scattering length of atoms in their ground state, and calculate the Franck–Condon coupling strengths. Reasonable agreement between theory and experiment confirms that the shift of scattering phase of the colliding atom pairs induced by a uniform magnetic field gives rise to substantial change in the density of atomic pairs in the short–range region.However, there are very few studies on the effect of uniform magnetic fields away from a Feshbach resonance position on the PA. These magnetic fields can be used to alter the atomic scattering length, which has a great effect on the PA rate2 molecule is produced in the outer well of the double–well potential of the 27. Figure 2. The number of the Cs atoms remaining in the trap is detected by the standard absorption imaging method. We identify the frequency location of the maximum trap–loss as a position of the PA resonance, the resonant frequency is obtained as ~ 11672.098 cm−1 by applying the Lorentz fitting to the observed PA spectrum. The binding energy of this rovibrational level is directly inferred to be Ebind ~ −70.085 cm−1 when referring to the The PA spectroscopy is measured by detecting the loss of Cs atoms optically trapped in a crossed ODT as a function of PA laser frequency. If the PA laser frequency is resonant with the energy difference between the scattering state of colliding atomic pairs and the bound excited state of diatomic molecule in a particular rovibrational level, the pure long-range state Csmentclass2pt{minim28. Thus, the dilute atomic sample makes it possible for the variation of We have investigated the influence of the magnetic field itself on the number of optically trapped atoms with a light field of an off-resonance frequency. For the Cs atoms in the hyperfine state 29. The relationship between the atom loss and the PA rate is described by the differential equation30, the exposure time The dependence of PA rate on the uniform magnetic field is investigated for a direct illustration to the manipulation of the PA using the uniform magnetic fields. The PA rate is known in terms of the on-resonance rate coefficient entclass1pt{minimaμK above the threshold is computed by using the analytical expressions of the square–well potential theory. The upper bound state wavefunction is computed with the standard Numerov algorithm using the potential function of the outer well of the 31.In Fig. 25. As has been already mentioned, the physically correct scattering wavefunction is presented by a mix of many channels formed by hyperfine components of the system. On the other hand, as is well known from the scattering theory32, the variation of the density of atomic pairs in short interatomic separations is strongly correlated with the asymptotic properties, such as the scattering length, which can be chosen as a governing parameter of the process, without much attention to the details of the short–range forces. This idea has been frequently exploited in the previous works34, where the simplified models of two coupled rectangular potentials were employed. The coupling of at least two channels looks to be important for a formation of the above–threshold Feshbach resonances, while the variation of atomic scattering wavefunction with the uniform magnetic field can be described within a simple model of a single–channel rectangular potential with an adjustable well depth as shown in Fig. The theory of laser–assisted resonant cold collisions denotes that the strength of PA of ultracold atoms is determined by the coupling between the continuum wavefunction of the initially colliding atomic pairs and the wavefunction of the excited bound molecules under a PA laser field35 we have chosen the radius of the model square–well potential to be the mean scattering length a0 securely covers the range of an upper bound state wavefunction of excited Cs2 molecule in the long-range Following refsB is chosen so as to assure the predicted dependence37 of the scattering length a on B0) and Gauss (G). This equation must be valid for the ground state The depth of the well for every magnetic field The global wavefunction of atoms in the entire region is constructed by the requirement for the wavefunction itself and its first derivative to be continuous at the switching point The scattering length After solving the latter equation for However, due to a periodicity of the tan function, the solution of this equation is not unique. The physical reason for this is that the desired variation of the scattering length is governed by the properties of a near–threshold bound or antibound level, independently of how many states lay below this near–threshold level. In most of our calculations we have chosen the shallowest well containing only one bound state level. We do the model calculations based on the above theory and Eq. To clarify the underlying physical mechanism, we discuss some intermediate results. Figure B = 18.6 G, at which a very small PA rate coefficient is observed in Fig. B = 117 G, the bound state level is very close to the threshold, its wavefunction is very similar to the scattering one, as shown in Fig. In Fig. entclass1pt{minimaThe widely used approximate expression for the scattering length via the binding energy of a near–threshold bound/antibound state is given asWe have compared this estimate to the accurate model scattering length in Fig. 38. After 10 ms of 3D DRSC, we obtain μK39. Subsequently, the cooled atoms are loaded into a large–volume crossed ODT operating at 1070 nm. The crossed ODT creates a confined potential of μK for Cs atoms. Due to a large anti–trapping potential induced by the gravity, the magnetic field gradient of 31.13 G/cm is produced using a pair of coils in the anti–Helmholtz configuration for the magnetically levitated loading40. Meanwhile, another pair of coils in the Helmholtz configuration provide the magnetic field of 75 G to cancel the resulting anti–trapping potential in the horizontal direction from the application of the magnetic field gradient.The experimental study on the manipulation PA of optically trapped ultracold Cs atoms using a uniform magnetic field is based on the trap–loss spectroscopy of the ultracold atomic sample at the changing magnetic field. To obtain an ultracold atomic sample, the three–dimensional (3D) degenerated Raman sideband cooling (DRSC) is employed to cool and polarize the Cs atoms in the desired state −1. The corresponding PA laser is obtained from a widely tunable continuous–wave (cw) Ti:sapphire laser, which yields an output power of 1 W with a narrow linewidth of ~75 kHz. The long–time frequency drift of the PA laser is less than 500 kHz by locking it to its self–reference cavity. After optical alignment and fiber coupling, a beam with the power of 108 mW is focused to a waist of μm, resulting in the intensity of 2. After 500 ms of the plain evaporation dominated by a three-body loss at the B of 75 G, the PA laser is switched on and the atomic cloud is illuminated for 100 ms.Once the atoms are prepared in the crossed ODT, PA is performed by illuminating the atoms with a certain detuning to the Cs B, and afterwards the PA of the Cs atoms is performed for 100 ms at this B field. The dependence of the PA rate on the magnetic field is determined by recording trap loss spectra at different magnetic fields. However, the variation of the magnetic field induces the shift of the resonant frequency between the scattering atomic and excited bound molecular states due to the Zeeman effect of the hyperfine state For investigating the effect of external uniform magnetic fields on the PA, we measure the PA of Cs atoms at different magnetic fields. Prior to switching on the PA laser focused on the Cs atoms, the magnetic field is ramped up or down over 30 ms to a magnetic field To summarize, we show that it is possible to use an external uniform magnetic field to manipulate the PA of ultracold atoms optically trapped in a crossed dipole trap, that, subsequently, makes it possible for enhancing the PA used to form ultracold molecules. The experimental results are reproduced with a reasonable quality by using the model of a single–channel square–well potential with the depth adapted to the magnetic field via the scattering length as a governing parameter. We present the clear physical mechanism for efficiently manipulating PA, the spatial density of near-threshold states in the short interatomic separation region can be manipulated by altering the scattering length that is related to the near-threshold bound and anti-bound states. Our result demonstrates that the one-channel square potential not only provides a tool for a reasonable modeling of the PA rate observed in our experiment, but can also serve as a very clear and well understandable illustration of the general near–threshold physics in a two–body quantum system.


Cardiac muscle tissue during relaxation is commonly modeled as a hyperelastic material with strongly nonlinear and anisotropic stress response. Adapting the behavior of such a model to experimental or patient data gives rise to a parameter estimation problem which involves a significant number of parameters. Gradient-based optimization algorithms provide a way to solve such nonlinear parameter estimation problems with relatively few iterations, but require the gradient of the objective functional with respect to the model parameters. This gradient has traditionally been obtained using finite differences, the calculation of which scales linearly with the number of model parameters, and introduces a differencing error. By using an automatically derived adjoint equation, we are able to calculate this gradient more efficiently, and with minimal implementation effort. We test this adjoint framework on a least squares fitting problem involving data from simple shear tests on cardiac tissue samples. A second challenge which arises in gradient-based optimization is the dependency of the algorithm on a suitable initial guess. We show how a multi-start procedure can alleviate this dependency. Finally, we provide estimates for the material parameters of the Holzapfel and Ogden strain energy law using finite element models together with experimental shear data. The personalization of computational models in cardiology is a key step toward making models useful in clinical practice and cardiac surgery. A computational model, once properly calibrated, has the potential to forecast cardiac function and disease, and can aid in planning treatments and therapies. To describe the mechanical function of the heart, the passive elasticity of the muscle tissue needs to be represented. Personalizing the effects of this elasticity in a computational model is typically accomplished by tuning a set of material parameters so that the output of the model fits observed data. Gradient-based optimization algorithms have successfully been used in the past to automatically perform the parameter tuning at an organ scale  is a Heaviside function with a jump at The incompressible Holzapfel and Ogden hyperelasticity model  to refer to a mode, with the index i referring to the normal of the face that is shifted, and j to the direction in which the shift occurs.In order to estimate the passive material parameters of the Holzapfel–Ogden model, we make use of a least squares objective functional. This functional defines a distance from the model output to the data points of the shearing experiment, and we seek the material parameter set i, j) in the experiment. Each G-point Gauss integration rule defined over i, j) with top face Our fit function is similar to that used in Schmid et al.  and is gEvaluating the inner loop of  solving  once forG Gauss integration points, rather than for all 250 recorded points for each shear mode, in order to greatly reduce the computational expense of evaluating Following Schmid et al. , we evals et al. .The use of Gauss integration is based on the observation that given by , is evalequation  from oneequation  always cWe represent each tissue sample of the shearing experiments by a three-dimensional cube On these geometries, we solve  and its Additionally, we model the case of a homogeneous deformation which corresponds to a linear displacement with a constant shear angle throughout the domain. Such a model can be represented by discretizing the cubes with a single layer of linear finite elements: the resulting displacement is completely determined by the prescribed boundary conditions. Figure The discrete variational formulation of the Euler–Lagrange equations is implemented using the FEniCS Project software x, y refer to Cartesian coordinates and t is a scaling parameter which we set to f which satisfied a pointwise form of Eq. We verified the computation of stresses in the finite element model by prescribing a homogeneous deformation and comparing the resulting numerically integrated top face shear stress values to analytically computed values. The analytic values were based on the calculations found in Holzapfel and Ogden ((iii)We confirmed the correctness of the adjoint gradients by considering the linearization of the functional Each of the finite element, adjoint, and optimization solvers have been carefully verified, separately and combined, as follows:Additionally, we verified the optimization solver by performing a synthetic data test. In this test we chose a target set of material parameters, Table We performed this test using our two models for deformation. The first model assumed a homogeneous shear angle through the material and the second model was a finite element model with a mentclass2pt{minimIn the following, we present the results of fitting the Holzapfel–Ogden strain energy law  using thAs the numerical solution of the nonlinear Euler–Lagrange equation  easily fThe material parameters of the Holzapfel–Ogden model have previously been estimated using a homogeneous deformation model . Doing this would increase the number of components of the gradient In such an application, a model parameter could be represented as a finite element function similarly to the displacement or hydrostatic pressure fields (equation  for eachequation  only neeWhen fitting material parameters to the Dokos experiment data, we were able to generate good initial guesses for the local optimization by progressively refining the mesh and using the optimal results from the previous coarser refinement level as an initial guess in the successive finer level. It would be more challenging to apply this technique using image-based ventricular geometries, due to the problem of accurately representing the geometry with few elements. As an alternative we propose the multi-start approach, which we have shown here to be accurate and viable using the Dokos experiment data.One issue that would arise in using the multi-start approach with image-based geometries would be the choice of the number of multi-start points; using less points is more computationally efficient, while using more is potentially more robust. Possible solutions are the use of optimal stopping criteria Boender and Kan  or more In this work, we have presented a new application of efficient gradient-based optimization methods in the context of estimating cardiac hyperelastic material parameters from experimental data. In particular, we have demonstrated how an adjoint solution can greatly speed up the evaluation of functional gradients. These methods have produced two new sets of material parameter values that yield simulated stress–strain curves that fit closely to orthotropic and transversely isotropic shear data. For future parameter estimation, studies using image-based geometries and a local search algorithm, multi-start or a similar method should be used in order to avoid suboptimal minima.


The characterization of heart dynamics with a view to distinguish abnormal from normal behavior is an interesting topic in clinical sciences. Here we present an analysis of the Electro-cardiogram (ECG) signals from several healthy and unhealthy subjects using the framework of dynamical systems approach to multifractal analysis. Our analysis differs from the conventional nonlinear analysis in that the information contained in the amplitude variations of the signal is being extracted and quantified. The results thus obtained reveal that the attractor underlying the dynamics of the heart has multifractal structure and the variations in the resultant multifractal spectra can clearly separate healthy subjects from unhealthy ones. We use supervised machine learning approach to build a model that predicts the group label of a new subject with very high accuracy on the basis of the multifractal parameters. By comparing the computed indices in the multifractal spectra with that of beat replicated data from the same ECG, we show how each ECG can be checked for variations within itself. The increased variability observed in the measures for the unhealthy cases can be a clinically meaningful index for detecting the abnormal dynamics of the heart. The various levels of this complexity and their variations, if properly discerned, will be useful in understanding the abnormalities that lead to many pathological cases10. A mathematical representation of the complexity using dynamical equations is most often not realizable due to the many interacting variables and uncertain parameters involved12. Hence the only possibility to study the dynamics in such situations to is to rely on information that can be deciphered from signals obtained from these systems15. Over the last few decades, several such physiological signals like EEG, ECG, fMRI etc have been subjected to different techniques available under the broad area of nonlinear time series analysis17.The complexity of many physiological rhythms originate from the underlying complex nonlinear dynamical processes20. This complexity is naturally reflected in the ECG signal that is an average response of the dynamics of the heart. But cardiac abnormalities or malfunctions can cause subtle changes or variations that reduce its complexity21. Therefore, it is important to quantify the changes in complexity due to abnormalities in terms of variations in its measures. However, such complexity related quantifiers have not yet reached the clinics for effective diagnostics and therapy. For this, we have to develop a unique way of characterizing its complexity that will help to distinguish healthy and pathological cases. Moreover it is important that the method should be effective with the short ECG data available in normal clinical practices. In the present work, we report a study in this direction where measures derived from the multi fractal spectrum of ECG signals can be used as a promising tool in the diagnosis of abnormalities of the heart.The dynamics of the human heart is well established as a complex system based on multiple approaches25 are on the peak to peak time intervals 27 and hence do not include the possible information content in the amplitude variations and shape of the waveforms. Also the multifractal analysis of ECG data reported are mostly using the method of Detrended Fluctuation Analysis (DFA)28 which quantifies the variability in the scaling of the fluctuations in data.In this context we note that most of the reported research in related analysisWe start with the hypothesis that even though ECG is an average response of the heart, the subtle variations in the actual shape of the ECG waveforms and their amplitude variations can be related to the variations in the underlying dynamics. Hence we follow a dynamical systems approach and try to reconstruct the attractor of the dynamics from the ECG waveform. We compute quantifiers that can capture the non-uniform distribution of points on this attractor that directly relate to its complexity and multifractal nature. Earlier studies based on multifractal analysis, using DFA or otherwise, mostly use the range of scales or width of the multifractal spectrum or its area as a characterizing measure indicating difference in complexity. In our study, we rely on at least four indices for a unique and complete characterization.35 using the fractal measures of the embedded attractor. These methods have found successful applications in diverse fields like astrophysics36, physiology37, atmospheric sciences39, geology40 and stock markets41. Among them, an important class of methods is related to the characterization of the complex dynamics using multifractal analysis42.Over the last few decades, a wide variety of methods have been developedWe present the embedded attractors using Singular Value Decomposition (SVD) technique and establish the nonlinear nature of their dynamics using surrogate analysis with Correlation dimension as a discriminating measure. We compute the indices that characterize the multifractal spectrum uniquely and show how their ranges or variations can distinctly distinguish healthy and unhealthy sets. Applying the analysis to a large number of ECG data sets, we group the data into healthy and unhealthy sets based on the values of the indices. Using supervised machine learning approach, we indicate how a new data can be assigned to either of these groups with high accuracy.R-R interval or HRV analysis because the actual amount of time (one to two minutes) for which the ECG needs to be recorded is smaller by orders of magnitude. This will increase its applicability for the normal daily clinical purposes.In addition, we repeat the analysis using a set of beat replicated data generated from each ECG and their multifractal indices are compared with those from the full ECG. The variability in these is found to increase in the presence of any abnormality compared to healthy cases. This variability itself can thus be another clinically meaningful index for diagnosis of abnormality in heart dynamics from ECG data. The analysis presented is also advantageous over 43 are pre-processed to make them suitable for the analysis (See Supplementary). Each dataset consists of ECG time series taken from six different chest electrodes or channels v1 to v6. As a preliminary analysis, we carry out the usual statistical and linear analysis and obtain the power spectra for all the data sets using the Fast Fourier Transform (FFT) algorithm. The frequency with the maximum power for each one of them and the distribution of these peak frequencies for all the subjects are summarized in Fig. The data of 97 unhealthy and 32 healthy subjects obtained from PhysioBank databases(tk). For a visual display of the resulting phase space structure or dynamical attractor, we use the technique of Singular Value Decomposition (SVD)45. (See Supplementary for details about embedding). In Fig. Hence we resort to methods of nonlinear analysis and reconstruct phase space structure of the system’s dynamics from its discretely sampled time series 46, the most effective among them being the set of fractal dimensions. In the following sections, we indicate how correlation dimension is used along with surrogate analysis to establish the nonlinear nature of the underlying dynamics and how the geometrical complexity of its structure is quantified using multifractal measures.Since the complex structure of the attractor inherently contains information about the complexity of its dynamics, it is important to characterize it quantitatively. Several quantifiers have been studied in this context over the yearsM with delay vectors constructed by splitting the discretely sampled ECG data with delay time τ. The dimension M is chosen as the value at which any fractal measure, like correlation dimension D2, saturates34. The distributions of the saturated D2 values for healthy and unhealthy groups is shown in Fig. D2 values are less than 4, for uniformity, we use M = 4 as the embedding dimension for all the signals.We recreate the phase space structure of the underlying dynamics in an embedding space of dimension 32. For this, we generate surrogate data using TISEAN package47. With correlation dimension as a measure, we plot in Fig. M. Since the values of D2 for surrogate data differ significantly from that of the original signal, we conclude that the signal comes from the underlying nonlinearity.Before undertaking any type of nonlinear analysis, it is important to first verify that the observed time series indeed results from an underlying nonlinear process. We check the nonlinear and deterministic nature of the underlying dynamics using a statistically rigorous method of surrogate analysisDq so that the non-uniformity in the distribution of points on the attractor becomes evident through different values of q. However the local scaling properties on the attractor are captured by a spectrum of singularities related to the probability measure on the course-grained attractor. Thus, if the attractor is covered by boxes of size r, the probability of points in the ith box scales as αi present is a measure of its complexity. The number of boxes with the same α scales as Dq,q) and (f(α),α) provide analogous characterizations and are related by Legendre transformations as35:Dq and then use equations (f(α) and α. An algorithmic approach to perform this has been given by Harikrishnan et al.35 and here we follow the same method with suitable modifications for ECG signals. The multifractal spectra thus computed is shown in Fig. α values or the width of the spectrum and the shape of the curves vary between the two groups. These variations can be quantified to extract information on the variations in the complexity of the corresponding attractors. This is done with the following mathematical form for f(α) curve:35, only four of the five parameters are independent. We choose the four parameters α1, α2, γ1 and γ2 to provide a unique characterization of f(α) spectrum of the multifractal.The reconstructed phase-space attractors for most of the complex systems like heart dynamics possess a multifractal structure which is characterized by a set of generalized dimensions mentclass2pt{minimα2 − α1, that measures the width of the f(α) spectrum provides the range of scaling indices. In Fig. We note that the non-uniform distribution of the points on the embedded attractor indicates the complexity of the heart dynamics revealed through the ECG signal. Therefore, the range of scales required to describe the probability of the distribution of the points over the whole attractor becomes a measure of its complexity. The difference α2 side mostly because this corresponds to the sparse regions of the attractor. This can be overcome to some extent by using longer datasets. However, in the present study, we use only the ECG data readily available in normal clinical practices. Therefore, while deriving conclusions from the results, we concentrate on the α1 and γ1 values that are associated with the dense regions of the attractor as well as α0 the value of α corresponding to the maximum of the f(α) curve.In principle, any of the above four parameters or any combinations of them should be a good indicator of the underlying complexity. For the data we have used, we find that the numerical error is more on the α1 and γ1 values for electrodes v1 to v6 obtained from the datasets from the two groups. The green circles in these plots represent the subjects identified as healthy in the PhysioNet database and the red squares represent the unhealthy ones. A few cases, for which the good fit could not be obtained have been discarded while plotting these parameter planes. For the purpose of visualization, we also show estimated kernel densities for the two groups as a background. As can be seen from these plots, multifractal analysis seems to have picked up almost every case correctly by separating the healthy and unhealthy cases into distinct clusters in α1-γ1 planes.In Fig. 48 to find out the regions in α1-γ1 planes corresponding to the two groups. The known group labels are used as a training data for the algorithm to identify healthy and unhealthy cases and then the algorithm is asked to divide the parameter plane into two regions. The regions so obtained for different channels are shown in Fig. α1-α0, where α0 is the α value corresponding to the maximum value of f(α) curve and perform a similar analysis. The resulting regions are shown in Fig. The separation of the two groups into two different clusters becomes useful only if it helps us to predict the group label  for a new unseen data. This is a standard problem in the theory of machine learning and we use a particular algorithm called a “support vector clustering” or SVC using the Radial basis function (RBF) kerneltp) and true negative rate (tn). True positive rate in this case is defined as the fraction of correctly identified healthy cases out of the actual healthy cases. Similarly, true negative rate is defined as the fraction of the correctly identified unhealthy cases out of the actual unhealthy cases. We then define the accuracy of the algorithm to be: Accuracy = tp × tn. This definition of the accuracy makes sure that both cases are predicted reasonably accurately since even if one of them is low the accuracy becomes low. In particular, if the algorithm labels all cases to be of the single group, the accuracy becomes zero. The value of the accuracy also depends on how the data is split and so, we average the value over ten random realizations of the splitting. These average accuracies, calculated for α1-γ1 planes, as a function of size of the training set for different channels are shown in Fig. The regions shown in Figs α1-γ1 values and then using its location in this parameter plane, we can predict the class the person belongs to with sufficient confidence.The results described in the previous section show that the multifractal analysis is extremely successful for separating healthy and unhealthy classes. Thus, given an ECG time series of a person that is not known to be in one or the other group a priori, we can calculate the corresponding 49. This is a finer level of analysis where the multifractal measures of a given ECG are compared with that of beat replicated data generated from a single beat of the same ECG. This would in a way compare each data within itself and the range of variations can be used effectively as a quantifier for normal and unhealthy states of the heart.We now present another approach where we invoke the concept of complexity variabilityα and γ. For the sake of brevity, we present the results for one of the indices α1. In Fig. α1 values for 20 randomly chosen subjects from each group. For comparison, we also show the original α1 value in each case (yellow circles). As can be seen, the actual α1 values in case of healthy subjects tend to coincide with the α1 values for beat replicated time series. On the other hand, the actual α1 values in case of unhealthy cases tend to be quite far from the mean of the replicated α1 values. To make this quantitative, we define δα1 to be the difference between the α1 value for the full time series and the mean of the replicated α1 values as For this, we extract 10 different randomly chosen beats from each signal (See Supplementary). Then we replicate each of these beats to get time series of approximately the same sizes as that of the original time series and perform multifractal analysis for each one of these to obtain the parameters mentclasspt{minimaWe report the results of a detailed multifractal analysis of discretized ECG data for two sets of healthy and unhealthy subjects. Our study establishes the highly complex fractal nature of a healthy heart, which gets reduced due to any abnormality in its functions. We could show that the measures derived from the multifractal spectrum can detect abnormalities in heart dynamics with a reasonable level of accuracy. The fact that this is achieved with short time ECG recordings of a significantly small amount of time enhances the scope for its applicability. Moreover the analysis is totally objective with supervised machine learning approach and the results are statistically significant.α2 − α1) in the multifractal spectrum. The presence of unhealthy conditions introduces differences due to defective functions, absence of influence of the autonomic nervous system or regulatory mechanisms that restrict adjustability or variability of heart dynamics. This results in less complexity, requiring only less number of scales to describe the distribution of points in its dynamical attractor in the reconstructed phase space. The reduction in complexity is thus a clear indicator of unhealthy condition and this is being quantified through the analysis reported here. Earlier work using HRV or DFA also capture the complexity in data using (α2 − α1) but our analysis use two additional quantifiers, γ1 and γ2 representing the shape of the multifractal curve. Since f(α) represents the scaling of the number of regions with the same α on the embedded attractor, these additional indices also have direct implications on the underlying dynamics and its complexity. The variations in this complexity, as carefully characterized in this work, can give clear indications of cardiac malfunctions. The novelty of our approach is that the subtle variations in the shapes of the beats are captured into readable quantities or values which will be much more reliable than conclusions derived from visual inspections. This information can be used to assess the status of health and risk level of the heart easily.The complexity of the ECG signal is understood as due to the adaptability of cardiac activity in response to internal or external stimuli, sympathetic modulations and self-regulations. This greater complexity for normal subjects corresponding to a healthy state for the heart is reflected in the larger values of (49.Our analysis, using beat replicated data, helps in comparing quantifiers of one ECG with variations within itself. We find the variance of the measures between actual ECG and the mean value of the same measure obtained from beat replicated data, is more for unhealthy cases. This increased variability can be another clinically meaningful index for detecting abnormalities. In this respect our results are similar to a recently reported study where the complexity variability is considered in the Lyapunov spectra, which is another measure of nonlinearityThe study is thus an advancement both in the basic understanding of the fractal nature underlying a complex system like heart and its unhealthy states as well as in designing clinically useful practical tools in effective diagnostics and therapy. Understanding and modeling of the intrinsic control mechanisms, nature of nonlinear feedback mechanisms etc will require further detailed studies. The approach of the theory of complex dynamical systems, following the present one, can certainly lead to a more complete description of the mechanisms underlying cardiac activity.www.physionet.org/physiobank/database/ptbdb.All the datasets used in this study are available at Supplementary Information


To explore patient‐reported complaints regarding communication and healthcare encounters and how these were responded to by healthcare professionals.A retrospective and descriptive design was used in a County Council in northern part of Sweden. Both quantitative and qualitative methods were used.The content of 587 patient‐reported complaints was included in the study. Descriptive statistical analysis and a deductive content analysis were used to investigate the content in the patient‐reported complaints.The results show that patients’ dissatisfaction with encounters and communication concerned all departments in the healthcare organization. Patients were most dissatisfied when they were not met in a professional manner. There were differences between genders, where women reported more complaints regarding their dissatisfaction with encounters and communication compared with men. Many of the answers on the patient‐reported complaints lack a personal apology and some of the patients failed to receive an answer to their complaints. Encounter is a concept related to the words meeting, appointment or relationship but diverges as the encounter regularly means more a personal contact between a few people that takes place planned or unplanned, that come across and get in touch with each other Westin, . Some heTo ensure a good healthcare encounter, there must be sufficient time for communication, enough resources and opportunities for patients and professionals to create a meaningful relationship, regardless of the duration of the encounter . The PAC is responsible for handling patients’ complaints and they act on behalf of the patients’ or their relatives and strive to solve the problems that have occurred together with the involved healthcare professionals  patients themselves during January 2010–December 2012. The chosen time period was based on that PAC stored 3 years of complaints at a time. For some complaints, parts of the patients’ records were attached. All identifying patient details have been omitted in the presentation of this study's results to protect the patients’ anonymity, in accordance with the Helsinki declaration. The patient‐reported complaints filed at the PAC were covered by confidentiality. The results of the study are therefore presented only at a group level and individuals cannot be identified.N = 1792) regardless of the area where the Patients’ Administrators had sorted them in, were read through. This reading resulted in that all (N = 625) reported complaints containing descriptions of dissatisfaction with encounters and communication were selected for the analysis. In 38 of the 625 selected reports, only a short note indicating the date of a phone call to the patient was found and thus these reports were excluded from the analysis. The remaining 587 complaints were included in the analysis.During the chosen time period, the PAC received 1792 patient‐reported complaints concerning issues related to the following areas: i) encounters and communication; ii) medical maltreatment and iii) organizational issues regarding rules/regulations. The administrators at the PAC sorted and classified the complaints in the file archive based on the above‐described areas. This sorting was part of the PACs normally classification of complaints and it was performed without a standardized system. To ensure that all complaints that contained dissatisfaction with encounters and communication were included in the analysis all submitted complaints  was used for the statistical analyses. Data in the patient‐reported complaints regarding gender, the type of organization, clinical department, reason for the complaint and the type of healthcare professionals who were the focus of the complaint, were extracted to a data template and thereafter included in the SPSS form. Descriptive statistics were used to describe the content and frequencies and a Pearson's Chi Square test was used to determine the relationships and significant differences between the patient's gender and the type of units and professions cited in the patient‐reported complaints.2.5The written text in the complaints was analysed in parallel with the statistical analysis, using deductive content analysis .3N = 587) each contained a written letter from a patient describing the situation that had occurred and indicating dissatisfaction with the healthcare encounter and/or communication. Each complaint also contained a summary written by the local PAC administrator as well as a checklist for actions to solve the situation. Furthermore, the reported complaints contained an answer from the healthcare professionals involved in the situation and a conclusion regarding how the report was handled and the outcome. Below presents a descriptive summary of the patient‐reported complaints characteristics and categories from the deductive content analysis in the two fields; encounters and communication. The qualitative findings are supported by quotations from the text in the complaints, written with italic style in the text.The patient‐reported complaints (3.1N = 195), followed by surgery (N = 171). The complaints described different groups of healthcare professionals who were the focus of the complaint and the most common professions the complaints focus on were physicians (N = 357), followed by healthcare managers (N = 100) and nurses (N = 79). Men's complaints were more often directed against physicians than were women's complaints (72% vs. 53%), while women were more likely than men to direct their complaints against healthcare managers (22% vs. 11%). Healthcare manager could be both a ward manager or a person in a higher management level not based in a particular ward or clinic area. Significant differences were found between the professional groups the complaints addressed and the patient's gender (p = .001)  of these were made by women. The 587 complaints concern all units in the healthcare organization and the clinical department that contained most complaints was consultation outpatient visits N = 95, folloN = 221) were most involved in complaints in hospital care followed by healthcare managers (N = 65) and nurses (N = 51). Significant differences were found between the different professional groups the complaints intended to address and the type of organization (p = .001) and clinical department (p = .001) the complaint reflect. An overview of the units and the professions that the complaints addressed is provided in Table The result further shows that physicians N = 21 were mp = .419 vs. .552). In the field communication (N = 333), most of the complaints were about the patients’ experiences of not being involved/lack of participation in the care (55%), followed by a lack of information and lack of possibilities for communication (45%). No significant differences were noted between women and men (p = .906 vs. .891).A description of the content, frequency and professions involved in the patient‐reported complaint is described in Table 3.23.2.1In the field encounters, two categories were identified; Lack of empathy and Non‐chalant treatment.The complaints often began with a summary of the reasons for the patients’ unhappiness with the meeting. Patients were most dissatisfied when they were not met in a professional manner. The complaints describe that inadequacies in meetings generated feelings of not being met with respect, not being understood and not being welcomed to the healthcare setting. Not being met with respect was described when healthcare professionals did not value the patient as a person. Another reason for reporting a complaint was that healthcare professionals could only attend to patients’ most necessary needs when patients found the healthcare environment stressful. The complaints described situations when the patients felt ignored by the healthcare professionals due to insufficient time throughout the caring encounter. One reported complaint described: “there was no time for healthcare professionals to listen to my story so I had to prioritize which needs I should present”. This meant that the patients were dissatisfied with the meeting as focus was only at one of their health instead of all their problems.The complaints gave also examples of how patients liked to be met by healthcare professionals such as through commitment and a genuine interest by being seen as an important person. In the complaints, the patients further expressed a desire for a resolution to the situation and to prevent it from happening again, either to themselves or to other patients. The patients’ need for justice was another important reason for many of the complaints. One patient perceived in the complaint that: “I had to wait longer than other patients for treatment or care”, another patient describe: “I got less examinations then others”.The complaints described situations when healthcare professionals had shown negative attitudes in their behaviour towards the patients. In some complaints, the patients were referred to as a diagnosis rather than as a person when healthcare professionals were talking among themselves, saying things such as “the broken leg”, “the painful lady” or “the mentally ill”. The patients describe in their complaints that these kinds of negative attitudes and bad behaviour affected their dignity. The patients expressed in the complaints that they would have become healthier sooner if they had been warmly greeted and seen as individuals in their encounters with healthcare professionals. The written text in the complaints indicated that it was unacceptable that the healthcare professionals engaged in this negative behaviour in their meetings with patients.Dissatisfaction with attitudes and/or negative behaviour in meetings was also described in situations where the patients perceived that they were not met in a professional manner. The complaints contained examples of caring situations where the patients received insufficient respect, such as a “lack of empathy” and “non‐chalant treatment from professionals who ignored their symptoms and illnesses”. Such complaints described how the patients felt lost and ignored in their meetings with healthcare professionals, which in turn led to anxiety. Examples of insufficient respect were also described in meetings when healthcare professionals talked about the costs of treatment and drugs rather than about the actual treatment of the patients’ symptoms and illnesses. One patient expressed in the written complaints that: “these kinds of attitudes and/or behaviours, where they were not met in a professional way, negatively affected their health”. As a result, the patients expressed in the complaints that their confidence in health care began to diminish.3.2.2In the field communication, two categories were identified; Not being involved in care and Answers to the patient's complaints.The complaints described that patients experience insufficient information: “I was not given an opportunity to receive adequate information or participate in decision‐making about my care”. Insufficient information was highlighted because of the language deficits of the provided care. The patients‐reported complaints contained examples of situations when the patients suffered due to the methods the healthcare professionals used to inform them. It was for example of situations where: “healthcare professionals use a medical terminology that I didn't understand” or “information was given during stressful circumstances with no time for questions”. The patients ask therefore in their complaints for more information that could explain their circumstances in a way they could understand.The complaints further indicated that the patients felt that they were not invited to participate in the communication about their treatment and care. One patient expressed in the complaints that: “it is difficult to take part in decision‐making about care alternatives when you not be invited”. The patients asked for more communication and their complaints gave examples of situations when the professionals provided information without taking care of the patient's individual needs. The content in the complaints describe that the patients asked for questions about their needs and personal conditions and an invitation for discussions of alternative treatments. One patient's complaints described: “I know best how I feel so they  should ask me”. The patient's complaints described further that healthcare professional lack interest about their situation and the patient‐reported complaints expressed the patients’ disappointments.p = .429).The administrators at the PAC clearly documented the procedure for how the complaints should be handled as well as the resulting outcomes, describing the way they contacted the patients by phone or mail to gather complementary information regarding the situations that had occurred. A checklist described how the administrators should further handle the complaints, for example, asking for the patient's record to get more information about the situation and contacting the involved healthcare professionals. The administrators at the PAC always requested an answer and response from the healthcare professionals concerned in the complaints, but responses were received in only 490 cases (83%) of the total 587 complaints. The distribution of answers in response to women's and men's complaints was relatively equal  of the answers were expressed in an understanding tone, such as “Dear Mrs. Svensson, thanks for your complaint. We understand your complaint and the described situation.” Furthermore, 58 answers (12%) were expressed in an apologetic manner, for example, “Dear Mrs. Jonsson, Thanks for your complaint. We apologize for the situation that occurred. We will investigate the situation that occurred and will return to you as soon as possible.” A frequent tone in the responses suggested that the healthcare professionals were not responsible for the situation, which, they explained, had occurred because the healthcare professionals had followed established healthcare routines; for instance: “Mrs. Larsson, Thanks for your complaint. The healthcare professional your complaint applies to has followed routines for the examination and treatment and they can therefore not be held responsible for the situation you are experiencing.” In 461 (94%) of the total 490 answers, the healthcare professionals showed no intention to act or correct the situation. The patient‐reported complaints also described that this lack of responsibility for the situation contributed to the patients’ feeling that they had been treated with disrespect.In 29 (5%) of the total 587 patient‐reported complaints, a successful handling of the situation was described. This occurred when the healthcare professionals involved in the situations contacted the patients and personally apologized to them. The healthcare manager was sometimes included in these personal meetings, to provide an opportunity for all invited parties to discuss the situation. The results of the meeting were documented in the patient‐reported complaints and describe that the patients were satisfied with the meetings when the healthcare professionals listened to them and their experiences. Furthermore, they were pleased that they had identified a solution together regarding how to have more caring encounters in the future. In other examples, the involved healthcare professionals who participated in follow‐up meetings had expressed their regret about the situations that had occurred and explained why the patient was treated inadequately. Another example of a case that was successfully handled was when the involved healthcare professional and the healthcare management met with the patient personally and apologized for the professional's lack of empathy.In 19 (3%) of the 587 patient‐reported complaints, the administrators at PAC had documented how the patients’ dissatisfaction with their healthcare encounters and communication should be used in the future to improve health care and, furthermore, become a part of the healthcare professionals’ continuing education to prevent similar situations from occurring with other patients.4This study explored patient‐reported complaints regarding communication and healthcare encounters and how these were responded to by healthcare professionals. The results indicate that the complaints concerned all departments in the healthcare organizations and were most common in hospital care. This corresponds with the results of Kline et al. , which iThe results that described satisfaction with encounters with physicians were based on receiving information through a dialogue that included both empathy and listening. When patients receive information about their health conditions, it is of great importance that the information includes empathy and an invitation to participate in care decision‐making (Skär & Söderberg, The results further show that the most common dissatisfaction with healthcare meetings involved being dissatisfied with professionals’ attitudes or approaches. The complaints described how the patients were ignored and treated with indifference. Uncaring behaviour affects patients’ dignity and thereby their health and well‐being (Eriksson, The results show further differences between genders, where women reported more complaints regarding their dissatisfaction with encounters and communication compared with men, which Schnitzer et al.  also notThe results show that many of the answers on the patient‐reported complaints lack a personal apology and that some of the patients not even received an answer to their complaints. This indicates that professionals often do not take responsibility for how they handle patients and behave in the context of health care. Research by Gallagher, Waterman, Ebers, Fraser, and Levinson  has showAn apology can have powerful effects for both the person offering it and the recipient and it contributes to improving the physician–patient relationship (Robbennolt, 4.1The limitations of this study are the subjective experiences reported by patients in the complaints and that data were collected from one single PAC in northern part of Sweden. However, a strength of this study was the number of complaints during a time period of 3 years included in the analysis. This retrospective and descriptive study included both a qualitative and quantitative design which resulted in a deep description of the findings. Furthermore, the analysis was conducted jointly and reviewed independently by both authors, which added rigour to the study (Creswell & Plano Clark, 5To conclude, this retrospective and descriptive study including both qualitative and quantitative approaches shows that patient‐reported complaints regarding provided care stem from asymmetric communication, where the patients are not met in accordance with their individual needs. From a person‐centred perspective, this can have a significant impact on patients’ satisfaction with healthcare encounters and experiences of quality of care. The results also revealed that not all patients received closure in the form of an answer or personal apology in response to their complaint. Transparency of the shortcomings in healthcare encounters could help patients to overcome negative experiences. These results stressed therefore that patient‐reported complaints should be used to identify why shortcomings that have been highlighted for several years persist, as well as, why healthcare professionals do not take responsibility for the complained‐about matter. However, more knowledge is needed about how healthcare organizations could address patient complaints to improve the quality of care.


We propose classical interferometry with low-intensity thermal radiation for the estimation of nonclassical independent Gaussian processes in material samples. We generally determine the mean square error of the phase-independent parameters of an unknown Gaussian process, considering a noisy source of radiation the phase of which is not locked to the pump of the process. We verify the sufficiency of passive optical elements in the interferometer, active optical elements do not improve the quality of the estimation. We also prove the robustness of the method against the noise and loss in both interferometric channels and the sample. The proposed method is suitable even for the case when a source of radiation sufficient for homodyne detection is not available. The development of quantum technology in various platforms of physics highly depends on the experimental ability to generate and detect nonclassical quantum processes for electromagnetic radiation. Nonclassical processes can change classical radiation to radiation incompatible with any mixture of classical waves2356789101113To estimate phase-sensitive squeezing, we need to probe the nonclassical processes by a suitable wavelength of radiation capable of classical interference. Nowadays, we can use narrow-band sources of radiation at various wavelengths to probe many new nonlinear processes in samples of different materials or soft matter. A standard narrow band source for spectroscopy typically exhibits thermal statistics with limited energy per time duration of the measurement. Many of such sources can be sufficient to build classical interferometry, for example, to estimate the relative classical phase introduced by the sampleOn the other hand, the estimation of squeezing in material samples has been limited to balanced homodyne detection (BHD)171921222324252627The problem of the optimal estimation of squeezing originates in the nineties30313234We are not trying to push these limits further, we are rather interested in what the minimal requirements of the estimation of squeezing are. Take as an example standard interferometry, where the optimal performance is achieved with a squeezed and displaced probeindependent squeezing processes. In general, it allows estimating all the main phase-independent characteristics of a Gaussian process . Simultaneously, we analyze the interferometry of a squeezing process with laser light far from the shot-noise-limit. We prove that active interferometry is not required: it does not improve the quality of estimation. We verify the robustness of interferometry if loss and noise are present in the channel. Our results largely relaxed the requirements of existing estimation schemes, thus, they can open many possibilities to estimate nonlinear phase-dependent processes in new materials and soft matter at wavelengths where high-quality laser light and homodyne detection is not available. A possible application could be quantum key distribution with macroscopic statesIn this paper we verify that classical interferometry with thermal light and limited intensity is sufficient to estimate φ), we couple the two modes on another optical element and use two intensity detectors to obtain N measurement data pairs.The investigated scheme see  is quiteWe decompose the Gaussian unitary process into three parts:S(ξ) = exp(1/2ξ2a†2 − 1/2ξ*2a2) is the squeezing operator (with ξ = weiα), R(Φ) = exp(iΦa†a) is the phase-shift operator, and D(γ) = exp(γa†−γ*a) is the displacement operator (with γ = deiβ). The squeezing can be described by the magnitude (q = ew) and by the direction (α) of the squeezing. The displacement can be described by the magnitude (d) and by the direction (β) of the displacement. The combination of these three transformations describes each possible Gaussian transformation of a given Gaussian state which preserves the mixedness of the original state.where θ is random: it is uniformly distributed on , resulting a rotationally symmetric state in the phase space . Let V denote the second moment of quadrature x or p of the random-phase source. Although the obtained formulas in the manuscript are valid for an arbitrary source, due to their practicability, during the numerical investigation we focus mainly on classical Gaussian states: we use a displaced thermal source with standard deviation R and displacement D, that is, a Gaussian state with mean value D = 0) and coherent states with random phase (R = 1). That is, for the general Gaussian source we have 〈x2〉 = 〈p2〉 = V = R2 + D2/2, which can be also derived as a simple function of the mean photon number of the source: The source is independent of the process, meaning their pumping can be physically independent, but more importantly, they can have different frequencies, and thus the phase of the source is not locked to the phase of the process. This means that the phase of the source relative to the phase of the Gaussian process is constantly changing and is unknown in any given moment. So we assume that its phase q and d can be estimated.Standard interferometry deals with the estimation of Φ. We are interested also in whether and how accurately parameters Note that the proposed interferometric scheme is not the simplest one. One could also apply the Gaussian process directly to the source and then measure the output of the process with an intensity measurement. Then the detector would have an average photon number ofV1 and V2, the value of q and d could be estimated. However, since we could obtain only two independent equations, if a noise or loss was present their estimation would not be feasible. They would be indistinguishable from the unknown Gaussian process, which would result in a biased estimation of the parameters .and by using two different sources with μ, while the second is a 50:50 beam splitter.In the first case we assume that both optical elements in our setup are beam splitters (BS). The first BS has a reflectivity of With the given parameters, it is easy to calculate the expected difference of the photon numbers:and also the expectancy of the sum of the photon numbers:VS = μV + 1 − μ and VR = μ + V − μV are the variances of the signal and the reference after the first beam splitter.where V) and the transmittance of the beam splitter (μ) are known, while regarding the parameters of the unknown Gaussian process  we have no prior knowledge. Then we can estimate the unknown process by using an unaltered (φ = 0) and an orthogonal (φ = π/2) reference. We have thenWe assume that the variance of the source (From which we obtain the estimatorsandwhereUsing (4) we obtain the estimatorLet us note that the described method is a straightforward extension of standard interferometry. Knowing the phase of the source is not necessary, it can be anything, the key issue is that for each signal-reference pair there should be a strong correlation in phase . The phase-shift is determined the same way as in the literature (checking the angle of the interference pattern), the magnitude of the displacement and the squeezing can be expressed by investigating also the magnitude of the photocurrents beside their angles.α and β are inaccessible in this scheme since the whole process is phase-insensitive. But that also means that there is no error caused by the phase-instability of the process or the source, which can be problematic in many cases .The angles R, D, N, μ or r), however, the figure of merit to use is not self-evident. The quantum Fisher information is a popular choice, however, since we have a minimalistic approach, we are far from its limit. The Fisher information associated with the given measurements would be better suited, however, we can calculate only the value for the normal approximation. The situation is similar if we want to calculate the variance of the estimators, moreover, the estimators are not unbiased, only asymptotically unbiased. So they are not optimal, since for a limited number of measurements either quantity will be imprecise. On other hand, for large values of N the approximation errors will be small, therefore they could be used for numerical optimization, and we can observe variances that are very close to the lower bound obtained from the Cramér-Rao inequality :M = 104) to draw the required statistic from the M numerically estimated values . While for the estimation of the phase-shift or the squeezing the dependence is not as strong, a large μ  ensures a better performance.In MSE~1/N (303132MSE~1/N2) one should use even in the phase sensitive case more sophisticated resources . Since our aim is to have a scheme as simple as possible, we are not investigating these types of extensions in the current manuscript.MSE~1/N . Let us We can see  that forr and ϕ. We assume that as the source, the angle of this two-mode squeezing is not locked to the unknown Gaussian process either. Therefore we can assume that ϕ is also random.Previously we considered only passive optical elements, here we investigate the case when one or both of the beam splitters are changed to an optical parametric amplifier (OPA). Let us assume that the OPA is fed by a pump, which results in a two-mode squeezing with parameters ϕ cancels it out completely. However, if we have two OPAs which have the same random ϕ parameter , then we can perform a similar analysis as in the BS case.An immediate consequence is that if we exchange only one BS for an OPA, then we do not see any interference pattern, since the randomness of Once again, we can calculate the expected difference in the photon numbers:and also the sum of the photon numbers:r1 and r2 are the parameters of the first and second OPA, respectively, and VS = cosh(r1)V + sinh(r1) and VR = sinh(r1)V + cosh(r1) are the variances of the signal and the reference after the first beam splitter. For simplicity, in the further discussion we will always assume that the two OPAs have the same parameter: r1 = r2 = :r.where q and d in a similar way as for passive interferometry.We can see that these equations are different from those obtained in the beam splitter case , but structurally consist of the same parts as r) should be neither too small nor too large; we have 1/N scaling; the coherent source is only slightly better than a thermal source; a stronger source improves the estimation of the phase-shift and the squeezing, but decreases the efficiency of the estimation of the modulation. The main difference is that for the investigated parameters passive interferometry in most cases outperforms the active counterpart . At first it may sound counterintuitive that using an outside source of energy (as a pump of OPA) to produce entanglement results in a worse estimation. But for example in the case of the phase-shift, it is known that the Fisher information using a BS is larger than by using an OPAr is limited, achieving the optimal working point , which will give use twice as many equations on the intensities. The total intensity Vi, so by using the difference of the intensities for the two sources The case of process-type errors see  is a litIt is no surprise that it is possible to estimate the phase-shift, we know from the theory of standard interferometry that it can be done even in the presence of errors. The estimation of the displacement can be a problem, since in a phase-insensitive setup it acts exactly as noise (both are only present in the increase of energy but not in the interference), so with limited measurement possibilities, it can happen that they are indistinguishable. However, the estimation of squeezing presents no difficulties, the difference is that the squeezing also changes the difference of photon numbers , not just the sum of photon numbers .Moreover, we can estimate the magnitude of the squeezing with both channel and process-type noise present only by using weak thermal sources . For thaV = 1000 had a 2–3 times lower variance than V = 10 . If we compare the values of MSE with the noiseless version, we can see that the estimation efficiency is worse if additional noise is present (as it is expected). Nevertheless, the characteristic of the lines is the same, both converge to zero with an error of the order of 1/N. That is, with a sufficient number of measurements we can estimate the magnitude of squeezing with arbitrary precision even with a weak thermal source and with different kinds of noises and losses present.Unlike the ideal case, when = 10 see , here thIn our work we investigated phase-insensitive measurements to obtain the estimators for an unknown general Gaussian unitary process. The common method for estimating only the phase-shifts is the well-known interferometry. We extended the standard method of interferometry to estimate squeezing and displacement without using more sophisticated tools .The efficiency of the estimation depends on numerous details. In general we can conclude that the estimation of the phase-shift is the most accurate, while the estimation of the displacement is the worst. This is not surprising since the phase-shift comes immediately from the interference pattern, while the effect of the displacement is only visible as an increase in the mean photon number. Our method can be applied to both coherent and thermal sources, and while the former yields more precise estimates the difference is not substantial. We examined the effect of including either beam splitters or optical parametric amplifiers in the interferometric scheme and constructed the estimators for both cases. We observed very similar behavior, with a slight advantage of passive elements.We also investigated the effect of loss and noise. In general, we can say that the estimators are not robust against these errors: only the phase-shift can be estimated well, the estimation of squeezing and modulation will be biased even for a relatively low level of losses or noise. However, if we know the structure of errors, we can estimate those too, and by incorporating these estimators we can already estimate the unknown process quite well. The only problem arises when we cannot estimate the error and the process independently, since then the noise and the displacement have the same effect due to the phase-insensitivity, hence they are indistinguishable. In spite of the displacement, the effect of squeezing is clearly visible in the interference pattern, so we can conclude that using our method it is possible to estimate the squeezing as well as the phase-shift in all possible scenarios very well even with both types of discussed errors present and weak thermal sources.How to cite this article: Ruppert, L. and Filip, R. Estimation of nonclassical independent Gaussian processes by classical interferometry. Sci. Rep.7, 39641; doi: 10.1038/srep39641 (2017).Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.