Completing the Cabrera Circle: deriving adaptable leads from ECG limb leads by combining constraints with a correction factor.

Objective.We present a concept for processing 6-lead electrocardiography (ECG) signals which can be applied to various use cases in quantitative electrocardiography.Approach.Our work builds upon the mathematics of the well-known Cabrera sequence which is a re-sorting of the six limb leads (I,II,III,aVR,aVL,aVF) into a clockwise and physiologically-interpretable order. By deriving correction factors for harmonizing lead strengths and choosing an appropriate basis for the leads, we extend this concept towards what we call the 'Cabrera Circle' based on a mathematically sound foundation.Main results.To demonstrate the practical effectiveness and relevance of this concept, we analyze its suitability for deriving interpolated leads between the six limb leads and a 'radial' lead which both can be useful for specific use cases. We focus on the use cases of i) determination of the electrical heart axis by proposing a novel interactive tool for reconstructing the heart's vector loop and ii) improving accuracy in time of automatic R-wave detection and T-wave delineation in 6-lead ECG. For the first use case, we derive an equation which allows projections of the 2-dimensional vector loops to arbitrary angles of the Cabrera Circle. For the second use case, we apply several state-of-the-art algorithms to a freely-available 12-lead dataset (Lobachevsky University Database). Out-of-the-box results show that the derived radial lead outperforms the other limb leads (I,II,III,aVR,aVL,aVF) by improving F1 scores of R-peak and T-peak detection by 0.61 and 2.12, respectively. Results of on- and offset computations are also improved but on a smaller scale.Significance.In summary, the Cabrera Circle offers a methodology that might be useful for quantitative electrocardiography of the 6-lead subsystem-especially in the digital age.

Benchmarking the Impact of Noise on Deep Learning-Based Classification of Atrial Fibrillation in 12-Lead ECG.

Electrocardiography analysis is widely used in various clinical applications and Deep Learning models for classification tasks are currently in the focus of research. Due to their data-driven character, they bear the potential to handle signal noise efficiently, but its influence on the accuracy of these methods is still unclear. Therefore, we benchmark the influence of four types of noise on the accuracy of a Deep Learning-based method for atrial fibrillation detection in 12-lead electrocardiograms. We use a subset of a publicly available dataset (PTB-XL) and use the metadata provided by human experts regarding noise for assigning a signal quality to each electrocardiogram. Furthermore, we compute a quantitative signal-to-noise ratio for each electrocardiogram. We analyze the accuracy of the Deep Learning model with respect to both metrics and observe that the method can robustly identify atrial fibrillation, even in cases signals are labelled by human experts as being noisy on multiple leads. False positive and false negative rates are slightly worse for data being labelled as noisy. Interestingly, data annotated as showing baseline drift noise results in an accuracy very similar to data without. We conclude that the issue of processing noisy electrocardiography data can be addressed successfully by Deep Learning methods that might not need preprocessing as many conventional methods do.

Analysis of a Deep Learning Model for 12-Lead ECG Classification Reveals Learned Features Similar to Diagnostic Criteria.

Despite their remarkable performance, deep neural networks remain unadopted in clinical practice, which is considered to be partially due to their lack of explainability. In this work, we apply explainable attribution methods to a pre-trained deep neural network for abnormality classification in 12-lead electrocardiography to open this "black box" and understand the relationship between model prediction and learned features. We classify data from two public databases (CPSC 2018, PTB-XL) and the attribution methods assign a "relevance score" to each sample of the classified signals. This allows analyzing what the network learned during training, for which we propose quantitative methods: average relevance scores over a) classes, b) leads, and c) average beats. The analyses of relevance scores for atrial fibrillation and left bundle branch block compared to healthy controls show that their mean values a) increase with higher classification probability and correspond to false classifications when around zero, and b) correspond to clinical recommendations regarding which lead to consider. Furthermore, c) visible P-waves and concordant T-waves result in clearly negative relevance scores in atrial fibrillation and left bundle branch block classification, respectively. Results are similar across both databases despite differences in study population and hardware. In summary, our analysis suggests that the DNN learned features similar to cardiology textbook knowledge.

Structured, Harmonized, and Interoperable Integration of Clinical Routine Data to Compute Heart Failure Risk Scores.

Risk prediction in patients with heart failure (HF) is essential to improve the tailoring of preventive, diagnostic, and therapeutic strategies for the individual patient, and effectively use health care resources. Risk scores derived from controlled clinical studies can be used to calculate the risk of mortality and HF hospitalizations. However, these scores are poorly implemented into routine care, predominantly because their calculation requires considerable efforts in practice and necessary data often are not available in an interoperable format. In this work, we demonstrate the feasibility of a multi-site solution to derive and calculate two exemplary HF scores from clinical routine data (MAGGIC score with six continuous and eight categorical variables; Barcelona Bio-HF score with five continuous and six categorical variables). Within HiGHmed, a German Medical Informatics Initiative consortium, we implemented an interoperable solution, collecting a harmonized HF-phenotypic core data set (CDS) within the openEHR framework. Our approach minimizes the need for manual data entry by automatically retrieving data from primary systems. We show, across five participating medical centers, that the implemented structures to execute dedicated data queries, followed by harmonized data processing and score calculation, work well in practice. In summary, we demonstrated the feasibility of clinical routine data usage across multiple partner sites to compute HF risk scores. This solution can be extended to a large spectrum of applications in clinical care.

Biomechanical analysis of temporomandibular joint dynamics based on real-time magnetic resonance imaging.

AIM: The traditional hinge axis theory of temporomandibular joint (TMJ) dynamics is increasingly being replaced by the theory of instantaneous centers of rotation (ICR). Typically, ICR determinations are based on theoretical calculations or three-dimensional approximations of finite element models. MATERIALS AND METHODS: With the advent of real-time magnetic resonance imaging (MRI), natural physiologic movements of the TMJ may be visualized with 15 frames per second. The present study employs real-time MRI to analyze the TMJ biomechanics of healthy volunteers during mandibular movements, with a special emphasis on horizontal condylar inclination (HCI) and ICR pathways. The Wilcoxon rank sum test was used to comparatively analyze ICR pathways of mandibular opening and closure. RESULTS: Mean HCI was 34.8 degrees (± 11.3 degrees) and mean mandibular rotation was 26.6 degrees (± 7.2 degrees). Within a mandibular motion of 10 to 30 degrees, the resulting x- and y-translation during opening and closure of the mandible differed significantly (10 to 20 degrees, x: P = 0.02 and y: P < 0.01; 20 to 30 degrees, x: P < 0.001 and y: P = 0.01). Rotation of both 0 to 10 degrees and > 30 degrees showed no significant differences in x- and y-translation. Near occlusion movements differed only for y-translation (P < 0.01). CONCLUSION: Real-time MRI facilitates the direct recording of TMJ structures during physiologic mandibular movements. The present findings support the theory of ICR. Statistics confirmed that opening and closure of the mandible follow different ICR pathways, which might be due to muscular activity discrepancies during different movement directions. ICR pathways were similar within maximum interincisal distance (MID) and near occlusion (NO), which might be explained by limited extensibility of tissue fibers (MID) and tooth contact (NO), respectively.

A Privacy Preserving Approach to Feasibility Analyses on Distributed Data Sources in Biomedical Research.

Funding agencies and field experts promote reuse of scientific data and biomaterial beyond the scope of the original project. The availability of research data, however, is limited by the interest of original authors as well as the privacy rights of the study participants, especially in the biomedical sciences. On the other hand, for an available data set to be a useful contribution to the scientific community, it has to be findable and accessible with reasonable effort. Therefore, using the R Shiny library, we designed and implemented a software for data discovery and feasibility analyses with compliance to regulatory and contractual regulations. Due to its genericity, it was successfully tested with heterogeneous data sets and ultimately applied to the data and biomaterial of the German Center for Cardiovascular Research (DZHK). The resulting tool - named the Feasibility Explorer - is publicly available and can be used by researchers to get an overview of data and biomaterial available in the DZHK and to select collectives in the process of submitting a usage application. To implement the rights of participants and original authors, data is integrated by querying the informed consent and not persistently stored. All calculations on the data are performed server-sided and only aggregated information is send to a client, whereas the extent of information was strictly limited to a necessary minimum that allows an applicant to assess whether an application is worthwhile.

Designing meaningful outcome parameters using mobile technology: a new mobile application for telemonitoring of patients with heart failure.

AIMS: Health data captured by commercially available smart devices may represent meaningful patient-reported outcome measures (PROMs) in heart failure (HF) patients. The purpose of this study was to test this hypothesis by evaluating the feasibility of a new telemonitoring concept for patients following initial HF hospitalization. METHODS AND RESULTS: We designed a cardio patient monitoring platform (CPMP) that comprised mobile iOS-based applications for patients' smartphone/smartwatch and the equivalent application on a physicians' tablet. It allowed for safe and continuous data transmission of self-measured physiological parameters, activity data, and patient-reported symptoms. In a prospective feasibility trial with 692 patient days from 10 patients hospitalized for newly diagnosed HF with reduced ejection fraction (mean left ventricular ejection fraction (LVEF) 26.5 ± 9.8%), we examined the CPMP during the first 2 months following discharge (69 ± 15 observation days per patient). The mean daily step count recorded by the mobile devices emerged as a promising new PROM. Its 14 day average increased over the study period (3612 ± 3311 steps/day at study inclusion and 7069 ± 5006 steps/day at end of study; P < 0.0001). It is unique for continuously reflecting real-life activity and correlated significantly with traditional surrogate parameters of cardiac performance including LVEF (r = 0.44; 95% CI 0.07-0.71; P = 0.0232), 6 min walk test (r = 0.67; 95% CI 0.38-0.84; P = 0.0002), and scores in health-related quality of life questionnaires. CONCLUSIONS: We provide the first patient monitoring platform for HF patients that relies on commercially available iOS/watchOS-based devices. Our study suggests it is ready for implementation as a tool for recording meaningful PROMs in future HF trials and telemonitoring.

The upper ankle joint: Curvature morphology of the articulating surfaces and physiological function.

PURPOSE: The curvature morphology of the articulating surfaces determines the physiological movement pattern. We quantitatively examined the curvature morphology of the tibiotalar articulating surfaces and specified their geometric contact patterns. METHODS: Geometrically equivalent cartographic nets were marked on the talar and tibial articulating surfaces of true-to-scale moldings of 20 human ankle joints (intervals of 5 mm) to relate corresponding articulating units of the surfaces. The corresponding contours of the net lines were compared, and the incongruity of articulating surfaces could thus be quantified locally. RESULTS: All tibial sagittal net lines represented circular arcs. Along the sagittal talar net lines, the curvature radii increased medially from anterior to posterior but decreased laterally. Each net line could be approximated by three circular arcs. Examining these three parts of the talar net lines, the anterior sagittal curvature radii increased from medial to lateral, whereas the posterior radii decreased. The tibial and talar transversal net lines were congruent. The articulation surfaces showed a transversal contact line in every dorsal/plantar joint position. The degree of local congruity was solely ascertained by the incongruity of the corresponding sagittal net lines. The maximal degrees of congruity were found laterally for dorsal flexion, laterally/centrally for neutral joint position, and centrally/medially for plantar flexion. CONCLUSIONS: By the transversal line contact, the contact area is broadened over the articulating surfaces from lateral to medial. In dorsal flexion, compressive loads are mainly transferred by lateral/anterior zones and in plantar flexion by medial/posterior zones of the articulating surfaces. Reconstruction of the transversal contact line is essential.

The thumb carpometacarpal joint: curvature morphology of the articulating surfaces, mathematical description and mechanical functioning.

PURPOSE: The purpose is to present a mathematical model of the function of the thumb carpometacarpal joint (TCMCJ) based on measurements of human joints. In the TCMCJ both articulating surfaces are saddle-shaped. The aim was to geometrically survey the shapes of the articulating surfaces using precise replicas of 28 TCMCJs. METHODS: None of these 56 articulating surfaces did mathematically extend the differential geometrical neighbourhood around the main saddle point so that each surface could be characterised by three main parameters: the two extreme radii of curvature in the main saddle point and the angle between the saddles' asymptotics (straight lines). RESULTS: The articulating surfaces, when contacting at the respective main saddle points, are incongruent. Hence, the TCMCJ has functionally five kinematical degrees of freedom (DOF); two DOF belong to flexion/extension, two to ab-/adduction. These four DOF are controlled by the muscular apparatus. The fifth DOF, axial rotation, cannot be adjusted but stabilized by the muscular apparatus so that physiologically under compressive load axial rotation does not exceed an angle of approximately ±3°. CONCLUSIONS: The TCMCJ can be stimulated by the muscular apparatus to circumduct. The mechanisms are traced back to the curvature incongruity of the saddle surfaces. Hence we mathematically proved that none of the individual saddle surfaces can be described by a quadratic saddle surface as is often assumed in literature. We derived an algebraic formula with which the articulating surfaces in the TCMCJ can be quantitatively described. This formula can be used to shape the articulating surfaces in physiologically equivalent TCMCJ-prostheses.

Compensation of skeletal Class III malocclusion by isolated extraction of mandibular teeth: Part 2: Skeletal, dentoalveolar and soft tissue parameters in comparison with nonextraction Class III therapies.

OBJECTIVES: To retrospectively compare two compensatory approaches taken in skeletal Class III patients during the main treatment stage, including a study group of multiband treatment plus isolated extraction of mandibular teeth and a control group of multiband treatment without extraction of teeth. PATIENTS AND METHODS: The extraction group included 22 (12 female, 10 male) patients receiving compensatory multiband treatment for a mean of 3.47 ± 1.14 years and 16.22 ± 1.92 years old at debonding. The nonextraction group included 24 (14 female, 10 male) patients undergoing multiband treatment for 2.76 ± 1.28 years and 15.38 ± 1.46 years old at debonding. Lateral cephalograms obtained at baseline and upon completion of active treatment were traced for skeletal, dentoalveolar, and soft tissue parameters. Welch and Wilcoxon tests were used to analyze intergroup differences (initial values, final values, initial-to-final changes) and within-group differences (p < 0.05). RESULTS: Upon completion of active treatment, the only significant intergroup differences were noted for U1NA and L1ML. Significant within-group changes over the courses of treatment were seen for SNB, MLNL, U1NA, U1NL, L1NB, L1ML, H-angle, ULipEL, and LLipEL (extraction group) or for SNB, ANB, individual ANB, Wits appraisal, U1NA, U1NL, H-angle, Naslab-a, ULipEL, and LLipEL (nonextraction group). Parameters that changed by significantly different amounts in both groups included Wits appraisal, L1NB, L1ML, and LLipEL. CONCLUSION: The added value of isolated extraction therapy basically lies in increasing the potential for retruding the lower incisor inclinations, so that compensatory treatment becomes an option even in selected patients presenting with adverse occlusal situations that would otherwise require orthognathic surgery. Given the successful outcomes in both groups, which had been established by Peer Assessment Rating (PAR) scores, it was possible to define the skeletal, dentoalveolar, and soft tissue characteristics of successful treatment more precisely than before.

The description of the human knee as four-bar linkage.

PURPOSE: We investigate the dependence of the kinematics of the human knee on its anatomy. The idea of describing the kinematics of the knee in the sagittal plane using four-bar linkage is almost as old as kinematics as an independent discipline. We start with a comparison of known four-bar linkage constructions. We then focus on the model by H. Nägerl which is applicable under form closure. METHODS: We use geometry and analysis as the mathematical methods. The relevant geometrical parameters of the knee will be determined on the basis of the dimensions of the four-bar linkage. This leads to a system of nonlinear equations. RESULTS: The four-bar linkage will be calculated from the limits of the constructively accessible parameters by means of a quadratic approximation. CONCLUSIONS: By adapting these requirements to the dimensions of the human knee, it will be possible to obtain valuable indications for the design of an endoprosthesis which imitates the kinematics of the natural knee.

The morphology of the articular surfaces of biological knee joints provides essential guidance for the construction of functional knee endoprostheses.

PURPOSE: In comparative examinations of kinematics of the knees of humans and pigs in flexional/extensional motion under compressive loads, the significant differential geometric essentials of articular guidance are elaborated to criticise the shaping of the articular surfaces of conventional knee-endoprostheses and to suggest constructional outlines that allow the endoprosthesis to adopt natural knee kinematics. Implantation is discussed with regard to the remaining ligamentous apparatus. METHODS: Twelve fresh pig knee joints and 19 preserved human knee joints were moved into several flexional/extensional positions. In each joint, the tibia and femur were repeatably caught by metal plates. After removing all ligaments, the tibia and femur were again caught in these positions, and their points of contact were marked on both articular surfaces. Along the marker points, a thin lead wire was glued onto each surface. The positions and shapes of the four contact lines were mapped by teleradiography. RESULTS: All contact lines were found to be plane curves. The medial and lateral planes were parallel, thus defining the joint's sagittal plane. In the human knee, as compared to the lateral, the medial femoral contact line was always shifted anteriorly by several millimetres. The tibial contact curve was laterally convex and medially concave. In the pig knees, the lateral and medial contact lines were asymmetrically placed. Both tibial curves were convex. CONCLUSIONS: Both knees represent cam mechanisms (with one degree of freedom) that produce rolling of the articular surfaces during the stance phase. Implantation requires preservation of the anterior cruciate ligament, and ligamentous balancing is disadvantageous.

Characteristic points and cycles in planar kinematics with application to the human gait.

PURPOSE: We present a novel method to process kinematical data typically coming from measurements of joints. This method will be illustrated through two examples. METHODS: We adopt theoretical kinematics together with the principle of least action. We use motion and inverse motion for describing the whole experimental situation theoretically. RESULTS: By using the principle of least action, the data contain information about inherent reference points, which we call characteristic points. These points are unique for direct and inverse motion. They may be viewed as centers of the fixed and moving reference systems. The respective actions of these characteristic points are analytically calculated. The sum of these actions defines the kinematical action. This sum is by design independent of the choice of reference system. The minimality of the kinematical action can be used again to select numerically one representative cycle in empirically given, approximately periodic motions. Finally, we illustrate the theoretical approach making use of two examples worked out, hinge movement and the sagittal component of the movement of a human leg during gait. CONCLUSIONS: This approach enables automatic cycle choices for evaluating large databases in order to compare and to distinguish empirically given movements. The procedure can be extended to three dimensional movements.

A caveat concerning center of resistance.

The center of resistance is a concept in theoretical orthodontics used to describe tooth movement under loads. It is commonly used to qualitatively predict tooth movement without recourse to complex equations or simulations. We start with a survey of the historical origin of the technical term. After this, the periodontal ligament is idealized as a linear elastic suspension. The mathematical formalism of vector and tensor calculus will clarify our reasoning. We show that a point such as the center of resistance basically only exists in two dimensions or in very special symmetric spatial configurations. In three dimensions, a simple counterexample of a suspension without a center of resistance is given. A second more tooth-like example illustrates the magnitude of the effects in question in dentistry. In conclusion, the center of resistance should be replaced by a newer and wider mathematical concept, the "center of elasticity," together with a limiting parameter, the "radius of resistance."

Does total disc arthroplasty in C3/C4-segments change the kinematic features of axial rotation?

We analyze how kinematic properties of C3/C4-segments are modified after total disc arthroplasty (TDA) with PRESTIGE(®) and BRYAN(®) Cervical Discs. The measurements were focused on small ranges of axial rotation (<0.8°) in order to investigate physiologic rotations, which frequently occur in vivo. Eight human segments were stimulated by triangularly varying, axially directed torque. By using a 6D-measuring device with high resolution the response of segmental motion was characterised by the instantaneous helical axis (IHA). Position, direction, and migration rate of the IHA were measured before and after TDA. External parameters: constant axially directed pre-load, constant flexional/extensional and lateral-flexional pre-torque. The applied axial torque and IHA-direction did not run parallel. The IHA-direction was found to be rotated backwards and largely independent of the rotational angle, amount of axial pre-load, size of pre-torque, and TDA. In the intact segments pre-flexion/extension hardly influenced IHA-positions. After TDA, IHA-position was shifted backwards significantly (BRYAN-TDA: ≈8mm; PRESTIGE-TDA: ≈6mm) and in some segments laterally as well. Furthermore it was significantly shifted ventrally by pre-flexion and dorsally by pre-extension. The rate of lateral IHA-migration increased significantly after BRYAN-TDA during rightward or leftward rotations. In conclusion after the TDA the IHA-positions shifted backwards with significant increase in variability of the IHA-positions after the BRYAN-TDA more than in PRESTIGE-TDA. The TDA-procedure altered the segment kinematics considerably. TDA causes additional translations of the vertebrae, which superimpose the kinematics of the adjacent levels. The occurrence of adjacent level disease (ALD) is not excluded after the TDA for kinematical reasons.

Influence of tibial slope asymmetry on femoral rotation in patients with lateral patellar instability.

PURPOSE: The geometry of the tibial plateau and its influence on the biomechanics of the tibiofemoral joint has gained increased significance. However, no quantitative data are available regarding the inclination of the medial and lateral tibial slope in patients with patellar instability. It was therefore the purpose of this study to evaluate tibial slope characteristics in patients with patellar dislocations and to assess the biomechanical effect of medial-to-lateral tibial slope asymmetry on lateral patellar instability. METHODS: Medial and lateral tibial slope was measured on knee magnetic resonance images in 107 patients and in 83 controls. The medial-to-lateral tibial slope asymmetry was assessed as the intra-individual difference between the medial and lateral tibial plateau inclination considering severity of trochlear dysplasia. The effect of tibial slope asymmetry on femoral rotation was calculated by means of radian measure. RESULTS: Severity of trochlear dysplasia was significantly associated with an asymmetric inclination of the tibial plateau. Whereas the medial tibial slope showed identical values between controls and study patients (n.s.), lateral tibial plateau inclination becomes flatter with increasing severity of trochlear dysplasia (p < 0.01). Consequently, the intra-individual tibial slope asymmetry increased steadily (p < 0.01) and increased internal femoral rotation in 20° and 90° of knee flexion angles in patients with severe trochlear dysplasia (p < 0.01). In addition, the extreme values of internal femoral rotation were more pronounced in patients with patellar instability, whereas the extreme values of external femoral rotation were more pronounced in control subjects (p = 0.024). CONCLUSION: Data of this study indicate an association between tibial plateau configuration and internal femoral rotation in patients with lateral patellar instability and underlying trochlear dysplasia. Thereby, medial-to-lateral tibial slope asymmetry increased internal femoral rotation during knee flexion and therefore might aggravate the effect of femoral antetorsion in patients with patellar instability. LEVEL OF EVIDENCE: III.

The influence of occlusal forces on force delivery properties of aligners during rotation of an upper central incisor.

OBJECTIVE: To determine the forces and moments delivered to a maxillary central incisor during rotation with aligners when a simulated occlusal force generated during swallowing acts on the appliance. MATERIALS AND METHODS: Five identical appliances were manufactured from four different starting materials (Erkodur 0.8 mm and 1.0 mm; Biolon 0.75 mm and 1.0 mm). An upper central incisor fixed in a measuring device was rotated around its central axis in 0.5-degree steps up to ±10 degrees with the appliance fixed in place. An occlusal force of 30 N generated during swallowing was simulated with a weight positioned on the appliance. For statistical analysis, the moments Tz (rotation) and forces Fz (intrusion) at a deflection of ±0.34 mm to the incisor edge (±5 degrees rotation) were tested. Means and standard deviations for Tz and median and 25% and 75% quartiles for Fz were calculated. An analysis of variance was performed. RESULTS: The simulated occlusal force increased the measured intrusive force Fz (maximum with a weight, -3.7 N [-3.7, -2.4]; minimum without a weight, -1.3 N [-1.4, -1.1]) and the rotary moment Tz (maximum with a weight, -50.8 Nmm [±0.8]; minimum without a weight, 18.2 Nmm [±0.9]) significantly in all cases (P < .01). This was found for all materials measured and for both directions of rotation. CONCLUSION: During rotation with aligners, a simulated occlusal force increases the intrusive force and the rotary moment. The biological adverse side effects of these phenomena remain unclear, especially in patients with periods of bruxism.

Identification of signal bias in the variable flip angle method by linear display of the algebraic Ernst equation.

A novel linear parameterization for the variable flip angle method for longitudinal relaxation time T(1) quantification from spoiled steady state MRI is derived from the half angle tangent transform, τ, of the flip angle. Plotting the signal S at coordinates x=Sτ and y=S/τ, respectively, establishes a line that renders signal amplitude and relaxation term separately as y-intercept and slope. This representation allows for estimation of the respective parameter from the experimental data. A comprehensive analysis of noise propagation is performed. Numerical results for efficient optimization of longitudinal relaxation time and proton density mapping experiments are derived. Appropriate scaling allows for a linear presentation of data that are acquired at different short pulse repetition times, TR << T1 thus increasing flexibility in the data acquisition by removing the limitation of a single pulse repetition time. Signal bias, like due to slice-selective excitation or imperfect spoiling, can be readily identified by systematic deviations from the linear plot. The method is illustrated and validated by 3T experiments on phantoms and human brain.

Exact algebraization of the signal equation of spoiled gradient echo MRI.

The Ernst equation for Fourier transform nuclear magnetic resonance (MR) describes the spoiled steady-state signal created by periodic partial excitation. In MR imaging (MRI), it is commonly applied to spoiled gradient-echo acquisition in the steady state, created by a small flip angle alpha at a repetition time TR much shorter than the longitudinal relaxation time T(1). We describe two parameter transformations of alpha and TR/T(1), which render the Ernst equation as a low-order rational function. Computer algebra can be readily applied for analytically solving protocol optimization, as shown for the dual flip angle experiment. These transformations are based on the half-angle tangent substitution and its hyperbolic analogue. They are monotonic and approach identity for small alpha and small TR/T(1) with a third-order error. Thus, the exact algebraization can be readily applied to fast gradient echo MRI to yield a rational approximation in alpha and TR/T(1). This reveals a fundamental relationship between the square of the flip angle and TR/T(1) which characterizes the Ernst angle, constant degree of T(1)-weighting and the influence of the local radio-frequency field.

Modeling the influence of TR and excitation flip angle on the magnetization transfer ratio (MTR) in human brain obtained from 3D spoiled gradient echo MRI.

Attempts to optimize the magnetization transfer ratio (MTR) obtained from spoiled gradient echo MRI have focused on the properties of the magnetization transfer pulse. In particular, continuous-wave models do not explicitly account for the effects of excitation and relaxation on the MTR. In this work, these were modeled by an approximation of free relaxation between the radiofrequency pulses and of an instantaneous saturation event describing the magnetization transfer pulse. An algebraic approximation of the signal equation can be obtained for short pulse repetition time and small flip angles. This greatly facilitated the mathematical treatment and understanding of the MTR. The influence of inhomogeneous radiofrequency fields could be readily incorporated. The model was verified on the human brain in vivo at 3 T by variation of flip angle and pulse repetition time. The corresponding range in MTR was similar to that observed by a 4-fold increase of magnetization transfer pulse power. Choice of short pulse repetition time and larger flip angles improved the MTR contrast and reduced the influence of radiofrequency inhomogeneity. Optimal contrast is obtained around an MTR of 50%, and noise progression is reduced when a high reference signal is obtained.