Cube Method Research Articles

4719585 Dividing cubes system and method for the display of surface structures contained within the interior region of a solid body

A system for displaying three dimensional surface structures employs the analysis of voxel elements defined by eight cubically adjacent grid locations which are associated with at least one physical property defined throughout a three dimensional body. The physical property measurements may be made with such systems as computerized tomographic x-ray systems, or magnetic resonance imaging devices. Surface structures are defined by a user selected threshold value such as distinct values for skin and bone In medical diagnostic applications. Selected voxels are subdivided and means are provided for generating signal values associated with intermediate additional grid locations. Means are also provided for generating surface normal vectors associated with these grid locations. The system of the present invention produces a sequence of signals representative of grid locations throughout the object being investigated together with associated surface normal vectors at these locations, the locations lying on a user selected surface. The system provides smooth, high resolution images which are particularly useful for medical diagnostic applications and is particularly useful with conventional display processor hardware used for electronic graphics display. The capabilities of the system additionally provide enhanced flexibility and speed for the convenience of interactive users.

Right-angled triangles in ancient China

The properties associated with the sides of a right-angled triangle had always been a subject of intriguing interest among mathematicians of ancient civilisations. The first and most outstanding of these properties is what is generally known in the West as the theorem of Pythagoras. This single theorem is considered by many historians of mathematics as one of the most profound concepts and an important milestone in the history of mathematics. In ancient China the study of the relationships of the sides of a right-angled triangle formed an important and substantial foundation of knowledge which had considerable influence on the mathematical thought and reasoning of early mathematicians. The knowledge led to the pursuit of other subjects such as similar triangles, measurement of distances, geometry involving triangles and circles, methods of extracting square and cube roots and methods of solving quadratic equations. A discussion on the right-angled triangle is found in the Zhoubi suanjing [a] (The arithmetic classic of the gnomon and the circular paths of heaven). This is the oldest existing Chinese mathematics classic and its date and authorship are unknown. It is generally regarded as being essentially a Han book stretching as far back as the 1st century BC. The last of the nine chapters in the Jiu zhang suanshu [c] (Nine chapters on the mathematical art) is devoted entirely to the right-angled triangle. Again nothing is known of the authors nor the exact date of this book which is generally placed between 100 BC and 100 AD. The bulk of the early knowledge on right-angled triangles is confined to the above mentioned two books and this is considerably and substantially enlarged by the commentaries of Zhao Shuang [d] and Liu Hui [e] on the relevant sections of the Zhou bi suanjing and the Jiu zhang suanshu respectively. Both Zhao and Liu were well-known commentators and were contemporaries living in the 3rd century AD. Zhao was sometimes known by his literary name Zhao Junqing [f]. A thorough analysis of a subject such as the relationship of the sides of a right-angled triangle would in general divide into three stages of development. The initial stage pertains to special numerical results obtained from particular

Improved technique for calculating X‐ray scattering intensity of biopolymers in solution: Evaluation of the form, volume, and surface of a particle

AbstractAn improved cube method has been developed for calculating the intensity of diffuse x‐ray scattering of macromolecules in solution using a certain set of their atomic coordinates. The technique is based on the ideas of B. Lee and F. M. Richards [(1971) J. Mol. Biol. 55, 374–400] and Richards [(1977) Annu. Rev. Biophys. Bioeng. 6, 151–176] on the possibility of estimating the molecular and accessible surface of a particle by “rolling” a sphere, simulating a water molecule, on its molecular surface. It is shown that this technique is more advantageous than earlier versions of the cube methods. The improved technique for calculating scattering curves was utilized for several globular proteins, and for the first time, reliable scattering curves were obtained for protein‐“bound” water complexes. In the case of globular proteins and tRNA, this technique has permitted a strict evaluation of their accessible surfaces, their volumes, and, apparently for the first time, their complete molecular surfaces.

Calculation of scattering curves for macromolecules in solution and comparison with results of methods using effective atomic scattering factors

An improved cube method was developed for the computation of X-ray scattering curves of macromolecules in solution. For double-helical DNA and RNA molecules the efficiency of this method is shown. The results are compared with curves calculated by effective atomic scattering factor methods. In the small-angle and in the wide-angle regions the improved cube method is superior to the effective atomic scattering factor methods. This was proved by the calculation of structure parameters and by a comparison with experimental scattering data. On the basis of the improved cube method, models with a reduced structure resolution are deduced for DNA and RNA molecules. The models consisting of the three scattering centres phosphate, sugar and base per nucleotide are equivalent in scattering to the real structure up to a scattering angle of about 0.15 rad for copper radiation.

Evaluation of the relation between the compressive strength of cement by the British Standard cube test and iso-rilem prism test

The compressive strength of cement mortar is principally determined by two methods, either by cubes or by prisms. In the cube method, the compressive strength is evaluated by directly subjecting the cube specimens to compressive load, where as in the prism method, the standard mortar prism (4×4×16cm) is first broken to two halves by flexural test, and then each half prism is subjected to compressive stress by distributing the axial compressive stress on the specified area on the two plane faces. This paper, aims at finding the relation between the compressive strength of the same cement, by these two methods of test.

Large-angle X-ray diffuse scattering, a new method for investigating changes in the conformation of globular proteins in solutions

A comparative analysis of the sperm-whale myoglobin structure in the crystal and in solution has been carried out with the technique previously formulated by the authors which uses the large-angle X-ray diffuse scattering for investigating the protein structure in solution. A `modified cube method', correctly taking into account the cavities within the protein molecule accessible to the solvent, is proposed for an accurate estimate of the solvent influence and for calculation of the scattering intensities. A comparison of the theoretical myoglobin scattering curves with the experimental scattering curve of this protein obtained by Stuhrmann shows rather noticeable quantitative divergences, which can be eliminated by a small increase in the distance between the `hairpin' GH and the other part of the protein molecule. It is also shown that several other physically reasonable shifts of helices (or groups of helices) do not lead to an agreement between the experimental and theoretical scattering indicatrices.

Test of reliability of echocardiographic estimation of left ventricular dimensions and volumes.

This test is based on the incompressibility of myocardium, which dictates that left ventricular wall volume remains constant throughout the cardiac cycle. The volumes occupied by the left ventricular cavity, by ventricular wall plus cavity, and hence by ventricular wall alone were estimated, both at end-systole and at end-diastole, from ecocardiographic measurements of cavity transverse dimension and wall thickness. Wall volumes were determined by assuming an ellipsoid shape (the major axis being predicted from aggression equations relating angiocardiographic and echocardiographic cavity dimensions) and also by the cube method. A discrepancy between systolic and diastolic wall volume estimates indicates either that the measurements of ventricular dimensions were unreliable or that the assumptions of ventricular geometry involved in the volume calculations were incorrect. Studies were made on 60 subjects. Using the ellipsoid formula, values for wall volume ranged from 66 to 719 ml; systolic and diastolic wall volumes correlated closely (r = 0-96, mean difference = 6-8 +/- 0-9 (SEM) %) supporting the reliability of the echocardiographic dimensions and estimates of cavity and wall volume. In the 12 patients with very large end-diastolic cavity transverse dimensions (6-5 to 8-6 cm) however, correlation was less good (r - 0-81, mean difference = 14-3 +/- 2-3 (SEM) 5). Using the cube method, which does not allow for the changing relation between minor and major cavity axes with increasing cavity size, wall volumes were greater (76 to 986 ml) but correlation was similar (r = 0-94, mean difference = 7-1 +/- 0-9 (SEM)%). Having established that it is possible to obtain close agreement between wall volumes determined at different points in the cardiac cycle, this test can be used to assess the reliability of echocardiographic left ventricular dimensions and volume estimates in individual subjects.

Noninvasive evaluation of ventricular hypertrophy in professional athletes.

Athletes often exhibit ECG findings which are considered to be abnormal. Therefore, we used noninvasive graphic methods to study 42 active professional male basketball players, ranging in age from 21 to 31 years, without clinically evident heart disease. Of the 42, 11 (25%) met the Romhilt-Estes ECG voltage criteria for left ventricular hypertrophy, and 12 (29%) satisfied VCG criteria for left ventricular enlargement; nine (21%) had left ventricular hypertrophy by both methods. In 33 subjects (79%) the 0.04 sec vector in the horizontal plane was anterior, and 29 of these exhibited one or more standard criteria for right ventricular enlargement; the ECG and VCG were concordant for right ventricular hypertrophy in 16 subjects (38%). Submaximal treadmill exercise tests (Bruce protocol) were normal in eight athletes, while in one subject ventricular premature beats occurred during the test. In 24 of 25 athletes (96%) from whom phonocardiograms were obtained a third heart sound was recorded, while in 14 (56%), a fourth heart sound was present. Of the 14 athletes who had a fourth heart sound, 12 (86%) had either ECG or VCG evidence of ventricular hypertrophy. Only four of 23 athletes had an increased cardiothoracic ratio (greater than .50) on routine chest X-ray. Ten athletes and ten control subjects matched for height, weight and body surface area had echocardiograms satisfactory for analysis. The left ventricular end-diastolic dimension in the athletes averaged 53.7 +/- 1.3 (SE) mm compared with a value of 49.9 +/- 0.7 mm in the control subjects (P less than 0.02), and was increased (greater than or equal to 56 mm) in four. Left ventricular posterior wall thickness averaged 11.1 +/- 0.6 mm, compared with a value of 9.8 +/- 0.5 mm in the control subjects (P less than 0.05), and was increased (greater than or equal to 11 mm) in six athletes. The right ventricular end-diastolic dimension averaged 20.8 +/- 1.1 mm compared with a value of 12.9 +/- 2.2 mm in the controls (P less than 0.004), and was increased (greater than or equal to 23 mm) in four athletes. No athlete or control subject exhibited paradoxical septal motion. In the athletes, ejection fraction (cube method) averaged 79 +/- 2.0% and mean Vcf averaged 1.13 +/- 0.04 circ/sec; these values did not differ from those of the control subjects. Thus, both right and left ventricular enlargement ("physiological hypertrophy") are often present in the well-trained athlete, but left ventricular performance remains normal in the basal state in such individuals. We condlude that these individuals represent a selected subgroup of subjects who are variants of normal.

Error in the Interferometric Measurement of Length by the Rotational Movement of a Mirror and Multipass Corner Cube Arrangement

To get high accuracy for measuring length, an interferometric measurement is usually used. In this case advancement of the mirror usually accompanies the rotational movement of the mirror, and this affects the proportionality between the advancement of the mirror and fringe displacement. If the rotational movement is small the deviation from proportionality is small, but if the multipass method is used, the angular deviation is also magnified, and the deviation becomes large. The best way to overcome the difficulty is to use a corner cube. A multipass method for a corner cube is described.

The Electrocardiogram and Vectorcardiogram in Congenital Heart Disease

This small volume, presenting the electrocardiographic findings in congenital malformations, is divided in two parts. Part 1 introduces the basic concepts of electrocardiography, then describes the normal electrocardiogram and vectorcardiogram (Grishman cube method) and the abnormalities characteristic of right, left, or combined hypertrophy and of right or left bundlebranch block. Part 2 presents a classification of cardiac malformations based on types and localization of hypertrophy, and on presence or absence of cyanosis. A description of the findings in the individual defects, as seen in the scalar, 12-leads electrocardiogram and in the vectorial loops, follows. The first part is an attempt to make the nonexperienced reader able to use the information given in part 2. The result is disappointing, for this introduction is inadequate for the beginner and superfluous for the expert. The detailed analysis of electrocardiographic findings in each malformation is repetitious rather than informative. This, I believe, could not

The Normal vectorcardiogram: A study of 150 normal adults

Vectorcardiograms were recorded using the cube method in 150 normal adults. Measurements of various displacements of the QRS loop were taken in the horizontal, frontal and sagittal planes. QRS-T angle and long axis angle were calculated. It was found that in the horizontal plane the maximal posterior displacement of the QRS loop increased significantly with age, while the anterior displacement remained constant. There were no significant changes with age in the maximal leftward displacement. The QRS-T angle increased significantly with age. In the frontal plane in younger subjects, the QRS loops were oriented more vertically, with a predominant clockwise direction. In the older patients the direction was mainly counterclockwise, and the loops were placed more horizontally. The QRS-T angle increased significantly with age. In the sagittal plane the measurements followed closely the corresponding displacements in the two other planes. It was shown that the normal vectorcardiogram changes with increasing age.

The ventricular gradient in space

1. 1. A simple, geometric method of determining the spherical coordinates of the ventricular gradient is presented. 2. 2. Electrocardiograms of eight normal subjects are thus analyzed, using the cube method of electrode placement. Mean magnitude, azimuth and elevation are found to be 47 microvolt-seconds, 16 ° and 52 °, respectively. 3. 3. The theory of the gradient is discussed. 4. 4. The role of the RS-T segment is elucidated and a method of evaluating its significance is suggested. 5. 5. Gradients are imposed on hypothetic muscle strips by means of a suitable equation, and the effects of change in rate and direction of activation on the form and area of the ventricular complex are demonstrated. It is concluded that these do not influence net area. 6. 6. A block diagram of an instrument for instantaneous determination of the gradient is included.

A method for recording spatial vectorcardiograms

Three perpendicular bipolar axes which constitute a spatial reference system have been described. The electrodes comprising the reference system are remote or equidistant. The angle subtended to the force measured in each instance was as great as the body would accommodate and not violate the distance necessary for remoteness. This reference system was compared in one hundred instances with the rectangular cube method used by Grishman and others. The chief difference in the two reference systems was found in the measurement of the anterior-posterior electromotive force. In each instance, using our modification, the anterior-posterior component was greater. We believe this change is due to the difference in the angle subtended by electrodes comprising this bipolar lead in the two systems. It is recognized that the electrode placement of this system, as with other spatial reference systems, may not always be considered remote. However, this factor is not inherently greater in the suggested reference system than other currently employed methods. Therefore, we suggest that the method described here may more closely reflect the true potential manifested in the heart.