Description:  This is tha authors' PDF version of an article published in Radiography© 2004. The definitive version is available at www.elsevierhealth.comPurpose The use of hand radiographs has both clinical and anthropometric applications. However, a method for converting standard bony points within the metacarpus to Cartesian co-ordinates does not exist. Methods A simple method for converting standard bony points of the second to fifth metacarpals to Cartesian co-ordinates is described for the first time. Results Using a small set of measurements and treating these with equations of known voracity, this method is accurate and allows the metacarpus to be interro¬gated via a much wider range of geometrical techniques than has so far been available. Conclusions This method allows naked-eye assessments to be supported or re¬placed by metrical evaluations. It is likely to have both clinical and anthropometric uses.University of Liverpool research development grant

Batschelet

Blincoe

Butler

Clark

Garn

George

Joseph

Lewis

Manning

Merrill

Parish

Park

Poznanski

Ulijaszek

Radiography

English

Lewis, Stephen J.

ChesterRep

       This work has been submitted to ChesterRep – the University of Chester’s online research repository  http://chesterrep.openrepository.com    Author(s): Lewis, Stephen J   Title: A Cartesian co-ordinate system for representing the second to fifth metacarpals in the human hand   Date: 2004   Originally published in: Radiography  Example citation: Lewis, S. J. (2004).  A Cartesian co-ordinate system for representing the second to fifth metacarpals in the human hand. Radiography, 10, 189-194   Version of item: Author’s post-print       Available at: http://hdl.handle.net/10034/64373 A Cartesian co-ordinate system for representing the second to fifth metacarpals in the human hand    Stephen Lewis Department of Biological Sciences, University College Chester, Parkgate Road, Chester CH1 4BJ, UK Tel.: +44-1244-375444; fax: +44-1244-392820.  E-mail address: s.lewis@chester.ac.uk.   Abstract   Purpose  The use of hand radiographs has both clinical and anthropometric applications. However, a method for converting standard bony points within the metacarpus to Cartesian co-ordinates does not exist. Methods  A simple method for converting standard bony points of the second to fifth metacarpals to Cartesian co-ordinates is described for the first time.  Results  Using a small set of measurements and treating these with equations of known voracity, this method is accurate and allows the metacarpus to be interrogated via a much wider range of geometrical techniques than has so far been available. Conclusions     This method allows naked-eye assessments to be supported or replaced by metrical evaluations. It is likely to have both clinical and anthropometric uses.    Introduction  Although the human hand receives attention from clinicians and anthropologists, the information obtained is usually somewhat limited. Previous work has focused on bone lengths, and sometimes widths, or the relative projection of key points to give digital or metacarpal formulae with the latter often relying more on anthroposcopy than metrical techniques.1-4 Yet the morphological study of the hand is important for clinical, biomechanical, ergonomic and anthropological reasons.  Clinical radiographs are an important source of anthropometric data5 including the hand. 6,7 In the past, the clinical use of hand radiographs has benefited from the work of anthropologists who have contributed metrical methods to aid diagnosis. 8-13  Although the digits, arranged as a series of hinged long bones, appear to lend themselves to little more than simple linear measurement, the metacarpals merit closer attention. By converting the points between which the metacarpal lengths have typically been measured14 into a series of Cartesian co-ordinates, a wider range of metrical and statistical options becomes available. These include the ability to determine the distances between any pair of points, angles between any set of three points and the areas bounded by any set of three or more points.   Methods  Standard dorsi-palmar projection radiographs of the hand are used.15, 16  For purposes of this method, only the second to fifth metacarpals are considered since the mobility of the first metacarpal prevents it from adopting a reliably consistent position during radiographic examination.  For all measurements and calculations, the origin (0,0) for this co-ordinate system is chosen as the point where the centre line of the shaft of third metacarpal crosses its base. This and all the other points between which measurements are taken are the same as those already adopted. 14 The metacarpal heads and bases are numbered as in Fig. 1. For consistency, all the metacarpal heads are given odd numbered co-ordinates and the bases are even numbered.  Rather than simply measuring the lengths of the second to fifth metacarpals between the points set out by Parish14 a form of triangulation based upon these points is adopted (Fig. 2). This triangulation allows the polar co-ordinates of each of the points at the heads and bases of the metacarpals to be determined. Polar co-ordinates take the form P(r,θ), where the point P is characterized by r, the distance of that point from the origin, and θ, the angle, relative to the horizontal axis, that that point subtends at the origin. The polar co-ordinates can then be converted to Cartesian co-ordinates using standard formulae.  The necessary distances are measured directly in Step 1, the angles are calculated in Step 2 and the conversion of polar to Cartesian co-ordinates is performed in Step 3, as described below.  Only points (x2,y2) to (x7,y7) need to be considered. Since the long axis of the third metacarpal lies in the plane of the y-axis, the x co-ordinate of point (x1-y1) is, by definition, zero and the y coordinate is the bone's measured length.   Step 1. Linear measurement  The following linear measurements are taken directly from the radiograph, in millimetres, using callipers accurate to 0.01 mm:  (a) The lengths of the second to fifth metacarpals (BC, AD, EF and GH in Fig.                                                      16  2) and the distances between the: (b) Second and third, third and fourth and third and fifth metacarpal heads (CD, DE and DG in Fig. 2) (c) Second and third, third and fourth and third and fifth metacarpal bases (AB, AF and AH in Fig. 2) (d) Base of the third metacarpal and the head of the second metacarpal (AC in Fig. 2) (e) Base of the third metacarpal and the head of the fourth metacarpal (AE in Fig. 2) (f) Base of the third metacarpal and the head of the fifth metacarpal (AG in Fig. 2)  These lengths constitute the sides of a series of triangles (Fig. 2).   Step 2. Calculation of angles  Given the lengths of the sides of the triangles constructed in Fig. 2, the internal angles may then be calculated. Using a version of the half-angle formula, six such angles are calculated. The individual equations (Eqs. (1A-1F)) used for each angle and the general form is given in Table 1. These angles and composites thereof provide the angular part of each polar co-ordinate.  Having measured the distances between the origin (0,0) and points (x2,y2) to (x7,y7) (Step 1) and calculated the angles each subtends at the origin relative to the horizontal axis (Step 2), all the elements of the polar co-ordinates of these points are known and can now be constructed (Table 2).   Step 3. Converting from polar to Cartesian co-ordinates  Having constructed the polar co-ordinates of the heads and bases of second to fifth metacarpals, they may now be converted to Cartesian co-ordinates using standard formulae. The general form and the individual equations for each conversion are given in Table 3.  Usefully, having made the original measurements in millimetres, the numerical values of the calculated co-ordinates are also in millimetres.    Results    Accuracy  As this method turns linear measurements into Cartesian co-ordinates, tests evaluating its accuracy should seek to explore how closely the calculated co-ordinates match the relevant points on the metacarpals. Thus, to test the closeness of these co-ordinates with the points they represent, one may plot them, full size on paper, place this on a light-table, overlay the original radiograph and align the respective points. The level of agreement between the original and the plotted points provides the degree of confidence one has in the method. When this is performed, that level of agreement has been found to be so great as to defy direct measurement.  However, it should be noted, that since this method uses mathematical calculations of known veracity, any errors that occur do not result from the calculations themselves but are those typical of the measuring processes.17 Thus, provided the initial measurements are made accurately, this method is independent of further error.  This set of co-ordinates constitutes all the points that characterize the extent of the second to fifth metacarpals. Further recourse to the original radiographs is unnecessary and additional measurement error is thereby avoided.   Applications  Having calculated this series of co-ordinates, it is possible to determine the distances between any two of the eight points characterized, the angles between any three of these points and the areas between any set of three or more points using standard equations in Cartesian geometry.   The angles by which the second, fourth and fifth metacarpals diverge from the central axis of the third metacarpal and their relative degree of divergence from each other, can now be determined without the need to draw projection lines on the original radiographs, as has previously been the case.  Points (x2,y2) to -(x7,y7) are no longer merely points between which simple linear measurements are made but now have a precise two-dimensional spatial distribution relative to the origin- (The distribution of point (x1,y1) lies along the y-axis only.) Thus, population and sexual dimorphic comparisons between the spatial distribution of respective pairs of co-ordinates can now be made using statistical techniques such as Hotelling's two-sample T2 test.18  Once calculated, it is also a simple process to standardize all co-ordinates to a given metacarpal length. Standardization for size allows shape differences to be more readily appreciated statistically or by plotting wire diagrams of these metacarpals (similar to that used in Fig. 2). This is useful in morphological growth studies or when following the progress of degenerative changes such as those occurring in rheumatoid arthritis.  As presented, this method relates to the human metacarpus but, in principal, it may also be applicable to the metacarpus (and, indeed, the metatarsus) of other species. Thus, intra- and inter-species comparisons are also possible.  This method also allows metacarpal formula (the relative degree of projection of the heads of the metacarpals) to be represented by the odd numbered y co-ordinates. Where the phalangeal lengths for each finger are also known, these may be added to these y co-ordinates to give a metrical version of digital formula (the relative degree of projection of the tips of the fingers). As this may not include the width of the joint spaces or other soft tissues, this does not correspond to the sizes to be expected in a live hand.   Conclusions  It is already known that a number of genetic syndromes are associated with alteration in skeletal geometry of the hand. For example, shortening of the fourth metacarpal (brachymetacarpia-4), also known as Metacarpal or Archibald's Sign, which occurs in a number of clinical and sub-clinical genetic conditions.19 Frequently, assessment of such cases relies on visual judgement. With a more sensitive metrical tool it may be possible to refine such judgements. Using a few simple measurements, quickly and easily entered into a computer spreadsheet application where the calculations can be performed rapidly and reliably, a more detailed impression may be obtained.  This method converts simple linear measurements, taken without the need for specific orientation of the original radiograph, to a system of orthogonal co-ordinates so that the region between the second to fifth metacarpals can be subjected to a greater range of analysis using Cartesian geometric and statistical techniques. It is a low cost, relatively simple, yet effective method. Using proven mathematical formulae, this method is accurate and trustworthy. It allows the collection of a body of data that more accurately depicts the metacarpus by introducing a spatial distribution where previously only individual lengths were taken and their separation ignored. The co-ordinate database generated can be readily re-worked, as new ideas arise, without having to take new measurements or resorting to the original radiographs (should they still be available) and the introduction of further error avoided.   Acknowledgements  I thank Mrs. K.D. Eaton, Superintendent Radiographer, and Dr J.G. Williams, Consultant Radiologist of Ysbyty Gwynedd for allowing access to clinical radiographs upon which this method was developed and I thank Mrs. A. Lewis for her help in the preparation of this manuscript. This work was supported by a University of Liverpool research development grant.   References  1. Blincoe H. Significant types of hands related to distal extent of digits. Anat Rec 1959;133:527-35. 2. Blincoe H. Significant hand types in women according to relative lengths of fingers. Am J Phys Anthropol 1962; 20:45-8. 3. George R. Human finger types. Anat Rec 1930;46:199-204. 4. Manning J. Digit ratio: a pointer to fertility, behaviour, and health. New Brunswick: Rutgers University Press; 2002. 5. Lewis S. Some research possibilities in diagnostic radiography. Radiography 1998;4:205-9. 6. Lewis S. Morphological aspects of male and female hands. Ann Hum Biol 1996;23:491-4. 7. Lewis S. Metacarpophalangeal pattern profile analysis of a sample drawn from a North Wales population. Ann Hum Biol 2001;28:589-93. 8. Butler M, Meaney F, Kaler S. Metacarpophalangeal pattern profile analysis in clinical genetics: an applied anthropometric method. Am J Phys Anthropol 1986;70:195-201. 9. Garn SM, Hertzog KP, Poznanski AK, Nagy JM. Metacarpophalangeal length in the evaluation of skeletal malformation. Radiology 1972;105:375-81. 10. Joseph MC, Meadow SR. The metacarpal index of infants. Arch Dis Child 1969;44:515-6. 11. Poznanski AK. The hand in radiologic diagnosis. Philadelphia: W.B. Saunders Co; 1984. 12. Poznanski AK. Useful measurements in the evaluation of hand radiographs. Hand Clin 1991;7:21-36. 13. Poznanski AK, Garn SM, Nagy JM, Gall JC. Metacarpophalangeal pattern profiles in the evaluation of skeletal malformations. Radiology 1972;104:1-11. 14. Parish JG. Radiographic measurements of the skeletal structure of the normal hand. Br J Radiol 1966;39: 52-62. 15. Clark K. Positioning in radiography. Oxford: Butterworth-Heinemann; 1991. 16. Merrill V. Merrill's atlas of radiographic positions and radiologic procedures. St. Louis: Mosby; 1999. 17. Ulijaszek S, Lourie J. Intra- and inter-observer error in anthropometric measurement. In: Ulijaszek S, Mascie-Taylor C, editors. Anthropometry: the individual and the population. Cambridge: Cambridge University Press; 1994. p. 30-55. 18. Batschelet E. Circular statistics in biology. London: Academic Press; 1981. p. 129-54. 19. Park E. Radiological anthropometry of the hand in Turner's syndrome. Am J Phys Anthropol 1977;46:463-70.     Figure 1 – Outline of the metacarpus showing co-ordinate points        Table 1 – Formulae for calculating required angels   General form All angles are determined using the sine version of the ‘half-angle formula’ for angle A in triangle ABC:       Table 2 – Polar co-ordinates, in the form p(r,θ), of each of the Cartesian co-ordinates            Table 3 – Formulae for converting polar to Cartesian co-ordinates   General form    

and inter-observer error in anthropometric measurement.

Circular statistics in biology.

Digit ratio: a pointer to fertility, behaviour, and health.

Human finger types.

Merrill's atlas of radiographic positions and radiologic procedures.

Metacarpophalangeal length in the evaluation of skeletal malformation.

Metacarpophalangeal pattern profile analysis in clinical genetics: an applied anthropometric method.

Metacarpophalangeal pattern profile analysis of a sample drawn from a North Wales population.

Metacarpophalangeal pattern profiles in the evaluation of skeletal malformations.

Morphological aspects of male and female hands.

Positioning in radiography.

Radiographic measurements of the skeletal structure of the normal hand.

Radiological anthropometry of the hand in Turner's syndrome.

Significant hand types in women according to relative lengths of fingers.

Significant types of hands related to distal extent of digits.

Some research possibilities in diagnostic radiography.

The hand in radiologic diagnosis.

The metacarpal index of infants.

Useful measurements in the evaluation of hand radiographs.

A Cartesian co-ordinate system for representing the second to fifth metacarpals in the human hand

Crossref

https://chesterrep.openrepository.com/bitstream/10034/64373/24/lewis-cartesian.pdf

A Cartesian co-ordinate system for representing the second to fifth metacarpals in the human hand

Abstract

Similar works

Full text

Available Versions

ChesterRep

Crossref