116 NUMERICAL METHODS IN CHEMICAL ANALYSIS [Proc. Soc. Analyt. Chem.

The Use of Radioactive Markers in the Study of Mineral Metabolism with Special Reference to the Alkaline Earths

BY G. E. HARRISON (Radiobiological Research Unit , Havwell, Didcot, Berks.)

THE speaker began by referring to the uptake, excretion and extracellular concentrations of calcium in adult man in the light of our knowledge two or three decades ago. He pointed out that absorption of calcium, its secretion from blood to gut and the bone turnover, although recognised, could not then be quantitatively evaluated.

Dr. Harrison then described an experiment on a healthy male adult in which single intravenous doses of radioactive markers for calcium, strontium, barium and radium were administered over a period of about 2 months. Serial blood samples were taken following each injection, also a continuous collection of urine and stools. In addition frequent measure- ments were made of the total body content of the different nuclides.

From these experimental results the speaker showed that secretion, absorption, kidney and intestinal clearance rates could be derived for each nuclide, and that the bone turnover rates for these four elements could also be obtained. This characterisation of the separate rate processes was a unique result of the application of tracer methods.

Numerical Methods in Chemical Analysis The following is a summary of the paper presented at a meeting of the Western Section

The audience held on March 15th, 1967, and reported in the April issue of Proceedings (p. 48). included senior pupils from Taunton schools.

Numerical Methods in Chemical Analysis

BY A. L. GLENN (The School of Pharmacy, University of London, 29-39 Brunswick Square, London, W.C. 1)

PRESENT-DAY analytical chemistry is no longer the Cinderella subject it once was and now offers enormous scope to the prospective entrant, who can be assured of a life-time of interest- ing work. The big change occurred around 1950 when several physico-chemical methods, which for two or three decades had been the monopoly of a few specialist research workers, suddenly blossomed forth into general use. A future chemical historian might well describ: this trend as a revolution in methods of data acquisition, and then remark that in the first two decades of the upheaval, inadequate methods of data processing had frustrated much of the data potentially available from the new methods. The current wastage of available data is

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July, 19673 NUMERICAL METHODS I N CHEMICAL ANALYSIS 117

reminiscent of the early days of tea, when Englishmen rejected the infusion and consumed the extracted leaves! Nevertheless, being armed with facts about our own future, our historian might then continue that 1970 had seen the start of a second revolution in which analysts had become increasingly aware of the virtues of numerical methods and of the value of computers in applying them.

A slide rule belongs to the analogue class, and to that extent is a humble relative of the advanced electrical analogue computer. In a similar way, a large digital computer like Atlas can be regarded as a development of the desk calculator, and this in turn as a development of the abacus used by the ancients. The greater potential accuracy of the digital, relative to the analogue computer, is evident from a comparison of abacus with slide rule, the latter involving errors of interpola- tion which do not arise with the abacus.

In using a digital computer, the analyst must translate his ideas into a source program written in a language such as Algol or Fortran and then on to tape together with the numerical data. Subsequent compilation of source program into object program (machine instructions), calculation and printing out of results is a matter entirely for the computer itself and the experts who manage it. Despite its extreme speed and versatility, a modern digital computer is, none-the-less, a moron that plods steadily down a list of instructions, placing a literal interpretation upon each one. I t consumes data and prints out results in a similarly unimagi- native way. Writing a program, therefore, calls for common sense organisation and careful attention to detail rather than mathematical insight, and for that reason non-mathematicians can program with confidence. Moreover, beginners can now profit from two recent Tutor Texts by T. G. Scott, which constitute an excellent basis for subsequent use of a practical corn put er language.

A chief analyst might, for example, increase his productivity by programming sections of past experience which, a t present, form the basis of time-consuming personal decisions that cannot be delegated to less experienced staff. Furthermore, tabletop computers, such as the Olivetti Programma 101 costing about Ll500, are now taking the sting out of statistical arithmetic, so removing a major hindrance to the use of statistics in analytical chemistry.

The view that statistics is irrelevant to analytical chemistry will almost certainly disappear as increasing numbers of analysts gain personal experience of the power of statistical methods. Although few analysts would deny the value of statistics in assessing sets of data that are sufficiently large to boggle the mind, they are apt to place unjustifiable faith in their ability to judge small sets by “just looking at the figures.”

The author’s steadily diminishing confidence in such judgements recently received a body blow in terms of the performance of a group of 24 research workers for whom the assess- ment of small sets of data is a frequent operation. The group was asked to consider the sets of results in Table I and to make three separate judgements as to whether the mean for set “A” differed significantly from that for set “B.” To minimise group influences, answers were given anonymously and in writing.

Computers may be divided into two main classes-analogue and digital.

Analytical chemistry provides abundant opportunity for the use of computers.

TABLE I PERSONAL JUDGEMENT VERSUS STATISTICAL JUDGEMENT

Agreement with the Judgement “A” Sets “R” Sets Student’s t (2 sided) statistical conclusion

1 68 (P = 0.05) 2 agreed

11.0 (insignificant) 16 disagreed 6 abstained

9 agreed 3.1 (significant) 11 disagreed

4 abstained

11 agreed 91 2.0 (insignificant) 4 disagreed

9 abstained

Moreover, with regard to the performance of individuals, and assessing scores according to the number of

75, 77,

74, 78,

95, 86, 94, 1,; 88, 91, 92, I. 73, 70,

74, 71,

84, 82, 90,

87, 91, 88,

I., 2

3

Results for the whole group appear in the last column of Table I .

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118 NUMERICAL METHODS IN CHEMICAL ANALYSIS [Proc. SOC. Amdyt. Chewz.

agreements with statistical conclusions, 8 scored 0, 10 scored 1, and 6 scored 2, out of a maximum of 3 points. I t seems fair to conclude that personal judgement leads to mutual disagreement and vacillation! Statistics, on the other hand, leads to uniform judgement, for given correct arithmetic and the use of the same level of probability all members of the above group would have reached complete agreement oil all three judgements.

Most applications of computers to analytical chemistry are likely to be highly specific to the needs of particular techniques, laboratories or analysts. One such application that the author has recently come across concerns the correction of ultraviolet absorption spectra, in which, starting from a simple set of measurements that take account of instrument para- meters, it is necessary to correct photometric and wavelength scales and then allow for slit function and stray light, respectively. To correct each observed point in the absorption spectrum necessarily entails so much arithmetic as to render the procedure impractical without the use of a digital computer. Nevertheless, the considerable advantages to be reaped from such corrections are already apparent from the work of R. N. Jones in the infrared region.

The present imbalance between data acquisition and data processing can be seen in several of the newer methods. For example, the continued use of simple “analytical arith- metic,” entirely adequate to the needs of 1945, frustrates techniques like absorption spectro- photometry by limiting the analytically relevant observations to two or three wavelengths a t most. The rest of the data potentially available is thus ignored through failure to make use of certain techniques of numerical analysis, which in the computer age are no longer the laborious chores they once were.

Orthogonal functions provide one means for exploiting the “whole” of the data by arithmetic which, to obtain a result, rarely involves more than 5 minutes’ operation of a desk calculator.

where E(A) denotes the ordinate value (e.g., extinction) corresponding with the abscissa value (e.g., wavelength), A, the latter belonging to a set of (n + 1) equally seated values a t which the orthogonal polynomials, Po(h), P,(A), P2(A), etc., are each defined. These polynomials re- present a series of fundamental curve shapes that may be scaled up or down by the appropriate coefficients (Po, el, p 2 , etc.) and then summed to reproduce the value of any E(A) within the set.

In applying this method to the determination of an absorbing compound in the presence of irrelevant absorption, each coefficient, like E(A) itself, represents the sum of contributions from background and absorbing compound, respectively. However, the ratio of the two contributions varies from one coefficient to another, and by suitable choice of wavelengths i t is usually possible to find one particular coefficient (e.g., p 2 ) to which the background contributes a negligible fraction of the whole. That coefficient is then proportional to the concentration of the absorbing compound. Further, as any background must contribute to Po and probably to p , also, the analytical result is always based upon p , or some higher coefficient.

Because of the orthogonality of the polynomials and the resultant mutual independence of the coefficients, calculation of the required coefficient entails no more than simple arith- metic. Moreover, each coefficient is an exact linear function of the set of E(A) used in its calculation. Hence, a given coefficient is exactly proportional to the concentration of absorb- ing compound, provided the latter obeys Beer’s law, and the background contributes nothing to the coefficient.

Working on the 270-mp band of phenol, A. M. Wahbi has shown that coefficients bear the anticipated proportionality to concentration, and are measurable (when n = 11) with about the same precision as individual extinctions. He has also been able to demonstrate the power of the method in discounting that part of the irrelevant absorption in cod liver oil, which is removed by saponification during the assay of vitamin A. The histogram (Fig. 1) presents results obtained for 31 different batches of oil, the abscissa referring to the result obtained on the gross oil and expressed as a percentage of the result obtained after saponifica- tion.

The unshaded rectangles refer to batches for which the ratio, E300mp/E328mp gave a clear indication of the need for prior saponification, and of the remaining oils only one fell outside the limits, &5 per cent. of the result obtained after saponification. By the Morton and Stubbs method, on the other hand, results on gross oils were mostly between 10 and 40 per cent. higher than those obtained after saponification.

Any experimental curve can be broken down (expanded) in the following way. E(4 = @ o P o ( 4 + PlPl(4 + @ 2 P 2 ( 4 + * - * + @ n p n ( 4

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July, 19671 MOSSBAUER SPECTROSCOPY 119

I10

I

- 5 95 I00 I05

Percentage (of result obtained after saponification)

Fig. 1 . Assay of vitamin A in cod liver oil without saponi- fication (1 2-point orthogonal polynomials)

Careful choice of conditions, a pre-requisite to good results, is apt to involve a good deal of arithmetic so that a digital computer is a great help in designing the assay process. Once developed, however, a desk calculator is entirely adequate to the evaluation of individual assay results. In the assay of vitamin A, time required for calculation is small compared with that expended in solution preparation, cell filling and extinction measurement. The total however is but a small fraction of the time required for the official B.P. assay and has been reduced even further by combining a recording spectrophotometer with a simple analogue computer, to obtain a direct indication of the orthogonal function coefficient a t the end of a 3-minute scan of the gross oil’s spectrum.

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