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References (10)
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S. Salmon, B. Smith (1970)
Immunoglobulin synthesis and total body tumor cell number in IgG multiple myeloma.The Journal of clinical investigation, 49 6
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Clonal origin of human tumors.Annual review of medicine, 30
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A. Laird (1965)
Dynamics of Tumour Growth: Comparison of Growth Rates and Extrapolation of Growth Curve to One CellBritish Journal of Cancer, 19
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P. Sullivan, S. Salmon (1972)
Kinetics of tumor growth and regression in IgG multiple myeloma.The Journal of clinical investigation, 51 7
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L. Mårtensson (1963)
ON " A KEY POINT OF MODERN BIOCHEMICAL GENETICS "The Lancet, 281
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Dynamics of Tumour GrowthBritish Journal of Cancer, 18
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Estimation of tumor cell kill from Gompertz growth curves.Cancer chemotherapy reports, 59 2 Pt 1
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XXIV. On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies. In a letter to Francis Baily, Esq. F. R. S. &cPhilosophical Transactions of the Royal Society of London
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L. Norton, R. Simon, H. Brereton, A. Bogdén (1976)
Predicting the course of Gompertzian growthNature, 264
- Publisher
- Wiley
- Copyright
- 1977 Blackwell Publishing Ltd
- ISSN
- 0960-7722
- eISSN
- 1365-2184
- DOI
- 10.1111/j.1365-2184.1977.tb00316.x
- Publisher site
- See Article on Publisher Site
Abstract
G. F . B R U N T O NA N D T . E . W H E L D O N Department o Clinical Physics and Bio-Engineering, f f West o Scotland Health Boards, Glasgow, Scotland (Received 9 February 1977; revision received 24 March 1977) ABSTRACT A method is presented for predicting the complete growth pattern of human IgG multiple myeloma from a few initial measurements of tumour cell number. This permits identification of patterns of undisturbed neoplastic growth in individual patients whose therapy cannot ethically be withheld for long periods. Since the unperturbed growth pattern is a necessary part of the information required for the prediction of individual patient response to alternative treatment schedules, the prospects for improved treatment of human multiple myeloma by optimal scheduling of drug administration are greatly increased. Human multiple myeloma (Salmon & Smith, 1970; Sullivan & Salmon, 1972), like most experimental tumours in laboratory animals (Laird, 1964, 1965; Simpson-Herren & Lloyd, 1970) exhibits a growth pattern characterized by an exponentially declining rate of growth. Such patterns can often be described mathematically by the Gompertz equation (Gompertz, 1825; Lloyd, 1975) which can be written either in the differential form: -.- dN(t) =a-pln[*] 1 N(t) dt No) or in the integrated form: where N ( t ) is, in this case, the number of tumour cells present at time I , N ( 0 ) the number of cells present at time zero, and a and p are two positive, independent parameters. Correspondence: Mr G. F. Brunton, Department of Clinical Physics and Bio-Engineering, West of Scotland Health Boards, 1 1 West Graham Street, Glasgow G4 9LF, Scotland. 59 I G.F. Brunton and T. E. Wheldon Very recently, however, it has been reported that the parameters a and p may not, in fact, be independent. In a study of two experimental tumours (a murine B 16 melanoma and a rat mammary carcinoma), Norton et al. (1976) observed that, although both a and p varied from one animal to another, the two parameters were positively though nonlinearly correlated so that high values of a were always associated with high values of p, and vice versa. From a knowledge of no more than tumour size and its instantaneous rate of change at any moment, this observation allowed these workers to identify the growth pattern for individual animals. Here we report a similar finding in human IgG multiple myeloma which, likewise, permits the prediction of the complete tumour growth pattern from a few initial measurements. Parameter (day -' ) FIG. 1. Quantitative relation between Gompertz parameters a and p for eleven human cases of IgG multiple myeloma in whom observation of undisturbed tumour growth was possible over long periods of time. A linear regression gives: a = (28.5 f 0.6)p - (0.0008 f 0.009); r = 0.906. Tumour cells in IgG multiple myeloma synthesize a characteristic immunoglobin whose homogeneity provides strong evidence for the monoclonal origin of this neoplasm (Martenssen, 1963; Fialkow, 1976). Assay of this immunoglobin permits serial monitoring of total malignant cell number (Salmon & Smith, 1970; Sullivan & Salmon, 1972), and Sullivan & Salmon (1972) have reported eleven patients in whom it was possible to monitor tumour growth sufficiently well to identify the Gompertz parameters a and /3 in each case. As a result, they were able to predict ultimate patient response to chemotherapy very shortly after treatment had commenced. However, this approach is not possible in many cases because it requires the protracted observation of undisturbed tumour growth without therapeutic intervention. Figure 1 shows a plot of a against p for the eleven cases in the Sullivan-Salmon series. As is apparent from this diagram, the two parameters are not independent but linearly related, Human multiple myeloma growth pattern the coefficient of correlation being extremely high ( r = 0-996). The relation can be expressed as: a = k/3 (3) the proportionality constant, k, being found to have the value; (28.5 k 0.6).* It was found that the intercept of the regression line of a against /with the a axis, viz. -0.0008+ l 0.009 day-’, was not significantly different from zero. Consequently, equation ( I ) can now be written: which is a reduced form of the Gompertz equation now having a as the only adjustable parameter. Also, as the evidence suggests that myeloma is a monoclonal neoplasm, N ( o ) in equation (2) can be set to unity so that the integrated form becomes: N(f)=exp 28.5 1 -exp { [ -- 2:5)1) Since protracted observation is not necessary for estimation of N ( t ) , d N (t)/dr and hence, by rearrangement of equation (4), a; the growth pattern can now be completely predicted in patients whose therapy cannot ethically be withheld for long periods. Moreover, for human multiple myeloma, the regrowth pattern of surviving tumour cells during treatment with melphalan has been shown to be identical to the original growth pattern of the untreated neoplasm (Sullivan & Salmon, 1972). In general, where the growth and regrowth patterns of a treated neoplasm are the same, identification of the original growth pattern enables the observed pattern of tumour regression during therapy to be resolved into two components: one due to cell kill by the therapeutic agent and the other due to regrowth of the surviving cells between fractions. Hence it is possible to determine the cell kill produced by a single administration of the agent. While the similarity of growth patterns before and after therapeutic cell killing certainly cannot be assumed for every tumour-drug combination, it has been reported for a number of experimental tumours treated by several agents (Lloyd, 1975) and may not be uncommon. These considerations raise the possibility of determining the ‘in vivo’ tumour cell survival curves for each individual patient and of predicting patient response not only for the treatment schedule chosen initially but also for any other tolerable schedule using the same agent. ACKNOWLEDGMENT One of us, T.E.W., was supported by an M.R.C. Grant.
Journal
Cell Proliferation – Wiley
Published: Nov 1, 1977