Meting van immuniteit tegen Toxoplasmose met behulp van vrije curven
Authors: Nass C. A. G. 
Published: April 2008

Free‐hand curves in estimating the potency of human sera against Toxoplasma1)

The observations of 70 human sera, selected on clinical and epidemiological grounds, where the only material available. Three doses of every serum, with ratios 1: 5: 25, were subjected to the “Sabin‐test”, counting the number of killed Toxoplasmas out of 50, exposed to every dose.

The chief problem was the estimation of the active dose in every medium serum dose. The percentage kill was used as response y (table I). For the active doses x, a logarithmic scale was used, with the origin at the median letal dose and the dilution I: 5 as unity. The three active doses of every serum were named x–I, x and x + I. The problem was solved by fitting three parallel free‐hand curves, Y (X – I), Y (X) and Y (X + I), to the 3 × 14 mean responses y of five subsequent sera, when ranked in order of magnitude of Σy (table 2, figure 3). The x of every single serum was estimated in principle by shifting the responses y (x – I), y (x) and y (x + I) along the abscis till the condition Σy=ΣY was satisfied; and in practice by reading from a previous y prepared double scale with Σy and x

The chief difficulty in drawing figure 3, the unknown value of x, was overcome by means of two accessory figures. In figure I, ȳ (x̄) is plotted on the abscis for the points and Y (X) for the curves. The points in this figure are not in the right position because the condition Σy =ΣY is not satisfied. Their shifting to the right position, or the substitution of their abscis ȳ (x̄) by Y (¯), was rather cumbersome, because every correction of the curves would imply corrections of the rrbscisses. Therefore figure 2 was drawn, with Σȳ=ΣY plotted on the abscis, to avoid the need of shifting.

Because the curves of figure 3 should be parallel, the curves of figures 1 and 2 shorild satisfy the following condition: Of any rectangle, parallel to the axes, with two diagonal summits on the medium curve and one summit on the upper curve, the fourth summit should ly upon the lower curve. In each figure a set of such rectangles is drawn, yielding the positions of x = ‐4, ‐3, ‐2, ‐1, 0, 1, 2, which were used for the construction of figure I. Every correction in one of the curves, caused corresponding. corrections in the other curves. The three graphs were jointly accomplished.

The most remarkable feature about figure 3 is an upper asymptote at about 82%. It is suggested that about 18% of the Toxoplasmas would be colored after death, or that the active component would have a maximum concentration, corresponding to some chemical balance in the staining mixture, which would be survived by 18% of the Toxoplasmns.

The differences y — Y have only two degrees of freedom for every serum. One of this was used to construct an orthogonal component Xb, designed to absorb individual differences of dose response curves. The remaining component xa was used for error variance. These components appeared to be independent (figure 4).

The following significant results were found:

I. The variance of Xa is somewhat larger than would follow from the binomial distribution of y.

2. The variance of xb is larger than that of xa.

3. The distribution of x b is negatively skew (table 3).

4. A serial correlation exists among the signs of Xb (table 4).

5. A serial correlation exists among the values of xb2 (table 5).

It is apparent that the individual dose response curves do not possess the same shape. So the curves of figure 3 represent some kind of a mean dose response curve and the Toxoplasma killing properties cannot be fully described by a single parameter x. Xb may be considered as an estimate of a second parameter.