Bulletin of Mathematical Btology, Vol. 59, No. 5, pp. 897-910, 1997 Elsevier Science Inc. 9 1997 Society for Mathematical Biology 0092-8240/97 $17.00 + 0.00 S0092-8240(97)00035'9 A NEW M E A S U R E OF XENOBIOTIC TOXICITY TO T H E FIRST-LINE H U M A N D E F E N C E SYSTEM F R O M T H E TIME-RESOLVED P H A G O C Y T E LUMINESCENCE 9 BONAWENTURA KOCHEL* and WALDEMAR SAJEWICZ Department of Toxicology, Wroclaw University of Medicine, PL-50417 Wroclaw, Poland (E.mail: lbs@highscreen.int.pan.wroc.pl, Kochel@polbox.corn) A new measure of toxicity based on stochastic modelling of single photon-counting processes, representing time-resolved phagocyte luminescence of xenobiotic-perturbed human neutrophils, has been constructed. The stochastic measure of toxicity has been verified by the QSAR method, and then compared and contrasted with the traditional toxicity measure used in bio- and chemiluminescent research. Phenol and benzene homologues were chosen as perturbers due to their importance from the viewpoint of ecotoxicology and occupational medicine. 9 1997 Society for Mathematical Biology 1. Introduction. Chemiluminescent measurements of the xenobiotic-induced perturbation of human phagocyte activity have been widely used over the last years due to the sensitivity of a single photon-counting technique and the importance of the role played by phagocytes as a first-line defence system. A quantity called "the integral intensity of light" (or "the integral of total counts over the measuring period") has traditionally been used in such measurements since 1972 (Allen et al., 1972; DeChatelet et al., 1982; Cheung et al., 1983; Slawinska and Slawinski, 1985; Van Dyke and Castranova, 1987; Lock and Dahlgren, 1988; Slawinski, 1990; Allen, 1990). This measure is sometimes accompanied by other measures such as "the value of the peak" (or "the peak height," "the intensity of the peak, .... the maximum intensity"), "the average light intensity," and "the time to reach the peak" (or "the nodal time"); however, the term is frequently used. For a few years, the need for a more sensitive measure than the integral value of luminescence response has been signalled (e.g. Ristola and Repo, 1990). During the period of research, time-resolved luminescence was recorded by the single photon-counting technique, and resulting single photon-counting time series (PCTSs) were analysed by autoregressive-integrated moving *Author to whom correspondence should be addressed. 897 898 B. KOCHEL AND W. SAJEWICZ average (ARIMA) models (Box and Jenkins, 1976; Kashyap and Rao, 1976). The perturbation in phagocytosis has been evaluated, and a new toxicity measure has been constructed starting from a memory function approach (MFA) to PCTSs (Kochel, 1990a, 1990b, 1992). A new stochastic measure of toxicity, constructed on the basis of an IMA(0, 1, 1) model for the descending stage of the PCTS, was compared and contrasted with a "traditional" measure of toxicity based on the integral intensity of light. The fMLP-stimulated neutrophils, isolated according to Boyum (1968), were perturbed by such xenobiotics as phenol and some benzene homologues--all important in ecotoxicology and occupational medicine. Experimental and biochemical details have been given elsewhere (Kochel, 1992; Sajewicz, 1993). The validity of both the stochastic and traditional toxicity measures has been estimated using methods based upon quantitative structure-activity relationships (QSAR) (Hansch and Leo, 1979; Karcher and Devillers, 1990). 2. Phagocyte Luminescence of Native and Perturbed Neutrophiis. A time-resolved phagocyte luminescence of either native or perturbed neutrophils has a form of nonstationary discrete stochastic processes. Each single realisation of the process was recorded as an {n(t)} PCTS composed of the numbers of photons n(t), detected via photoelectrons, in time intervals of (t, t + Ate), where At c is the counting time, separated by the dead time A t a ( = At c --- 1 sec). In each of the PCTSs, an ascending stage was followed by a descending one. To avoid any possible single-instance variability of phagocyte activity, each given {nper(t)} process was considered in relationship to the {r/nat(t)} process (for the same instance) as a reference process. In the case of dimethyl sulfoxide (DMSO)-perturbed neutrophils, the phagocyte luminescence of nature neutrophils was used as a reference. However, in the case of other toxicants (dissolved in DMSO), the luminescence of DMSO-perturbed neutrophils was used as a reference. An example of PCTS is shown in Fig. 1. It represents the {nnat(t)} process corresponding to native human neutrophils during phagocytes. The process is nonstationary. For all of the PCTSs, the inequality VtE[1,N]nper(t) ~ nnat(t) holds true; the subscripts "per" and "nat" refer, respectively, to perturbed (by DMSO, benzene, toluene, o-xylene, p-xylene, p-xylene or phenol) or unperturbed phagocyte luminescence. It is shown in Fig. 2, where the descending stages of the {n(t)} processes are compared. The finding that the xenobiotics used in the experiment inhibit phagocytosis points to different biochemical mechanisms in comparison with those involved in phenomena of necrotic or degradative radiation known so far (Perelman and Tarusov, 1966; Ruth, 1979; Gurvitsch and Livanova, 1980), of gluten-modified neutrophil lumi- NEW MEASURE OF XENOBIOTIC TOXICITY 899 4; Figure 1. Time-resolved phagocyte luminescence of fMLP-stimulated native human neutrophils represented by a single photon-counting time series {n(t): t = 1 , 2 , . . . , N}. n(t) denotes the number of photoelectrons detected in a time interval (t, t +/xtr at Ate --- 1 sec, n(t) ~ [2426, 22995] cps and N is the number of the n(t) values in the PCTS (N = 455). nescence (Roccatello et al., 1990), of emission from perturbed cell/tissue cultures (Reiber et al., 1988) or of luminescence from dying organisms (Slawinski and Kochel, 1990; Slawinski, 1990) where the emission is enhanced. However, there are also known phenomena of luminescence inhibition, for instance, inhibition of neutrophil phagocytosis by cytochalasin B (Roschger et al., 1990). On the basis of the {nnat(t)} and {nper(t)} processes, a difference process {ndif(t)} is defined as {ndif(t)} := {nnat(t)} -- {nper(t)}. 7 n(t s \ m6t- Figure 2. Descending stage of the single photon-counting time series corresponding to fMLP-stimulated neutrophils perturbed by benzene (1), phenol (2), toluene (3), o-xylene (4), p-xylene (5), or DMSO (6), and to unperturbed (native) ones (7). All the PCTSs are in the same scale. 900 B. KOCHEL AND W. SAJEWICZ An ascending stage of the {ndif(t)} process, which can be described by an ARIMA(1, 2,1) model, the 0 and ~0 parameters of which have been observed to vary as toxicant-dependent quantities in the ranges [0.79, 0.90] or [-0.55, -0.40], respectively, does not allow for the construction of a measure of perturbation due to the large (as much as 25%) values of SD(q~)/q~, the details of which have been omitted herein. As a consequence, in appears that the ascending stage, most likely due to its short duration and an unstable behaviour (leading to a weak repeatability), is not as appropriate for the stochastic evaluation of the toxicant-dependent changes as the descending stage, which gives much more information owing to a longer duration and lesser variation of the latter. Only the descending stage therefore will be considered within the flamework of a stochastic approach. There is also formal, mathematical reason for that limitation: on the basis of the A R I M A approach, a unification of the IMA(0, 1, 1) and ARIMA(1, 2, 1) models, corresponding to the descending and ascending stages, respectively, is impossible. Therefore, a model parameter, common for those two stages which would be useful in constructing a measure of perturbation for the whole process, does not exist. A new stochastic measure, however, will be compared with the traditional measure determined for the descending stage, as well as for the whole process. 3. IMA(0,1,1) Model of the Descending Stage of Phagocyte Luminescence. The descending stage of the {nail(t)} process can be modeled as an IMA(0,1, 1) process (Kochel, 1992): (1 --B)ndif(t) = (1 -- O'B)a(t) (1) where a(t) is a value of the white noise process {a(t)} at a moment t, and 0 is a parameter, 0 ~ (0", 1). 0* is the parameter of the IMA(0, 1, 1) model for the {nnat(t)} process, 0* _+ SD(0*) = 0.30 + 0.06. A variance of the white 2 noise o-a(t) has been expressed, according to earlier work (Box and Jenkins, 1976), by the 0 parameter and by a variance of the ((1 - B)ndif(t)} process: 2 O'a(')- Or(1 -B)ndif(t) 1 "~ 0 2 (2) The 0 parameter was determined by minimising the conditional sum of squares of the residuals, S[ Ola(O) = 0] = ~N=lo~z(t), where a(t) = half(t) ndif(t -- 1) + 0" a ( t -- 1). The IMA(0,1,1) model of the descending stage of the {na~f(t)} process was positively verified by the use of the Q-test of goodness of fit, Q = ( N 1) E~= K 1 r~(t)(~-), 2 at a 0.05 significance level (Box and Jenkins, 1976). r~(t)(~') NEW MEASURE OF XENOBIOTIC TOXICITY 901 Table 1. The model parameter, conditional sum of squares of the residuals, and the Q-test of goodness of fit for the IMA(0,1,1) model of the descending stage of the {ndif(t)} process Toxic agent DMSO o-xylene p-xylene Toluene Benzene Phenol IMA(0, 1, 1) parameter 0_+ SD(0) a Conditional sum of squares of residuals S( O l a(O) = O) Q-test of goodness of fit Qb 0.80 +_0.03 0.72 + 0.04 0.68 + 0.04 0.60 + 0.05 0.59 + 0.05 0.59 + 0.05 10.81.106 9.86.106 8.24.106 9.64.106 8.36- 106 10.66.106 30.139 26.850 20.432 30.137 29.984 30.110 aSD = ~(1 - 0 2 ) / ( N - 1). bCritical value of the Q-test is x2.95(19) = 30.144. is the autocorrelation function of the {odD} residual series. Q obeys a X2-probability distribution with K - 1 degrees of freedom. The Q-test takes values shown in Table 1. Another verification was made by using the test of a cumulative periodogram (Kashyap and Rao, 1976; Box and Jenkins, 1976) for the residual series {a(t)} resulting from fitting the IMA(0,1,1) model at 0 = 0.68 to {nail(t)}. The results can be seen in Fig. 3. C(f ) 1 ,o"" oo f B 8.5 Figure 3. Cumulative periodogram C = C ( f ) of the {a(t)} residual PCTS resulted from fitting the IMA(0,1,1) model with 0 = 0.68 to the descending stage of the (rtdif(t)} process. Neutrophils were perturbed by p-xylene. A 95% Kolrnogorov-Smirnov confidence interval for white noise has been marked with dashed lies. 902 B. KOCHEL AND W. SAJEWICZ 1 0 - .............. " ........ " ' - + - ; " ....... ' " ..................... ' '" ....................... I" |': -1 Figure 4. Partial autocorrelation function ~(r, r) of the descending stage of the differentiated {ndtf(t)} process, corresponding to neutrophils perturbed by pxylene. The empirical (solid lines) and theoretical MA(1) (dashed lines) values are shown against a background of the double standard deviation (dotted lines) for white noise autocorrelations. A partial autocorrelation function ~(r, r) of the MA(1) model - 0~(1 - 02) q~(r,r)= 1-02~+1) ' r=l,2,... (3) is compared with an empirical partial autocorrelation function of the {(1-B)ndif(t)} process in Fig. 4. The example refers to neutrophils perturbed by p-xylene. The partial correlation function takes negative values that vanish to zero with a raise of the time lag r. A spectral density function g(f), i.e. a normalised power spectrum, of the MA(1) process (Kochel, 1990b) 2(1 + 02 - 20 cos2~-f) g(f) = ( N - 2)(1 + 0 2) ' ZE [0,0.5], ~.,g(f) / 1 (4) is compared in Fig. 5 with the Fourier spectrum of the {(1--B)ndif(t)} process. Once again, neutrophils perturbed by p-xylene are shown as an example. 4. Stochastic M e a s u r e s of Perturbation and Toxicity. A n IMA(0, 1, 1) process, which describes the PCTS recorded in the time intervals (t, t + At c) separated by the same intervals corresponding to a dead-time intervals At~ ( = Atc) of the recorder, has a memory time T~ (Kochel, 1990a, 1990b): [ ln(1-~) 2 ] ].Atc (S) NEW MEASURE OF XENOBIOTIC TOXICITY 903 9(s 0.041 s B 8.5 Figure 5. Comparison of the normalised empirical (vertical bars) and theoretical (continuous line) power spectra of the descending stage of the {ndif(t)} process. Neutrophils were perturbed by p-xylene. The empirical and theoretical spectra have been calculated for the {(1 - B)ndif(t)} or MA(1) processes, respectively. where 0 and 6 are the IMA(0, 1, 1) parameter and the determination level, 6 ~ [0, 1], respectively. The memory time means the time interval which includes that part of the whole history of the {n(t)} process influencing the present state n(t) at a given determination level. Since the memory time and the perturbation of a light-producing biosystem are inversely proportional (Kochel, 1990a, 1990b), one can expect, because the relationship between toxicity and perturbation is directly proportional, that as the memory time lessens, the toxicity increases. Starting from the premise of proportionality, toxicity r perturbation cz TM1, the following stochastic measures of perturbation and toxicity have been constructed on the basis of the 0 parameter of the IMA(0, 1, 1) process: 1-0 PC - - 1-0" 1 1-0 TC . . . . c 1-0" 100 [%] 100 [ % / m M ] . (6) (7) Since 0 ~ [0", 1], PC is a normalised quantity. By taking into account that the perturbation is a concentration-dependent extensive quantity, the toxicity coefficient, as defined by (7), can be considered as an intensive quantity. Values of the perturbation (PC) and toxicity (TC) coefficients o f the 904 B. K O C H E L A N D W. S A J E W I C Z Table 2. The memorytime, stochasticperturbation and toxicitycoefficientsfor the toxic agents affectingthe phagocyteluminescence;the quantitieshave been determined for the descendingstage of the {ndif(t)} processes Toxic agent DMSO o-xylene p-xylene Toluene Benzene Phenol Agent concentration Memoryt i m e c • SD(c) [mM] TO'99• SD(TM )a 21.0 + 1.0 + 1.0 • 2.5 • 5.0 + 0.20 + 0.1 0.1 0.1 0.1 0.1 0.01 40.3 • 27.0 • 22.9 • 17.0 • 16.5 • 16.5 • 6.9 4.8 3.7 3.0 2.8 2.8 Perturbation coefficient PC • SO(PC) b 28.6 40.0 45.7 57.1 58.6 58.6 + _ _ + • _ 5.0 6.7 7.0 8.7 8.8 8.8 Toxicity coefficient TC • SD(TC)c 1.4 + 0.3 40.0 + 7.8 45.7 + 8.3 22.9 ___3.6 11.7 + 1.8 292.9 + 46.1 a S D ( T M ) = - 2 S D ( 0 ) . ln(1 - 6 ) / ( 0 . lnZ0) 9A t C. b S D ( P C ) = 1 0 0 / ( 1 -- 0 " ) " ( S D 2 ( 0 ) + [(1 - 0 ) / ( 1 - 0")]2- S D 2 ( 0 , ) } 1 / 2 . CSD(TC) = 1 0 0 / [ c - ( 1 - 0 " ) ] - { S D 2 ( 0 ) + [SD2(c)/c 2 + S D 2 ( 0 " ) / ( 1 - 0 " ) 2 ] - ( 1 - 0)} 1/2. compounds affecting the phagocyte activity of neutrophils are compared with those of TM in Table 2. Indeed, as suggested, the stronger the perturbation, the lower the memory time. The above results indicate a certain toxicity hierarchy of the compounds used in the study. Confirmation of that hierarchy would therefore prove the accuracy of the definition of the stochastic toxicity coefficient. To verify the toxicity hierarchy, resulting from the stochastic toxicity coefficient, some parameters that connect a biological (including toxic) activity of compounds with their structure have been employed. They are widely used in QSAR research. 5. Verification of the Stochastic Measure of Toxicity. Since the beginning of this century, it has been known (Hansch and Leo, 1979) that the biological activity, including the toxic one, of many compounds depends on their chemical structure. At present, this forms the basis of quantitative structure-activity relationships (QSAR). Transport of a chemical by cell membranes to the site of action by diffusion is strongly dependent on its lipophilicity; therefore, this is related to its oil-water partition coefficient (Hansch and Leo, 1979; Karcher and Devillers, 1990). Consequently, the octanol-water partition coefficient (P) could be used as a measure for lipophilic properties of the compounds under study. Absorption into an organism, i.e. the uptake and transport, of a chemical usually increases with its partition coefficient. The molar refractivity (RM) , in turn, is useful in describing dispersion forces playing an important role in the interactions of small organic compounds with macromolecular antibodies. It results from the definition of RM and the Clausius-Mosotti equation interrelating an electron polarisability, electron polar•177 and molar refractivity (Hansch and Leo, 1979; Dearden, 1990). NEW MEASURE OF XENOBIOTIC TOXICITY 905 Table 3. Comparison of the stochastic toxicity coefficient TC of benzene homologues with their QSAR parameters Toxicants-benzene homologues Benzene Toluene o-xylene p-xylene Toxicity coefficient TC _+SD(TC) [%/raM] Logarithm of the partition coefficient a Molar refractivity Parachor log p b R Ma pr a 2.13 2.69 2.77 3.15 26.14 31.06 35.77 35.96 206.3 246.9 283.3 283.8 11.7 ___+1.8 22.9 __+3.6 40.0 __+7.8 45.7 __+8.3 aAccording to Hansch and Leo (1979). bCommon logarithm. Power regressions of T C on log P, RM and Pr take the forms TC(log P ) = (0.77 _+ 0.63). (log p)3.62+0.83, T C ( R M) = [(1.69 +_ 1.35). 10 -5 ] - R ~ 11+ 0.23, T C ( P r) = [(0.38 _+ 0.58)" 10 -s ] .p4.1o • 0.2S, r=0.95+0.05 (8a) r = 0.997 _+ 0.003 (8b) r = 0.995 _ 0.005, (8c) whereas analogous exponential regressions are TC(log P ) = (0.62 _+ 0.58). exp[(1.39 _+ 0.34) .log P ] , r = 0.95 _ 0.06 (9a) T C ( R M) = [(0.36 • 0.08). exp[(0.133 • 0.007)'RM], r = 0.997 _ 0.003 (9b) T C ( P r) = [(0.36 • 0.10)- exp[(0.017 -!-_0.001).RM], r = 0.997 _+ 0.004. (9c) O t h e r elementary T C = TC(log P ) regressions take r values not greater than 0.93. For T C = T C ( R M) or T C = T C ( P r) regressions, that value is equal to 0.98. T h e very high positive correlation between T C and RM indicates that the T C variability corresponds very well to the R M variability, and that T C parallels, via RM, dispersive or semi-polar interactions of benzene homologues in neutrophils. T h e high positive correlation of T C with log P indicates that the T C variability is in good correspondence with the variability of log P, and that T C reflects the increasing lipophilicity of the benzene h o m o l o g u e s followed by their increasing toxicity to neutrophils. 906 B. KOCHEL AND W. SAJEWICZ Since a linear regression correlation coefficient between the molar refractivity and the parachor is equal to r = 0.9997 _+ 0.0003, it is sufficient to consider only the trio TC, log P and RM in further calculations of partial and multiple correlation coefficients. Taking into account the power regressions of TC on log P and R M (in the form of InTC=B.e Ax, where X = log P, RM), In TC (as linearly dependent on X, InTC = In B + A .X) has to be considered in calculating partial and multiple correlations within the {ln TC, log P, R~t} system. According to stochastic systemic analysis (Kochel, 1993), the bilateral interrelations in the pairs In TC and log P, or In TC and RM, are described by the partial correlation coefficients equal to rlnTC logP.R = 0.967 + 0.065 or rlnTC R log P : 0.998 + 0.004, respectively. A multiple correlation coefficient, RlnTCqogp = 0.9998 + 0.0003, describes the multilateral interrelation of In TC with l~)~[h log P and R M. This means that the variability of In TC is explained in 99.96% by the variability of both log P and R M. A multiple regression of In TC on log P and R M takes the form : M 9 -- 9 " M' -- l n T C = - 1.015 + 0.264. log P + 0.111 "RM. 9 (10) The 70.36% or 29.64% contributions of log P or RM, respectively, to the variability of In TC result from (10) according to the mutual contribution coefficient as a systemic description of bilateral interrelations (Kochel, 1993). 6. Comparison of the Stochastic and Traditional Toxicity Measures. Since Allen's work on phagocyte luminescence in 1972, a relative luminescence CLre ~ has been used as a measure of perturbation: C L r e I :~-- I p e r t / I n a t (11) where 0 </pert </nat, I .'= EN= 1 n(t). /pert and /nat denote the integral intensity of light corresponding to perturbed or native neutrophils, respectively. As previously, n(t) is the number of photoelectrons detected within the interval (t, t + Ate) , and N is the number of data points in the photoncounting time series {n(t): t = 1, 2,..., N}. Therefore, on the basis of the relative luminescence CLr~~, a directly proportional normalised perturbation measure can now be defined as NCL .'= (1 - CLre~)" 100 [%], (12) which takes NCL = 100% at maximum perturbation, lp~rt = 0 ( C L r e 1 = 0), and the minimum NCL = 0% when the perturbation does not occur, /pert = / n a t ( C L r e l = 1). The following toxicity measure, based on NCL, is thus derived as TCL :-- 1- CLre 1 C 9100 [ % / m M ] (13) NEW MEASURE OF XENOBIOTIC TOXICITY 907 Table 4. The traditional perturbation (NCL) and toxicity (TCL) measures corresponding to the toxic agents affecting the phagocyte luminescence; NCLs and TCLs correspond to the descending stage (D) and the whole (AD) {ndif(t)}process Toxic agent DMSO o-xylene p-xylene Toluene Benzene Phenol aSD(NCL) = Normalised relative luminescence (desc. stage) NCLD _+ SD(NCLD)" [%] Normalised relative luminescence (whole PCTS) NCLAD _+ SD(NCLAD )a [%] Traditional toxicity coefficient (desc. stage) TCLD + SD(TCLD) b [%/raM] Traditional toxicity coefficient (wholePCTS) TCLD _+ 21.6 _+1.0 40.1 _+0.9 6.9 _+1.3 24.1 _+1.2 35.3 _+1.2 49.8 _+0.9 22.5 + 1.3 33.3 _+1.1 28.2 _+1.6 42.2 _+1.4 49.9 _+0.9 49.6 _ 1.0 1.0 _+0.0 40.1 ___4.1 6.9 _ 1.5 9.7 ___0.5 7.1 ___0.2 248.9 _ 25.8 1.1 _+0.0 33.3 _+3.5 28.2 + 3.3 16.9 _+0.8 10.0 _+0.2 247.8 + 27.9 SD(TCLAD )b [%/mM] 104(N/Inat)~/1 + (Ipert/lnat)2 at SD(n(t)) = 100 cps. bSD(TCL) = (1/c2)~/NCL2. SD2(c) + SDZ(NCL) . where c is the toxicant concentration expressed in raM. The T C L values have been calculated both for the descending stage of PCTSs (TCL D) and the whole PCTSs (TCLAo). 6.1. The descending stage of the PCTS. The (TCL o, log P) points (Tables 3 and 4) are approximated by power and logarithmic regressions at the correlation coefficient r +_ SD(r) = 0.17 + 0.49, whereas exponential and linear regressions do this at r _+ S D ( r ) = 0.13 _+ 0.50. On that basis, and owing to the high correlations among log P, RM and Pr, any further consideration of regressions of T C L D on R M or Pr can be omitted. Because of the lack of significant correlation between T C L D and log P on the one hand and the high correlation between the toxicity and lipophilicity on the other, T C L o cannot be accepted as an efficient measure of toxicity. 6.2. The whole PCTS (ascending and descending stages). A m o n g the elementary regressions approximating the (TCLAD , log P ) points, the highest correlations were f o u n d for power (r + SD(r) = 0.87 _+ 0.13) and exponential (r _+ SD(r) = 0.86 _+ 0.14) regressions. Both for power and exponential regressions of TCLAo on R M or Pr, the highest correlation, r _+ SD(r) = 0.99 _+ 0.02, was found. However, the power regression of TCLAo on log 2 P fits the (TCLAD , l o g P ) points worse by (rT2c_ ~oge - r~CL-log e/rTcL-~og ~')" 100 = 19.2% than the analogous regression of TC on log P. 908 B. KOCHEL AND W. S M E W I C Z The toxicity hierarchy resulting from the stochastic PC with the standard deviations regarded, phenol > p-xylene/> o-xylene > toluene > benzene > DMSO, is different from that corresponding to the traditional TCLo: phenol > o-xylene > toluene > benzene >/p-xylene > DMSO, as well as that for the traditional TCLAo: phenol > o-xylene >/p-xylene > toluene > benzene > DMSO. The differences in the xenobiotic toxicity evaluation do not consist just in a transposition between, for instance, o-xylene and p-xylene (as occurs in the case of a whole PCTS), but mainly in the higher multiple correlation between TC and the QSAR parameters of the xenobiotics in comparison with the analogous quantity for TCL and those parameters, as well as in the following. TCL D, in comparison to TC, leads to the underestimation of the phenol toxicity by (TC - TCLD)/TC- 100 = 15.0%, of p-xylene by 84.9%, of toluene by 57.6%, of benzene by 39.3%, and of DMSO by 28.6%. The toxicity of o-xylene is insignificantly overestimated by 0.3%. TCLAo, in comparison to TC, leads to the underestimation of the phenol toxicity by (TC-TCLAD)/TC. 100 = 15.4%, of p-xylene by 38.3%, of toluene by 26.2%, of benzene by 14.5%, of DMSO by 21.4%, and of o-xylene by 16.8%. Consequently, the TCLAD measure of toxicity defined over the whole luminescence process has been shown to be more consistent with the stochastic measure of toxicity, TC, than TCL D. However, in accordance with QSAR verification, performed with the use of benzene homologues, the traditional measure, regardless of the relationship with either the whole process or its descending stage, is considerably less appropriate than the stochastic measure. 7. Final Remarks. The proposed stochastic measure of toxicity is not restricted either to benzene homologues as a perturber/toxicant or to neutrophil phagocytic activity as a target for toxicants. Other luminescent processes generated by various biosystems influenced by a variety of different perturbers can be considered in this way. It is also unnecessary to fulfill the criterion of inhibition, i.e. ndif(t)> 0; then, toxicant-induced stimulation of luminescence instead of inhibition will be evaluated. 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