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Файл:Ординатура / Хирургия / Библиотека им академика М.И. Перельмана / Книга_5993_Библиотеки_им_академика_М_И_Перельмана
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20 Advances in Combinatorial Chemistry & High Throughput Screening, Vol. 1 Lars Carlsen
A
B
0
0,05
0,1
0,15
0,2
0,25
0,3
1234567891011
Abs olute Rank
Probability
0
0,2
0,4
0,6
0,8
1234567891011
Abs olute Rank
Probability
4
7
10
Fig. (5). Probability distribution of A: compounds 4 and 7 and B: compound 10 to occupy specific
absolute ranks (rank 1 and 11 is top and bottom rank respectively).
The compound, X (Fig. 4B) is comparable with compound 4, 7 and 10 and
incomparable with the remaining 7 compounds in the scen ario, the high number
of incomparisons being reflected in the relative broad probability distribution for
compound X (Fig. 6).
0
0,05
0,1
0,15
0,2
1234567891011
Absolute Rank
Probability
Fig. (6). Probability distribution of compound X to occ upy spe cific absolute ra nks.
The probability distribution of compounds X, 4, 7, 10 is depicted in Fig. (7)
strongly indicating that compound X must be located between compound s 4 and 7
and compound 10 (cf. Fig. 4B) [12].
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Assessing Chemicals Advances in Combinatorial Chemistry & High Throughput Screening, Vol. 1 21
0
0,2
0,4
0,6
0,8
1234567891011
Absolute ra nk
Probability
Fig. (7). Probability distribution of compound X in relation to compounds 4, 7 and 10 to occupy
specific absolute ranks.
The behavior of the single compounds under investigation on a combined basis,
e.g., taking all descriptors simultaneously into account was further elucidated
applying the concept of average rank, in the present case as calculated by the
approximate local linear order model (LPOMext) [37]. In Table 2 the average
rank calculated according to eqn. 4 is given together with the ‘uncertainty’, i.e.,
the range wherein the true/absolute rank are to be found.
Table 2. Average Ranks of the 11 Compounds Included in the Hasse Diagram Displayed in Fig. (4B)
Calculated by the Local Partial Order Model (LPOMext) [37]
Compound No. Average Rank (Rk
av
) Uncertainty
1 3,1 1...8
2 8,4 3...11
3 5,3 1...10
4 2,0 1...6
5 6,7 2...10
6 8,4 4...10
7 2,5 1...7
8 3,0 1...8
9 8,3 4...10
10 10,9 10...11
X 7,6 3...10
22 Advances in Combinatorial Chemistry & High Throughput Screening, Vol. 1 Lars Carlsen
The average ranks (Table 2) is in complete agreement with the above conclusion
that the new compound X would be located between the compounds 4 and 7 and
compound 10 (cf. Fig. 4B) [12]. Further the relative broad range (‘uncertainty’) of
the Rkav for X is in perfect agreement with the fact that X is comparable only to
4, 7 and 10 and thus a significant number of incomparisons prevails.
If the average ranks, Rk
av
, of two compounds are close, the two compounds will
on an average basis display similar characteristics based on the set of descriptors
applied. Thus, the study of Carlsen [12] disclosed (Table 2) that compound X
most closely resembles compounds 2, 6 and 9. Consequently, compound X has in
this way obtained an identity compared to the basis set of compounds, i.e.,
compounds 1-10.
3.2. Relative Descriptor Importance
To illustrate the relative importance of the descriptors used for a partial order
ranking, e.g., the sensitivity analysis, we return to the above described study on
the 45 anilines. The result of the sensitivity analysis is depicted in Fig. (8).
Fig. (8). Relative i mp o r tance of the 4 d escripto r s u sed to charact er i ze th e en viron mental beh av io u r
of the 45 aniline s.
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Assessing Chemicals Advances in Combinatorial Chemistry & High Throughput Screening, Vol. 1 23
From Fig. (8) it appear clear that in the present example the descriptors log Kow,
reflecting bioaccumulation and log VP, reflecting persistence apparently are the
most important for the overall ranking the individual relative weights being 0.39
and 0.36, respectively. The relative weights for the biodegradation potential
(BDP2) and the toxicity component, expressed as the PNEC value was found to
be 0.10 and 0.15, respectively. The results are in good agreement with recent
studies on the ranking of a series of PBT compounds [45].
3.3. Hierarchical Partial Order Ranking
Hasse diagrams are characterized by the presence of a number of comparisons and
incomparisons. Sørensen et al. [39] demonstrated that the actual number of
incomparisons is a result of interplay between the number of compounds and the
number of descriptors. Thus, increasing the number of descriptors will, for the
same number of compounds, increase the number of incomparisons. In a simple
example looking at 16 compounds applying 9 descriptors unambiguously shows
the lack of robustness, the Hasse diagram displays a complete anti-chain (Fig. 9).
Fig. (9). Hasse diagram visualizing the partial order ranking of 16 compounds applying 9
descriptors.
Carlsen [14, 46] showed that the adoption of the more elaborate approach,
hierarchical partial order ranking (HPOR) [14] might remedy this problem leading
to a robust model [39].
HPOR involves primary analyses of specific sets of connected descriptors, leading
to latent variables, meta-descriptors derived as th e average ranks. Subsequently,
the set of meta-parameters are used for a second partial order ranking analysis
leading to a final assessment. Thus, HPOR may advantageously be applied to
problems involving a high number of parameters and possibly a relatively low
number of element (here chemicals).
To illustrate the HPOR concept, let the above mentioned 9 descriptors
characterizing the 16 compounds could corr espond to the following: Persistence,
24 Advances in Combinatorial Chemistry & High Throughput Screening, Vol. 1 Lars Carlsen
Bioaccumulation potential, Toxicity, octanol water partitioning, vapor pressure,
Henry’s Law Constant, production volume, production cost and costs of
substitution, respectively. These 9 descriptors may now be collected in 3 sets, i.e.,
PBT characteristics, factors related to solubility and factors related to production
and possible substitution. Subsequent ranking of the 16 compounds based on the 3
sets of 3 descriptors, respectively are illustrated by the Hasse diagrams depicted in
Fig. (10), the corresponding meta-descriptors derived as average ranks according
to eqn. 4 are found in Table 3
Fig. (10). Partial order ranking of the 16 compounds applying based on the 3 sets of 3 descriptors,
e.g., representing A: Persistence, Bioaccumulation potential and Toxicity, B: octanol water
partitioning, vapor pressure and Henry’s Law Constant, and C: Production volume, Production
cost and costs of subst itution.
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Assessing Chemicals Advances in Combinatorial Chemistry & High Throughput Screening, Vol. 1 25
The subsequent ranking based on the 3 meta-descriptors (Table 3) is visualized in
the Hasse diagram in Fig. (10) and the final average ranking being found in Table
4.
Table 3. Meta-Descri p tors Derived as Averaged Ranks (c f. Fig. 9)
Compound No. Meta A (Rk
av
) Meta B (Rk
av
) Meta C (Rk
av
)
1 14,2 5,7 2,8
2 12,8 15,5 12,8
3 3,4 6,8 8,5
4 14,9 12,8 13,6
5 13,6 5,7 3,4
6 3,4 5,7 3,4
7 4,3 14,9 5,7
8 13,6 12,8 5,7
9 2,4 14,6 8,5
10 2,4 11,3 15,8
11 3,4 3,4 2,1
12 13,6 12,8 10,2
13 14,2 3,4 10,2
14 4,3 2,8 6,8
15 14,2 1,4 12,1
16 2,1 4,3 2,8
3.4. Accumulating Partial Order Ranking
The assessment of specific characteristics for chemical substances, like PBT
characteristics may advantageously be conducted applying the accumulating
partial order methodology [15]. In order to increase the basis for the eventual
assessment of the chemical under investigation several “tests” will often be
conducted. Thus, the combined - or accumulated - knowledge based on the results
from the test battery will constitute the eventual support for the eventual
assessment. One possibility to do so would be to adopt the concepts of Bayesian
statistics [47, 48] increasing the predictability of QSARs as a risk assessment tool.
However, applying this methodology to multicriteria analysis is not directly
intelligible. In contrast to this it appears that partial order ranking can be applied
in a straightforward way.
26 Advances in Combinatorial Chemistry & High Throughput Screening, Vol. 1 Lars Carlsen
Table 4. Final Averaged Ranks Following HPOR (cf. Fig. 10)
Compound No. Final Averaged Ranks (Rk
av
)
1 10,2
2 15,1
3 8,5
4 15,7
5 8,5
6 4,6
7 11,3
8 10,6
9 8,5
10 11,3
11 1,4
12 13,6
13 10,2
14 2,8
15 5,7
16 1,3
Fig. (11). Partial order ranking of the 16 compounds based on the 3 meta-descriptors given in
Table 3.
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Assessing Chemicals Advances in Combinatorial Chemistry & High Throughput Screening, Vol. 1 27
The applicability of APOR was demonstrated in a study inve stigating a series of
compounds to establish their possible classification as PBT compounds [15] based
on 5 different tests. To be classified a s a PBT compound, the compound must b e
environmentally persistent (P), bioaccumulating (B) and toxic (T) [49].
Consequently, based a single test for a given compound can be classify the
compound as PBT only if P, B and T, respectively, are found “true”, i.e., all three
descriptors, p
i
, b
i
and t
i
, (vide supra) are assigned the value 1. The Hasse diagrams
of the single tests will obviously be equal to that of the event space (Fig. 2), the
single elements in the diagram reflecting equivalent classes of compounds.
Obviously the analysis, based on only one single test is rather rigid as the single
compounds are denoted PBT (S = 1), i.e., all three demands are fulfilled or not (S
= 0), where one or more of the demands are not fulfilled, respectively (cf. eqn. 4).
The APOR approach follows partially the r ecently published evaluation method
METEOR [50, 51], where the descriptors are step-wise aggregated; hence the
effect of weights can be controlled and traced back. The advantage of APOR and
METEOR is that there is an enrichment of order relations. The disadvantage of
both methods is that now some descriptors can be considered as substitutes for
others.
In this case only compounds placed in the top level of the Hasse diagram, i.e., the
(1, 1, 1) equivalent class will in this case be depicted as PBT's. A much more
subtle picture develops following accumulation of the results obtained through the
further tests, which is obvious from the Hasse diagrams displayed in Fig (12)
corresponding to accumulating data from 2, 3, 4 and 5 tests, respectively, the
results being adopted from a recent study including 30 compounds subjected to 5
tests (for illustrative purposes the corresponding data wer e randomly generated)
[15].
The enrichments of the Hasse diagrams (Fig. 12) compared to the event space
(Fig. 2) are obvious. T he accumulated descriptors may take any value between 0
and M.
The analyses of the 4 partial order ed rankings visualized by the Hasse diagrams
(Fig. 12) are summarized in Table 5, reflecting the much more subtle approach to
28 Advances in Combinatorial Chemistry & High Throughput Screening, Vol. 1 Lars Carlsen
the evaluation of PBT substances. The average ranks of the single compounds as
well as the probabilities, ranging between 0 and 1 (cf. eqn. 4), for a given
compound to be classified as PBT.
Fig. (12). Hasse diagrams following consecutive inclusion of Tests-2 to Test-5: A: After Tests 1
and 2, B: After Tests 1, 2 and 3, C: After Tests 1, 2, 3 and 4, and D: After Tests 1, 2, 3, 4 and 5.
4. DISCUSSION
The above discussion has illustrated how partial order ranking methodologies
advantageously may be applied as decision support tools in relation to the
assessment of chemicals. However, it is important to stress that partial order
ranking only constitutes as a decision support help, as the techniques
unambiguously suffer from potential drawbacks.
Obviously the number of incomparisons constitutes a major problem applying
partial order ranking. This is typically a result of the interplay between the nu mber
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Assessing Chemicals Advances in Combinatorial Chemistry & High Throughput Screening, Vol. 1 29
Table 5. Ranking Analysis of 30 Arbitrary Compounds Plus Artificially Introduced Maximum and
Minimum References After Accumul ation
After Tests 1 and 2 After Test s 1, 2 and 3 After Te s ts 1, 2, 3 and 4 After Tes ts 1, 2, 3, 4 and 5
ID
HD Level
a
Rk
av
b
S
c
HD Level
a
Rk
av
b
S
c
HD Level
a
Rk
av
b
S
c
HD Level
a
Rk
av
b
S
c
max 1 2.0 1.000 1 1.0 1.000 1 1.0 1.000 1 1.0 1
1 2 4.3 0.500 2 4.4 0.333 2 5.2 0.281 2 3.1 0.288
2 3 13.2 0.000 4 15.7 0.148 4 13.2 0.094 2 4.1 0.192
3 6 29.7 0.000 6 29.5 0.037 6 30.7 0.031 7 30.6 0.032
4 4 21.2 0.000 4 15.7 0.148 5 23.4 0.063 4 12.2 0.144
5 1 2.0 1.000 3 9.0 0.296 3 11.0 0.188 2 2.9 0.288
6 4 23.4 0.125 6 29.5 0.037 4 19.3 0.125 5 27.2 0.096
7 5 28.1 0.000 6 29.5 0.037 6 30.7 0.031 6 28.1 0.048
8 4 23.4 0.125 3 9.0 0.296 2 5.2 0.281 2 4.7 0.256
9 4 23.4 0.125 4 15.7 0.148 4 19.3 0.125 4 17.6 0.144
10 5 24.8 0.000 6 27.8 0.000 6 29.7 0.000 7 29.5 0.000
11 7 32.0 0.000 7 30.7 0.000 6 30.7 0.031 6 27.8 0.048
12 4 23.4 0.125 6 29.5 0.037 6 30.3 0.031 5 28.9 0.048
13 5 28.7 0.000 6 29.5 0.037 6 30.7 0.031 5 23.1 0.096
14 4 23.4 0.125 5 19.8 0.074 4 19.3 0.125 5 27.2 0.096
15 4 18.6 0.000 3 18.9 0.000 4 15.4 0.094 5 23.1 0.096
16 3 10.4 0.250 3 11.8 0.111 3 9.0 0.125 2 11.0 0.120
17 3 8.6 0.250 3 5.2 0.222 2 11.0 0.125 5 18.0 0.128
18 2 4.7 0.500 2 2.9 0.444 2 4.3 0.281 2 3.9 0.256
19 2 4.7 0.500 2 3.1 0.333 2 6.0 0.188 3 8.8 0.192
20 3 9.6 0.250 4 15.7 0.148 4 19.3 0.125 4 14.7 0.144
21 3 9.6 0.250 4 15.7 0.148 3 7.5 0.188 3 8.8 0.192
22 4 22.0 0.000 3 7.1 0.222 3 11.0 0.188 3 9.4 0.192
23 5 28.1 0.000 6 29.5 0.037 5 27.5 0.063 5 27.2 0.096
24 4 18.9 0.000 5 26.4 0.000 2 11.0 0.125 2 12.4 0.120
25 4 23.4 0.125 3 9.0 0.296 2 4.3 0.281 3 6.3 0.216
26 4 18.6 0.000 4 24.8 0.000 5 19.8 0.047 5 23.1 0.096
27 4 22.0 0.000 4 12.4 0.111 3 20.6 0.063 5 27.0 0.032
28 4 18.9 0.000 4 18.9 0.000 2 11.0 0.125 2 12.4 0.12
29 2 4.0 0.500 2 3.7 0.333 2 3.3 0.281 4 12.2 0.144
30 3 10.4 0.250 2 3.0 0.444 2 3.9 0.250 3 11.0 0.192
min 7 32.0 0.000 8 32.0 0.000 7 32.0 0.000 8 32.0 0
a
cf. Hasse diagram Fig. (1),
b
calculated by eqn. 3,
c
calculat ed by eqn. 4
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