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A graph-theoretical tool for priority setting of chemicals
Chemosphere,Vol.27,No.9, pp 1729-1736, 1993 Printed in Great Brilain
A GRAPH-THEORETICAL
0045-6535/93 $6.00 + 0.00 Pergamon Press Ltd.
TOOL FOR PRIORITY SETTING OF CHEMICALS
R. Briiggemann*, B. Miinzer GSF-Forschungszentrum fiir Umwelt und Gesundheit Ingolst~idter Landstr. 1 8042 Neuherberg Federal Republic of Germany (Receivedin Germany 5 May 1993; accepted21 July 1993)
ABSTRACT Ranking of chemicals according to their environmental hazard is an important step in any risk assessment. Instead of using a ranking index which aggregates different substance properties a graph theoretical (Hasse-diagrams) tool is presented in this paper, which preserves almost all features of the ranking index but shows also additional insights useful for priority setting.
KEYWORDS Hazard potential of chemicals; priority setting; Hasse-diagrams.
INTRODUCTION In many cases it is necessary to select chemicals according to their potential environmental hazard. This often is the basis for further investigations, which may be time consuming and expensive. Therefore a priority setting procedure which first ranks the chemicals and helps to decide which chemicals are of high priority, may save time and money.
* To whom reprint requests should be sent.
1729
1730
The problem Chemicals have different aspects which we generally call p l"'"'Pn" These aspects may contribute in different ways to the hazard potential. However for the ranking, a map • is to be introduced which assigns the tuple (Pl ..... Pn) to an index tp, which is used as an order index to perform the ranking. The ranking is based on the relation (q0, ~). Sometimes the idea to construct ~ is simply given by qualitative arguments. For example to introduce cp according to the hazard potential for marine environments [1], two aspects are considered: (1)
The higher the concentration c of a chemical within the water body, the higher the hazard potential. Therefore:
(p
(2)
~
C
The higher the ecotoxicity T of a chemical the higher the hazard potential. Therefore: cp -
T
To relate c and T to purely chemical substance properties, the following proportionalities are considered. (3)
c ~ 1/H
under constant environmental conditions. H the Henry law coefficient (atm. m3/mol)
(4)
T-
1/LC50 under constant ecosystems conditions. LC50 the letal concentration for 50 % of the test species (rag/])
The hazard potential cp as function of c and T may be given by any monotonously increasing function with respect to c and to T or with respect to 1/H and 1/LC50" For example, as suggested in a paper by Fiedler [1] cp was defined as follows: cp: = 1/(H" LCso ) Thus the needed map ~, a special realization of many others is given. Within the paper [1] qDis called F, the "hazard factor". Although the dimension F is somewhat strange, the construction does its job, and a ranking can be performed. 23 pesticides were ranked according to their LC50- and H-values aggregated to F. The following list gives the chemicals and the LC50-. und H-values. They are already ordered according to their F-values.
1731
Table 1: Marine environmental hazard potential of pesticides
No:
1
substance
- 10log LC50 (mg/1)
Endrin
- 10log H
(atm" m3/mol)
4.22
8.12 9.74
2
Captafol
1.68
3
Prochloraz
0.00
10.9
4
Lindane
3.00
7.29
5
Dieldrin
2.10
8.12
6
DDT
1.1
7.08
7
Chlordane
2.85
5.03
8
Captan
2.47
5.88
9
Pentachlorophenol
0.77
6.18
10
Diuron
- 0.60
7.59
11
Propanil
0.45
5.88
12
Bromophos
1.30
5
13
Trichlorophenol, 2,4,6-
1.00
4.81
14
Dicofol
1.28
4.38
15
Aldrin
1.75
2.88
16
Hexachlorobenzene
1.30
3.29
17
2,4-DB
- 0.30
4.87
18
Trichiorophenol, 2,4,5-
- 0.49
4.77
19
Dichlorophenol, 2,4-
- 0.87
3.96
20
2,4-D
0.00
2.85
21
MCPA
- 2.365
3.52
22
Tetrachlorophenoi, 2,3,4,6
0.602
1.17
23
Allyl chloride
-1.00
2.49
Advantages of using a ranking index q~are: .
Lucidity
*
Simplicity in calculation
Disadvantages are *
Arbitrariness of the construction. Why not for example cp : = 1/(LC50) a • (H) b a,b some arbitrary exponents unequal to 1 and > 0.
.
Loss of information, because • maps a multidimensional tuple to a one-dimensional index. A high value of F as in [1] may come from an extraordinary low H-value, or (exclusively) from an extraordinarily low
1732
value of LC50, or finally from moderately low values of both H and LC50.
Hasse-diagram-technique as an alternative The application of Hasse-diagrams is well described in the literature [2-5]. However a simple example may be the best way to show, how Hasse-diagrams are used. Considering for example four arbitrarity selected chemicals of table 1 (No. 8, 12, 14, 15), which are described by two attributes log(1/LC50) and log(l/H). A graph is drawn with the objects in mind: This means in our context, four chemicals are drawn as four points within the diagram (graph theoretical: "vertices" or "knots"). These four chemicals are connected by lines (graph theoretical: "edges"), if it is valid:
(5)
Chemical (j) -: chemical (i): ~,
Each attribute ofj is lower or equal than that ofi. Or in short hand
notation: j .~ i : '~ PIXj ": PIXi for all considered attributes Ix = 1.... 1 if 1 attributes are to be considered. Here:
I =
2
These lines have an orientation, which could be denoted by small arrows, or more clearly by drawing within the figure the point (knot), representing the chemical (j) lower than that of chemical (i). (6)
Furthermore the diagram can be simplified, if lines which result from transitivity of the "-:"-relation are omitted. This means: Let be k
.~ j
.~ i then:
line (k,j) between k andj line (j,i) between j and i and because of the transitivity: line (k,i) between k and i, which however is no___~tdrawn. (7)
For reasons of clearness, the knots are drawn as symmetrical as possible. In general some arbitrariness of the location of the knots still may remain. But according to rule 5 the essential information is given by the existence or non existence of lines.
(8)
If hazard potentials are to be presented, it is a good policy, to locate knots as high as possible in order to be on the conservative side. (On the assumption that high values of attributes indicate a high contribution to the hazard potential.)
With the rules (5) and (8) the Hasse-diagram representing the so called "partial order set" (poset) of the four chemicals (short-hand notation: P4 -- {8, 12, 14, 15}, -: )) can be drawn
1733
8
12
I
J\
high values of log (1/LC50) and log (l/H) = high hazard potential
15
14
low values of log (1/LC50) and log (1/1-I) = low hazard potential
Fig. 1: Hasse diagram of the poset P4 (see text)) Because of rule 7 the diagram has three levels. Level 1 includes chemical 8 as the most hazardous one. Level 2 includes chemicals 12 and 15. These chemicals have both lower values in all their attributes than chemical 8. Chemical 12 has higher values than chemical 14. However both chemicals of this level (chemical 12 and 15) cannot be compared with each another, because the effect potential of chemical 12 has a lower value than chemical 15, but the exposure hazard potential of chemicals 12 is higher than that of chemical 15. Analogous arguments set up for chemicals 15 and 14 (see below). The location of chemical 15 within the figure obeys the rules 7 and 8. Level 3 includes chemical 14 as the chemical with the lowest environmental hazard compared with the two chemicals 12 and 8. However, it cannot be compared with chemical 15. Its exposure potential is higher than that of 15, but the ecotoxicity is slightly lower. Therefore, without any deterministic model which relates the LCvalues with the environmental concentration, it may be better to show this incomparability than to hide it by aggregating the two attributes to a single value tp. Note that the line connecting 8 and 14 would be correct, but it is omitted, according to rule 6. Clearly, chemicals which are connected by lines, like 14, 12 and 8 are also ranked by the same order by all single ranking indices, which are strictly monotonous functions of F. Therefore the incomparabilities like those between 12 and 151 14 and 15 give the interesting additional information beyond the ranking performed with the help of an index q~.
The full Hasse-diagram of 23 pesticides Figure 2 shows the full Hasse-diagram of the 23 pesticides, which is drawn by application of rules 5 to 8.
1734
Fig. 2: Full Hasse-diagram of all 23 chemicals of table 1
1735
The most hazardous chemicals are 1, 2 and 3, because there are no further chemicals which have higher values with respect to both attributes. Their low identifying numbers show that using the factor F they also are ranked high. However, these three chemicals are incomparable: Chemical 1 is a "maximal" (this notation comes from lattice theory and is immediately understood by optical inspection), because of its high ecotoxicity. Chemical 2 and 3 however are on the top of the diagram (maximals too) because of their rather high log (1/H)-value, representing a high exposure hazard potential and moderately low toxicity values. Chemical 3, having a value log (1/LC50) = 0 but the highest value of log (I/H) under all 23 chemicals is a maximal, but it represents only few comparable chemicals. This means, only few chemicals are reachable downward from chemical 3 having values lower than log(1/LC50 ) = 0 an_.__d_dlog(l/H) = 10.90 (namely 7 chemicals). This set of objects which are reachable from a given object in downwards direction is called "down set" [4]. The second level contains the chemical 4 and 5, which again are not comparable to each other and beyond this incomparable with the maximals: chemical 2, and chemical 3. This fact shows that incomparabilities do not appear only between chemicals of the same level but also between different levels. Looking downward, the "minimals" of the diagram are the chemicals 21 and 23. They are the chemicals with the lowest environmental hazard potential. However they are incomparable. This illustrates the fact that different data structures characterizes these two chemicals. Table 1 shows the reason: Chemical 21 has a very low value with respect to its ecotoxicity but a rather high value with respect to its exposure potential. Chemical 23 shows the reverse effects. Another minimal is chemical 22. It cannot be compared with 21 and 23 because its toxicity is higher than that of 21 and 23. But its exposure hazard potential is remarkably lower than that of chemicals 21 and 23. By rule 8, the chemical 22 is located at a high level. Finally "chains" may be discussed. Chaines are sequences of chemicals (in general objects) which are connected by lines and following the lines in strictly downwards - or upwards - direction. For example 2 1 - 1 9 - 1 8 - 1 7 - 1 1 - 9 - 6 - 4 - 1
or 2 2 - 1 6 - 1 2 - 7 - 4 - 1
isachain.
By rule (5) every object of a chain is comparable to every other object. The fact that many different chains can be found, going from the minimals to the maximals again demonstrates the incomparabilities, which are hidden by the use of (one dimensional) ranking indices.
DISCUSSION Some disadvantages of Hasse-diagrams should be mentioned: Small differences without physical significance are treated as real ones. This however can be avoided, using scores instead of the physical values. Sometimes some objects (here: chemicals) are incomparable to all others. Here the interpretation will be difficult.
1736
*
The diagram may be overburdened by a complex system of lines. Hereto the author and E. Halfon have developed appropriate tools, which will be published in the near future.
However there are some definite advantages: *
The ranking by Hasse-diagrams gives almost all information using a ranking index q~.
*
Hasse diagrams show additionally that different reasons lead to their location within a priority setting i.e.: Incomparabilities show the reverse order of attributes which would not be discovered otherwise.
*
Data structures are outlined by analyzing the down sets. For example all chemicals are easily identified for which 10log LCso < 0 and at the same time 10log H < 10.9.
REFERENCES [1]:
H. Fiedler: Nachrichten aus Chemie, Technik und Laboratorium 1_, 88 - 89, 1990
[2]:
E. Halfon: Environ. Science Technol. ~
[3]:
E. Halfon, M.G. Reggiani: Environ. Science Technol. 2_.0,01173 - 1179
[4]:
R. Briiggemann, J. Altschuh: Sci. Tot. Environ. 109/110, 41 - 57, 1991
[5]:
B.A. Davey, H.A. Priestley: Introduction to Lattices and Order. Cambridge University Press, Cambridge, 1990
600 - 609, 1989
A GRAPH-THEORETICAL
0045-6535/93 $6.00 + 0.00 Pergamon Press Ltd.
TOOL FOR PRIORITY SETTING OF CHEMICALS
R. Briiggemann*, B. Miinzer GSF-Forschungszentrum fiir Umwelt und Gesundheit Ingolst~idter Landstr. 1 8042 Neuherberg Federal Republic of Germany (Receivedin Germany 5 May 1993; accepted21 July 1993)
ABSTRACT Ranking of chemicals according to their environmental hazard is an important step in any risk assessment. Instead of using a ranking index which aggregates different substance properties a graph theoretical (Hasse-diagrams) tool is presented in this paper, which preserves almost all features of the ranking index but shows also additional insights useful for priority setting.
KEYWORDS Hazard potential of chemicals; priority setting; Hasse-diagrams.
INTRODUCTION In many cases it is necessary to select chemicals according to their potential environmental hazard. This often is the basis for further investigations, which may be time consuming and expensive. Therefore a priority setting procedure which first ranks the chemicals and helps to decide which chemicals are of high priority, may save time and money.
* To whom reprint requests should be sent.
1729
1730
The problem Chemicals have different aspects which we generally call p l"'"'Pn" These aspects may contribute in different ways to the hazard potential. However for the ranking, a map • is to be introduced which assigns the tuple (Pl ..... Pn) to an index tp, which is used as an order index to perform the ranking. The ranking is based on the relation (q0, ~). Sometimes the idea to construct ~ is simply given by qualitative arguments. For example to introduce cp according to the hazard potential for marine environments [1], two aspects are considered: (1)
The higher the concentration c of a chemical within the water body, the higher the hazard potential. Therefore:
(p
(2)
~
C
The higher the ecotoxicity T of a chemical the higher the hazard potential. Therefore: cp -
T
To relate c and T to purely chemical substance properties, the following proportionalities are considered. (3)
c ~ 1/H
under constant environmental conditions. H the Henry law coefficient (atm. m3/mol)
(4)
T-
1/LC50 under constant ecosystems conditions. LC50 the letal concentration for 50 % of the test species (rag/])
The hazard potential cp as function of c and T may be given by any monotonously increasing function with respect to c and to T or with respect to 1/H and 1/LC50" For example, as suggested in a paper by Fiedler [1] cp was defined as follows: cp: = 1/(H" LCso ) Thus the needed map ~, a special realization of many others is given. Within the paper [1] qDis called F, the "hazard factor". Although the dimension F is somewhat strange, the construction does its job, and a ranking can be performed. 23 pesticides were ranked according to their LC50- and H-values aggregated to F. The following list gives the chemicals and the LC50-. und H-values. They are already ordered according to their F-values.
1731
Table 1: Marine environmental hazard potential of pesticides
No:
1
substance
- 10log LC50 (mg/1)
Endrin
- 10log H
(atm" m3/mol)
4.22
8.12 9.74
2
Captafol
1.68
3
Prochloraz
0.00
10.9
4
Lindane
3.00
7.29
5
Dieldrin
2.10
8.12
6
DDT
1.1
7.08
7
Chlordane
2.85
5.03
8
Captan
2.47
5.88
9
Pentachlorophenol
0.77
6.18
10
Diuron
- 0.60
7.59
11
Propanil
0.45
5.88
12
Bromophos
1.30
5
13
Trichlorophenol, 2,4,6-
1.00
4.81
14
Dicofol
1.28
4.38
15
Aldrin
1.75
2.88
16
Hexachlorobenzene
1.30
3.29
17
2,4-DB
- 0.30
4.87
18
Trichiorophenol, 2,4,5-
- 0.49
4.77
19
Dichlorophenol, 2,4-
- 0.87
3.96
20
2,4-D
0.00
2.85
21
MCPA
- 2.365
3.52
22
Tetrachlorophenoi, 2,3,4,6
0.602
1.17
23
Allyl chloride
-1.00
2.49
Advantages of using a ranking index q~are: .
Lucidity
*
Simplicity in calculation
Disadvantages are *
Arbitrariness of the construction. Why not for example cp : = 1/(LC50) a • (H) b a,b some arbitrary exponents unequal to 1 and > 0.
.
Loss of information, because • maps a multidimensional tuple to a one-dimensional index. A high value of F as in [1] may come from an extraordinary low H-value, or (exclusively) from an extraordinarily low
1732
value of LC50, or finally from moderately low values of both H and LC50.
Hasse-diagram-technique as an alternative The application of Hasse-diagrams is well described in the literature [2-5]. However a simple example may be the best way to show, how Hasse-diagrams are used. Considering for example four arbitrarity selected chemicals of table 1 (No. 8, 12, 14, 15), which are described by two attributes log(1/LC50) and log(l/H). A graph is drawn with the objects in mind: This means in our context, four chemicals are drawn as four points within the diagram (graph theoretical: "vertices" or "knots"). These four chemicals are connected by lines (graph theoretical: "edges"), if it is valid:
(5)
Chemical (j) -: chemical (i): ~,
Each attribute ofj is lower or equal than that ofi. Or in short hand
notation: j .~ i : '~ PIXj ": PIXi for all considered attributes Ix = 1.... 1 if 1 attributes are to be considered. Here:
I =
2
These lines have an orientation, which could be denoted by small arrows, or more clearly by drawing within the figure the point (knot), representing the chemical (j) lower than that of chemical (i). (6)
Furthermore the diagram can be simplified, if lines which result from transitivity of the "-:"-relation are omitted. This means: Let be k
.~ j
.~ i then:
line (k,j) between k andj line (j,i) between j and i and because of the transitivity: line (k,i) between k and i, which however is no___~tdrawn. (7)
For reasons of clearness, the knots are drawn as symmetrical as possible. In general some arbitrariness of the location of the knots still may remain. But according to rule 5 the essential information is given by the existence or non existence of lines.
(8)
If hazard potentials are to be presented, it is a good policy, to locate knots as high as possible in order to be on the conservative side. (On the assumption that high values of attributes indicate a high contribution to the hazard potential.)
With the rules (5) and (8) the Hasse-diagram representing the so called "partial order set" (poset) of the four chemicals (short-hand notation: P4 -- {8, 12, 14, 15}, -: )) can be drawn
1733
8
12
I
J\
high values of log (1/LC50) and log (l/H) = high hazard potential
15
14
low values of log (1/LC50) and log (1/1-I) = low hazard potential
Fig. 1: Hasse diagram of the poset P4 (see text)) Because of rule 7 the diagram has three levels. Level 1 includes chemical 8 as the most hazardous one. Level 2 includes chemicals 12 and 15. These chemicals have both lower values in all their attributes than chemical 8. Chemical 12 has higher values than chemical 14. However both chemicals of this level (chemical 12 and 15) cannot be compared with each another, because the effect potential of chemical 12 has a lower value than chemical 15, but the exposure hazard potential of chemicals 12 is higher than that of chemical 15. Analogous arguments set up for chemicals 15 and 14 (see below). The location of chemical 15 within the figure obeys the rules 7 and 8. Level 3 includes chemical 14 as the chemical with the lowest environmental hazard compared with the two chemicals 12 and 8. However, it cannot be compared with chemical 15. Its exposure potential is higher than that of 15, but the ecotoxicity is slightly lower. Therefore, without any deterministic model which relates the LCvalues with the environmental concentration, it may be better to show this incomparability than to hide it by aggregating the two attributes to a single value tp. Note that the line connecting 8 and 14 would be correct, but it is omitted, according to rule 6. Clearly, chemicals which are connected by lines, like 14, 12 and 8 are also ranked by the same order by all single ranking indices, which are strictly monotonous functions of F. Therefore the incomparabilities like those between 12 and 151 14 and 15 give the interesting additional information beyond the ranking performed with the help of an index q~.
The full Hasse-diagram of 23 pesticides Figure 2 shows the full Hasse-diagram of the 23 pesticides, which is drawn by application of rules 5 to 8.
1734
Fig. 2: Full Hasse-diagram of all 23 chemicals of table 1
1735
The most hazardous chemicals are 1, 2 and 3, because there are no further chemicals which have higher values with respect to both attributes. Their low identifying numbers show that using the factor F they also are ranked high. However, these three chemicals are incomparable: Chemical 1 is a "maximal" (this notation comes from lattice theory and is immediately understood by optical inspection), because of its high ecotoxicity. Chemical 2 and 3 however are on the top of the diagram (maximals too) because of their rather high log (1/H)-value, representing a high exposure hazard potential and moderately low toxicity values. Chemical 3, having a value log (1/LC50) = 0 but the highest value of log (I/H) under all 23 chemicals is a maximal, but it represents only few comparable chemicals. This means, only few chemicals are reachable downward from chemical 3 having values lower than log(1/LC50 ) = 0 an_.__d_dlog(l/H) = 10.90 (namely 7 chemicals). This set of objects which are reachable from a given object in downwards direction is called "down set" [4]. The second level contains the chemical 4 and 5, which again are not comparable to each other and beyond this incomparable with the maximals: chemical 2, and chemical 3. This fact shows that incomparabilities do not appear only between chemicals of the same level but also between different levels. Looking downward, the "minimals" of the diagram are the chemicals 21 and 23. They are the chemicals with the lowest environmental hazard potential. However they are incomparable. This illustrates the fact that different data structures characterizes these two chemicals. Table 1 shows the reason: Chemical 21 has a very low value with respect to its ecotoxicity but a rather high value with respect to its exposure potential. Chemical 23 shows the reverse effects. Another minimal is chemical 22. It cannot be compared with 21 and 23 because its toxicity is higher than that of 21 and 23. But its exposure hazard potential is remarkably lower than that of chemicals 21 and 23. By rule 8, the chemical 22 is located at a high level. Finally "chains" may be discussed. Chaines are sequences of chemicals (in general objects) which are connected by lines and following the lines in strictly downwards - or upwards - direction. For example 2 1 - 1 9 - 1 8 - 1 7 - 1 1 - 9 - 6 - 4 - 1
or 2 2 - 1 6 - 1 2 - 7 - 4 - 1
isachain.
By rule (5) every object of a chain is comparable to every other object. The fact that many different chains can be found, going from the minimals to the maximals again demonstrates the incomparabilities, which are hidden by the use of (one dimensional) ranking indices.
DISCUSSION Some disadvantages of Hasse-diagrams should be mentioned: Small differences without physical significance are treated as real ones. This however can be avoided, using scores instead of the physical values. Sometimes some objects (here: chemicals) are incomparable to all others. Here the interpretation will be difficult.
1736
*
The diagram may be overburdened by a complex system of lines. Hereto the author and E. Halfon have developed appropriate tools, which will be published in the near future.
However there are some definite advantages: *
The ranking by Hasse-diagrams gives almost all information using a ranking index q~.
*
Hasse diagrams show additionally that different reasons lead to their location within a priority setting i.e.: Incomparabilities show the reverse order of attributes which would not be discovered otherwise.
*
Data structures are outlined by analyzing the down sets. For example all chemicals are easily identified for which 10log LCso < 0 and at the same time 10log H < 10.9.
REFERENCES [1]:
H. Fiedler: Nachrichten aus Chemie, Technik und Laboratorium 1_, 88 - 89, 1990
[2]:
E. Halfon: Environ. Science Technol. ~
[3]:
E. Halfon, M.G. Reggiani: Environ. Science Technol. 2_.0,01173 - 1179
[4]:
R. Briiggemann, J. Altschuh: Sci. Tot. Environ. 109/110, 41 - 57, 1991
[5]:
B.A. Davey, H.A. Priestley: Introduction to Lattices and Order. Cambridge University Press, Cambridge, 1990
600 - 609, 1989