Application of the Theory of Graphs to the Optimization of Chromatographic Separation Schemes for Multicomponent Samples Desire L. Massart, Conny Janssens, Leonard Kaufman,’ and Robert Smits Pharmaceutical Institute, Vrije Universiteit Brussel, Paardenstraat 67, B 1640 Sint Genesius Rode, Belgium THE CHROMATOGRAPHIC SEPARATION of mixtures of several metal ions into the individual components can usually be achieved in several ways. Extensive schemes have, for example, been published by Aubouin and Laverlochbre (1) and by many others (2-11). These schemes often include extraction, precipitation, and distillation steps as well. When a combination of several metal ions for which no published separation method is available, has t o be separated, a lot of work is often needed to find a n acceptable flow scheme and the scheme found is not necessarily the best or fastest possible. The analytical chemist has t o his disposition a n enormous quantity of data such as distribution or extraction coefficients (in reversed phase partition chromatography). His problem is first to find flow schemes which will permit the separation in question and then, among these possibilities, to choose the best or fastest method. The number of possible combinations which have t o be considered and the number of data from which these possibilities have to be extracted, call for a computer method. Massart et al. (12) described the use of computers in the elaboration of separations of inorganic ions by ion exchange and, more particularly, their use as a n aid in documentation through the constitution of data and literature reference files. A method for the choice of the optimal procedure for the separation of two ions was described. Moreover, the Italian coauthors of this article reported on a program for the choice of possible methods for the separation with inorganic filters of one ion from several others. However when one applies their method t o ionexchange separations, one does not always obtain a useful method and when a possibility is found, it is not necessarily the best one. Until now, no procedure for the separation of more than two ions from each other was published. We have found that it is possible t o represent such separations as graphs and to find the optimal (shortest) path through these networks using algorithms designed for such a purpose. 1 Present address, Centre of Operational Research and Statistics, Vrije Universiteit Brussel, Terhulpse Steenweg 160. B-1050 Brussels, Belgium.
( 1 ) G. Aubouin and J. Laverlochkre, Rripport C.E.A., No. 2359 (1 963). (2) J. C. Ricq. J . Radioaiiul. Ciiem., 1, 443 (1968). (3) G. H. Morrison, J. T. Gerard, A. Travesi, R. L. Currie, S. F. Peterson, and N. M. Potter, ANAL.CHEkt., 41, 1633 (1969). (4) P. 0. Wester, D. Brune, and K. Samsahl, J . Appl. Radiut. Isotopes, 15, 59 (1964). (5) V. V. Moiseew: R. A. Kuznetzov, and A . I. Kalinin, Verres Rr:fi.., 5, 374 (1963). ( 6 ) P. Van den Winkel, A. Speecke, and J. J. Hoste, “Nuclear Activation Techniques in the Life Sciences,” IAEA. 1967, p 159. (7) K. Samsahl, Amilysr (Loiidon), 93, 101 (1968). (8) F. Girardi and R . Pietra, ANAL.CHEM., 35, 173 (1963). (9) A. Fourcy, N . Neuburger, C. Garrec, A. Fen, and J. P. Garrec, Prnc. Modcrii Trriids Acticurioii Aiiafysis, 1968, 160. ( I O ) J. S. Fritz and G. L. Latwesen, Trrlo/itu, 17, 81 (1970). ( 1 1 ) F. De Corte, A. Speecke, and J. Hoste, J . Radioamd. Chenz., 8,
287 (1971). (12) D. L. Massart, J. Hoste, F. Girardi, G. Guzzi, G. Di Cola, G. Aubouin, and E. Junod, J . Chronzatogr., 45, 453 (1969).
DATA SEARCH .4ND CONSTRUCTION OF A WEIGHTED GRAPH Let us consider the separation of three metal ions, A, B, and C and the successive steps which can be carried out t o achieve their separation: Step 1. The three elements are brought together on an ion-exchange column. Notation used in the graph (Figure 1): ABC//. Step 2. There are two possibilities: (a) Elution of one element. This element is now definitively separated from the others. This leads to one of three situations, AB//C, AC//B, or CB//A. (b) Elution of two elements together. This leads to A//BC, B//AC, or C//AB. Step 3. (a) If in step 2, a situation such as AB//C was reached, one now has to elute either A or B. The result from AB/;C is Ai/B/C or B//A/C. This notation means that B and C or A and C are now definitively separated and eluted. (b) If in step 2 , a situation such as A//BC was reached, one first has to absorb the unseparated group BC on another column. In the meantime, one elutes A. The elution of A takes less time and attention than the transfer of BC (which is possibly, and even probably, accompanied by a change of solvent in which the elements are present). Therefore the time taken by such a step is determined by the time needed for the transfer. A situation A//BC will lead t o BC//A. Then the separation of BC must be carried out as described under (a), thereby leading t o BI/C,IA or C//B :A. Step 4. Elution of the element which remains o n the column. This is the last step since the three elements are now definitively separated and eluted (//A/B/C). A graph containing the possible situations as nodes and the way in which they are related as edges (or arcs) can be constructed. This is illustrated in Figure 1 . To obtain the shortest path from ABCI! t o j/A/B/C, one must be able to assign a value to each edge, i.e., one must construct a weighted graph. This value must correspond t o the shortest time in which it is possible to achieve the separation represented by the arc in question. In ion exchange, the time needed for a separation step is proportional to the distribution coefficient, D. We considered that a reasonably fast separation of sufficient quality is achieved when all the ions in the group which is eluted have D-values 6 5 and when all the retained ions have a D 3 10. The time in which it can be carried out is proportional to the D-value for the last eluted of the elements of the D 6 5 group. When the separation is not possible, a very large time ( = 900) is accorded to the corresponding edge. When the time is determined by a transfer from one column to another, it is a t first sight more difficult t o assign a value. The important aspect in a transfer operation is not so much that it takes u p time but that it is an effort needing additional attention. This is undesirable and one will usually prefer a separation on one column t o a separation on two columns. Therefore a trans-
2390 * ANALYTICAL CHEMISTRY, VOL. 44, NO. 14, DECEMBER 1972
fer is penalized by according a value of 10 to it, i.e., twice the maximum time taken by a separation. For each edge the fastest separation possible must be found. This is carried out by searching through a data file. To test the procedure, a data file containing D-values, for cation exchange only, of 23 ions (UOZ2+,Th4+, Zr4?, Ce3&,Mo6+, V5+, Fe3+, A13+, G a g + ,In3- Bi3+,Mg2+,Ca2+,Pb2+,ZnZr, CdZ+, Cu2+, Mn2+, Co*-, NiZC,Sr2+,Na+, and K+) in 308 different mobile phases was put on punched cards and a computer program was developed using the criteria described above to search this file for the fastest methods permitting the different separations necessary to construct the graph. GRAPHICAL SOLUTIONS
UE/C
A4EC
x*
f
/
\
Figure 1. Graph for the separation of A, B, and C
The shortest pathway in this weighted graph can be obtained by using an algorithm. Until now we made use of two algorithms, namely those of Ford and Dijkstra. Ford’s algorithm can be used to obtain a graphical solution. The procedure is illustrated by the graph of the separation of Ca, Co, and Th (see Figure 2). Mark each node starting with x1 for CoCaTh// and ending with x l l for ,!/Co,’Ca/Th. Assign a value 0 to X I and y , = x1 $- 1 (xl, x,) to each node x, directly linked to XI,1 (xl, x,) being the value of edge (xo, x,). In this way values of 3, 5 , 900, 0, 900, and again 900 are assigned to XZ,x 3 , x4, x s , X6, and x7, respectively. Assignvaluesy, = y n 1 (n,m) to all the nodes directly linked to all the x,’s to which a value has already been assigned. Continue in this way until all the nodes have a value. In this example, one would for example assign the values 15, 915, 900, and 902 to xg, xg,x10,and xll, respectively. Having achieved this, one checks the graph to see whether it is possible to assign a smaller value to some of the nodes. In this graph, this is, for example, possible with xg. The value 15 first assigned to it is a result of adding the value of l ( 6 , 8) to x6,in C U S I I , 15 0 = 1 5 . By adding f ( 5 , 8) to x 5 one, however, obtains a smaller value 0 2 = 2. This is continued until one is satisfied that the smallest possible value has been found for each node. The smallest pathway is then found by retracing the steps which have led to the final value of x l l . It appears that the pathway (xo,x 5 ) , (x:, XS), (xs, xll) is the shortest. This means that one first elutes Co, retaining Th and Ca with a mobile phase in which D (Co) = 0. According to our file, these are: 90% acetic acid, 1.2N HC1; 90% acetone, 0.6-0.9 or 1.2N HCl; 90% tetrahydrofurane, 0.9 or 1.2N HCl. The HCl concentrations are final concentrations and the data have been obtained from an article by Korkisch (13). After that, one elutes Ca while Th remains on the column. The fastest method to do this has a D (Ca) = 2. The mobile phase in question is 4 N HNO, (14). Finally, one elutes Th with a D = 4. The mobile phase is 0.5M (NH4)&04-0.025MH&04 (15). The graphical method can also be applied to mixtures of 4 and sometimes 5 ions. Some simplifications have however to be made because the graphs become too complex, even for mixtures of 4. This can be illustrated by the separation of Th, Co, Ca, and Fe. When one applies the method to 4 ions, 38 nodes are obtained which can be divided in 8 levels. Representative examples in each level are : ThCaCoFe//, Th//CaCoFe, CaCoFellTh, ThCa//CoFe, Th//Co/CaFe,
0 1
+
+
+
( 1 3 ) J. S. Korkisch and S. S. Ahluwalia, Tuluutu, 14, 155 (1967). (14) F. W. E. Strelow, R . Rethemeyer. and C. J. C. Bothma, ANAL. CHEW.37, 106 (1965). ( 1 5 ) K . Kaa.abuchi, T. Ito, and R . Kuroda, J . Chromarogr.. 39, 61 (1968).
I131
mco//cn
Figure 2.
Weighted graph for the separation of Co, Ca, and Th
ThCa//Co/Fe, Th//Ca/Co/Fe, and /jTh/Ca/Co/Fe. The total number of edges is 70. By first omitting the edges that have a value equal to 900 and the nodes that cannot be reached except by edges having such a value, only 24 nodes and 43 arcs remain. The solution reached has a value of 6, meaning that no transfer is necessary. This will often be the case for ‘small numbers of ions and therefore one can also first try a graph in which transfers are forbidden. Furthermore, omit1:ing the edges and nodes with values of 900 or more, leads in this case to a n extremely simple graph containing 8 nodes and 10 edges. With some luck, it is possible to obtain relatively simple graphs for up to 6 ions. The no-transfer procedure is however only possible for a limited number of ions and as soon as this number reaches 6, the separation without transfer becomes less probable. A special but important case is the separation of one ion from a number of other ions. In this case, the fate of the other ions becomes unimportant as soon as they have been separated from the ion in question. Therefore a number of nodes, necessary in the general scheme, must not be considered in the one-element scheme. In this case, graphs for mixtures of 5 or 6 ions can be handled without much difficulty. In general, therefore, one remarks that the graphical method i!j limited as t o the number of ions which can be handled. The upper limits can be fixed at 5 (6 in special cases) for general schemes and at 8 for one-element schemes. COMPUTER CALCULATION OF THE SHORTEST PATH
When the graphical solution method is not possible, a computer solution becomes necessary and, since the data-file search
ANALYTICAL CHEMISTRY, VOL. 44, NO. 14, DECEMBER 1972
0
2391
Number of elements
2 3 4 5 6 7 8 9 10 11
Table I. Number of Nodes of the Graph, Obtained with Different Assumptions Number of nodes, C Multi-element scheme ~. One-element scheme General 1 unseptd group Eq 2 and 1 2 unseptd groups Eq 2 and 6 No transfer Eq 4 Transfer Eq 3 No transfer Eq 5 4 4 4 4 3 3 7 11 11 5 11 8 38 38 38 16 15 9 152 152 137 32 31 17 675 675 480 64 63 33 3264 3159 1619 128 127 65 17008 14908 5290 256 255 129 ... ... ... 512 511 257 ... ... 1024 1023 513 ... 1025 . I .
is already carried out by computer, this seems preferable, even for cases where graphical solution is easily possible. Several classical algorithms (Ford, Dijkstra, Dantzig, Whiting-Hillier) have been taken into consideration. Since according t o Hitchner (16) and Govaerts (17), the most economical for computer calculation is the algorithm of DijkstraLoubal, a computer program integrating data search and solution by this algorithm is now in preparation. When using the classical algorithms, a matrix notation is usually necessary. A graph containing rn nodes leads t o a m x nz matrix. In our situation (the computer center makes use of a CDC-6400 computer) this limits the application of such methods to about 200-300 nodes. The applicability of these methods is therefore very limited. More sophisticated methods such as the Dijkstra-Loubal algorithm do not necessarily use marrix notation and can be used for graphs with 10002000 nodes depending upon the density of the network. Since the graphs which are obtained in this case are of a rather low density, one may presume that the upper limit is 2000 nodes. To know the maximal number of elements that can be handled by these methods, one has t o calculate how many nodes can be expected. Suppose n elements are present. These distribute between the resin phase and the eluate, so that p elements remain on the resin and n - p go into the eluate. There are C,”-possible ways of doing this. However for each combination of elements in the resin phase there are several possibilities in the eluate: (a) The elements can be divided into a number of groups. (b) These groups can consist of different combinations of elements. In general the elements can be separated into k groups. These are arranged according t o the number of elements n, in each group : The total number of combinations in the eluate B(n - p ) of n - p elements divided into k groups, with k = 1, 2 , . . . . (n - p ) and 0 < n - p < n then is: B(E
- p)
=
k
C n i = n - p i=l
where A(nl, n2, . . . nn) is the product of I’actorials of the number of groups with the same amount of elements. When n - p = n or n - p = 0, then B(n - p ) = 1. The total number of combinations in resin and eluate is then n
c = p = o C,” . B(n
For the calculation of the number of nodes in a one-element scheme, first consider the separation of A from a mixture ABCD. Fifteen nodes occur in the graph, namely those representing the situations ABCD//, ABC//, ADC//, ABD//, //ABC, //ABD, //ACD, AB//, ACII, AD//, //AB, //AC, //AD, A//, and //A. This means that all the combinations of A with, respectively, three, two, one, and zero elements out of three possible have to be taken into account, as well in the resin as in the mobile phase (with the exception of the mixture of the four ions in the latter). For a mixture of n ions, the following formula is obtained:
c=
( c
p=n-1
2
cp,-l
p=O
1-
1
=
2”
-
1
(3)
Both schemes can be simplified, if one considers that transfers are not to be allowed (see also the graphical solutions). For the general scheme, this also means that no unseparated groups can be permitted in the eluate and that only the combinations in one phase have to be considered, since to each combination in one phase corresponds one and only one combination in the other phase. For this scheme, the elements ABCD and the resin phase, this means that the nodes are ABCD//, ABC//, ACD,I/, ABDII, BCDi!, AB//, AC//, AD//, BC//, BD//, CD//, A//, B//, C//, D//, and /I. The last node is the terminal node (in full //A/B/C/D). The total number of combinations is thus given by the sum of all the combinations of four, three, two, one, and zero out of four elements. Therefore: (4) p=o
For the one-element scheme, the same initial mixture and A being the element which has to be separated, the nodes are ABCD//, ABC//, ACD/;, ABD//, AB//, AC//, AD//, A//, and //. The first eight nodes represent all the combinations of A with three, two, one, zero elements. To this the terminal node // has to be added so that
(16) L. E. Hitchner, Nut. Bur. Starid. (US.),rep. AD 685619, University of California, Berkeley, November 1968. (17) J. Govaerts, Thesis, Vrije Universiteit, Brussel, 1971. 2392
(2)
- p)
ANALYTICAL CHEMISTRY, VOL. 44, NO. 14, DECEMBER 1972
/n-1
\
As discussed before, the possibility of using a no-transfer procedure will become more and more improbable for the general scheme (and indeed also for the one-element scheme) when the number of elements grows. Much more probable is the possibility of using a procedure in which only situations with one unseparated group in the eluate are permitted. Then in Equations 1 and 2 , one must add the condition n2 1. Alternatively one can also reason that the ( n - p ) elements in the resin must be divided over one unseparated group, containing ng elements and n, groups of one element. Taking into account that n, n, = n - p a n d 0 n, 6 n - p , the number of combinations of (n - p ) elements contained in one group of ni- elements is Cnk(,-p) and the total number of combinations for n - p elements (0 < n - p < n) is
