On the potential of two- and multi-dimensional separation systems

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  • Fresenius J Anal Chem (1995) 352:625-632 I:resenius' Journal of

    @ Springer-Verlag 1995

    On the potential of two- and multi-dimensional separation systems Michel Martin

    Ecole Sup6rieure de Physique et Chimie Industrielles, Laboratoire de Physique et M~canique des Milieux H~t6rog~nes (URA CNRS 857), 10, rue Vauquelin. F-75231 Paris Cedex 05, France

    Received: 27 December 1994/Revised: 4 February 1995/Accepted: 11 February 1995

    Abstract. A probability model of zone overlapping in n-dimensional (n-D) separation systems has been de- veloped. The probability that all sample components are separated is given as a function of the number of compo- nents, m, and of the peak capacity, no, of the n-D system. Application to 1-D separations provides the same expres- sion as that previously obtained with a more rigorous peak overlapping model, in the limit of large m and no and of low saturation of the separation space. The major result is that the probability of total resolution of the sample decreases exponentially with the square of the number of sample components and with the reciprocal of the peak capacity, whatever the dimension of the separ- ation system. In addition, a simple general relationship is obtained between this probability and the probability to separate one or a few components of interest from all other sample components. It is found that, for a given number of components and a given peak capacity, these probabilities slightly depend on the dimension of the separation system, which indicates that the peak capacity is not a fully universal index of characterization of the resolving power. The peak capacity required to separate all sample components at a given probability level in- creases with the square of the number of components. Accordingly, the individual peak capacity per dimension does not increase as fast as m when the dimension of the system exceeds 2.

    Introduction

    Tremendous progress has been accomplished during the last three decades in the development of analytical zonal separation methods (gas chromatography, liquid chromatography, capillary electrophoresis, field-flow fractionation). The demands for the performances of these methods have been steadily increasing as they have been applied to samples of increasing complexity. However, in

    Dedicated to Professor Dr. h.c. mult. J.F.K. Huber on the occasion of his 70th birthday

    spite of the level of performances reached, the separation capability of the individual methods is still insufficient to handle highly complex samples, especially those from environmental or biological origin. The limitations of these individual methods are better understood since a decade or so, owing to the development of various statistical models of the overlapping of peaks in multi- component chromatograms [1 3] which have been re- cently reviewed [4].

    In order to overcome these limitations, various more or less sophisticated procedures of coupling of separ- ation/sample preparation methods have been attempted. They include heartcutting and column switching methods, variations in the mode of coupling chromato- graphic columns [5], or on-line coupling of gas and liquid chromatographic columns [6]. Planar two-dimensional separations have been performed using different retention mechanisms in liquid chromatography [7] or gel elec- trophoresis [8, 9]. More recently, methods have been developed to perform two-dimensional separations by microcolumn coupling in a so-called comprehensive mode that mimic planar separations [10 14]. Some of these methods have recently been reviewed [15] and clas- sified according to their intrinsic characteristics [16].

    The main driving force in developing these two- or multi-dimensional separation methods arises from the fact that their separation power can be significantly in- creased over that of one-dimensional systems. This was quantitatively expressed by Giddings in terms of the peak capacity, no, of the separation system, a convenient index of the overall separation power [17, 18]. Basically, the peak capacity of a two-dimensional system is of the order of the product of the peak capacities of the two individual one-dimensional contributing systems. Therefore, even if one of these two 1-D (for one-dimensional) peak capaci- ties is modest (i.e. of a few units), the resulting 2-D peak capacity will reach a value which would have been im- possible or hard to obtain with an individual 1-D system. Indeed, in order to increase by a factor of, let say, 5 the peak capacity of a column, one must multiply its plate number, i.e. its length at constant plate height, by 25. This will frequently not be possible due to technological or time limitations. Although the 1-D peak capacity of

  • 626

    a given separation system can vary in a wide range de- pending on the operating conditions, it can be useful to have in mind some typical values: 25 in thin-layer chromatography, 50-70 in isocratic liquid chromatogra- phy, 200-500 in gradient elution liquid chromatography, up to 2000-5000 in temperature programming capillary gas chromatography, 500-5000 (?) in capillary elec- trophoresis (in the latter case, this depends on the extent of the relative retention time interval which can be effec- tively used). Although some loss may occur in the peak capacity obtained in one direction when the separation is performed in the second one, these values and the multi- plicative law for nc allow to get rough estimates of the peak capacities which could be reached in multi-dimen- sional separations.

    Using a Poisson 2-D zone overlap model, Davis has computed the expected numbers of singlets, doublets and triplets in 2-D separations [19]. This model was later refined and extended to any multiplet [20] by taking profit of a model developed by Roach [21]. The Roach model was tested by computer simulations in 2-D [22], then extended to n-D for obtaining the expected numbers of any kind of multiplet as well as their sum which is the expected number of "peaks" in terms of the mean "den- sity" (or saturation factor) of components in the separ- ation space [23].

    Resulting expressions are very useful to interpret or predict the outcome of a n-D separation. In the present work, one uses a different but complementary statistical perspective to estimate the potential of multi-dimensional separations. Namely, one is mainly looking for the prob- ability that all the m components of a complex sample mixture are separated. In the present context, a n-D separation system is understood as a system for which the whole separation space can be occupied by the sample components. In practice, such a multi-dimensional system has, up to now, been realized only for 2-D, for instance in two-dimensional thin-layer chromatography or two-di- mensional electrophoresis, by introducing the sample in a corner of the rectangular plate, developing the separ- ation in one direction, then in the second (orthogonal) direction by means of a different separation mechanism. One can imagine to physically extend this separation scheme to 3-D using a parallelepipedal configuration, but not to more than 3-D. Another possibility to implement a 2-D system is to use a set of two columns operated by different mechanisms or different separation methods, to collect fractions eluting from the first column and inject- ing them sequentially onto the second column, possibly after some concentration step. Let q be the number of fractions so collected. If the q fractions collected from the first column are small enough (in practice of the size of about one unit of peak capacity) and all are reinjected onto the second column, the overall result of the develop- ment of the q separations in the second column will, after some appropriate reconstruction, mimic that which would be obtained in a continuous planar two-dimen- sional separation system. This comprehensive mode of operation [-10-14] can, in principle, be extended to 3-D by collecting, for all q runs, small effluent fractions from the second column and reinjecting them separately onto a third column operating by a separation mechanism or

    principle different from those of the two first columns. One easily imagines that this scheme can, in principle, be extended to n-D by using n different columns. In order for this approach to be effective, it is essential that the separ- ation in a given column is not correlated to that in any of the n - 1 other columns, hence that the n separation mechanisms or separation methods involved are mutually independent.

    Theory Whatever the dimension of the zonal separation system and physico-chemical nature of the individual (one-di- mensional) separation methods, the representation of the separation is called a diachorismogram, from the greek word 81~Z~plc~ocy for separation [24, 25]. It generally gives the concentration vs. the position in the n-D separ- ation space. It does not matter for the present purpose whether the individual axes of the separation space have dimensions of time, length or of any other appropriate unit suitable for describing the location of the compo- nents (e.g., retention index).

    Hypotheses of the model

    One is not concerned here with the relative amount of the various components (i.e. by the concentration axis) as one focuses on the position of each component zone in the separation space. However, it is assumed, for the sake of simplicity, but without loosing generality in the trends, that the space occupied by each sample component in n-D hyperspace is a hyperball (of hyperspherical shape) and that the hypervolumes Of all components are equal. Since the model presented applies whatever the dimension of the separation system, the general terms hyperball and hypervolumes are used in the following, remembering that a hyperball is a segment in l-D, a disk in 2-D, a real sphere in 3-D, . . . .

    Let ro be the radius of each individual component zone (for a 1-D separation, r0 is half the segment length) and Vo its hypervolume, ro is related to the standard deviation, cy, of the concentration profile along an individual direction. The exact relationship between ro and ~ depends on the level of resolution required for ascertaining that two components do not overlap (for instance, ro = 2cy corresponds to a unit resolution). Since all component zones are assumed to have the same hypervolume, a given component is separated from all others if no other component center is included in an exclusion hyperball of radius 2ro around its center. Let mo be the hypervolume of this exclusion hyper- ball and V the total hypervolume of the separation space. In this model, V is supposed to be large enough compared with Vo for the shape of the separation space to be irrelevant and edge effects negligible. One has:

    mo= 2nVo (1)

    where n is the dimension of the separation space.

  • 627

    Peak capacity of the separation system

    The peak capacity, n .... in n-D is defined, as in l-D, as the maximum number of separated zones which can be ac- commodated in the separation space with the required resolution between two adjacent zones along the line of their centers. In l-D, one has:

    V n~, 1 = - - (2 )

    V0

    In that case, V is the length of the separation space, i.e. the length of the retention interval between the first and last component peaks, Vo the length of a given peak. By extension, the peak capacity in n-D is defined as:

    V no,, = d). - - (3)

    v0

    Although this definition would be simplified by defining n,, as in Eq. (2), as was done by Davis [19, 23], it is found necessary here to introduce a numerical coefficient d?,, smaller than 1, which may vary with the dimension n of the separation space, in order to respect the isotropic (hyperspherical) shape of the component zones. Indeed, for n > 1, there is not a unique way to define no. For instance, in 2-D, if one uses a square grid for accommo- dating the individual components disks, (~2 is equal to re/4 = 0.785 since x/4 is the ratio of the area of the disk to the area of the square within which it is included. If a compact 2-D lattice model is used, then d2a becomes closer to 1 and equal to rex/3/6 = 0.907. In 3-D, for a cubic grid or a compact hexagonal lattice, ~3 is equal to re/6 = 0.524 or rex/2/6 = 0.740, respectively.

    Peak capacity in n-D in terms of peak capacities of individual dimensions

    Defining the peak capacity by means of the hypercubic grid method evoked above (i.e. square grid in 2-D, cubic grid in 3-D, ... ) has the advantage that the n-D peak capacity, n .... is easily expressed in terms of the individual 1-D peak capacities of the constituting dimensions, no, 1,i, as;

    .L nc, n = |1 no, l,i (4)

    i=l

    In this equation, it is implicitly assumed that the separ- ations in the n directions are mutually independent, as discussed above. Indeed, if, for instance, for a 2-D system, the separations in the two directions were totally corre- lated (as might be the case, for instance, for two gas chromatographic columns of similar polarity), the result- ing peak capacity would be, at most, the sum of the individual peak capacities instead of their product. Then, the advantage of such a correlated 2-D system would not be larger than that of an increase of the length of the separation space in a single direction.

    In order to obtain the simple result expressed by Eq. (4), qbn must be selected as the ratio of the hyper- volume, Vo, of the hyperball to the hypervolume, (2r0) n, of

    the hypercube which includes it, which gives [26]: ~n/2

    ~n -- 2n/ln C(n/2) (5)

    For this reason, the hypercubic grid definition of no, n and (~n is assumed in the following. Corresponding values of d~2 and ~3 have been given above. For larger dimensions, one gets, for instance, 04 = re2/32 = 0.308, qb5 = ~2/60 = 0.164 and d~6 = rc3/384 = 0.081. If the peak capacities of all individual dimensions are equal to nc, l,i,a, Eq. (4) becomes:

    nc, n = n~n,l,ind (6)

    One notes that the selection of the hypercubic grid basis for defining no, n may not just be a question of convenience but corresponds to the correct definition of the peak capacity in some multi-dimensional strategies using multi-column or so-called "comprehensive" ap- proaches.

    Probability that a component is a singlet

    One considers a sample mixture containing m compo- nents, distributed within the separation space in such a way that each component hyperball has an equal probability to be located at any position. A given component will be separated from a second one if the center of the zone of this second component does not fall within the exclusion volume determined by the first one. The probability that this happens is 1 - c%/V. The prob- ability, P1, that this first component is a singlet, i.e. that it is separated from all the other m - 1 sample compo- nents, is then:

    v>m exp( ,m I,V) ,7, The second equality holds when ~0o/V is much smaller than 1, which is generally the case for multi-dimensional separations. PI is also the ratio of the expected number of singlets to the number of sample components. Combina- tion with equations 1 and 3 allows to express it in terms of m and no,n:

    (m--1)~

    This equation is identical to those equations already given in 1-D [1], as well as in 2-D with d?2 = 1 [19] and with ~2 = re/4 [27] and in 3-D with qb3 = To/6 [27]. It becomes identical to that obtained by Davis (equation 10 of ref. [22] for v = 1) when taking d~n = 1.

    Probability that all components are separated

    One is looking for the probability, Pro, that all m compo- nents are separated. Let now assume that one has pre- viously performed a separation of the m - 1 first sample components and let Pm-1 be the probability that these m- 1 components are all separated. Adding the ruth

  • 628

    component to the sample and performing the separation in identical conditions, one then has:

    Pm = Pm-tP[,mlm - 1] (9)

    where P[,mlm - 1] is the probability that the mth com- ponent is separated once the m - 1 first components are separated. Repeating this process for a m - 1 mixture once the first m - 2 components are separated, one gets:

    Pm-1 = Pm-2P[ ' m -- 1 Im -- 2] (lO)

    Repeating this iterative process for decreasing values of the component number, one gets:

    Pm : I ] P[-ili -- 1] (11) i--2

    P[,ili - 1] is the probability that the center of the hyper- ball of the ith component is not located within the overall exclusion hypervolume,

  • This equation is easily understood in 1-D. Indeed, imag- ine the components represented by points are distributed along an horizontal line. A given component is separated from its left neighbour if the distance between them is larger than 2to. Let P be the probability, it is so. This component will be a singlet if its right neighbour is also at a distance larger than 2r0. The probability to be a singlet is then P1 = p2. Now, in order for all components to be singlets, the (m- 1) intervals between adjacent compo- nents must all be larger than 2r0. The probability that it is so is then Pm= pm 1 ~ pm= p1~/2, as given by Eq. (21). It is tess obvious that this property also holds for n > 1. It appears to be a kind of paradigm of multi-dimensional separations.

    Required peak capacity The peak capacity, n ....... q{l~, required to have a given probability, P1, to separate a single sample component is given from Eq. (8) by:

    m n ...... q(*~ = ~"2= ln(1/P,---~ (22)

    This peak capacity increases with the number of sample components. If one now wishes to have a given probabil- ity, Pro, to separate all sample components, this requires a peak capacity, n ...... q(m), obtained from Eq. (20) as:

    m 2 n ...... q(m) = qb,2"- 1 _ _ (23)

    In (1/Pm)

    It is noteworthy that, whatever the dimension of the system, it increases with the square of the component number. Moreover, the ratio of these two required peak capacities is completely independent of the dimension of the separation system. It depends only on m and on the stated probability levels:

    n ...... q{m) _ m lnPm (24) nc ,n , req( t ) 2 lnPi

    When Pm and P1 are selected to be equal, then this ratio becomes equal to m/2.

    Results and discussion

    The main theoretical result obtained is expressed by Eq. (20) which gives the probability that all m sample components are separated as a function of m and of the n-D peak capacity. Figure 1 shows the variation of Pm with log min 2-D for various values of no,> Pm is seen to decrease rapidly with increasing m. Whatever the peak capac i ty , Pm decreases from 0.9 to 0.5 then 0.1 when m increases by a factor 2.6 then 4.7. If one desires a rather high probability to separate all components, m must be rather low. For instance, even with a peak capacity, no, 2, as large as 104 , the component number must be lower than 18 in order to have a probability of separating them exceeding 95%, while a 66-component sample can be resolved with a probability of 50%. Increasing the

    629

    1

    Pm 0.8

    0.6

    0.4

    0.2

    l ls

    \ 21s 5 ; .s

    log m

    Fig. l. Probability, Pro, that all m sample components are separated vs. log m, in 2-D. From left to right curve: 2-D peak capacity, n~.2 = 103; 104; 10s; 106; 107

    1'

    Pm 0.8.

    0.6

    0.4

    0.

    3 4 5 6

    log nc,2

    Fig. 2. Probability, Pro, that all m sample components are separated vs. log no. 2, in 2-D. From left to right curve: number of sample components, m = i0; 30; 100; 300; 1000

    dimension of the separation system, and hence the peak capacity, one obviously increases these figures, for in- stance in 3-D with no.3 = 106, a sample of 575 compo- nents can be totally resolved at a 50% probability level. With an individual peak capacity of 100 for every addi- tional dimension, this sample component number rises to 5300, 51300 and 518 000 for 4-D, 5-D and 6-D, respective- ly, making apparent the tremendous separation power of high-dimension separation systems.

    The probability Pm of total resolution is shown to increase with the logarithm of the peak capacity in Figs. 2 and 3 for 2-D and 3-D, respectively, for various values of the sample component number. These figures are similar but not identical. Whatever the dimension of the separ- ation system, P~ increases from 0.1 to 0.5 or 0.9 when the peak capacity is multiplied by a factor 3.3 or 22, respec- tively.

    The fact that the curves of Figs. 2 and 3 are not exactly identical is made more apparent in Fig. 4 which shows the variation of the probability to totally resolve a 100-com- ponent mixture with nc,n for various values of the dimen- sion of the separation system, n. One notes that these curves would be identical, but shifted by the same amount along the log no,n axis for another value of m. In the n = 1

  • 5.5.

    log ne,2,req(m)

    5-

    i 4

    Pm

    0.8

    0.6

    0.4[

    0.2

    3 4 5 6 log nc,3

    Fig. 3. Probability, Pm, that all m sample components are separated vs. log no, 3, in 3-D. From left to right curve: number of sample components, m = 10; 30; 100; 300; 1000

    1

    Pm

    0.8

    0.6

    0.4

    0.2

    log ne,n

    Fig. 4. Probability, Pro, that all m sample components are separated vs. log he, n, for a given value of the number of sample components, m = 100. From upper to lower curve: dimension of the separation space, n = 1; 2; 3; 4; 5

    to 5 range covered in Fig. 4, the larger is the dimension n, the smaller is Pro. This characteristic is not linked to the specific hypercubic grid definition of no,n. Indeed, the difference between curves at various n would be enhanced if n~,n was defined as V/vo (in which case (~, = 1 whatever n). According to Fig. 4, it thus appears that if, in order to separate a 100-component mixture, one has the choice between e.g., l-D, 2-D and 3-D systems all having the same overall peak capacity (e.g. no = 20000); one will get the best chance to achieve this goal by using the 1-D system rather than 2-D or 3-D systems or the 2-D rather than the 3-D system. Indeed, the respective probabilities for l-D, 2-D and 3-D would be 0.61, 0.46 and 0.35. This indicates that the peak capacity is not a fully universal index of characterization of the resolving power of a sep- aration system.

    Although this lack of universality is not crucial, the fundamental reason of this behaviour is not entirely clear. It is related to the fact that, given m and no, the probabil- ity, P1, for a component to be a singlet is lower in 2-D than in 3-D and in 3-D than in 4-D. This is understood by the fact that the number of directions of interference of a component with another one increases with the

    4.5

    4

    3.5

    2.

    115 ~ 21s

    630

    3Is

    log m

    Fig. 5. Logarithm (to base 10) of the 2-D peak capacity, no, z,~oq{m), required to separate all m sample components at a given probability level, Pm, VS. log m. From upper to lower curve, probability level, Pm = 0.9; 0.5; 0.1

    dimension of the separation system. Analysis of the fac- tors 2nqbn and 2n-id)n, which appears in Eqs. (8) and (20), shows that they steadily increase (hence P1 and Pm stead- ily decrease) with increasing n if the peak capacity is defined in such a way that d?n = 1. However, if d?n is given by Eq. (5), which corresponds to the hypercubic grid definition of no, these factors appear to have a maximum (P1 and Pm have a minimum) for n = 5. Then, systems with n > 10 provide higher Pm values than 1-D systems, given m and no.

    Figure 5 is a log-log plot of the 2-D peak capacity required to separate m components with a given probabil- ity in 2-D versus m, for three different values of the fixed probability. According to Eq. (23), these curves are straight lines of a slope equal to 2. These curves would be similar, although slightly shifted, in a n-D system. In all cases, doubling the sample complexity would imply to multiply the peak capacity by a factor of 4. Such an increase in no can be obtained by increasing the individual peak capacity of one of the dimensions of the separation system. Alternatively, the peak capacity of several indi- vidual dimensions can be simultaneously increased. When the n individual peak capacities of the dimensions of a n-D system are simultaneously increased in the same proportion, the requirement per dimension for total resolution when increasing the sample complexity becomes less drastic since it has to increase as m 2/n. Thus, when, doubling the sample complexity, the individual nc per dimension must double a 2-D system and increase by a factor smaller than 2 for systems of higher dimen- sions.

  • 631

    Pk

    0.6-

    0.4-

    _4 3 l _l 2 ~ "~- -2.5 -ii.5 -I -0.5 o15 log m/ne,2

    Fig. 6. Probability, Pk, that k sample components are separated vs. log m/nc.2 (for k ~ m) in 2-D. From left to right curve: k = 1, 2, 5, 10

    In some cases, the analyst is not interested, in the separation of all components of a complex sample, but instead in the separation of one or a few key components of the mixture. The probability of separating a given component is expressed by Eq. (8). The probability, Pk, to separate k components can be derived from Eq. (8) as:

    Pk = pk = p2k/m (k ~ m) (25)

    These equations are correct when k is much smaller than m, so that the probability that the exclusion hyperballs of each of the k components of interest interfere is negligibly small. As k increases, Pk becomes larger than the value given by Eq. (25). It can be expressed as p~k (or p2~k/m) with varying between 1 and 1/2 and approaching 1/2 when k becomes close to m. Figure 6 shows the variation of Pk with the saturation factor, m/no, 2, in 2-D in the limit )v = 1. The curves would be slightly shifted along the abscissa axis for n-D systems with n ~ 2. They appear to drop relatively rapidly with increasing saturation factor. In order to have 90% chance to separate the k compo- nents of interest, the saturation must be lower than 3.4% for k = 1 and 0.67% for k = 5. The limiting saturation appears to vary as 1/k.

    Conclusion

    There are various methods of implementing such separ- ations; the model was primarily developed assuming that the separation proceeds continuously in every dimension of the system. This physically restricts its dimensionality to 2-D and 3-D systems. However, the use of multi- column systems in a comprehensive mode allows to ap- proach this continuous mode. In addition, in principle, it allows to extend the separation process to more than 3-D systems. The theoretical model developed above allows, in spite of its crude assumptions, to envision the capability of multi-dimensional separation systems. In practice, it allows the separation scientist to estimate whether or not, and in which operating conditions, a given one-dimen- sional separation system can fulfill the separation objec- tive (i.e. the full resolution of all sample components or the

    isolation of one or a few components of interest). This depends on the sample complexity and on the achievable peak capacity. In addition, when a two- or multi-dimen- sional separation system has to be used, the present model allows to compute the overall peak capacity required to solve the separation problem and thus it gives an indica- tion about the operating conditions to be selected in each separation direction in order to obtain this required peak capacity.

    While it is not easy to predict a significant increase of the present level of resolving power of the individual separation methods (except possibly for capillary elec- trophoresis if its very high efficiency can be coupled with a large relative retention time interval and for field-flow fractionation if it can be miniaturized to improve its efficiency), their combination in a multi-dimensional arrangement can significantly extend the overall resolv- ing power. The above model expresses the capability of n-D systems in terms of probability on the basis of a random distribution of the components in the separation space. If the validity of this hypothesis has been ques- tioned for 1-D chromatographic systems, it should be more easily accepted for n-D systems since the effective implementation of such systems requires that the separ- ation order in one dimension is not correlated to that in another dimension.

    In liquid chromatography, optimization studies of the mobile phase composition has made realizable the total separation of 20 components in systems with peak capaci- ties of the order of 50-70, for which the probability of performing this separation can be computed from Eq. (20) to be around 0.1%. If separations with this level of prob- ability can be performed in multi-dimensional systems, one can expect to totally resolve about 200 components with nc = 104 and 2000 components with nc = 10 6. Al- though these numbers can change slightly with the dimen- sion of the system, their order of magnitude remains the same. The latter case corresponds nearly to the maximum possibility of 2-D systems. Indeed, in the best case, when coupling a good LC column operated in gradient elution conditions (for which no,1 is of the order of 500) with an excellent capillary GC column (no,1 = 5000 in temper- ature programming conditions), at most a peak capacity, n. 2, of 2.5 x 106 is achieved. For more complex samples, one would have to perform separations with larger than 2-D systems.

    It must be stressed that these numbers correspond to the complexity of the sample for which the resolution of all components is looked for. In some instances the num- ber of peaks observed in multidimensional separations is larger, and even quite larger, than the figures given above. This does not contradict the conclusions derived above. Indeed, if large numbers of peaks (or "spots" in planar 2-D separations), which in any case have to be smaller than the peak capacities of the systems, are observed, this most probably indicate that the number of components contained in the samples are still larger and, possibly, very much larger than these numbers of observed peaks, espe- cially if the fraction of the total separation space occupied by these peaks is not very small.

    Finally one believes that quantitative statistical mod- els such as the one developed above should prove useful

  • 632

    for the rational development and optimization of multi- dimensional separations.

    Acknowledgements. Fruitful discussions on the probability of overlap- ping in multi-dimensional systems with Laurent Limat and St+phane Roux are gratefully acknowledged.

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