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Pharmaceutical Research, Vol. 21, No. 5, May 2004 ( 2004) Analysis of Pharmacokinetics,
gies such as DNA arrays and proteomics, which are now ca-pable of simultaneously measuring the expression of thou- Pharmacodynamics, and
sands of genes from single samples, the data sets have become Pharmacogenomics Data Sets Using
Data visualization, however, has not been extensively in- VizStruct, A Novel Multidimensional
vestigated in the context of the pharmaceutical sciences. Vi- Visualization Technique
sualization, using graphs and other techniques, is an efficientaid for finding structures, features, patterns, and relationshipsin a data set. However, multidimensional data sets, whicharise in many pharmaceutical applications, present special vi- Kavitha Bhasi,1 Li Zhang,2 Aidong Zhang,2 and
sualization challenges because they cannot be easily repre- Murali Ramanathan1,3
sented in interpretable two-dimensional or three-dimensionalgraph formats. Good interactive, multidimensional visualiza-tion tools can provide additional perspectives that assist the Received December 3, 2003; accepted February 9, 2004 user to understand large complex data sets at an intuitive Purpose. Data visualization techniques for the pharmaceutical sci-
level, facilitate subsequent hypothesis generation, and en- ences have not been extensively investigated. The purpose of this hance the data mining experience. In this report, we investi- study was to evaluate the usefulness of VizStruct, a multidimensional gate a visualization tool, VizStruct, which is capable of map- visualization tool, for applications in pharmacokinetics, pharmacody- ping multidimensional data to two dimensions, and assess its usefulness for several key pharmaceutical applications.
Methods. The VizStruct tool uses the first harmonic of the discrete
Fourier transform to map multidimensional data to two dimensions
for visualization. The mapping was used to visualize several published
DERIVATIONS AND RESULTS
pharmacokinetic, pharmacodynamic, and pharmacogenomic datasets. The VizStruct approach was evaluated using simulated popula-tion pharmacokinetics data sets, the data from Dalen and colleagues The Mapping
(Clin. Pharmacol. Ther. 63:444–452, 1998) on the kinetics of nortrip-
tyline and its 10-hydroxy-nortriptyline metabolite in subjects with
VizStruct is a projection that maps the n-dimensional differing number of copies of the CYP2D6, and the gene expression vectors in the input data to two-dimensional points (3–5). If profiling data of Bohen and colleagues (Proc. Natl. Acad. Sci. USA the vector x[n] ס (x[0], x[1], . . . , x[n − 1]), represents a data
100:1926–1930, 2003) on follicular lymphoma patients responsive and
item in n-dimensional space, Rn, its mapping to a point
nonresponsive to rituximab.
Results. The VizStruct mapping preserves the key characteristics of
F (x[n]) in the complex plane C is given by:
multidimensional data in two dimensions in a manner that facilitatesvisualization. The mapping is computationally efficient and can be used for cluster detection and class prediction in pharmaceutical data F ͑x͓n͔͒ = ͚x͓j͔e−2␲ijրn
sets. The VizStruct visualization succinctly summarized the salient similarities and differences in the nortriptyline and 10-hydroxy-nortriptyline pharmacokinetic profiles in subjects with increasing In equation above, i ס √−1 and the complex exponential number of CYP2D6 gene copies. In the simulated population phar- has the effect of dividing the circle of display into equally macokinetic data sets, it was capable of discriminating the subtle spaced sectors. The real and imaginary components of differences between pharmacokinetic profiles derived from 1- and F (x[n]) are used for creating the two-dimensional mapping.
2-compartment models with the same area under the curve. The two- Because the real and imaginary components representation of dimensional VizStruct mapping computed from a subset of 102 in- a complex number is mathematically equivalent to its ampli- formative genes from the Bohen and colleagues data set effectivelyseparated the rituximab responder, rituximab nonresponder, and tude (R) and phase (␾) representation, the VizStruct map- ping can be plotted either on a Cartesian real and imaginary Conclusions. The VizStruct approach is a computationally efficient
axis plot or on a polar plot; we use both plots in this report.
and effective approach for visualizing complex, multidimensional The mapping F (x[n]) is equivalent to the first harmonic of
data sets. It could have many useful applications in the pharmaceu- the discrete Fourier transform (DFT), which allows the fast Fourier transform algorithm to be used for computation. The KEY WORDS: microarray; pharmacodynamics; pharmacogenomic
VizStruct mapping is equivalent to the geometric projection modeling; pharmacokinetics; visualization algorithms.
technique used in radial visualization algorithms (3,4); how-ever, our use of the first Fourier harmonic, which facilitates INTRODUCTION
computation and provides insights into the underlying prop- Many pharmaceutical applications result in large multi- erties of the mapping, is novel (5).
dimensional data sets. With the advent of genomic technolo- The VizStruct mapping preserves the correlation rela- tionships between vectors in the input and the output spaces.
For example, all vectors of the form (a,a, . . . , a) are mapped Department of Pharmaceutical Sciences, State University of New to the center of the unit circle. If two vectors X and Y share
York at Buffalo, Buffalo, New York 14260, USA.
geometric similarity, that is, Y
Department of Computer Science, State University of New York at ס aX where a is scalar, they
Buffalo, Buffalo, New York 14260, USA.
will map to a radial line. All vectors with the same pattern will 3 To whom correspondence should be addressed. (e-mail murali@ map to the same line, and those that are similar map to a 0724-8741/04/0500-0777/0 2004 Plenum Publishing Corporation Bhasi et al.
Application of the VizStruct Visualization Approach
ies because parallel coordinates representations of drug con- in Pharmacogenetics
centration-time profiles get cluttered and difficult to interpretfor large PK data sets. Here, we present results from a popu- As a pharmacogenetics case study, we used the data from lation PK simulation case study, in which two groups differing Dalen et al., who demonstrated that the plasma concentra- in clearance (clearance values of 5 volume units/time, and 7.5 tion-time profiles of the antidepressant, nortriptyline, and its volume units/time) were generated. For both groups, the dose metabolite, 10-hydroxynortriptyline, were dependent on the and the volume of distribution of the central compartment number of copies of the cytochrome P450 2D6 isoform (CYP were set to 100 mass units and 5 volume units, respectively. A 2D6), which is involved in its metabolism (1). The subjects clearance-parameterized, one-compartment model was used, either had no functional copies or 1, 2, 3 or, in one case, 13 and the ADAPT pharmacokinetic and pharmacodynamics copies of the CYP 2D6 gene. The findings, summarized in systems analysis program (6) was used to simulate 100 indi- Figs. 1A and 1B, show that individuals with more CYP 2D6 vidual PK profiles for each group assuming a log-normal dis- copies metabolize the drug more extensively, and the result- tribution for the clearance and volume of distribution of the ant metabolite levels are higher compared to subjects with Figure 2A shows that the two groups differing in clear- The polar plots in Figs. 1C and 1D are obtained using the ance are well separated by the mapping from their sampled VizStruct mapping. The results in Figs. 1C and 1D show that kinetic profiles. The mean values of the “amplitude” are VizStruct succinctly captures the essential aspects of the ki- greater for the low clearance group because their mean con- netics in Figs. 1A and 1B, respectively. The inverse relation- ship between the kinetics of nortriptyline and its metabolite(subjects with more CYP 2D6 genes produce lower nortrip-tyline levels and higher levels of 10-hydroxynortriptyline) arerepresented by the mappings in Figs. 1C and 1D. The subjectswith two and three copies of CYP 2D6 show similar nortrip-tyline kinetics in Fig. 1A and are mapped close to one anotherby VizStruct in Fig. 1C. The geometric similarity of the kineticprofiles is reflected in the radial arrangement of points in Figs.
1C and 1D.
These findings demonstrate the feasibility of using Viz- Struct as an intuitive visualization approach because it is ca-pable of representing key features of multidimensional datasets succinctly in two dimensions.
Application of the VizStruct Visualization Approach in
Population Pharmacokinetics
We reasoned that novel visualization tools could have a significant impact on population pharmacokinetics (PK) stud- Fig. 1. (A) and (B) show the kinetics of nortriptyline and its metabo-
lite, 10-hydroxynortriptyline, in subjects with 0, 1, 2, 3, or 13 copies of
Fig. 2. Two different views, (A) a polar plot and (B) a linear plot of
the CYP 2D6 gene (1). The drug and metabolite concentrations are a simulated population pharmacokinetics data set. A one- in nM. (C) and (D) are the polar plot representations of the VizStruct compartment model was used: the CL was set to 7.5 volume units/ mapping. The amplitude and phase of the first Fourier harmonic are time for one group and 5.0 volume units/time for the other; the vol- shown. The numbers shown against the curves (A and B) and the ume of distribution was 5 volume units. A log-normal distribution points (in C and D) are the number of CYP 2D6 copies.
was assumed for clearance and volume of distribution.
VizStruct for Visualizing Complex Data Sets
To further challenge the capabilities of VizStruct, we C ס Ae−␣t + Be−␤t; all the curves have the same area under simulated the challenging PK data set shown in Figs. 3A and the curve (AUC0→ϱ), which causes the curves to overlap sig- 3B. The curve shown in open circles is the simulation for n nificantly. For the one-compartment model, the dose was set one-compartment model, and the remaining four curves are to 100 mass units; the volume of distribution was set to 5 simulations from two-compartment models differing slightly volume units; the clearance was set to 10 volume/time units.
in the A, ␣ and B, ␤ values of the bi-exponential relationship Together, these parameters result in degradation rate con-stant (K) of 2 time−1, an initial concentration of 20 mass/volume units, and in a AUC0→ϱ of 10 mass·time/volume units.
The two-compartment simulations in Fig. 3 all had AUC0→ϱof 10 mass·time/volume units. The A and B, in the bi-exponential relationship were, respectively, set to 5 and 10mass/volume units for all the curves. The ␣ were 1, 2, 2.5, and3 time−1 units with corresponding ␤ values of 2, 1.33, 1.25, and1.20 time−1 units, for the curves marked, respectively, withfilled circles, open squares, filled squares, and open trianglesthat are discernible in Fig. 3B. Figure 3A shows that on linearaxes, the data points are virtually superimposed on one an-other; the differences become apparent only in Fig. 3B be-cause of the use of the logarithmic axes. Figure 3C shows thatthe four PK profiles are well separated in the VizStruct map-ping. The two-compartment profiles are clustered with eachother and well separated from the one-compartment profile.
These findings suggest that VizStruct visualization may proveuseful for exploring large pharmacokinetic data sets.
Application of the VizStruct in Assessing Treatment
Responses from Gene Expression Profiles
Gene expression profiling with microarrays provides si- multaneous measurements of thousands of messenger RNAsand yields large multidimensional data sets that are a chal-lenge to visualize. Bohen et al. measured the pretreatmentgene expression profiles in follicular lymphomas of patientsreceiving rituximab, a monoclonal antibody directed againstthe CD20 protein found on B cells (2). These authors sug-gested that the gene expression patterns of lymphomas thatwere nonresponsive to rituximab therapy were more similarto those of control lymphoid tissues than lymphomas thatresponded to therapy. We used the significance analysis ofmicroarrays (SAM) algorithm (7) to identify an informativesubset of 102 genes from the data set and projected the 102-dimensional vectors for each sample to 2 dimensions using theVizStruct algorithm (Fig. 4). The VizStruct projection sepa-rated the partial/complete responders (open circles) from thenonresponders (filled circles) and normal controls (triangles).
One nonresponder sample was not separated from the partial/complete responder cluster. Additionally, the proximity andradial location of the nonresponder cluster relative to normalcontrols indicate that the nonresponder group was somewhatmore similar to the normal control group than the partial/complete responder group. Thus, the findings obtainedthrough visualization are generally consistent with the con-clusions of Bohen et al. (2).
DISCUSSION
In this report, we presented case studies for several phar- maceutical applications ranging from pharmacogenetics, Fig. 3. (A) shows the simulated data from a one-compartment model
population pharmacokinetics, and pharmacogenomics for (open circles) and three different two-compartment simulations which the VizStruct multidimensional visualization approach (square, triangle, filled circle). (B) shows the same data as (A) onlogarithmic y-axis. In (A) and (B), the concentrations are in arbitrary units. (C) shows the VizStruct mapping of the PK data set from (A).
The pharmacogenetics case study of the Dalen et al. data Bhasi et al.
data; that is, each dimension is plotted along a separate axis.
The parallel coordinates approach gets easily cluttered andincreasingly difficult to interpret when used with large datasets with modest levels of noise. The MDS approach is thecurrent gold standard for multidimensional visualization. InMDS, the presentation in two dimensions is optimized to pre-serve a specific aspect of the relationship, for example, theEuclidean distance, block distance, or rank relationships be-tween the points in the N-dimensional space. In practice,MDS encompasses a class of methods, depending on thestress function being optimized. For many pharmaceutical ap-plications, Sammon’s nonlinear mapping is probably the mostappropriate MDS method because it normalizes the distancesin the stress function to distances in the original N-dimensional space. Generally, Sammon’s mapping improvesvisualization of data sets containing a wide dynamic range ofvalues. Despite providing results that are mathematically op-timal in some sense, MDS (Sammon’s mapping included) isnot ideal primarily because the incremental addition of evena single point requires a complete repetition of the optimiza- Fig. 4. This figure shows the results from projecting the expression
tion procedure and possible extensive reorganization of all levels of the 102-gene subset of informative genes from the data of the previously mapped points to new locations. It is compu- Bohen et al. (2). The 11 partial/complete responder samples are tationally intensive with time complexity of O(n2), where n is shown in open circles; the nonresponders are shown in filled circles, the number of points, because of the time-consuming function and the 4 normal tissues are shown with the open triangles.
evaluations and iterations required for optimization. Viz-Struct, in contrast, is computationally efficient with time com- set was kept intentionally simple so that the findings from plexity of O(n log n) and does not require any rearrange- VizStruct could be easily compared to the interpretations ob- ments to accommodate incremental points.
tained by visual inspection of the time profiles. Therefore, this In conclusion, VizStruct is effective, flexible and versa- example should considered as a feasibility study rather than a tile. The method is also computationally efficient and has rigorous demonstration of VizStruct capabilities because it sound theoretical underpinnings. It may prove a useful ap- could be argued that the original data from Dalen et al. could proach for multidimensional data visualization in many phar- ACKNOWLEDGMENTS
that the use of VizStruct was unnecessary. To further chal-lenge the capabilities of VizStruct, we examined other, more This work was supported in part by Grant RG3258A2 difficult data sets in the remaining case studies.
from the National Multiple Sclerosis Society. Support from We examined two pharmacokinetic data sets that were the National Science Foundation (Research Grant 0234895) structured using simulations but included more difficulty than and the National Institutes of Health (P20-GM 067650) is also the pharmacogenetics case study. In pharmacokinetics, expo- nential decay processes are commonly used for modeling, and REFERENCES
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Source: http://www.cse.buffalo.edu/DBGROUP/bioinformatics/papers/PR04.pdf