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Analysis of Pharmacokinetics, Pharmacodynamics, and

Pharmacogenomics Data Sets Using VizStruct, A Novel

Multidimensional Visualization Technique

Kavitha Bhasi1, Li Zhang2, Aidong Zhang2, and Murali Ramanathan1,3

1 Department of Pharmaceutical Sciences, State University of New York at Buffalo, Buffalo, New York 14260,

USA

2 Department of Computer Science, State University of New York at Buffalo, Buffalo, New York 14260, USA

Abstract

Purpose—Data visualization techniques for the pharmaceutical sciences have not been extensively

investigated. The purpose of this study was to evaluate the usefulness of VizStruct, a

multidimensional visualization tool, for applications in pharmacokinetics, pharmacodynamics, and

pharmacogenomics.

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 pharmacokinetic, pharmacodynamic, and pharmacogenomic data sets. The VizStruct

approach was evaluated using simulated population pharmacokinetics data sets, the data from Dalen

and colleagues (Clin. Pharmacol. Ther. 63:444−452, 1998) on the kinetics of nortriptyline and its

10-hydroxy-nortriptyline metabolite in subjects with differing number of copies of the CYP2D6, and

the gene expression profiling data of Bohen and colleagues (Proc. Natl. Acad. Sci. USA 100:1926

−1930, 2003) on follicular lymphoma patients responsive and nonresponsive to rituximab.

Results—The VizStruct mapping preserves the key characteristics of multidimensional data in two

dimensions in a manner that facilitates visualization. The mapping is computationally efficient and

can be used for cluster detection and class prediction in pharmaceutical data sets. The VizStruct

visualization succinctly summarized the salient similarities and differences in the nortriptyline and

10-hydroxynortriptyline pharmacokinetic profiles in subjects with increasing number of CYP2D6

gene copies. In the simulated population pharmacokinetic data sets, it was capable of discriminating

the subtle differences between pharmacokinetic profiles derived from 1- and 2-compartment models

with the same area under the curve. The two-dimensional VizStruct mapping computed from a subset

of 102 informative genes from the Bohen and colleagues data set effectively separated the rituximab

responder, rituximab nonresponder, and control subject groups.

Conclusions—The VizStruct approach is a computationally efficient and effective approach for

visualizing complex, multidimensional data sets. It could have many useful applications in the

pharmaceutical sciences.

Keywords

microarray; pharmacodynamics; pharmacogenomic modeling; pharmacokinetics; visualization

algorithms

3 To whom correspondence should be addressed. (e-mail murali@acsu.buffalo.edu).

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Published in final edited form as:

Pharm Res. 2004 May ; 21(5): 777–780.

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INTRODUCTION

Many pharmaceutical applications result in large multidimensional data sets. With the advent

of genomic technologies such as DNA arrays and proteomics, which are now capable of

simultaneously measuring the expression of thousands of genes from single samples, the data

sets have become even larger.

Data visualization, however, has not been extensively investigated in the context of the

pharmaceutical sciences. Visualization, using graphs and other techniques, is an efficient aid

for finding structures, features, patterns, and relationships in a data set. However,

multidimensional data sets, which arise in many pharmaceutical applications, present special

visualization challenges because they cannot be easily represented in interpretable two-

dimensional or three-dimensional graph formats. Good interactive, multidimensional

visualization tools can provide additional perspectives that assist the user to understand large

complex data sets at an intuitive level, facilitate subsequent hypothesis generation, and enhance

the data mining experience. In this report, we investigate a visualization tool, VizStruct, which

is capable of mapping multidimensional data to two dimensions, and assess its usefulness for

several key pharmaceutical applications.

DERIVATIONS AND RESULTS

The Mapping

VizStruct is a projection that maps the n-dimensional vectors in the input data to two-

dimensional points (3-5). If the vector x[n] = (x[0], x[1], . . . , x[n – 1]), represents a data item

in n-dimensional space, Rn, its mapping to a point F1(x[n]) in the complex plane C is given

by:

(1)

In equation above,

display into equally spaced sectors. The real and imaginary components of F1(x[n]) are used

for creating the two-dimensional mapping. Because the real and imaginary components

representation of a complex number is mathematically equivalent to its amplitude (R) and phase

(ϕ) representation, the VizStruct mapping can be plotted either on a Cartesian real and

imaginary axis plot or on a polar plot; we use both plots in this report. The mapping F1(x[n])

is equivalent to the first harmonic of the discrete Fourier transform (DFT), which allows the

fast Fourier transform algorithm to be used for computation. The VizStruct mapping is

equivalent to the geometric projection technique used in radial visualization algorithms (3,4);

however, our use of the first Fourier harmonic, which facilitates computation and provides

insights into the underlying properties of the mapping, is novel (5).

and the complex exponential has the effect of dividing the circle of

The VizStruct mapping preserves the correlation relationships between vectors in the input and

the output spaces. For example, all vectors of the form (a,a, . . . , a) are mapped to the center

of the unit circle. If two vectors X and Y share geometric similarity, that is, Y = aX where a is

scalar, they will map to a radial line. All vectors with the same pattern will map to the same

line, and those that are similar map to a narrow region around the line.

Application of the VizStruct Visualization Approach in Pharmacogenetics

As a pharmacogenetics case study, we used the data from Dalen et al., who demonstrated that

the plasma concentration-time profiles of the antidepressant, nortriptyline, and its metabolite,

10-hydroxynortriptyline, were dependent on the number of copies of the cytochrome P450 2D6

isoform (CYP 2D6), which is involved in its metabolism (1). The subjects either had no

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functional copies or 1, 2, 3 or, in one case, 13 copies of the CYP 2D6 gene. The findings,

summarized in Figs. 1A and 1B, show that individuals with more CYP 2D6 copies metabolize

the drug more extensively, and the resultant metabolite levels are higher compared to subjects

with few copies.

The polar plots in Figs. 1C and 1D are obtained using the VizStruct mapping. The results in

Figs. 1C and 1D show that VizStruct succinctly captures the essential aspects of the kinetics

in Figs. 1A and 1B, respectively. The inverse relationship between the kinetics of nortriptyline

and its metabolite (subjects with more CYP 2D6 genes produce lower nortriptyline levels and

higher levels of 10-hydroxynortriptyline) are represented by the mappings in Figs. 1C and 1D.

The subjects with two and three copies of CYP 2D6 show similar nortriptyline kinetics in Fig.

1A and are mapped close to one another by VizStruct in Fig. 1C. The geometric similarity of

the kinetic profiles is reflected in the radial arrangement of points in Figs. 1C and 1D.

These findings demonstrate the feasibility of using VizStruct as an intuitive visualization

approach because it is capable of representing key features of multidimensional data sets

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) studies because parallel coordinates representations of drug

concentration-time profiles get cluttered and difficult to interpret for large PK data sets. Here,

we present results from a population PK simulation case study, in which two groups differing

in clearance (clearance values of 5 volume units/time, and 7.5 volume units/time) were

generated. For both groups, the dose and the volume of distribution of the central compartment

were set to 100 mass units and 5 volume units, respectively. A clearance-parameterized, one-

compartment model was used, and the ADAPT pharmacokinetic and pharmacodynamics

systems analysis program (6) was used to simulate 100 individual PK profiles for each group

assuming a log-normal distribution for the clearance and volume of distribution of the central

compartment.

Figure 2A shows that the two groups differing in clearance are well separated by the mapping

from their sampled kinetic profiles. The mean values of the “amplitude” are greater for the low

clearance group because their mean concentrations are greater.

To further challenge the capabilities of VizStruct, we simulated the challenging PK data set

shown in Figs. 3A and 3B. The curve shown in open circles is the simulation for n one-

compartment model, and the remaining four curves are simulations from two-compartment

models differing slightly in the A, α and B, β values of the bi-exponential relationship C =

Ae−αt + Be−βt; all the curves have the same area under the curve (AUC0→∞), which causes the

curves to overlap significantly. For the one-compartment model, the dose was set to 100 mass

units; the volume of distribution was set to 5 volume units; the clearance was set to 10 volume/

time units. Together, these parameters result in degradation rate constant (K) of 2 time−1, an

initial concentration of 20 mass/volume units, and in a AUC0→∞ of 10 mass·time/volume units.

The two-compartment simulations in Fig. 3 all had AUC0→∞ of 10 mass·time/volume units.

The A and B, in the bi-exponential relationship were, respectively, set to 5 and 10 mass/volume

units for all the curves. The α were 1, 2, 2.5, and 3 time−1 units with corresponding β values

of 2, 1.33, 1.25, and 1.20 time−1 units, for the curves marked, respectively, with filled circles,

open squares, filled squares, and open triangles that are discernible in Fig. 3B. Figure 3A shows

that on linear axes, the data points are virtually superimposed on one another; the differences

become apparent only in Fig. 3B because of the use of the logarithmic axes. Figure 3C shows

that the four PK profiles are well separated in the VizStruct mapping. The two-compartment

profiles are clustered with each other and well separated from the one-compartment profile.

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These findings suggest that VizStruct visualization may prove useful for exploring large

pharmacokinetic data sets.

Application of the VizStruct in Assessing Treatment Responses from Gene Expression

Profiles

Gene expression profiling with microarrays provides simultaneous measurements of thousands

of messenger RNAs and yields large multidimensional data sets that are a challenge to

visualize. Bohen et al. measured the pretreatment gene expression profiles in follicular

lymphomas of patients receiving rituximab, a monoclonal antibody directed against the CD20

protein found on B cells (2). These authors suggested that the gene expression patterns of

lymphomas that were nonresponsive to rituximab therapy were more similar to those of control

lymphoid tissues than lymphomas that responded to therapy. We used the significance analysis

of microarrays (SAM) algorithm (7) to identify an informative subset of 102 genes from the

data set and projected the 102-dimensional vectors for each sample to 2 dimensions using the

VizStruct algorithm (Fig. 4). The VizStruct projection separated the partial/complete

responders (open circles) from the nonresponders (filled circles) and normal controls

(triangles). One nonresponder sample was not separated from the partial/complete responder

cluster. Additionally, the proximity and radial location of the nonresponder cluster relative to

normal controls indicate that the nonresponder group was somewhat more similar to the normal

control group than the partial/complete responder group. Thus, the findings obtained through

visualization are generally consistent with the conclusions of Bohen et al. (2).

DISCUSSION

In this report, we presented case studies for several pharmaceutical applications ranging from

pharmacogenetics, population pharmacokinetics, and pharmacogenomics for which the

VizStruct multidimensional visualization approach can prove useful.

The pharmacogenetics case study of the Dalen et al. data set was kept intentionally simple so

that the findings from VizStruct could be easily compared to the interpretations obtained by

visual inspection of the time profiles. Therefore, this example should considered as a feasibility

study rather than a rigorous demonstration of VizStruct capabilities because it could be argued

that the original data from Dalen et al. could be visualized simply by plotting the tmax (time at

which the maximal concentration Cmax occurs) against the Cmax, and that the use of VizStruct

was unnecessary. To further challenge the capabilities of VizStruct, we examined other, more

difficult data sets in the remaining case studies.

We examined two pharmacokinetic data sets that were structured using simulations but

included more difficulty than the pharmacogenetics case study. In pharmacokinetics,

exponential decay processes are commonly used for modeling, and VizStruct has an excellent

ability to succinctly represent PK profiles with a wide range of half-lives. For example, the

unit bolus—which can be viewed as an extremely fast, first-order process—is mapped to

amplitude R = 1, and phase ϕ = 0, while a constant function—which does not change with time

and can therefore be viewed as an extremely slow, first-order process—is mapped to R = 0 and

ϕ = 0. VizStruct is also relatively insensitive to noise because the discrete Fourier-transform

sum-product, which underlies the mapping, reduces the effects of noise.

The case study of the gene expression profiles reported by Bohen et al., which had 102 genes,

highlights the ability of VizStruct to handle large multidimensional data sets (2). Despite the

large number of genes and simplicity of the visualization, we found that the VizStruct approach

was capable of recapitulating the key findings of Bohen et al., which were arrived at using a

hierarchical clustering algorithm.

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The VizStruct approach differs distinctively from competing visualization methods such as

parallel coordinates and multidimensional scaling (MDS). Most pharmaceutical scientists use

the parallel coordinates approach for visualizing the data; that is, each dimension is plotted

along a separate axis. The parallel coordinates approach gets easily cluttered and increasingly

difficult to interpret when used with large data sets with modest levels of noise. The MDS

approach is the current gold standard for multidimensional visualization. In MDS, the

presentation in two dimensions is optimized to preserve a specific aspect of the relationship,

for example, the Euclidean distance, block distance, or rank relationships between the points

in the N-dimensional space. In practice, MDS encompasses a class of methods, depending on

the stress function being optimized. For many pharmaceutical applications, Sammon's

nonlinear mapping is probably the most appropriate MDS method because it normalizes the

distances in the stress function to distances in the original N-dimensional space. Generally,

Sammon's mapping improves visualization of data sets containing a wide dynamic range of

values. Despite providing results that are mathematically optimal in some sense, MDS

(Sammon's mapping included) is not ideal primarily because the incremental addition of even

a single point requires a complete repetition of the optimization procedure and possible

extensive reorganization of all the previously mapped points to new locations. It is

computationally intensive with time complexity of O(n2), where n is the number of points,

because of the time-consuming function evaluations and iterations required for optimization.

VizStruct, in contrast, is computationally efficient with time complexity of O(n log n) and does

not require any rearrangements to accommodate incremental points.

In conclusion, VizStruct is effective, flexible and versatile. The method is also computationally

efficient and has sound theoretical underpinnings. It may prove a useful approach for

multidimensional data visualization in many pharmaceutical sciences applications.

ACKNOWLEDGMENTS

This work was supported in part by Grant RG3258A2 from the National Multiple Sclerosis Society. Support from the

National Science Foundation (Research Grant 0234895) and the National Institutes of Health (P20-GM 067650) is

also gratefully acknowledged.

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