Functional mapping of drug responsewith pharmacodynamic–pharmacokinetic principlesKwangmi Ahn1, Jiangtao Luo1, Arthur Berg1, David Keefe2 and Rongling Wu1

1 Center for Statistical Genetics, Pennsylvania State University, Hershey, PA 17033, USA2 Department of Obstetrics and Gynecology, New York University, Langone Medical Center, New York, NY 10016, USA

Opinion

Recent research in pharmacogenomics has inspired ourhope to predict drug response by linking it with DNAinformation extracted from the human genome. How-ever, many genetic models of drug response do notincorporate biochemical principles of host–drug inter-actions, limiting the effectiveness of the predictivemodels. We argue that functional mapping, a compu-tational tool aimed at identifying genes and geneticnetworks that control dynamic traits, can help explainthe detailed genetic architecture of drug response byincorporating pharmacokinetic and pharmacodynamicprocesses. Functional mapping is particularly powerfulin determining the genetic commonality and differencesof drug efficacy vs. drug toxicity and drug sensitivity vs.drug resistance. We pinpoint several future directions inwhich functional mapping can be coupled with systemsbiology to unravel the genetic and metabolic machineryof drug response.

Challenges and opportunities in pharmacogeneticsThere is tremendous variability in drug response amongdifferent individuals who receive the same drug treatment[1]. Because genes play a central role in determining thisvariability, pharmacogenetics or pharmacogenomics, thestudy of the genetics or genomics of drug response, has nowemerged as one of the most widely accepted disciplines inmodern medicine [2–4]. The development of pharmacoge-netics has been further stimulated by the completion of thehuman genome sequence in 2005 and its derivative, theHapMap Project, together with rapid improvements ingenotyping analysis [5]. These technologies allow geneti-cists to elucidate the detailed genetic architecture of drugresponse at the nucleotide level. For example, genome-wide association studies (GWAS) of single nucleotide poly-morphisms have identified genes that encode the action ofFood and Drug Administration-approved drugs includingthiazolidinediones and sulfonylureas (type 2 diabetes) [6],statins (lipid levels) [6] and estrogens (bone density) [7]. Inshedding light on the genetic basis of complex traits,GWAS will have an increasing impact on the identificationof genes for drug response.

Corresponding authors: Keefe, D. (david.keefe@nyumc.org);Wu, R. (rwu@hes.hmc.psu.edu).

306 0165-6147/$ – see front matter � 2010 Elsevier Ltd. All rights reserv

By integrating the dynamic characteristics of drug–bodyinteractions [8], we argue that the genetic control mech-anisms of drug response can be studied in an effective way,which could yield a breakthrough that cannot be achievedusing current pharmacogenetic approaches based on asingle measurement of drug response. Drug response isunderscored by two mutually related dynamic processes,pharmacokinetics (PK), which describes what an organismdoes to a drug, and pharmacodynamics (PD), whichdescribes what a drug does to an organism. To make acomprehensive description of drug response, a series ofdynamic measurements should be taken at multiple dis-crete time points or states. Empirical analyses of PK andPD data suggest that the time-dependent change of drugconcentration in the body typically follows a biexponentialcurve [9,10], whereas drug effects increase with drug dosein a sigmoid manner (i.e. the Emax curve) [11]. Thesemathematical models that quantify pharmacodynamic/pharmacokinetic reactions can be integrated into a generalframework of functional mapping [12–14], pioneered by R.Wu, to better study the interplay between genetic actionsand drug response.

Pharmacogenetic studies of drug response are chal-lenged by biochemical complexity in which various path-ways interacts with each other through a hierarchic andcross-sectionalmanner [15]. To better study the behavior ofeach pathway, drug response can be modeled as a dynamicsystem, in which pathways are first assayed by analyticapproaches and then integrated by synthetic approaches.Observed associations among these pathways could bedriven by common genetic factors or can result from phys-iological interactions. It is of fundamental importance toknow whether these pathways share common geneticdeterminants, which genetic factors are specific to differentpathways and what is the nature of the non-genetic inter-actions among these pathways.

In this article, we show that functional mapping offersan unprecedented potential to study the genetic and geno-mic architecture of drug response by dissecting it into PD-and PK-governed pathways. We will first outline the bio-logical merit of functional mapping and explain how it canbe used to identify genes responsible for drug response.Then, we will show how functional mapping is imple-mented to a dynamic system of drug response to addressfundamental questions about the genetic control of drug

ed. doi:10.1016/j.tips.2010.04.004 Trends in Pharmacological Sciences 31 (2010) 306–311

Opinion Trends in Pharmacological Sciences Vol.31 No.7

efficacy and drug toxicity, drug–body interactions andresponse rhythms. Examples will be presented, althoughit is not our goal to review these in a comprehensivemanner; rather we intend to identify the challenges thatmust be overcome and their potential solutions throughfunctional mapping. In Box 1, we describe a statisticalframework for functional mapping.

Functional mappingBiological merit

All biological traits experience a dynamic or developmentalprocess. Traditional approaches for genetic mapping ofdynamic traits have been to associate markers with phe-notypes for different ages or stages of development and tocompare the differences across these stages, or to usemultiple-trait mapping where the same character isrepeatedly measured at different times. In either case,these approaches do not capture the dynamic structureand pattern of the process, which greatly limits the scope of

Box 1. Functional mapping

Functional mapping is founded on finite mixture models, a

statistical model that represents a heterogeneous population as a

mixture of homogeneous components, where each component

follows a certain distribution and the components aggregate to yield

the mixture distribution. Mixture models are employed to model

genotypic segregation of specific genes that determine dynamic

traits such as drug response [47]. For a dynamic trait, each subject i

is described by a curve that is observed by a finite set of

measurements with T time points, arrayed in yi=[yi(1),. . ., yi(T)].

According to the mixture models, each curve is assumed to have

arisen from one of the components, where each component is

modeled by a multivariate (usually normal) distribution. Assuming

there are J genotypes that contribute to the variation among

different curves, such a mixture model is expressed as

y� pðyjv;w; hÞ ¼XJ

j¼1

v j f jðyi ; ’ j ; hÞ (1)

where v=(v1,. . . ,vJ) are the mixture proportions (i.e. genotype fre-

quencies) which are constrained to be non-negative and sum to

unity; w=(w1,. . . ,wJ) are the component- (or genotype-) specific

parameters, with wj being specific to component j; h are parameters

which are common to all components; and fj(yi;wj,h) is a multivariate

normal distribution with mean vector

u j ¼ ½u j ð1Þ; . . . ;u jðT Þ� (2)

and covariance matrix

X¼

s21 � � � s1T

..

.} ..

.

sT 1 � � � s2T

(3)

Traditional genetic mapping approaches estimate all JT time-

specific means in the mean vector [2] and T(T+1)/2 variances and

covariances in the matrix in Equation 3. Instead of estimating all

these parameters, functional mapping estimates the parameters

that model the mean structure and covariance structure based on

biological and statistical principles. In particular, genotype-specific

parameters, w, are used to model the time-dependent changes of

genotypic means, whereas common parameters, h, are used to

model the structure of the longitudinal covariance matrix. The

detailed description of functional mapping used to map the dynamic

change of genetic control is given in Ma et al. [12]. Wang et al. [16]

and Zhao et al. [48] extended functional mapping to model the

temporal pattern of genetic variance and correlation between

bivariate dynamic traits that jointly change as a function of the

same independent variable.

inference about its genetic architecture. To overcome theselimitations, functional mapping, that is the integration ofthe mathematical aspects of the biological mechanismsand processes of the trait with the statistical geneticmapping framework, is the natural way to approach thegenetics of dynamic traits by mapping the underlyingquantitative trait loci (QTLs) [12–14]. The combinationof statistical modeling, genetics and developmental biologyis able to address many questions, such as the patterns ofgenetic control over development, the duration of QTLeffects, as well as what causes developmental trajectoriesto change or stop changing. Functional mapping provides aquantitative and testable framework for assessing theinterplay between gene action and development.

The genetic dissection of a biological process throughfunctional mapping will not only identify genes thatregulate the final form of the trait but also characterizethe dynamic pattern of genetic control in time. Further-more, functional mapping capitalizes on the parsimoniousmodeling of longitudinal mean–covariance structures and,therefore, substantially increases the statistical power androbustness of genetic mapping.

Application

As a demonstration, we used bivariate functional mapping,as shown in Wang et al. [16], to detect QTLs for drugresponse with a small data set collected from the AIDSClinical Trials Group (ACTG)Protocol 315 [17]. In this trial,53 HIV-1-infected patients were periodically measured fortheirHIV-1andCD4+cell contents ondays0, 2, 7, 10, 14, 21,28 and weeks 8 and 12 after the treatment of highly activeantiretroviral therapy (HAART) [18]. According to previousresearch, we used a biexponential function to model HIV-1cell dynamics [17] and a quadratic function to model CD4+dynamics [19]. Functional mapping was used to analyze theACTGdatabyestimatinggenotype-specificparameters thatdefine these dynamic curves and testing their differencesamong different genotypes. A major QTL was detected forHIV and CD4+ cell dynamics [20]. The estimated curveparameters for the three genotypes at this major QTLgraphically illustrate pronounced differences in HIV-1(Figure 1a) and CD4+ profiles (Figure 1b).

Functional mapping has the power to detect QTLs thatpleiotropically or separately affect HIV-1 and CD4+dynamics. Hypothesis tests indicated that this detectedQTL exerts a significant pleiotropic effect on both function-valued traits. The favorable allele (denoted by A) thatcauses a rapid decrease in viral loads and a rapid increasein CD4+ cell counts occurs with rare frequency; its fre-quency is estimated as 0.231, compared with 0.769 for theunfavorable alternative (denoted by a). The heterozygote(Aa) and homozygote (aa) for the unfavorable alleletogether account for an overwhelming majority of patients(0.93). The detected QTL operates in an overdominantmanner. In both the cases of HIV-1 and CD4+ dynamics,the heterozygote seems to have an unfavorable effect onthe progression to AIDS. Patients who carry such a geno-type display a smaller decay rate of viral loads in long-livedand/or latently infected cell compartments (Figure 1a) andsmaller increase rates in CD4+ cell counts (Figure 1b) thanthe patients who carry the other two genotypes.

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Figure 1. Load trajectories of HIV-1 virions (a) and CD4+ cell counts (b) for 46

patients from ACTG Protocol 315. Seven patients were removed from the analysis

because they seem to be outliers. The log-likelihood ratio (LR) test statistic (58.6)

for the full model (there is a major QTL) over the reduced model (there is no major

QTL) was calculated, and it is 40% greater than the critical threshold at the

significance level of 0.01 (41.8) determined from simulated data under the null

hypothesis of no major QTL. Three genotypes, AA, Aa and aa, at the detected QTL

have different curves of drug response, which together fit well those for all

subjects (shown in yellow). HIV-1 cell dynamics was fitted by a biexponential

function, V ðtÞ ¼ P1e�l1 t þ P2e�l2 t , where V(t) is the total amount of virions at time

t, P1 and P2 are baseline values, and l1 and l2 are the two-phase viral decay rates of

productively infected cells and long-lived and/or latently infected cells,

respectively. CD4+ dynamics was fitted by a quadratic function, C(t)=b0+b1t+b2t2,

where C(t) is the count of CD4+ T cells at time t, b0 is the baseline value, and b1 and

b2 are the first- and second-order slopes of quadratic function, respectively. The

autoregressive first-order model was used to model the covariance structure for

functional mapping with this longitudinal data set [16].

Box 2. Differential equations for drug response

Pharmacokinetics: Wang et al. [22] show that the one-compartment

model with the first-order absorption and Michaelis–Menten

elimination can be described by a coupled system of ordinary

differential equations (ODEs)

dQ

dt¼ �kAQ

dC

dt¼ kAQ

V� V maxC

km þ C

8><>: (4)

where time t is the independent variable varying between 0 and

300 min, Q is the amount (mg) of drug in the gastrointestinal tract, C

is the plasma drug concentration (mg/l), V is the distribution volume,

ka is the decay rate of the drug, km is the Michaelis–Menten constant,

Vmax is the maximum rate of the reaction. According to the Michaelis–

Menten equation that describes the initial rate of reaction as a

function of the substrate concentration, the reaction reaches a certain

maximum rate as the substrate concentration increases.

Pharmacodynamics: The one-compartment indirect response

model that describes how a drug stimulates the production of the

effect is expressed [22] as

dC

dt¼ �Cl

VC

dR

dt¼ kin 1þ EmaxC

EC50 þ C

� �� koutR

8>><>>:

(5)

where time t is the independent variable varying between 0 and

300 min, C is the plasma drug concentration (mg/l), R is drug

response that can be viewed as any physiological phenotypes related

to drug efficacy and toxicity, V is the distribution volume, Cl is the

clearance rate, kin and kout are the zero-order and the first-order rate

constant for production and loss of an effect, respectively, and Emax

and EC50 are the maximum effect of the drug and the drug concen-

tration producing 50% of the maximum stimulation, respectively.

Opinion Trends in Pharmacological Sciences Vol.31 No.7

Functional mapping for a pharmacokinetic andpharmacodynamic systemBiochemical networks in a system for drug response con-sist of chemical reactions, such as association, dissociation,degradation and synthesis. The dynamics of biochemicalnetworks obeys the rules of mechanics that regulate theirability to organize movement and biological functions.Given the complexity of a biological system due to massiveamounts of interactions, we need a sufficiently sophisti-cated analysis that can deal with such complexity. Math-ematical models using kinetic theory provide a robustapproach for describing a complex system that encom-passes principles of non-equilibrium statistical mechanics.The models for system dynamics based on the generalizedBoltzmann equation can describe and predict the popu-lation dynamics of several interacting elements [21].

When a drug is administered into a patient’s body, theconcentration of the drug will decay with time due to drugabsorption, distribution and elimination (i.e. metabolism)and excretion [8]. PK is the study of the rate processesobserved over time for a given dosage of a drug. Absorbedby the body, the drug will trigger its effect on the body,which is described by PD. Whereas PK concerns the time-course of drug concentration, PD studies the time-courseand intensity of drug action or response.

PK and PD reflect many biochemical properties of drug–

body interactions directly or indirectly related to drugeffects. One way to model and predict the dynamic behaviorof these two processes is to view the body as a system with

308

several interconnected compartments through which thedrug is distributed and absorbed at certain rates. Thismethod can quantify distributional behaviors of a drugbetween the plasma and other tissues or organs in the body.Such a compartmental analysis has benefitted from amodelbuilt by ordinary differential equations (ODEs) thatdescribes a continuous biological process. Numerous ODEshave been derived to describe changes in drug concen-trations and action in each compartment at a rate pro-portional to the flow of blood to that compartment [22].Box 2 provides such an example in which two coupledsystems of ODEs are used to define a one-compartmentmodelwith thefirst-order absorptionandMichaelis–Mentenelimination, and a one-compartment indirect responsemodel when the drug stimulates production of the effectused, respectively [21].

Luo et al. [23] developed a mathematical algorithm tosolve ODE parameters that define functional mapping.Luo et al.’s approach can be used to estimate genotype-specific ODE parameters that define the system. By testinggenotype-specific differences in these parameters, func-tional mapping can empower geneticists to identify thegenetic control of PK and PD.

Drug–body interaction modelsThe flow of drugs among different organs could affect thepharmacological (favorable or adverse) effects of the drugs.To characterize the elimination of a drug called warfarinfrom the plasma and its excretion in the bile, a physiologi-cal flow model was formulated by a system of ODEs for

Figure 2. Differential equations for a phyniohpiral flow model of different organs. V=volume of organ; subscripts: P=plasma, M=muscle, S=skin, K=kidney, G=gut, L=liver,

Q=blood flow to subscripted organ, C=concentration in organ; R=tissue-to-plasma ratio, r=rate in liver and bile sections, 1, 2 and 3. Redrawn from Ref. [24].

Figure 3. Efficacy–toxicity curves for different patients. Red bars denote drug

efficacy at different doses, whereas blue bars denote drug toxicity at different

doses. The doubled arrows indicate the width of window for the safe use of a drug.

The drug can be used safely with a wider window in (a) than in (b) Redrawn from

Rosania’s (2008) lecture notes at the University of Michigan.

Opinion Trends in Pharmacological Sciences Vol.31 No.7

each organ including plasma, liver, kidney, muscle, skin,gut and bile, aimed to model the elimination of warfarinfrom the plasma and its excretion in the bile [24]. Themodel is based on organ weights or volumes, rates of bloodflow to the organs and the ratio (R) of the concentrations ofdrugs in the different tissues to the concentration inplasma (Figure 2). These parameters contained in thedifferential equations (Figure 2) describe the distribution,action and elimination of the drug in different organs.

A system of differential equations for each organ(Figure 2) can be organized into the framework of func-tional mapping, allowing the test of how genes control theflow of drugs into different organs in a dynamic system.Luo et al. [23] provided the derivation of algorithms forestimating mathematical parameters and the stabilitytests of these parameter estimates. Many of these typesof studies are conducted in experimental animals such asthe rat because the kinetic data and tissue concentrationsrequired cannot be obtained safely in man.

Drug efficacy and drug toxicityThe same medications often produce remarkably differenteffects among patients with the same illness. It is possiblethat a specific medication produces both the desired thera-peutic effects (efficacy) and a risk for an adverse drugresponse (toxicity) both of which vary based on an individ-ual genetic makeup [25]. In Figure 3, the first patient (a)displays greater drug efficacy but lower drug toxicity, thushaving a wider window for the safe use of the drug,compared with the second patient (b).

Mathematical equations have been established tomodelthe PK and PD reactions of drug efficacy and drug toxicity(Box 3). For example, a system of differential equations

309

Box 3. Kinetic equations for drug efficacy vs. toxicity

The following differential equations can be used to specify the PK

processes of drug reactions in different compartments, healthy

tissue and tumor [26]

dP

dt¼ �lP þ iðtÞ

VdC

dt¼ �mC þ P

dD

dt¼ �nC þ P

(6)

where P is the free Pt plasma concentration, C is the total Pt con-

centration in the healthy tissue, D is the total Pt concentration in the

tumor, i(t) is the intravenous drug infusion flow (mg/h) at time t, V is

the distribution volume (ml), l, m, n are the diffusion parameters

calculated after the half-life (ln2/half-life) of the drug in each compart-

ment.

The PD process in the healthy tissue is described as drug toxicity

by

f ðC ; tÞ ¼ F

�C

C50

�gS

1þ�

CC50

�gS

�1þ cos2p

�ðt � fSÞ

T

��(7)

where gS is the Hill coefficient, C50 is the half-saturation concen-

tration, T (24 h) is the period of drug sensitivity variation, fS is the

maximum toxicity phase (h), F is the half-maximum toxicity [26].

By contrast, the PD process in the tumor is described as drug

efficacy by

gðC ; tÞ ¼ H

�D

D50

�gT

1þ�

DD50

�gT

�1þ cos2p

�ðt � fT Þ

T

��(8)

where gT is the Hill coefficient, D50 is the half-saturation concen-

tration, T (24 h) is the period of drug sensitivity variation, fT is the

maximum efficacy phase (h), H is the half-maximum efficacy [26].

Opinion Trends in Pharmacological Sciences Vol.31 No.7

was used to quantify the flows of oxaliplatin, a type of drugused to treat human colorectal cancer, between differentcompartments [26]. After intravenous or intraperitonealinjection, oxaliplatin diffuses according to order 1 kineticsfirstly in the plasma, then to healthy tissues and to tumors.Oxaliplatin triggers its efficacy and toxicity by inhibitingcell population growth in tumoral and healthy issues,respectively. The mean drug activity of oxaliplatin in thehealthy and tumor cell populations are described by agroup of non-linear equations [26].

The PK and PD processes of drug efficacy and drugtoxicity can be organized into functionalmapping. The rolesof genes in coregulating these processes can be estimatedand tested. The specific questions to be addressed include (i)Are there the same or different genes that control drugefficacyanddrug toxicity? (ii)Are there the sameordifferentgenes that control the PK and PD of drug efficacy as well asthe PK and PD of drug toxicity? (iii) Are the relevant genesexpressed at the individual single nucleotide polymorphismor haplotype level? (iv) How can we determine the bestwindow (Figure 3) for a drug to be safely used in a specificpatient based on his/her genetic makeup?

Drug chronotherapyIn several studies of drug response, so-called clock geneswere found to affect patients’ circadian rhythms throughclock-controlled transcription factors [27–29], holding agreat promise for the determination of an individualized

310

optimal body time for drug administration based on apatient’s genetic makeup. It has been suggested that drugadministration at the appropriate body time can improvethe outcome of pharmacotherapy by maximizing potencyand minimizing the toxicity of the drug [27,30], whereasdrug administration at an inappropriate body time caninduce severe side effects [31]. In practice, body time-de-pendent therapy, termed chronotherapy [27,28,32], can beoptimized by implementing the patient’s genes that controlexpression levels of his/her physiological variables duringthe course of a day.

The molecular bases of circadian rhythms have beenclarified during the past decade by experimental advances,first in Drosophila and Neurospora, and more recently incyanobacteria, plants and mammals [33,34]. In view of thelarge number of variables involved and of the complexity offeedback processes that generate oscillations, mathemat-ical models are necessary to comprehend the transitionfrom simple to complex oscillatory behavior and to delin-eate the conditions under which they arise [35].

Future directions: functional mapping merged intosystems biologyTraditional genetic mapping detects genes for drugresponse based on a direct genotype–phenotype relation-ship, whereas functional mapping does so through linkingbiochemical influences to the outcome of drug response inPK and PD pathways. Functional mapping has beenextended to incorporate environmental signals into themodel, testing genotype�environment interaction effectson biological processes [36,37]. Given that the endpointoutcome of drug response is the consequence of the mutualcoordination of multiple factors, such as biochemical,developmental, environmental and stochastic error [38–

42], functional mapping should be expanded to model aweb of interactions among these factors. The predictivemeasurement is then used with genetic, biochemical,developmental and environmental parameters obtainedfrom a subject to predict whether that subject will respondto the particular drug or treatment regimen.

Current functional mapping models are based on adirect relationship between genotype and phenotypethrough statistics, leaving intermediate steps from geno-type to phenotype unknown. As a fundamental rule toform, control and direct every aspect of a phenotype, thecentral dogma of biology, DNA!mRNA!enzyme (inacti-ve)!enzyme (active)!metabolite(s)!metabolism!cellu-metabolism!cellular physiology!phenotype, subjectedto continuous addendums and modifications in the recentpast, can be incorporated into functional mapping. Withsuch incorporation, we are in a better position to unravelthe mechanistic and regulatory process of how genes con-trol the phenotype of drug response.

As the cost of methods for measuringmRNA, protein andother indicators continues to fall, it becomes reasonable todesign experiments for functionalmapping that capture thedynamic processes of phenotypic formation of drug effects ontimescales. With these data, a new discipline, systemsbiology, arises from the study of the transcriptome (theset of RNA transcripts) and the metabolome (the entirerange of metabolites taking part in a biological process).

Opinion Trends in Pharmacological Sciences Vol.31 No.7

Other omes (sets) that might also be of interest include theinteractome (complete set of interactions between proteinsor between theseand othermolecules), the localizome (local-ization of transcripts, proteins, etc.) or even the phenome(complete set of phenotypes) of a given organism [43,44]. Infuture research, we need to develop a new functional map-ping model for reconstructing biological networks by incor-porating interactome, localizome and phenome related todrug response. In particular, the model is capable of map-ping specific QTLs that control transcriptional (eQTL),proteomic (pQTL) and metabolomic (mQTL) expressions[45] and their interaction networks among these differenttypes of QTLs. Linking functional mapping with systemsbiology holds new opportunities to unravel the genetic anddevelopmental machinery of patients’ responses to medi-cations and push forward the frontier of personalized medi-cine [46].

AcknowledgementsWe are grateful to Dr. Zuoheng Wang for analyzing the ACTG data usinga bivariate functional mapping model. This work is supported by NationalInstitute of Health American Recovery and Reinvestment Act (NIHARRA) grant 09095.

Conflict of interestThe authors declare no competing financial interests.

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