Stochastic modeling of systems mapping in pharmacogenomics

Abstract

As a basis of personalized medicine, pharmacogenetics and pharmacogenomics that aim to study the genetic architecture of drug response critically rely on dynamic modeling of how a drug is absorbed and transported to target tissues where the drug interacts with body molecules to produce drug effects. Systems mapping provides a general framework for integrating systems pharmacology and pharmacogenomics through robust ordinary differential equations. In this chapter, we extend systems mapping to more complex and more heterogeneous structure of drug response by implementing stochastic differential equations (SDE). We argue that SDE-implemented systems mapping provides a computational tool for pharmacogeneticor pharmacogenomic research towards personalized medicine.

1. Introduction

Substantial interindividual variability exists in drug response and toxicity for most patient populations. Often, a proportion of patients does not respond, responds only partially, or encounters adverse drug reactions (ADRs) to drugs [1]. In some circumstance, drug concentrations in plasma can differ by more than 600-fold between two patients of the same weight treated with the same drug dosage. Although this variation can be attributed to differences in genetic, pathophysiological, or environmental factors, it should be ultimately determined by a drug's physicochemical (e.g., passive diffusion) and biochemical properties (e.g., interactions with metabolizing enzymes and transporters) [2], controlled jointly by genes and the environment [3]. In general, genetic variants are found to account for 15–30% of inter-individual variability in drug metabolism and response, but for certain drugs, genetic variants can account for up to 95% of interindividual differences in drug disposition and effects [1,4–6]. All these discoveries have led to a prevailing perspective that the choice of drugs and drug dosage needs to be matched to the genetic makeup of individual patients.

It has been recognized that genetic factors affect the response of individuals to a drug by altering its absorption, distribution and metabolism, and interactions with its target within the pharmacokinetic and pharmacodynamic machinery [3]. Pharmacokinetics (PK) describes the time course of drug absorption, distribution, metabolism, and excretion after a drug is administered into the body, whereas pharmacodynamics (PD) refers to the relationship between drug concentration at the site of action and the resulting effect, including drug efficacy and drug toxicity [6–10]. As a theory that can reveal the mechanisms and processes of drug reactions in the body, PK/PD principles have been applied to the safe and effective therapeutic management of drugs for an individual patient [3]. By treating drug–body interactions as a dynamic system, a number of ordinary differential equations (ODEs) have been established to characterize the dynamic behavior of PK/PD processes [9,11,12]. These ODEs have been instrumental for the prediction of drug response from the physiology and biochemistry of drug action [3]. More recently, Wu and group have incorporated PK/PD-related ODEs into their newly developed systems mapping model [13–18] aimed to identify the genetic determinants of drug response through its underlying PK/PD processes [19], providing an analytical tool to unravel the genetic architecture of drug response.

Since ODEs are derived under the assumption that observed kinetics and dynamics are driven through internal deterministic mechanisms, their applications may be limited in real pharmacological processes in which random fluctuations affect the relationship between drug effect and drug concentration [20]. Tremendous efforts have been made to extend the deterministic models to stochastic differential equations (SDEs) where one or more of the terms are modeled as a stochastic process [21]. D'Argenio and Park [22] and Ramanathan [23,24] are among the first who incorporated random fluctuations into the PK/PD models. More sophisticated stochastic models that include multiple compartments, non-linear or time-inhomogeneous absorption or elimination have been available through collective work of many mathematical biologists [25–29]. The approaches for estimating the curve parameters that specify a group of SDEs using observed data at discrete time points have been proposed [30]. Donnet et al. [31] developed a Bayesian method to estimate SDE-based mixed model parameters using PK/PD data, typical of high sparsity and subject-dependent measure schedule. A review of the estimation of SDEs in PK/PD models is given by Donnet and Samson [21].

The purpose of this study is to incorporate PK/PD-related SDEs into the framework of systems mapping by which the genetic control mechanisms of drug response can be elucidated more precisely by integrating random fluctuations. We implement an EM algorithm to estimate genotype-specific SDE parameters that characterize the process of drug reactions. The model allows the interplay between genes and PK/PD processes to be tested in a quantitative way. We performed computer simulation to investigate the statistical properties of SDE-embedded systems mapping and compared with the previously developed ODE-based models. The new model will provide a useful tool to characterize drug response-related genes toward precision drug delivery.

2. Stochastic systems mapping

2.1. Stochastic coupled PK-PD models

In PK/PD modeling frequently using noisy data, SDEs, as stochastic state space models, are more advantageous over ODEs by including a diffusion term and thereby allowing a better fit of observed data. For parameter estimation, SDE models are powerful for analyzing serially correlated residuals. Several stochastic PK/PD models have been available, some of which only consider the PK or PD model stochastic whereas the others treat both models as stochastic [20,25,26]. This article does not intend to provide a review of stochastic PK/PD models, rather than devise a systems mapping model of drug response based on a representative PK/PD model.

Tornøe et al. [26] derived a stochastic first-order elimination one-compartment PK model and an indirect response PD model, which is coupled in the form,

$$\{\begin{array}{c}d{C}_t=-k_e{C}_{t}dt+{\gamma }_{C}d{B}_{1t}\\ d{R}_t=[k_{\text{in}}-k_{\text{out}}(1+\frac{C_t}{E{C}_{50}+C_t}\left)R_t\right]dt+{\gamma }_{R}d{B}_{2t}\end{array}$$ (1)

where C_t is the state variable for the plasma concentration of a drug at time t; R_t is the state variable for the PD response at time t; k_e is the elimination constant, EC₅₀ is the drug concentration causing 50% of maximal stimulation; γ_C and γ_R are two diffusion coefficients; and B_1t and B_2t are two independent standard Wiener processes. The PD parameters include k_in and k_out, the zero- and first-order rate constant for production and loss of an effect, respectively [9]. Parameters in the models (1) are (k_e,k_in,k_out,EC₅₀,γ_C,γ_R) = (θ,γ), where θ = (k_e,k_in,k_out,EC₅₀) are the parameters corresponding to the drift terms and γ = (γ_C,γ_R) are the parameters corresponding to the diffusion terms.

2.2. Clinical design

The study design in clinical pharmacogenetics is described in Wu and Lin [32]. Assume that a random sample of n subjects is drawn from a natural population at Hardy-Weinberg equilibrium. All these samples are genotyped for DNA markers throughout the genome. After a drug is administrated, these subjects are measured for the plasma concentration of drug (C) (mg/l), and the pharmacologic response of drug (R) such as blood pressure or heart rate at a series of time points. Considering the reality of pharmacological trials, we allow the phenotypic measures of the subjects to be taken at irregular, unequally-spaced intervals, with subject-dependent measurement times. Let (t₁, …, t_Ti) denote the time points used for subject i. The observed concentration (C) and effect (R) data for this subject is expressed as ${\mathbf{Y}}_i=({\mathbf{Y}}_i^C,{\mathbf{Y}}_i^R)=(Y_i^C(t_1),\dots ,Y_i^C(t_{T_i}),Y_i^R(t_1),\dots ,Y_i^R(t_{T_i}))$. Our hypothesis is that there exist specific genes, or quantitative trait loci (QTLs), that control the PK/PD process through a system of SDEs (1). These underlying QTLs can be mapped by using molecular markers that are associated with them.

In order to focus on the description of systems mapping, we assume no effects due to covariates such as race, sex, life style among others by data standardization. In theory, it is straightforward to incorporate covariate effects into systems mapping, although this needs extensive computing capacity.

2.3. Likelihood and estimation

If specific QTLs exist to affect the PK/PD system (1), the parameters (θ,γ) that specify the system should be different among QTL genotypes. Genetic mapping uses a mixture model-based likelihood to estimate QTL genotype-specific parameters from the mapping population. Assuming that multiple QTLs are involved, leading to J QTL genotypes, this likelihood is expressed as

$$L({\mathbf{Y}}^C,{\mathbf{Y}}^R)=\prod _{i=1}^n\left[{\mathit{\omega }}_{1|i}{f}_1\right({\mathbf{Y}}_i^C,{\mathbf{Y}}_i^R;{\mathbf{\theta }}_1,{\mathbf{\gamma }}_1,\mathbf{\Psi })+\dots +{\mathit{\omega }}_{J|i}{f}_J({\mathbf{Y}}_i^C,{\mathbf{Y}}_i^R;{\mathbf{\theta }}_J,{\mathbf{\gamma }}_J,\mathbf{\Psi }\left)\right]$$ (2)

where $({\mathbf{Y}}^C,{\mathbf{Y}}^R)=({\mathbf{Y}}_1^C,{\mathbf{Y}}_1^R,\dots ,{\mathbf{Y}}_n^C,{\mathbf{Y}}_n^R)$is the joint vector of phenotypic values for the PK/PD responses; ω_j|i is the conditional probability of QTL genotype j (j = 1,…, J) given the marker genotype of subject i, and $f_j({\mathbf{Y}}_i^C,{\mathbf{Y}}_i^R;{\mathbf{\theta }}_j,{\mathbf{\gamma }}_j,\mathbf{\Psi })$ is a multivariate normal distribution with expected mean vector for subject i that belongs to QTL genotype j,

$$({\mathbf{\mu }}_{j|i}^C,{\mathbf{\mu }}_{j|i}^R)\equiv ({\mu }_{j|i}^C(t_1),\dots ,{\mu }_{j|i}^C(t_{T_1}),{\mu }_{j|i}^R(t_1),\dots ,{\mu }_{j|i}^R(t_{T_1}\left)\right)$$ (3)

and covariance matrix for subject i,

$${\mathbf{\sum }}_i=\left(\begin{array}{cc}{\mathbf{\sum }}_i^C& {\mathbf{\sum }}_i^{CR}\\ {\mathbf{\sum }}_i^{RC}& {\mathbf{\sum }}_i^R\end{array}\right)$$ (4)

with ${\mathbf{\sum }}_i^C$ and ${\mathbf{\sum }}_i^R$ being (T_i × T_i) covariance matrices of time-dependent C and R values, respectively, and ${\mathbf{\sum }}_i^{CR}={\mathbf{\sum }}_i^{RC}$ being a (T_i × T_i) covariance matrix between the two variables.

For a natural population, the conditional probability of QTL genotype given a marker genotype (ω_j|i) is expressed in terms of haplotype frequencies or linkage disequilibria between markers and QTLs [33]. In the density function $f_j({\mathbf{Y}}_i^c,{\mathbf{Y}}_i^R;{\mathbf{\theta }}_j,{\mathbf{\gamma }}_j,\mathbf{\Psi })$ for systems mapping, we capitalize on SDEs (1) to model QTL genotype-specific mean vectors with PK/PD parameters (θ_j,γ_j) (j = 1,…, J). Meanwhile, we choose a parsimonious and flexible approach to model the covariance structure using a set of parameters contained in Ψ. These approaches include autoregressive models, antedependence models, autoregressive moving average models, and nonparametric and semiparametric approaches [34,35].

In previous systems mapping models [13-19], the EM algorithm was implemented to estimate marker-QTL haplotype frequencies using a close-form expression derived by Wang and Wu [33]. The fourth-order Runge-Kutta algorithm is used to estimate the parameters that define differential equations for individual QTL genotypes, whereas the parameters that model the covariance structure are estimated by the simplex algorithm. Thus, a hybrid approach that combines the EM, fourth-order Runge-Kutta and simplex algorithms is implemented to solve the mixture likelihood (2). Other methods for parameter estimation include Liu and Wu's [36] Bayesian parametric approaches and Das et al.'s [37] Bayesian nonparametric approach.

Here, we implemented an EM algorithm coupled with the Kalman filter procedure [38] for the maximum likelihood estimation of the parameters. We express the observed values of PK/PD variables at a time t_τ as

$$\begin{array}{c}{\mathbf{Y}}_i^c(t_{\tau })={\mu }_{\tau }^C+{\epsilon }_{i\tau }^C\\ {\mathbf{Y}}_i^R(t_{\tau })={\mu }_{\tau }^R+{\epsilon }_{i\tau }^R\end{array}$$ (5)

where residual errors ${\epsilon }_{i\tau }^C\sim N(0,{\sigma }_C^2)$ and ${\epsilon }_{i\tau }^R\sim N(0,{\sigma }_R^2)$. now, we reformulate the density function in likelihood Eq. (2) as $f_j({\mathbf{Y}}_i^C,{\mathbf{Y}}_i^R;{\mathbf{\theta }}_j,{\mathbf{\gamma }}_j,{\sigma }_C^2,{\sigma }_R^2)$ by assuming no autocorrelations for residual errors. According to model Eq. (5), the autocorrelations for the PK/PD variables among different time points are reflected in the system of SDEs Eq. (1).

Let ${\mathbf{Y}}_{i\tau }=(Y_i^C(t_{\tau }),Y_i^R(t_{\tau }))$ denote the phenotypic values of PK/PD variables at time point t_τ and ${\overrightarrow{\mathbf{Y}}}_{i\tau }=(Y_i^C(t_1),Y_i^R(t_1),\dots ,Y_i^C(t_{\tau }),Y_i^R(t_{\tau }))$ denote the phenotypic values at time points from t₁ to t_τ. Then we express $f_j({\mathbf{Y}}_i^C,{\mathbf{Y}}_i^R;{\mathbf{\theta }}_j,{\mathbf{\gamma }}_j,{\sigma }_C^2,{\sigma }_R^2)$ as

$$f_i({\mathbf{Y}}_i^C,{\mathbf{Y}}_i^R,{\mathbf{\theta }}_j,{\mathbf{\gamma }}_j,{\sigma }_C^2,{\sigma }_R^2)=P({\mathbf{Y}}_{i1})\left(\prod _{\tau =2}^{T_i}P\right({\mathbf{Y}}_{i\tau }|{\stackrel{\rightharpoonup }{\mathbf{Y}}}_{i,\tau -1}))=P({\mathbf{Y}}_{i1})(\prod _{\tau =2}^{T_i}\frac{exp(-\frac{1}{2}{\mathbf{e}}_{i\tau }^T{\mathbf{V}}_{i,\tau |\tau -1}^{-1}{\mathbf{e}}_{i\tau })}{2\pi \sqrt{det\left({\mathbf{V}}_{i,\tau |\tau -1}\right)}})$$ (6)

where e_iτ = Y_iτ − E(Y_iτ| Y⃗ _i,τ−1) and V_{i,τ|τ − 1} = Var(Y_iτ| Y⃗ _i,τ−1).

Here, the prediction error e_iτ and conditional covariance V_{i,τ|τ − 1} can be computed using the Kalman filtering procedure, whereas the EM algorithm is implemented to estimate marker-QTL haplotype frequencies.

2.4. Hypothesis tests

The first question to address is to test whether there are specific QTLs for the PK or PD processes. This can be tested by calculating the likelihood ratio (LR), expressed as

$$\text{LR}=-2(ln{L}_0-ln{L}_1)$$ (7)

where the L₀ is the likelihood under the null hypothesis, i.e., there are no QTL, expressed as H₀: (θ_j,γ_j) ≡ (θ,γ) for j = 1,…, J, respectively, and the L₁ is the likelihood under the alternative hypothesis, i.e., there are QTLs, expressed as H₁: not H₀. The LR value calculated using Eq. (7) is compared with the critical threshold determined empirically from permutation tests.

In the PK/PD models, different parameters have different mechanistic means. Systems mapping allows the genetic control of each parameter or parameter combination to be tested. In practice, it is interesting to test whether there are pleiotropic QTLs that jointly control both PK and PD processes. This can be tested by the two hypotheses as follows:

$$H_0:(k_{ej},{\gamma }_{Cj})\equiv (k_e,{\gamma }_C)\text{vs}.H_1:(k_{ej},{\gamma }_{Cj})\ne (k_e,{\gamma }_C)\text{for at least one}j$$ (8)

$$\begin{array}{c}{H}_0:(k_{\text{in}j},k_{\text{out}j},E{C}_{50j},{\gamma }_{Rj})\equiv (k_{\text{in}},k_{\text{out}},E{C}_{50},{\gamma }_R)\text{vs}.\\ H_1:(k_{\text{in}j},k_{\text{out}j},E{C}_{50j},{\gamma }_{Rj})\ne (k_{\text{in}},k_{\text{out}},E{C}_{50},{\gamma }_R)\text{for at least one}j\end{array}$$ (9)

If the null hypotheses of both Eqs. (8) and (9) are rejected, this means that pleiotropic QTLs exist to control these two processes.

SDE parameters (k_e,k_in,k_out,EC₅₀,γ_C,γ_R) = (θ,γ) contain drift parameters θ = (k_e,k_in,k_out,EC₅₀) and diffusion parameters γ = (γ_C,γ_R). Whether pleiotropic QTLs exist to affect these two types of parameters can be tested by

$$H_0:{\theta }_j\equiv \theta \text{vs}.H_1:{\theta }_j\ne \theta \text{for at least one}j$$ (10)

$$H_0:{\gamma }_j\equiv \gamma \text{vs}.H_1:{\gamma }_j\ne \gamma \text{for at least one}j$$ (11)

If both null hypotheses in Eqs. (10) and (11) are rejected, there should be pleiotropic QTLs for drift and diffusion processes of PK/PD responses.

The likelihood for systems mapping Eq. (2) is general, which allows multiple QTLs to be characterized. QTLs may govern PK/PD processes through additive, dominant and epistatic effects, all of which can be tested by formulating appropriate null hypotheses.

3. Computer simulation

We performed simulation studies to investigate the statistical properties of systems mapping used to identify QTLs for PK/PD responses. We make a simplified assumption, i.e., the PK/PD process is controlled by a single QTL, although it is likely that multiple QTLs are involved, operating in a complex manner. A random sample of 500 subjects from an equilibrium human population was simulated, in which a marker of alleles M and m is associated with a PK/PD QTL of alleles A and a. The population frequencies of alleles are assumed as 0.6 for M, 0.4 for m and 0.7 for A, 0.3 for a. The two loci have the linkage disequilibrium of 0.05. These subjects are assumed to receive the administration of a drug to treat a disease. The PK/PD phenotypic data, the plasma concentration of drug (C) (mg/l) and drug effect (R) at 6 time points after a drug is administered, were simulated by summing time-dependent genotypic means for QTL genotypes, AA, Aa and aa, determined by Eq. (1) and the residual errors with variances (Eq. (5)). Two different levels of heritability, 0.1 and 0.4, are considered when the residual variances are determined.

Table 1 gives the maximum likelihood estimates of SDE parameters, residual variances, marker and QTL allele frequencies and marker-QTL linkage disequilibrium obtained from a joint analysis of the EM algorithm and Kalman filter procedure. Note that in this simulation study we add the diffusion terms, γ_C and γ_R, to the residual variances. We also present the maximum likelihood estimates using ODEs in Table 1. Systems mapping provides reasonably good estimates of all parameters with a modest sample size. Increasing heritability can improve the accuracy and precision of parameter estimation dramatically, suggesting that precise measurements of PK/PD phenotypes are crucial for better estimation of parameters. Compared to ODEs, the SDE-based models provide more accurate estimate of model parameters. The estimated SDE parameters were used to draw dynamic curves of PK and PD responses over time as well as dynamic relationships between these two processes (Figs. 1 and 2). It is found that the estimates curves are broadly consistent with the true curves obtained from given SDE values, confirming the precision of systems mapping. PK/PD responses can be better estimated when the underlying heritability increases from 0.1 (Fig. 1) to 0.4 (Fig. 2). Our model allows the test of a number of clinically meaningful hypotheses about pleiotropic control expressed in Eq. (8) vs. Eq. (9) and Eq. (10) vs. Eq. (11). In this particular example of simulation, we found that the same QTL pleiotropically affects both simulated PK and PD responses.

Table 1.

Maximum likelihood estimates (MLE) of SDE parameters, residual variances, marker and QTL allele frequencies, and marker-QTL linkage disequilibrium from a simulated natural population of 500 subjects. The means of the estimates and their standard errors (SE) are calculated from 200 simulation replicates.

			MLE (SE)
			H2 = 0.4				H2 = 0.1
Parameter	Genotype	True	SDE		ODE		SDE		ODE
ke	AA	1.00	1.019	(0.013)	1.012	(0.018)	1.006	(0.035)	1.028	(0.037)
	Aa	0.80	0.772	(0.015)	0.780	(0.023)	0.807	(0.028)	0.759	(0.035)
	aa	0.50	0.514	(0.009)	0.518	(0.017)	0.525	(0.024)	0.516	(0.029)
kin	AA	1.00	0.962	(0.031)	1.029	(0.038)	1.036	(0.058)	0.968	(0.064)
	Aa	1.50	1.539	(0.035)	1.527	(0.052)	1.495	(0.081)	1.531	(0.079)
	aa	2.00	1.907	(0.047)	1.916	(0.064)	1.951	(0.098)	1.926	(0.107)
kout	AA	0.10	0.103	(0.003)	0.092	(0.011)	0.099	(0.015)	0.093	(0.018)
	Aa	0.30	0.304	(0.005)	0.291	(0.013)	0.288	(0.022)	0.286	(0.025)
	aa	0.20	0.209	(0.005)	0.212	(0.009)	0.190	(0.017)	0.215	(0.031)
EC50	AA	5.00	5.243	(0.316)	5.195	(0.427)	5.140	(0.528)	5.283	(0.564)
	Aa	3.00	3.043	(0.283)	3.052	(0.358)	3.077	(0.450)	3.142	(0.518)
	aa	1.00	1.047	(0.179)	0.949	(0.221)	0.980	(0.396)	1.053	(0.430)
γC		0.10	0.108	(0.008)	–		0.112	(0.025)	–
γR		0.50	0.516	(0.017)	–		0.519	(0.041)	–
σC		0.08	0.085	(0.004)	0.076	(0.005)	–		–
		0.20	–		–		0.226	(0.027)	0.217	(0.026)
σR		0.72	0.717	(0.012)	0.724	(0.015)	–		–
		1.76	–		–		1.699	(0.081)	1.825	(0.093)
p		0.60	0.592	(0.005)	0.595	(0.005)	0.605	(0.016)	0.593	(0.018)
q		0.70	0.693	(0.013)	0.708	(0.014)	0.684	(0.037)	0.712	(0.039)
D		0.05	0.065	(0.008)	0.061	(0.009)	0.068	(0.019)	0.059	(0.017)

Fig. 1.

Genetic variation in PK and PD processes estimated from a couple of simulated dynamic phenotypic variables with heritability of 0.1 by systems mapping. Broken lines are the estimated PK and PD curves for three genotypes at a hypothesized QTL, AA (black), Aa (red) and aa (green), which are broadly consistent with the true curves indicated by solid lines. (A) PK process in which drug concentration decreases with time. (B) PD process in which drug effect increases with time. (C) Cycle limit of PK and PD response.

Fig. 2.

Genetic variation in PK and PD processes estimated from a couple of simulated dynamic phenotypic variables with heritability of 0.4 by systems mapping. Broken lines are the estimated PK and PD curves for three genotypes at a hypothesized QTL, AA (black), Aa (red) and aa (green), which are broadly consistent with the true curves indicated by solid lines. (A) PK process in which drug concentration decreases with time. (B) PD process in which drug effect increases with time. (C) Cycle limit of PK and PD response.

4. Discussion

The identification of specific genes involved in drug response has been one of the hottest and fastest growing areas in pharmacogenetics and pharmacogenomics toward personalized medicine [39–45]. While traditional genetic analysis based on the direct association between marker genotypes and static drug response lacks a mechanistic characterization of the relationship between genes and drug effect or toxicity, a dynamic model, called functional mapping, has emerged as a powerful tool to unravel the underlying genetic mechanisms by integrating the physiology of drug actions [46–48]. The fundamental idea of functional mapping is to incorporate the mathematical aspects of PK or PD responses, allowing the interplay between genes and biochemical steps of PK or PD to be identified [8–10].

As two different processes of drug response, PK and PD are linked in a body–drug interaction system where ordinary differential equations (ODEs) can be used to quantify their dynamic relationship [12]. Fu et al. [19] implemented these PK/PD-related ODEs into a systems mapping framework by which genes for individual steps and pathways toward final drug efficacy or drug toxicity can be identified in a quantitative way. In practical clinical trials, PK/PD variables may not be measured under a completely controlled condition, inevitably leading to imprecision of pharmacological data. Also, PK/PD data are often highly sparse and unbalanced due to irregular measurement intervals. ODE modeling is less powerful to handle these heterogeneous data. This challenge can be addressed by a different type of models, stochastic differential equations (SDEs), that embrace more complex variations in the dynamics [20,25,26]. This article presents a first attempt to implement SDEs into systems mapping for mapping PK/PD genes or QTLs.

All biological systems, including pharmacological processes, evolve under stochastic forces. The advantage of SDEs lies in its capacity to model random influences by taking into account the subsystems of the real world that cannot be sufficiently isolated from effects captured by ODEs [21]. Systems mapping implemented with SDEs can particularly model and investigate the influence of random noise in the PK/PD dynamics through genes. In pharmacological studies, both internal and external factors may lead to the erratic behavior of drug reactions. SDE-implemented systems mapping can model variations due to these factors, enhancing the model predictions and results interpretation. Our computer simulation has shown that this approach may be useful for the identification of pharmacological QTLs in practice. Maximum likelihood approaches are implemented as model-fitting algorithms of systems mapping, but other approaches, such as weighted least squares or Bayesian, can also be used in helping select the best-fit model to real data sets.

We have focused on the modeling of genetic influences on drug response in a traditional Mendelian manner, although we have integrated it with the physiology of drug reactions in the body. This is not adequate to illustrate a complete picture of the genetic architecture of drug response. First, individual QTLs may trigger their effects through not only additive and dominant actions, but also imprinting actions. The identification of imprinting effects due to epigenetic marks requires genotype information of the parents of subjects under study [49]. Second, epigenetic modifications resulting from DNA methylation, histone modification and chromatin remodeling have been thought to play an important role in affecting variation in drug response [50]. Third, genes affect final phenotypes of drug response by perturbing the abundances of transcripts, proteins, and metabolites that form a sequential order of biochemical pathways toward cell physiology and drug reactions [51]. Thus, an integrative approach of systems biology and our systems mapping model will provide an unprecedented opportunity to find and construct cellular regulatory networks causing drug effects. Taken all these together with the analysis and modeling of data from genome-wide association studies [44,52,53], systems mapping with appropriate extensions will help to drive the drug discovery processes, predict rare adverse events, and catalyze the practice of precision medicine.