Drugs that inhibit the reverse transcriptase (RT) activity of Human Immunodeficiency Virus (HIV) are widely used to treat HIV infection. RT is an ideal target for antiviral HIV therapy because it is the key required for HIV replication. Non-nucleoside reverse transcriptase inhibitors (NNRTIs) inhibit RT activity by selectively binding RT at a hydrophobic binding pocket adjacent to the polymerase active site. Efavirenz (EFV) is an commonly used NNRTI to treat HIV infection 
but patients can develop resistance to this drug because of the development of mutations in the NNRTI binding cite which in turn inhibits NNRTI binding 
and can lead to resistance mutations such as K101E, K103N, Y188C, G190S, G190A 
, and L100I 
Understanding the pharmacodynamic properties associated with the development of NNRTI resistant mutations is vital for devising treatment strategies for HIV. In pharmacodynamics, the drug-target interaction can be modeled by:
denotes the drug and
the target (usually enzyme). Drug efficiency is primarily determined by the drug target binding affinity. In pharmacodynamic studies, the drug target affinity is usually assessed by comparing dose response curves: the stronger the drug binds target, the steeper the curve is.
Therefore, a critical index of the dose response curve, the half-maximal inhibitory concentration (IC
), is commonly used to compare the binding affinities of drugs to the same target. IC
represents the concentration of a drug that is required for 50% of maximal inhibition in vitro. IC
are the concentrations corresponding to 25% and 75% inhibition, respectively. The dose response curve usually has the steepest portion in the middle. Thus, using IC
, rather than IC
, minimizes the random error for estimation, making IC
the preferred measure of drug affinity.
To estimate the IC
value, several nonlinear functions have been commonly used, for example,
is the drug concentration,
is the percentage of inhibition at this concentration, and
is a shape parameter. Other parametric models include the complementary log-log model for asymmetric quantal response data, and the two-parameter Weibull model for carcinogenic experiments 
An important feature of the function (1) is that as the value of
, reflecting that a drug's inhibitory potential changes from none to full inhibition as the drug concentration increases (). As per (1), a steeper dose response curve corresponds to a smaller IC
; for a given IC
, the curve shape is depicted by
(). In addition, this function curve has the steepest portion in the middle, which is a characteristic of a sigmoidal dose response relationship.
Figure 1 Dose response curves for various IC (A) or shape parameters, (B).
The advantages of this function are that (a) it is symmetrically about the IC
; (b) it is monotonic, which equals that the target protein has an inhibitory binding site only, and the drug has an inhibitory effect only; and (c) IC
can be easily estimated under certain conditions. On the other hand, these advantages become restrictive when some conditions fail, for instance, if observations are not monotonic. As a consequence, the models can produce remarkably biased estimates or not even fit the observations. For example, Bliss's beetle data show that symmetry is not a required feature of a dose response curve 
. Another example is that a viral mutation occurs before the drug concentration reaches a certain level, such as in the following example.
A recent study on HIV mutations conferring resistance to NNRTI found that the monotonicity relation does not always hold 
. The dose response was determined as proportion reduction in HIV replication at a given NNRTI dose relative to viral replication in the absence of drug. As seen in , the study shows that replication of HIV mutation M230L was promoted when the concentration of EFV is lower than 70 nM. Similarly, our data example shows that increasing the concentration of EFV stimulates the replication of HIV K101E+G190S mutant strain when EFV concentrations are below 2000 nM (). It has also been reported that EFV stimulation of the K101E+G190S double mutant strain can be abolished by the presence of additional M41L+T215Y mutation 
. A potential explanation for this nonsigmoidal dose response relationship is that dimerization is essential for a fully functional RT. For the double mutant K101E+G190S strain, at low concentrations EFV can enhance the dimerization of the two subunits of RT without interfering with the binding of the incoming nucleotide during DNA polymerization.
Dose response curves for viral replication of various HIV mutations at different EFV concentrations.
As the real dose response relationship is nonmonotonic in our example, our preliminary analysis indicates that traditional estimation methods for the sigmoidal model (1) fitting lead to results that either do not converge or are remarkably biased, which may reach an erroneous conclusion. Thus, appropriate estimation of IC
for this type of dose response relationship poses statistical challenges. If we use model (1) to fit data when the dose response pattern is nonmonotonic, the fit is poor, and the estimated IC
values are not reliable because the fact that lower EFV concentrations can enhance replication of an HIV mutant strain is ignored.
Thus, to appropriately estimate the pattern of observations and then estimate IC
, we developed a robust modeling strategy to test whether: (i) our model fitting is comparable to monotonic parametric models such as model (1) when the observed data are monotonic; and (ii) our model fitting yields reasonable estimates when the data pattern is nonmonotonic and monotonic parametric models, such as model (1), does not work.
The rest of this paper is organized as follows. Section 2 briefly introduces monotonicity testing, our model, estimation, and test methods. Section 3 gives simulation results, including p-values of the monotonicity test. Section 4 presents extensive analysis of our real data example, including estimated IC
s using the proposed model and model (1) when it is appropriate.