Appendix. An Iterative Algorithm for a Pharmacokinetics-based Adherence Metric
The proposed adherence measure was based on a comparison of the average plasma
concentration levels of nevirapine estimated under the ideal and actually observed dosing times,
during the study time period. To estimate the average nevirapine concentrations in individual
patients, we used an iterative algorithm.
Considering the standard one compartment model for the Cp level in the ith subject, we
estimated the Cp levels at pre-selected time points t j , where j=0, 1, 2,… J i . Herein, we chose
the time points at hourly intervals and at all the recorded nevirapine dosing times. For the ith
patient, we used the first recorded dosing time as the starting point ( t 0 =0), and the last time point
in the observation period as t Ji , where t Ji was allowed to vary from subject to subject. We then
estimated the plasma concentration of nevirapine in the ith subject at time t j , Cp[i,t], for all the
pre-selected time points t j , j=0,1, 2, … J i . Following the multi-dose one compartment model
(28), we considered the iterative algorithm below.
Algorithm
1. Let d [i, j ] be the dose amount of the medication that the ith subject took at time t j . If
the patient did not take his medication at the time, d [i, j ]  0 . At time t 0 =0, the cumulated dose
D[i,0] is the first dose taken d [i,0] , and the plasma concentration is
Cp[i,0] 
F  d [i,0]  k a
[exp( ke  t0 )  exp( k a  t0 )]  0,
V [i ]( k a  ke )
(1)
where F is the population based bioavailability of the medication, V[i] is the volume of
distribution adjusted for the ith patient’s weight, ka and ke are the first order absorption and
elimination rates, respectively.
2. At all later time points t j , j=1, 2, …, J i , we first calculate the cumulated dose
D[i, j ]  D[i, j  1] exp  k a (t j  t j 1 )   d [i, j ].
(2)
Then we write the plasma concentration at time t j as
Cp[i, j ]  Cp[i, j  1] exp  ke (t j  t j 1 ) 

F  D[i, j  1]  k a
exp[ ke (t j  t j1 )]  exp[ ka (t j  t j1 )].
V [i]( k a  ke )
(3)
We used the above iterative algorithm to estimate the hourly plasma concentration levels
based on the dosing events recorded by the MEMS records. Specifically, we calculated the
values of Cp[i, j ] in hourly increments following the first dosing event for the entire duration of
the study period. We then averaged the estimated hourly Cp[i, j ] in each study subject for the
mean hourly plasma concentration for the study period. We denoted this average as Cpave[i] for
the ith subject, i.e. Cpave [i ] 
1
J 'i
J 'i
 Cp[i, j ] , where J '
j 1
i
was the number of hours contained in the
subject’s MEMS log. Therefore, Cpave[i] can be viewed as an approximation of the mean plasma
concentration of nevirapine during the study period.
Similarly, we calculated the intended plasma concentration level under the assumption of
perfect adherence to the prescribed dose and frequency of administration. We denoted the
average of hourly Cp values under the perfect adherence as Cp'ave [i] .
We use the ratio R= Cpave[i] / Cp'ave [i] to quantify patient’s adherence to nevirapine.
Illustration
To illustrate the use of the above algorithm, we estimate nevirapine concentration in the plasma
at times t 0 = 0, t1 = 1.0, t 2 =1.2, and t3 =2.0 hours in an iterative fashion under the assumption
that a patient took the prescribed medication at time t 0 =0 and t 2 =1.2 at a constant dose amount
d.
1. At time t 0 = 0, the patient took the medication at dose d, so d [i,0]  d , and the cumulated
dose D[i,0]  d [i,0]  d . From Equation (1), we have Cp[i,0]  0.
2. At time t1 = 1.0, the patient did not take the medication, so d [i,1]  0 . From Equation (2)
we derived the cumulated dose D[i,1]  d  exp( ka ) at this time, and from Equation (3)
we calculated the plasma concentration
Cp[i,1] 
F  d  ka
[exp( ke )  exp( k a )],
V [i ]( k a  ke )
which followed exactly the standard one compartment model.
3. Similarly at time t 2 =1.2, since the subject took another dose of the medication,
d [i,2]  d . The cumulated dose at this time point was calculated from Eq (2) as
D[i,2]  d  exp( 1.2ka )  d . From Equation (3), we had
Cp[i,2] 
F  d  ka
exp( ke )  exp( k a )   exp( 0.2ke )
V [i ]( k a  ke )


F  d exp( k a )  k a
exp( 0.2ke )  exp( 0.2k a ) 
V [i ]( k a  ke )
F  d  ka
exp( 1.2ke )  exp( 1.2k a ) 
V [i ]( k a  ke )
which again followed the standard one compartment model.
3. At time t3 =2.0, we had d [i,3]  0 because the patient did not take the medication at this
time point; the cumulative dose was D[i,3]  d  [exp( 2ka )  exp( 0.8ka )]. From
Equation (3), we calculated the plasma concentration at this time point as
Cp[i,3] 
F  d  ka
exp( 1.2ke )  exp( 1.2k a )  exp( 0.8ke )
V [i ]( k a  ke )


F  d  (exp( 1.2k a )  1)  k a
exp( 0.8ke )  exp( 0.8ka ) 
V [i ]( k a  k e )
F  d  ka
exp( 2.0ke )  exp( 2.0k a ) 
V [i ]( k a  ke )

F  d  ka
exp( 0.8ke )  exp( 0.8ka ) ,
V [i ]( k a  ke )
where the first term in the above expression represented the portion of plasma concentration
contributed by the first dosing event at time t 0 =0, while the second term represented the portion
contributed by the second dosing event at t 2 =1.2. This showed how the iterative algorithm was
able to accommodate the cumulative drug effect in multiple dose regimens.