What should patients do if they miss a dose of medication? A probabilistic analysis
WWhat should patients do if they miss a dose ofmedication? A probabilistic analysis
Elijah D. Counterman ∗ Sean D. Lawley † February 11, 2021
Abstract
Medication adherence is a major problem for patients with chronic dis-eases that require long term pharmacotherapy. Many unanswered ques-tions surround adherence, including how adherence rates translate intotreatment efficacy and how missed doses of medication should be han-dled. To address these questions, we formulate and analyze a mathe-matical model of the drug level in a patient with imperfect adherence.We find exact formulas for drug level statistics, including the mean, thecoefficient of variation, and the deviation from perfect adherence. Wedetermine how adherence rates translate into drug levels, and how thisdepends on the drug half-life, the dosing interval, and how missed dosesare handled. While clinical recommendations should depend on drug andpatient specifics, as a general principle we find that nonadherence is bestmitigated by taking double doses following missed doses if the drug has along half-life. This conclusion contradicts some existing recommendationsthat cite long drug half-lives as the reason to avoid a double dose aftera missed dose. Furthermore, we show that a patient who takes doubledoses after missed doses can have at most only slightly more drug in theirbody than a perfectly adherent patient if the drug half-life is long. Wealso investigate other ways of handling missed doses, including taking anextra fractional dose following a missed dose. We discuss our results inthe context of hypothyroid patients taking levothyroxine.
Adherence to medication is the extent to which patients take medications asprescribed by their healthcare providers [1]. It is well-documented that non-adherence (or noncompliance [2]) is a major problem, resulting in over 100,000preventable deaths and $
100 billion in preventable health care costs per year inthe United States alone [1]. In fact, the World Health Organization noted that ∗ Department of Mathematics, University of Utah, Salt Lake City, UT 84112 USA. † Department of Mathematics, University of Utah, Salt Lake City, UT 84112 USA( [email protected] ). a r X i v : . [ q - b i o . Q M ] F e b increasing the effectiveness of adherence interventions may have a far greaterimpact on the health of the population than any improvement in specific medi-cal treatments” [3, 4]. Nonadherence is especially prevalent and problematic inpatients with chronic diseases that require long term pharmacotherapy [5]. Asformer US surgeon general C Everett Koop famously observed, “Drugs don’twork in patients who don’t take them” [6].Many outstanding questions surround adherence and how it relates to ther-apeutic outcomes. Adherence is often reported as the percentage of doses ofmedication actually taken by the patient over a specified time [1]. How does anadherence percentage p translate into treatment efficacy? How much worse is,for example, p = 70% compared to p = 85%? How much adherence is neededfor full treatment benefits? How can clinicians increase patient adherence? Arethere protocols to increase treatment benefits in spite of poor adherence?While the causes of nonadherence vary, a significant portion of nonadher-ence stems from patients simply forgetting to take their medication [7,8]. Whatshould a patient do if they miss a dose of medication? Although patients com-monly ask this question, they often do not receive adequate instructions forwhat to do when a dose is missed [9–11].To address these questions, we formulate and analyze a mathematical modelof the drug level in a patient with imperfect adherence. Mathematical model-ing is especially well-suited to investigate these questions, given the ethics ofclinical trials that force patients to miss doses of medication. To model im-perfect adherence, we assume that the patient takes their medication at only agiven percentage of the prescribed dosing times. Doses are missed at random,and thus the drug level in the body is random. We find exact mathematicalformulas for statistics of this model, including the average drug level, the druglevel coefficient of variation, and how the drug level deviates from a patientwith perfect adherence. These statistics are obtained as explicit functions ofthe adherence percentage, the drug half-life, and the prescribed dosing interval(i.e. the time between scheduled doses). Furthermore, we determine how thesestatistics depend on how the patient handles missed doses, including the casethat they skip missed doses and the case that they take double doses followingmissed doses.From a mathematical standpoint, the random variables that model the druglevel in our model generalize infinite Bernoulli convolutions [12–16]. The studyof infinite Bernoulli convolutions has a rich history in the pure mathematics lit-erature, dating back to Erd˝os and others in the 1930s [17–19]. Infinite Bernoulliconvolutions typically have very irregular distributions, including singular dis-tributions supported on a Cantor set [18]. Infinite Bernoulli convolutions alsoarose in the pharmacokinetic models in [20, 21]. Our analysis of the generalizedinfinite Bernoulli convolutions that arise in our model relies on the theory ofrandom pullback attractors [22–26].From the standpoint of pharmacology, there are several practical results ofour analysis. First, we provide quantitative estimates of how an adherencepercentage p translates into statistics of drug levels in the body, and how thesestatistics depend on the drug half-life t half , the dosing interval τ , and how missed2oses are handled. Further, these estimates show how the effects of nonadher-ence can be lessened by drugs with half-lives that are long compared to thedosing interval, i.e. t half (cid:29) τ . While clinical recommendations should dependon drug and patient specifics, as a general principle we find that the effects ofnonadherence are best mitigated by taking double doses following missed dosesif t half (cid:29) τ , whereas missed doses should be skipped if t half (cid:28) τ . This conclu-sion contradicts some existing recommendations that cite long drug half-livesas the reason to avoid a double dose after a missed dose. Since double dosesare sometimes avoided due to concern that they may cause toxic drug levels,we provide an upper bound for the highest possible drug level in the body. Wefind that a patient who takes double doses after missed doses can at most haveonly slightly more drug in their body compared to a perfectly adherent patientif t half (cid:29) τ . We also investigate other ways of handling missed doses, includ-ing taking an extra half dose following a missed dose, which we find is mostappropriate when t half ≈ τ .The rest of the paper is organized as follows. We formulate and analyzethe mathematical model in the Methods section. In the Results section, weexplore the pharmacological implications of the mathematical analysis. Sincethese pharmacological implications depend on rather complicated mathematics,we also provide an intuitive explanation for our results in this section. TheDiscussion section concludes by describing related work, model limitations, andfuture directions. We also discuss our results in the context of hypothyroidpatients taking levothyroxine. The Appendix collects some technical points andthe proofs of the theorems. Our model builds on the classical pharmacokinetic model of extravascular (oral)administration in a single compartment with first order kinetics [27, 28]. In thestandard model, the drug concentration, c , in the body at time s > c d s = k a gV − k e c , (1)where k a and k e are the respective rates of absorption and elimination, V is thevolume of distribution, and g is the drug amount at the absorption site. Theamount g satisfies the ODE, d g d s = − k a g + I ( s ) , (2)where I ( s ) describes the drug input.For most drugs administered extravascularly in conventional dosage forms,the absorption rate is much larger than the elimination rate, meaning k a (cid:29) e (see [27, 29–33]). In this parameter regime, the solution of (1) is well-approximated by the solution tod c d s = I ( s ) V − k e c, (3)which is the standard model for intravascular administration with first orderelimination. In this paper, we assume k a (cid:29) k e and thus consider the simplermodel in (3) rather than the system in (1)-(2). Suppose a patient is instructed to take a dose of size
D > τ > I perf ( s ) = DF (cid:88) n ≥ δ ( t − nτ ) , (4)where F ∈ (0 ,
1] is the bioavailability fraction and δ denotes the Dirac deltafunction. Solving (3)-(4) yields the following well-known formula for the drugconcentration at time s ≥ c perf ( s ) := DFV N ( s ) (cid:88) n =0 e − k e ( s − nτ ) , (5)where N ( s ) + 1 is the number of dosing times elapsed by time s , N ( s ) := sup { n ≥ n ≤ s/τ } . If t = s − N ( s ) τ ∈ [0 , τ )denotes the time elapsed since the most recent dosing time, then (5) can bewritten as c perf ( s ) = α t/τ DFV N ( s ) (cid:88) n =0 α n , (6)where we have defined the dimensionless constant α := e − k e τ ∈ (0 , , (7)which is the fraction of a dose that remains in the body after one dosing interval.If the patient continues their perfect adherence for a long time, then it iseasy to see from the form in (6) that the drug concentration approaches thefollowing function, c perf ( N τ + t ) → C perf ( t ) := α t/τ DFV A perf as N → ∞ , (8)4here t ∈ [0 , τ ) is the time since the last dose and A perf := (cid:88) n ≥ α n = 11 − α . In pharmacokinetics, it is common to measure the drug exposure over a singledosing interval by the so-called “area under the curve,” which for this case ofperfect adherence is AUC perf := (cid:90) τ C perf ( t ) d t = DFV k e . (9) To model patient nonadherence, we suppose that the patient occasionally missesa dose. Specifically, at each dosing time, the patient “remembers” to take theirmedication with probability p ∈ (0 , − p . Mathematically, let { ξ n } n be a sequence of independent and identicallydistributed (iid) Bernoulli random variables with parameter p , meaning ξ n = (cid:40) p, − p. (10)Hence, ξ n = 1 means that the patient takes their medication at the n th dosingtime.If Df n ≥ n th dosing time, then the druginput is I ( s ) = DF (cid:88) n ≥ δ ( t − nτ ) f n , (11)and solving (3) with I ( s ) in (11) yields the drug concentration in the patient, c ( s ) = α t/τ DFV N ( s ) (cid:88) n =0 α N ( s ) − n f n . (12)Notice that (12) reduces to (6) if f n = 1 for all n . We take f n = 0 , if ξ n = 0 , which means the patient does not take any medication when they forget. How-ever, we allow for the possibility that f n > , if ξ n = 1 , which means that the patient may take more than a single dose to make upfor prior missed doses. In general, we allow f n to be a function of the history { ξ i } i ≤ n , and we refer to a choice of f n as a “dosing protocol.”5he simplest dosing protocol is for the patient to merely take a single doseif they remember, which means f single n := (cid:40) ξ n = 0 , ξ n = 1 . (13)We refer to (13) as the “single dose” protocol. Another common dosing protocolis for the patient to take a double dose to make up for a missed dose at the priordosing time, which means f double n := ξ n = 0 , ξ n = 1 , ξ n − = 1 , ξ n = 1 , ξ n − = 0 . (14)We refer to (14) as the “double dose” protocol. Notice that in the doubledose protocol, the patient never takes more than two doses at a time, evenif they missed more than one previous dose. Our analysis below covers otherdosing protocols, but we are primarily interested in comparing the single doseand double dose protocols in (13)-(14). As a technical aside, we are ultimatelyinterested in the large time behavior of c ( s ) in (12), and thus the values of f n in (12) for small n are irrelevant. In particular, the fact that the definition of f double0 in (14) depends on ξ − is immaterial.Figure 1a illustrates how the drug level in the body evolves in time. Theblack dotted curve describes the perfectly adherent patient, and the red dashedcurve and blue solid curve describe patients with imperfect adherence followingthe single dose and double dose protocols, respectively. We set the initial drugconcentration equal to C perf (0) in this illustration. For the case of perfect adherence, the drug concentration at large time is de-scribed by C perf ( t ) in (8). Analogously, for the case of imperfect adherence, weprove below that the drug concentration converges in distribution at large time, c ( N τ + t ) → d C ( t ) as N → ∞ , (15)where t ∈ [0 , τ ) is the time since the last dosing time and C ( t ) is a certain randomfunction given below. In particular, C ( t ) describes the drug concentration in apatient who has been taking the drug for a long time with adherence p ∈ (0 , (cid:90) τ C ( t ) d t. (16)We emphasize that C ( t ) and AUC are random since patient adherence is mod-eled by a random process. 6 a) 0 1 2 3 4 5 6 7 80 . . . . . . C ( t ) / C p e r f ( ) perfectsingle dosedouble dosedouble dose max (b) 0 . . . . . Z = AUC / AUC perf e m p i r i c a l d e n s i t y perfectsingle dosedouble dosedouble dose max Figure 1: (a) The black dotted curve depicts how the drug concentration in aperfectly adherent patient evolves in time, and the gray shaded region is thearea between the peaks and troughs of the perfectly adherent patient. The reddashed curve and blue solid curve describe patients with imperfect adherencefollowing the single dose and double dose protocols, respectively. The bluedashed line depicts the largest possible drug concentration for the double doseprotocol (see Theorem 6). (b) The distribution of the relative drug level Z =AUC / AUC perf = C ( t ) /C perf ( t ) for the single dose protocol (red) and the doubledose protocol (blue) obtained from stochastic simulations. The black dottedvertical line at Z = 1 describes the perfectly adherent patient, and the bluedashed vertical line describes the largest possible drug level for the double dosepatient (see Theorem 6). In both plots, p = 0 . α = 0 . We measure the effects of nonadherence by comparing the drug levels in a patientwith imperfect adherence to the drug levels in a patient with perfect adherence.It is natural to quantify this in terms of the drug exposure ratio AUC / AUC perf or the drug concentration ratio C ( t ) /C perf ( t ). It turns out that these two ratiosare the same, as we prove below that Z := AUCAUC perf = C ( t ) C perf ( t ) , for all t ∈ [0 , τ ) . (17)Therefore, we can study the effects of nonadherence (in terms of the relativedrug exposure or drug concentration) by studying the single random variable Z . Figure 1b plots the distribution of Z for the single dose protocol (red)and the double dose protocol (blue) obtained from stochastic simulations (thedistribution is obtained from 10 realizations of c ( N τ + t ) with N = 100).We study Z primarily in terms of the following three statistics. First, wedefine the mean, µ := E [ Z ] = E [AUC]AUC perf , (18)7hich compares the average drug level to the perfectly adherent patient. Wefurther define the deviation,∆ := (cid:112) E [( Z − ] = (cid:113) E (cid:2) (AUC − AUC perf ) (cid:3) AUC perf , (19)which measures how the drug levels deviate from the perfectly adherent patient.In statistics, (19) is called the relative root-mean-square deviation or relativeroot-mean-square error. We also compute the coefficient of variation of Z , butwe find that ∆ is a better measure of the effects of nonadherence. Finally, sincedosing protocols in which the patient takes more than a single dose at a timemay cause drug levels to rise too high, another useful statistic is the largestpossible drug level compared to the perfectly adherent patient, λ := sup ξ Z = sup ξ AUCAUC perf , (20)where sup ξ denotes the supremum over patterns of the patient remembering orforgetting to take their medication (i.e. { ξ n } n in (10)).We point out that the statistics µ , ∆, and λ in (18)-(20) are dimensionless,and thus they are independent of the units used to measure drug amounts,concentrations, time, etc. Furthermore, these statistics depend only on α in (7),the adherence percentage p ∈ (0 , f n . In addition,(17) implies that the values of µ , ∆, and λ are unchanged if AUC and AUC perf are replaced by C ( t ) and C perf ( t ) for any t ∈ [0 , τ ). In this section, we analyze the mathematical model formulated above. We beginwith some general probabilistic theory.
Let { ξ n } n ∈ Z be a bi-infinite sequence of iid Bernoulli random variables as in (10)(it is convenient to allow the index n to vary over positive and negative integers).The dose taken at dosing time n may depend on the patient’s behavior at time n and the prior m dosing times for some given memory parameter m ≥
0. Towardthis end, let { X n } n ∈ Z be the history process, X n = ( ξ n − m , ξ n − m +1 , . . . , ξ n − , ξ n ) ∈ { , } m +1 , (21)which records whether or not the patient remembered at dosing time n and theprior m dosing times. It is immediate that { X n } n ∈ Z is an irreducible discrete-time Markov chain on the state space { , } m +1 [34]. In particular, let P = { P ( x, y ) } x,y ∈{ , } m +1 ∈ R m +1 × m +1 (22)8enote the transition probability matrix of the Markov chain { X n } n ∈ Z withentries defined by P ( x, y ) = P ( X = y | X = x ) , x, y ∈ { , } m +1 , where x ∈ { , } m +1 denotes the vector, x = ( x − m , x m +1 , . . . , x − , x ) ∈ { , } m +1 , and y ∈ { , } m +1 is denoted analogously. The definition of { ξ n } n ∈ Z thenimplies that the entries of P are P ( x, y ) = p if y = 1 , ( x − m +1 , . . . , x ) = ( y − m , . . . , y − ) , − p if y = 0 , ( x − m +1 , . . . , x ) = ( y − m , . . . , y − ) , . Furthermore, the definition of { ξ n } n ∈ Z implies that the distribution of X n is π ( x ) := P ( X n = x ) = p s ( x ) (1 − p ) m +1 − s ( x ) > , n ∈ Z , x ∈ { , } m +1 , (23)where s ( x ) ∈ { , , . . . , m + 1 } is the number of 1’s in x , s ( x ) := m (cid:88) k =0 x k . A dosing protocol f n = f ( X n ) is any function f : { , } m +1 (cid:55)→ [0 , ∞ ) . (24)While we are most interested in the single dose and double dose protocols in(13) and (14), we also investigate a few other protocols. First, consider the“boost” dosing protocol, f boost n := ξ n = 0 , ξ n = 1 , ξ n − = 1 , b if ξ n = 1 , ξ n − = 0 , (25)in which the patient takes a standard single dose of size D plus a “boost” doseof size bD if they missed the prior dose, for some b ≥
0. Notice that the boostprotocol reduces to the single dose protocol if b = 0 and the double dose protocolif b = 1. Another protocol is the “triple dose” protocol, f triple n := ξ n = 0 , ξ n = 1 , ξ n − = 1 , ξ n = 1 , ξ n − = 0 , ξ n − = 1 , ξ n = 1 , ξ n − = 0 , ξ n − = 0 , (26)9n which the patient takes a double dose to make up for a single missed doseand a triple dose to make up for two or more consecutive missed doses. Finally,consider the “all dose” protocol in which the patient takes all of their misseddoses, f all n := (cid:40) ξ n = 0 ,k + 1 if ξ n = 1 , ξ n − = 0 , . . . , ξ n − k = 0 , ξ n − k − = 1 . (27)The all dose protocol does not fit into the framework of (21), and thus analternative analysis is developed in the Appendix.For a dosing protocol f , a real number a ≥
0, integers M ≤ N , and time t ∈ [0 , τ ), define the random variable C M,N ( a, t ) := α t/τ DFV (cid:16) α N − M +1 a + N (cid:88) n = M α N − n f ( X n ) (cid:17) , (28)which is the drug concentration if time t has elapsed since dosing time n = N ,where a ≥ n = M −
1. We areinterested in the drug concentration after a long time, which corresponds totaking N → ∞ in (28). We will see that this limiting distribution is independentof a and M .Since { X n } n ∈ Z is a stationary sequence, we have that C M,N ( a, t ) = d C − ( N − M ) , ( a, t ) , for integers N ≥ M, (29)where = d denotes equality in distribution. Define C ( t ) := lim N →∞ C − ( N − M ) , ( a, t ) = α t/τ DFV A, for t ∈ [0 , τ ) , (30)where A := ∞ (cid:88) n =0 α n f ( X − n ) . (31)The function f must be bounded since the state space { , } m +1 is finite, andthus the Weierstrass M-test ensures that C ( t ) exists almost surely, and it isimmediate that C ( t ) does not depend on M ∈ Z or a ≥
0. Random variables ofthe form in (30)-(31) are sometimes called random pullback attractors becausethey take an initial condition (in this case, a ≥
0) and pull it back to the infinitepast [22–26].Therefore, (29) and (30) imply that for any M ∈ Z and a ≥
0, the randomvariable C M,N ( a, t ) converges in distribution to C ( t ) as N → ∞ [35], which wedenote by C M,N ( a, t ) → d C ( t ) , as N → ∞ . (32)10ince f is bounded, C M,N ( a, t ) can be bounded by a nonrandom constant in-dependent of N , and thus (29), (30), and the Lebesgue dominated convergencetheorem ensure the convergence of every moment of C M,N ( a, t ), E [( C M,N ( a, t )) j ] → E [( C ( t )) j ] , as N → ∞ for all j > . (33)Summarizing, the large N distribution and statistics of C M,N ( a, t ) are indepen-dent of a ≥ M ∈ Z , and we can study them by studying the distributionand statistics of C ( t ).Furthermore, it is immediate from the definitions in (8) and (30) that Z := AUCAUC perf = C ( t ) C perf ( t ) = AA perf , for all t ∈ [0 , τ ) . (34)Therefore, studying how drug levels are affected by imperfect adherence amountsto studying Z . Note that E [ Z ] = (1 − α ) E [ A ] and E [ Z ] = (1 − α ) E [ A ] since A perf = 1 / (1 − α ).For the single dose protocol in (13), the analysis of Z is straightforward sinceelements of the sequence { f ( X n ) } n ∈ Z are independent in this special case. Thefollowing theorem computes statistics of Z for a general dosing protocol. Weprove the theorem in the Appendix. Theorem 1.
The first and second moments of Z are E [ Z ] = (cid:88) x f ( x ) π ( x ) , (35) E [ Z ] = 1 − α α (cid:16) (cid:88) x f ( x )2 u ( x ) − (cid:88) x ( f ( x )) π ( x ) (cid:17) , (36) where (cid:80) x denotes the sum over all x ∈ { , } m +1 , π is in (23) , and u := ( I − αP (cid:62) ) − v, (37) where I ∈ R m +1 × m +1 is the identity matrix, P (cid:62) is the transpose of P in (22) ,and v ∈ R m +1 is the vector with entries v ( x ) = f ( x ) π ( x ) for x ∈ { , } m +1 .Remark . The random variable C ( t ) in (30) generalizes an infinite Bernoulliconvolution [12]. If we let C single ( t ) denote C ( t ) in the case that f is the singledose protocol in (13), then an infinite Bernoulli convolution is merely a shiftand rescaling of C single ( t ),Θ = ∞ (cid:88) n =0 α n (2 ξ n −
1) = 2 C single ( t ) − C perf ( t ) α t/τ DFV . Dating back to Erd˝os and others in the 1930s [17–19] and continuing in morerecent years [12–16], mathematicians have studied the distribution of Θ. Thoughthe definition of Θ is quite simple, its distribution is often quite irregular anddepends very delicately on the parameters α and p .11 a) 0 0.2 0.4 0.6 0.8 100 . . . . . p µ = E [ Z ] all missed dosestriple dosedouble dosesingle dose (b) 0 0.2 0.4 0.6 0.8 100 . . . . . p p E [ Z ] all missed dosestriple dosedouble dosesingle dose Figure 2: The plots compare the mean (panel (a)) and second moment (panel(b)) computed from stochastic simulations (square markers) to the exact ana-lytical formulas as functions of the adherence p for various dosing protocols. Wetake α := e − k e τ = 0 . We now use Theorem 1 to compute pharmacologically relevant statistics. Recallthat µ = E [ Z ] in (18) compares the average drug level to the perfectly adherentpatient. Corollary 3.
Using superscripts to denote the dosing protocol, we have that µ single = p, µ double = p + p (1 − p ) ,µ boost = p + bp (1 − p ) , µ triple = 3 p − p + p , µ all = 1 . In Figure 2a, we plot µ as a function of p for various dosing protocols. Noticethat nonadherence causes a reduction in the average drug levels for the single,double, and triple dose protocols since µ single , µ double , and µ triple are all strictlyless than 1 for all p ∈ (0 , µ single < µ double < µ triple < µ all , for all p ∈ (0 , . However, notice that while µ double is much larger than µ single , the additionalincrease is relatively small for the triple and all dose protocols, as µ double ≈ µ triple ≈ µ all if the adherence p is not too small.Applying Theorem 1 also yields explicit formulas for E [ Z ] for the variousdosing protocols, which are plotted in Figure 2b. Finding these formulas merelyrequires solving the system of algebraic equations in (37). We do not presentthese formulas as they are fairly complicated, but they are used in the deviation∆ and the coefficient of variation c v presented below.The curves in Figure 2a and Figure 2b use the analytical formulas for E [ Z ]and E [ Z ] and the squares markers are results from stochastic simulations with α = 0 .
85. In particular, the square markers are obtained from 10 independent12ealizations of C M,N ( a,
0) with a = M = 0 and N = 100. The simulation resultsagree with the exact analytical results.To measure the variability in drug levels that stems from imperfect adher-ence, we introduce the coefficient of variation of Z , c v := (cid:112) E [( Z − E [ Z ]) ] E [ Z ] , (38)which is defined as the ratio of the standard deviation to the mean. Notice that(17) implies that the coefficient of variation of Z is equal to the coefficient ofvariation of AUC or C ( t ) for any t ∈ [0 , τ ). Applying Theorem 1 gives explicitformulas for the coefficient of variation for the single dose and double doseprotocols, which we give in the following corollary (the formulas for the otherdosing protocols are omitted for brevity). Corollary 4.
Using superscripts to denote the dosing protocol, we have that c singlev = (cid:114) − α α (cid:112) p (1 − p ) ,c doublev = c singlev (cid:112) − p + p − p (2 − p ) α. The coefficient of variation measures the variability induced by nonadher-ence by measuring how drug levels deviate from their average value. From apharmacological standpoint, a small coefficient of variation is desirable. How-ever, a small coefficient of variation does not necessarily imply that the effectsof nonadherence are small. Indeed, the coefficient of variation vanishes if thepatient never takes their medication ( p = 0).Hence, a more useful statistic for measuring the effects of nonadherence ishow drug levels deviate from the drug levels of a perfectly adherent patient,which is the deviation ∆ defined in (19). Applying Theorem 1 gives explicitformulas for the deviation ∆ for the single dose and double dose protocols, whichwe give in the following corollary. The deviations for other dosing protocols aregiven in the Appendix. Corollary 5.
Using superscripts to denote the dosing protocol, we have that ∆ single = (cid:114) − p α (cid:112) − α (2 p − , (39)∆ double = (cid:114) − p α (cid:112) p + (1 − p + 2 p ) α + 2 p (2 − p ) α . (40)For certain drugs, it is important to ensure that the dosing protocol cannotcause the drug level in the patient to rise too high. We thus consider λ in(20), which is the largest possible drug level compared to the perfectly adherentpatient. The following theorem calculates λ for the dosing protocols above.13 heorem 6. Using superscripts to denote the dosing protocol, we have that λ single = 1 , λ double = 21 + α ,λ boost = max (cid:110) , b α (cid:111) , λ triple = 31 + α + α , λ all = ∞ . Notice that if we set b = α in the boost protocol in (25), then λ boost = 1 andthus Theorem 6 ensures that a patient following the boost protocol with b = α will never have more drug in their body than the perfectly adherent patient. We now explore some pharmacological implications of the analysis above. Recallthat α := e − k e τ , where τ is the dosing interval and k e is the drug eliminationrate. Since elimination rates are often expressed in terms of half-lives, we notethat the drug half-life, t half >
0, is related to the other parameters via α = 2 − τ/t half , t half = (cid:18) ln ln α (cid:19) τ = ln 2 k e . (41)Hence, in the following a “long drug half-life” means t half is long compared to τ , and thus α is large (i.e. α is near 1). Similarly, a “short drug half-life” means t half is short compared to τ , and thus α is small.Recall that AUC perf in (9) denotes the drug exposure for a patient withperfect adherence, and AUC in (16) denotes the drug exposure for a patientwith adherence p ∈ (0 , − p fraction oftheir doses. To compare AUC to AUC perf , we consider the deviation ∆ definedin (19), which measures how AUC deviates from AUC perf on average (which isequivalent to measuring how C ( t ) deviates from C perf ( t ) on average by (17)).Explicit formulas for the deviation ∆ were found in (39)-(40) in Corollary 5for the single dose and double dose protocols, and are denoted respectively by∆ single and ∆ double . We begin by considering the single dose protocol. In Figure 3a, we plot ∆ single as a function of α for different patient adherence levels p . As expected, ∆ single decreases as the patient adherence p increases. Furthermore, ∆ single decreasesas α increases, and ∆ single approaches its minimum value as α → single → − p as α → . These properties can be seen from Figure 3a and equation (39).Importantly, Figure 3a shows that the effect of patient nonadherence, asmeasured by the deviation ∆ single from perfect adherence, depends critically on14 a) 0 0.2 0.4 0.6 0.8 100.10.20.30.40.50.6 α = 2 − τ/t half d e v i a t i o n ∆ s i n g l e f o r s i n g l e d o s e p = 0 . p = 0 . p = 0 . (b) 0 0.2 0.4 0.6 0.8 100 . . . . . . α = 2 − τ/t half d e v i a t i o n ∆ single dosedouble dose Figure 3: Deviation ∆ from perfect adherence. Panel (a) plots the deviation∆ single for the single dose protocol as a function of α for different adherencepercentages p . Panel (b) compares the deviations ∆ single and ∆ double for thesingle and double dose protocols with p = 0 . α . For example, notice that the horizontal line in Figure 3a at ∆ single = 0 . single curves for the three different levels of patient adherenceconsidered (namely, p = 0 . p = 0 .
8, and p = 0 . p and small α and a patient with low adherence p and large α can have the same deviation from the perfectly adherent patient.Put another way, the effects of nonadherence can be lessened by increasingthe α value of the drug (i.e. increasing the half-life t half or decreasing the dosinginterval τ ) without changing the patient’s actual adherence p . This result isinline with previous analysis, as it is commonly noted that drugs with longhalf-lives tend to be more “forgiving” of missed doses [36]. This analysis thusquantifies drug “forgiveness.” For other measures of drug forgiveness, see [37–39]. To compare the deviations from perfect adherence for the single dose and doubledose protocols, in Figure 3b we plot ∆ single and ∆ double as functions of α . Thisfigure shows that∆ single < ∆ double if α is small (i.e. short half-life) , and ∆ double < ∆ single if α is large (i.e. long half-life) . (42)We set p = 0 . p yield similar results. Indeed,the formulas in (39) and (40) imply the small α limits,lim α → ∆ single = (cid:112) − p < lim α → ∆ double = (cid:112) − p , α limits,lim α → ∆ double = (1 − p ) < lim α → ∆ single = 1 − p. (43)In practical terms, (43) means that if α is large and the patient has adherence of p = 0 .
9, then the deviation from perfect adherence is roughly 10 times smallerfor the double dose protocol compared to the single dose protocol.While we have shown (42) for large α , it follows from (39)-(40) that it isactually the case that∆ double < ∆ single if and only if α > α c := 25 − p + (cid:112) p − p + 25 . (44)It is straightforward to check that the critical value α c always lies in the interval, α c ∈ (0 . , .
5) for all p ∈ (0 , . Therefore, α > . double < ∆ single , and α > . t half > τ . These results imply that if α > α c , then a patient can effectively increasetheir adherence by merely following the double dose protocol rather than thesingle dose protocol. That is, by following the double dose protocol with actualadherence p , they can have the same deviation from perfect adherence as theywould have by following the single dose protocol with a higher adherence p + .To calculate p + , suppose the patient has actual adherence p . We then find thevalue of p + which satisfies ∆ single | p + = ∆ double | p , (45)where ∆ single | p + and ∆ double | p denote setting the adherence equal to p + and p in the respective formulas in (39)-(40). Solving (45) yields the “effectiveadherence” p + ∈ (0 ,
1) as a function of α and the actual adherence p . Note that(44) implies that p + > p if and only if α > α c .In Figure 4a, we plot p + as a function of p for different values of α . Thisfigure shows that the effective increase in adherence by following the doubledose protocol is quite substantial, especially if α is close to 1. For example, if α = 0 .
8, then a patient with actual adherence of only p = 0 . p + = 0 .
8, and a patient with actual adherence of p = 0 . p + = 0 .
92. Notice that p + = p if α = α c .Summarizing, this analysis suggests that (i) the single dose protocol is bestwhen t half (cid:28) τ and (ii) the double dose protocol is best when t half (cid:29) τ . Conclu-sion (ii) contradicts some common dosing recommendations. Indeed, long drughalf-lives are sometimes stated as the reason to avoid the double dose protocol16 a) 0 0.2 0.4 0.6 0.8 100 . . . .
81 actual adherence p e ff ec t i v e a dh e r e n ce p + α = 0 . α = 0 . α = 0 . α = α c (b) 0.5 0.6 0.7 0.8 0.9 15%10%15%20%25%30%35% α = 2 − τ/t half o v e r s h oo t f o r d o ub l e d o s e p r o t o c o l max. possible overshoot λ double − λ typ − Figure 4: Panel (a) plots the effective adherence p + obtained by following thedouble dose protocol rather than the single dose protocol as a function of theactual adherence p . For a patient following the double dose protocol, panel (b)shows how their drug levels could rise above the levels in a perfectly adherentpatient.in favor of the single dose protocol (for example, see recommendations for per-ampanel [10] and lamotrigine sodium valproate [11]). However, we have shownthat drugs with long half-lives are precisely the drugs for which patients couldbenefit from taking a double dose following a missed dose. Taking a double dose is sometimes avoided due to concern that it may cause atoxic drug level in the body. For a patient following the double dose protocol,Theorem 6 provides an upper bound to how their drug exposure (AUC double )could compare to the perfectly adherent patient (AUC perf ). Indeed, Theorem 6ensures that AUC double
AUC perf < λ double = 21 + α . (46)Note that (17) ensures that the same bound holds for the relative drug concen-trations (that is, C double ( t ) /C perf ( t ) < λ double for any t ∈ [0 , τ )).We plot the maximum possible “overshoot” λ double − α . Importantly, λ double approaches 1 for large α , which means thatthe possible overshoot from following the double dose protocol vanishes for drugswith long half-lives. In practical terms, (46) means that if α = 0 .
8, then thedrug level is at most 11% greater than the perfectly adherent patient, and if α = 0 .
9, then the drug level is at most 5% greater than the perfectly adherentpatient.Furthermore, it is extremely rare for a patient to have a drug level nearthe theoretical upper bound in (46) if α is large. Indeed, the upper bound in1746) is approached only by a patient that alternates exactly between taking andmissing the scheduled doses for many dosing intervals if α is large. A moretypical overshoot occurs in the following way. If the patient has been takingtheir medication as prescribed for a long time, then the concentration in theirbody time t ∈ [0 , τ ) after a dose is roughly the same as the perfectly adherentpatient, which is C perf ( t ). Then, if they miss one dose and take a double doseat the following dosing time, then the drug concentration time t ∈ [0 , τ ) afterthe double dose is (compared to C perf ( t )) λ typ := α C perf ( t ) + 2 α t/τ DFV C perf ( t ) = 1 + (1 − α ) < λ double , (47)which is shown in Figure 4b.Summarizing, if α is large, then a patient following the double dose protocolcannot have much more drug in their body than the perfectly adherent patient,where the precise upper bound is in (46). Furthermore, it is rare for the druglevel to approach the upper bound in (46) if α is large, and the more typicalovershoot is in (47). How much patient adherence is needed for the patient to obtain full treatmentbenefits? The adherence threshold p ≥ . p . To illustrate, suppose patient p = 0 .
85 and α = 0 .
2, whereas patient p = 0 . α = 0 .
8. Therefore, by the standard definition in (48), patient single1 ≈ .
33 for patient , and ∆ single2 ≈ .
29 for patient . Hence, the supposed “non-adherent patient” (patient p = 0 .
85 and p = 0 .
75, but now suppose they both have α = 0 .
8. If patient single1 ≈ .
19 for patient , and ∆ double2 ≈ .
14 for patient . Therefore, the drug levels in the “non-adherent patient” are again closer to theperfectly adherent patient than the “adherent patient.”
We now consider more complicated dosing protocols. Notice that in the doubledose protocol, the patient never takes more than two doses at a time, even ifthey missed two or more consecutive prior doses. A more aggressive protocol isthe “triple dose” protocol in (26) in which the patient takes a double dose tomake up for a single missed dose and a triple dose to make up for two or moreconsecutive missed doses. An even more aggressive protocol is the “all dose”protocol in (27) in which the patient takes all of their missed doses. As anotherexample, consider the “fractional” dosing protocol, f frac n := ξ n = 0 , ξ n = 1 , ξ n − = 1 , α if ξ n = 1 , ξ n − = 0 , (49)in which the patient takes an extra large fractional dose if they missed one ormore prior doses. The reasoning behind the size of this extra dose is that if thepatient had taken their prior dose, then the fraction of that prior dose remainingin their body at the next dosing time would be α . Note that (49) is a specialcase of the boost protocol in (25) with b = α .These protocols may be impractical, as (26)-(27) require the patient to keepfairly detailed records and (49) requires the ability to take a fractional dose.Nevertheless, it is interesting to consider the implications of these dosing pro-tocols. In the Appendix, we obtain exact analytical formulas for the deviation∆ for these different dosing protocols (see (66), (69), and (70)). In Figure 5,we plot ∆ for these protocols and for the single and double dose protocols asfunctions of α (we set p = 0 . α is large, then the deviation ∆ is smallest for the triple dose andall dose protocols. Indeed, formulas (69) and (70) imply thatlim α → ∆ triple = (1 − p ) , lim α → ∆ all = 0 , where the superscript indicates the corresponding dosing protocol. Hence, onemight recommend the triple dose protocol or even the all dose protocol if α is large. However, while ∆ double is much less than ∆ single for large α , thefurther reductions in the deviation ∆ for the triple dose and all dose protocolsare comparatively much smaller. Furthermore, compared to the double dose19 . . . . . . α = 2 − τ/t half d e v i a t i o n ∆ single dosedouble dosetriple doseall dosefractional dose Figure 5: Deviation ∆ from perfect adherence for various dosing protocols.protocol, the triple dose and all dose protocols come with the costs of (i) beingmore complicated and (ii) allowing for a higher possible drug concentration inthe body (see Theorem 6).Next, notice in Figure 5 that the fractional dose protocol in (49) results in adeviation ∆ frac that is near minimal for all values of α ∈ (0 , α ranges from 0 to 1. Furthermore, it isnoteworthy that a patient following the fractional dose protocol is assured tonever have too much drug in their body. Indeed, Theorem 6 implies that thedrug exposure AUC frac for a patient following the fractional dose protocol isbounded above by the exposure for the perfectly adherent patient,AUC frac ≤ AUC perf . Note that (17) ensures that the same bound holds for drug concentrations (i.e. C frac ( t ) ≤ C perf ( t ) for all t ∈ [0 , τ )). Therefore, if a patient is able to takefractional doses, then the fractional dose protocol (i) yields a small deviation ∆and (ii) ensures that the patient cannot have more drug in their body than theperfectly adherent patient (regardless of α and p ).Of course, the fractional dose protocol is similar to the single dose protocolif α is small, and it is similar to the double dose protocol if α is large. Inparticular, the fractional dose protocol differs significantly from both the singleand double dose protocols only in the case that α ≈ .
5, which means t half ≈ τ .Therefore, this analysis suggests that (i) the single dose protocol is best when t half (cid:28) τ , (ii) the double dose protocol is best when t half (cid:29) τ , and (iii) the “1.5dose” protocol is best when t half ≈ τ , where the 1.5 dose protocol means the20atient takes an extra half dose to make up for a missed dose, f half n := ξ n = 0 , ξ n = 1 , ξ n − = 1 , . ξ n = 1 , ξ n − = 0 . (50)From a practical standpoint, the 1.5 dose protocol may often be feasible toimplement (if a standard dose is two pills, then the patient takes three pills ifthey missed their prior dose). Since (50) is a special case of (25) with b = 0 . . , is bounded above byAUC . ≤ (cid:40) AUC perf if α ≥ . , . α AUC perf if α < . . Hence, a patient following the 1.5 dose protocol will never have much more drugin their body than the perfectly adherent patient if t half ≈ τ . We have found that the single dose protocol is best when t half (cid:28) τ and thedouble dose protocol is best when t half (cid:29) τ . These results relied on rathertechnical mathematical analysis. The purpose of this section is to provide anintuitive explanation for these results. We begin by plotting stochastic simulations of the drug concentration in thebody as a function of time. In Figure 6a, we set p = 0 . α = 0 .
25 (meaning t half (cid:28) τ ) and plot the concentration under perfect adherence (black dottedcurve), and for imperfect adherence for the single dose protocol (red dashedcurve) and double dose protocol (blue solid curve). The shaded gray highlightsthe region between the peaks and troughs for perfect adherence. While thisis just one particular realization of the missed doses (the patient happens tomiss doses at the first and fourth dosing times), it nevertheless illustrates thatthe curve for the patient with perfect adherence is better approximated by thesingle dose protocol than the double dose protocol. Indeed, the single dose anddouble dose protocols both undershoot the perfect adherence case when a doseis missed, but the double dose protocol then overcompensates when the patienttakes their next dose. This is further illustrated in Figure 6b, which plots thesame scenario but for a longer time period.In Figures 6c and 6d, we plot the same curves as in Figures 6a and 6bexcept in the case that α = 0 .
95 (meaning t half (cid:29) τ ). In this case, the doubledose protocol approximates perfect adherence much better than the single doseprotocol. While the double dose protocol curve does rise above the perfectadherence curve, it is only by a few percent. In contrast, the single dose protocol21 a) 0 1 2 3 4 5 6 7 800 . . . . . . . C ( t ) / C p e r f ( ) perfectsingle dosedouble dose (b) 0 10 20 30 40 50 6000 . . . . . . . C ( t ) / C p e r f ( ) perfectsingle dosedouble dose (c) 0 1 2 3 4 5 6 7 80 . . . . .
91 dosing times C ( t ) / C p e r f ( ) perfectsingle dosedouble dosedouble dose max (d) 0 10 20 30 40 50 600 . . . . .
91 dosing times C ( t ) / C p e r f ( ) perfectsingle dosedouble dosedouble dose max Figure 6: Stochastic simulations of drug concentration time courses. In panels(a) and (b), we set α = 0 .
25 (meaning t half (cid:28) τ ), with panel (b) plotted fora long time period. Panels (c) and (d) are the same as (a) and (b), except α = 0 .
95 (meaning t half (cid:29) τ ). The adherence is p = 0 . α = 0 .
25 in Figure 7a and α = 0 .
95 in Figure 7b. Thedistributions are computed from 10 realizations of C M,N ( a, t ) with a = M = 0and N = 100.It is again evident from Figure 7 that the single dose protocol best approxi-mates the perfectly adherent patient when α is small (i.e. short drug half-life),whereas the double dose protocol best approximates the perfectly adherent pa-tient when α is large (i.e. long drug half-life). Notice also from Figure 7b thatit is very rare for the double dose protocol to ever result in a drug level muchlarger than the perfectly adherent patient. The phenomena seen above can be explained with a simple calculation. Supposethe patient has been taking the drug as prescribed for a long time and so the22 a)0 0 . . . . . . . Z = AUC / AUC perf e m p i r i c a l d e n s i t y perfectsingle dosedouble dosedouble dose max (b) 0 . . . . Z = AUC / AUC perf e m p i r i c a l d e n s i t y perfectsingle dosedouble dosedouble dose max Figure 7: The distribution of the drug exposure for the single dose protocol(red) and the double dose protocol (blue) obtained from stochastic simulations.We take p = 0 . α = 0 .
25 in (a) and α = 0 .
95 in (b). In bothplots, the black dotted vertical line at AUC / AUC perf = 1 describes the perfectlyadherent patient, and the blue dashed vertical line describes the largest possibledrug level for the double dose patient (see Theorem 6).drug concentration time t ∈ [0 , τ ) after a dose is C perf ( t ). Suppose the patientthen misses one dose and remembers to take the drug at the following dosingtime. Under the single dose protocol, the concentration time t ∈ [0 , τ ) after thesingle dose is ρ single ( t ) := α C perf ( t ) + α t/τ DFV = (1 − α (1 − α )) C perf ( t ) , where we have used that C perf ( t ) = α t/τ DFV A perf and A perf = 1 / (1 − α ). Alter-natively, under the double dose protocol, the concentration time t ∈ [0 , τ ) afterthe double dose is ρ double ( t ) := α C perf ( t ) + 2 α t/τ DFV = (1 + (1 − α ) ) C perf ( t ) . For small α , we have that ρ single ( t ) ≈ (1 − α ) C perf ( t ) , ρ double ( t ) ≈ (2 − α ) C perf ( t ) , if α is near 0 , which means the single dose protocol puts the patient slightly below the desired C perf ( t ), but the double dose protocol puts the patient at almost twice C perf ( t ).However, for large α , we have that ρ single ( t ) ≈ (1 − (1 − α )) C perf ( t ) ,ρ double ( t ) = (1 + (1 − α ) ) C perf ( t ) , if α is near 1 , which means that while the single dose protocol puts the patient below C perf ( t ),the double dose protocol puts the patient above C perf ( t ) by a much smalleramount. In practical terms, if α = 0 .
9, then the single dose patient undershoots C perf ( t ) by about 10%, whereas the double dose patient overshoots C perf ( t ) bya mere 1%. 23 Discussion
We have formulated and analyzed a mathematical model to investigate hownonadherence to medication affects drug levels in the body. We computed phar-macologically relevant statistics of the drug levels in the body, thus providingquantitative descriptions of the effects of nonadherence, and how these effectsdepend on the adherence percentage p , drug half-life t half , dosing interval τ ,and how missed doses are handled (i.e. the dosing protocol). In agreement withprevious results [1], we found that drug levels are less affected by missed dosesif the half-life is long compared to the dosing interval, and we quantified thiseffect. As a general principle, we found that nonadherence is best mitigated bytaking double doses following missed doses if the drug half-life is long comparedto the dosing interval (i.e. t half (cid:29) τ ). Furthermore, in this scenario we foundthat taking double doses following missed doses cannot cause the drug level torise much above the desired level. Although long drug half-lives are sometimesstated as the reason to avoid a double dose after a missed dose, we have shownthat drugs with long half-lives are precisely the drugs for which patients couldbenefit from taking a double dose after a missed dose.As an application of these results, consider the synthetic form of thyrox-ine known as levothyroxine [41]. Levothyroxine is the standard treatment forhypothyroidism, which is one of the most common diseases in the world andaffects up to 5% of the global population [42]. Levothyroxine pills are used toreplace missing thyroid hormone in hypothyroid patients and are usually takenonce daily for the remainder of the patient’s life [42]. Hence, the dosing intervalis τ = 1 day. The half-life of levothyroxine for hypothyroid patients is between9 and 10 days [43, 44], and therefore setting t half = 9 days yields α = 2 − τ/t half ≈ . . Since this α value is close to 1, our results imply that a hypothyroid patienttaking levothyroxine with imperfect adherence can make the drug levels in theirbody much closer to the levels in a perfectly adherent patient by followingthe double dose protocol rather than the single dose protocol. That is, if thepatient realizes that they missed a dose, then they should take the missed doseas soon as possible, even if that means taking a double dose (see below for moreon delayed doses). These results conflict with common recommendations forlevothyroxine, which advise patients to skip any dose that is delayed by morethan 12 hours [45–48]. However, some sources recommend a double dose oflevothyroxine after a missed dose (see Chapter 376 of [49]), and indeed takinga double dose is recognized as safe (see Chapter 36 in [50]).Furthermore, although following the double dose protocol may cause thedrug levels in the patient to rise above the levels in a perfectly adherent patient,the maximum possible overshoot for levothyroxine is less than 4% since λ double = 21 + α ≈
21 + 0 . < . .
24n addition, it would be very rare for a patient to have drug levels near thismaximum, as this maximum corresponds to a patient missing doses every otherday for many days. Indeed, the typical overshoot is about one half of one percentfor this α value (see (47)).Our model assumes that the drug absorption rate k a is much faster thanthe drug elimination rate k e . This is true for most drugs administered orallyin conventional dosage forms [27, 29–33], including levothyroxine. Indeed, forhypothyroid patients taking levothyroxine, the time to maximum concentration, t max , is only 3 hours [51], whereas the elimination half-life is t half = 9 days[43, 44]. Using (41) and the relation [27], t max = ln( k a /k e ) k a − k e , implies that k e /k a < . t half (cid:29) τ , then our analysis implies that taking the missed dose at the follow-ing dosing time (i.e. a double dose) is preferable to skipping the missed dose.However, if taking the missed dose at the following dosing time is superior toskipping the dose, then it is clear that taking the missed dose before the fol-lowing dosing time (i.e. a late dose) is even more superior. Summarizing, if t half (cid:29) τ , then our analysis implies that a missed dose should be taken as soonas possible, even if that entails taking a double dose at the next dosing time.Several important prior works have used mathematical modeling to investi-gate the effects of medication nonadherence. Li and Nekka developed stochasticmodels of the effect of medication nonadherence on patient drug levels [52, 53].The models in [52, 53] allow the drug to be administered at irregular times,and the authors obtained analytical formulas for drug level statistics. In a se-ries of papers [20,21], another group of authors developed a variety of stochasticpharmacokinetic models, including ones that allow for variation in dosing times,dose amounts, and elimination rates. The discrete time model proposed in [20]is essentially identical to the model in the present paper in the special case ofthe single dose protocol. These prior works did not analyze different protocolsfor handling missed doses. Ma [54] analyzed the mean first passage time for thepatient’s drug level to reach a therapeutic range for various ways of handlinga missed dose assuming that the patient never misses two or more consecutivedoses. Numerical simulations of computational models have also been useful forunderstanding the effects of nonadherence for specific drugs [55], especially forantiepileptic drugs [56–63] and antipsychotic drugs [64, 65].Naturally, our model neglects various pharmacological details. We have de-veloped a simple model aimed at addressing patients remembering or forgettingto take their medication. However, nonadherence is a dynamic process andpatients exhibit a variety of patterns of nonadherence [5], including extended“drug holidays” [66] and “white-coat adherence” [67]. Furthermore, while wehave addressed some aspects of delayed doses, a more detailed model would25xplicitly allow patients to take medication at times that vary continuously.Another source of stochasticity is that pharmacokinetic parameters vary be-tween patients, which has been modeled by analyzing a population of patientswith a distribution of parameters [20, 21].To conclude, medication nonadherence is a complex and multi-faceted prob-lem, and steps toward its alleviation require contributions from a variety ofdisciplines. Mathematical modeling is a valuable tool in this endeavor, espe-cially given the ethics of clinical trials that require sporadic dosing. Further,mathematical models can disentangle the effects of various factors and quicklyinvestigate the efficacy of possible interventions. Moving forward, we antici-pate that mathematical modeling and analysis will play an important role inunderstanding and alleviating the effects of medication nonadherence. A Appendix
In this appendix, we collect some technical points and the proofs of the theorems.In particular, we prove Theorem 1 (Appendix A.1), consider an alternativehistory process (Appendix A.2), consider more complicated dosing protocols(Appendix A.3), and prove Theorem 6 (Appendix A.4).
A.1 Proof of Theorem 1
Define A := ∞ (cid:88) n =0 α n f ( X − n +1 ) . The definition of A in (31) and the stationarity of { X n } n ∈ Z imply that A = d A = αA + f ( X ) , (51)where = d denotes equality in distribution. The invariance relation in (51) playsa key role in our analysis.Taking the expectation of (51) and rearranging implies that E [ A ] = E [ f ( X )]1 − α , (52)where we have used that A = d A . We note that (52) can also be obtained bytaking the expectation of (31). Combining (23), (34), and (52) gives (35) inTheorem 1.Squaring (51), taking expectation, and rearranging implies that E [ A ] = 11 − α (cid:16) α E (cid:2) Af ( X ) (cid:3) + E (cid:2) ( f ( X )) (cid:3)(cid:17) , (53)26here we have again used A = d A . By definition of expectation, we have that E (cid:2) ( f ( X )) (cid:3) = (cid:88) x ( f ( x )) π ( x ) . (54)Computing E [ Af ( X )] is more challenging since A and X are in general corre-lated if m ≥ E ∈ { , } denote the indicator function on an event E , meaning1 E := (cid:40) E occurs , . Decomposing E [ Af ( X )] based on the value of X gives E [ Af ( X )] = (cid:88) x E [ Af ( X )1 X = x ] = (cid:88) x f ( x ) E [ A X = x ] . (55)Multiplying (51) by the indicator function on the event X = x , taking expec-tation, and using that ( A, X ) = d ( A , X ) yields E [ A X = x ] = E [ A X = x ] = α E [ A X = x ] + f ( x ) π ( x ) , x ∈ { , } m +1 . (56)Using the tower property of conditional expectation [68], it follows that E [ A X = x ] = (cid:88) y E [ A X = x X = y ] = (cid:88) y E [ A X = y ] P ( y, x ) , (57)where P is the transition matrix in (22). Combining (56) and (57) yields thefollowing system of linear algebraic equations for E [ A X = x ], E [ A X = x ] = α (cid:88) y E [ A X = y ] P ( y, x ) + f ( x ) π ( x ) , x ∈ { , } m +1 . (58)If we define the vectors u, v ∈ R m +1 by u ( x ) := E [ A X = x ] , v ( x ) := f ( x ) π ( x ) , then (37) solves (58). Note that the Perron-Frobenius theorem guarantees that I − αP (cid:62) in (37) is invertible since I − αP (cid:62) = α ( α − I − P (cid:62) ) and α ∈ (0 , A.2 An alternative history process
The history process in (21) assumes that the patient remembers whether ornot they took their medication at the previous m ≥ { X n } n ∈ Z to encode how much time has passedsince the patient last took their medication. Specifically, for integers n ∈ Z and k ≥
1, define X n = ξ n = ξ n − = 1 ,k if ξ n = 1 , ξ n − = · · · = ξ n − k = 0 , ξ n − ( k +1) = 1 , − k if ξ n = · · · = ξ n − ( k − = 0 , ξ n − k = 1 . (59)In words, X n = 0 if the patient takes the drug at time n and n − X n = k ≥ n after missing the last k doses, and X n = − k ≤ − k th consecutive dose at time n .Given that { ξ n } n ∈ Z are iid as in (10), it follows that { X n } n ∈ Z is a discrete-time Markov chain on Z that evolves according to the following transition matrix P = { P ( x, y ) } x,y ∈ Z with P ( x, y ) := P ( X = y | X = x ). For x ∈ Z and x ≥ P ( x, y ) = p if y = 0 , − p if y = − , , P ( − x, y ) = p if y = x, − p if y = − ( x + 1) , . (60)It is straightforward to check that the distribution of X n is π ( k ) := P ( X n = k ) = (cid:40) p (1 − p ) k k ≥ ,p (1 − p ) | k | k ≤ − , n, k ∈ Z . (61)For this alternative history process { X n } n ∈ Z in (59), a dosing protocol is afunction f : Z → [0 , ∞ ). Since the state space of { X n } n ∈ Z (namely Z ) is infinite,we assume for technical reasons that dosing protocols cannot grow faster thanlinearly, 0 ≤ f ( k ) ≤ B | k | + B , k ∈ Z , (62)for some constants B , B > π in (61) ensure that all the moments of X n are finite. Furthermore, the bound in (62) ensures that the definition of A in(30) exists almost surely. To see this, note that (61) implies (cid:88) n ≥ P ( | X − n | ≥ n ) = (cid:88) n ≥ (cid:88) k ≥ P ( | X | = n + k ) ≤ (cid:88) n ≥ (cid:88) k ≥ p (1 − p ) n + k = 2 p < ∞ . Therefore, the Borel-Cantelli lemma [68] implies that there is an almost surelyfinite random integer N ≥ | X − n | < n for all n ≥ N . Hence, (62) im-plies that f ( X − n ) ≤ B n + B for all n ≥ N , and the almost sure existence of A in (30) follows, as well as the convergence in distribution in (32). Furthermore,the moment convergence in (33) follows from the Lebesgue dominated conver-gence theorem upon noting that we have almost sure convergence of momentsand using some simple bounds on ( C M,N ( a, t )) j .28he definitions of C M,N ( a, t ), A , and A and the analysis in the main textand in Appendix A.1 carry over directly to this definition of { X n } n ∈ Z if we usethe definition of P and π in (60) and (61). In particular, the formula for themean in (35) and the formula for the second moment in (36) hold. The benefitof the structure of P in (60) is that we can solve for u in (37) in closed form.To simplify the formulas for u , we take f ( − k ) = 0 for all k ≥
1, whichmeans the patient cannot take medication when they forget. Equation (58)then implies u ( − k ) = α (1 − p ) u ( − ( k − , k ≥ ,u ( k ) = αpu ( − k ) + f ( k ) π ( k ) , k ≥ ,u (0) = αp (cid:88) k ≥ u (0) + f (0) π (0) ,u ( −
1) = α (1 − p ) (cid:88) k ≥ u ( k ) . It is straightforward to solve these equations and obtain that u ( −
1) = α (1 − p )(1 − α (1 − p )) (cid:80) k ≥ f ( k ) π ( k )1 − α ,u ( − k ) = α k − (1 − p ) k − u ( − , k ≥ ,u ( k ) = α k p (1 − p ) k − u ( −
1) + f ( k ) π ( k ) , k ≥ . Plugging into (36) yields E [ A ] = 11 − α (cid:20) αp (1 − α (1 − p ))1 − α (cid:16) (cid:88) k ≥ f ( k ) π ( k ) (cid:17) (cid:88) k ≥ α k (1 − p ) k f ( k )+ (cid:88) k ≥ ( f ( k )) π ( k ) (cid:21) . (63) A.3 More complicated dosing protocols
We now work out the first and second moments of A for a few different choicesof the dosing protocol f . We begin by considering the history process in (21)with some given memory parameter m ≥
0. The simplest case is m = 0, whichcorresponds to the patient having no recollection of their behavior at prior dosingtimes. It is natural to suppose that the patient takes no medication when theyforget ( f (0) = 0) and that they take their normal dose when they remember( f (1) = 1). Using (35) and (36), we obtain in this case, E [ A single ] = p − α , E [( A single ) ] = p (1 + α (2 p − − α ) (1 + α ) . (64)29 more interesting case is m = 1, which allows the patient to potentiallytake a higher dose if they missed their prior dose. In this case, we need to specify f ( i, j ) for i, j ∈ { , } , where f ( i, j ) is the dose taken at the n th dosing timeif ξ n = j and ξ n − = i . Let f (0 ,
0) = f (1 ,
0) = 0 to impose that the patientmust miss their dose when they forget. Further, suppose f (1 ,
1) = 1, whichmeans the patient takes their normal dose if they remember and they did notmiss their prior dose. If f (0 ,
1) = 1 + b >
0, then we obtain the “boost” dosingprotocol in (25), and (35) and (36) yield E [ A boost ] = p (1 + b (1 − p ))1 − α , E [( A boost ) ] = p (1 − α ) (1 + α ) (cid:104) b ( p − α + 2 α ( p − p − b (1 − p )( α ( αp + p −
1) + 1) + α (2 p −
1) + 1 (cid:105) . (65)Using these formulas, it follows that the deviation (19) for the boost protocol is∆ boost = (cid:114) − p α (cid:112) α + p ( b + 2 α bp (1 + b − bp ) − α ( b ( b − p + 4) + 2)) + 1 . (66)Note that the cases b = 1, b = 2, b = 1 + α , and b = 1 . A gets more complicated for larger values of m . If m = 2, then we need to specify f ( i, j, k ) for i, j, k ∈ { , } , where f ( i, j, k ) isthe dose taken at the n th dosing time if ξ n = k , ξ n − = j , and ξ n − = i . We set f ( i, j,
0) = 0 for i, j ∈ { , } , and f (1 , ,
1) = 1 by the same reasoning as above.It follows then from (35) that the mean amount is E [ A ] = p (cid:2) f (1 − p ) + p ( − p ( f + f ) + f + f + p ) (cid:3) − α , (67)where we have set f ijk := f ( i, j, k ) to simplify notation. We can similarly use(36) to obtain a complicated, but explicit formula for E [ A ], which we omit forsimplicity. These formulas allow us to investigate dosing protocols in which thepatient takes even higher doses following two missed doses compared to a singlemissed dose. For example, setting f (0 , ,
1) = 1 , f (1 , ,
1) = 2 , f (0 , ,
1) = 3 , (68)yields the triple dose protocol in (26). Indeed, taking the values in (68) yieldsthat the deviation (19) for the triple dose protocol is∆ triple = (cid:114) − p α (cid:104) − p + 4 p + 2 α (1 − p ) p (( p − p + 3)+ 2 α p (( p − p + 3) + α (cid:0) − p + 11 p − p (cid:1) (cid:105) / . (69)30e now consider the alternative history process in (59) in order to considerthe “all dose” dosing protocol in (27). In this case, using standard results forsumming infinite series, (35) and (63) yield E [ A all ] = A perf , E [( A all ) ] = α (2 − p )(1 − p ) + α ( p ( p + 4) − − p + 2 p (1 − α ) (1 + α )(1 − α (1 − p )) d , ∆ all = (cid:114) − p α (cid:115) − α ) − p ( α (1 − p )) . (70) A.4 Proof of Theorem 6
For the single dose protocol, it is immediate that λ single = 1, and this corre-sponds to a patient who never misses a dose.For the double dose protocol, observe that if ξ n = 1 and ξ n +1 = 0 for all n ∈ Z , then A = − α , and thus λ double ≥ / (1 − α ) A perf = 21 + α . (71)This describes a patient who misses a dose at every odd dosing time, and thusalways takes a double dose at even dosing times.To see that λ double ≤ α , suppose that the patient has concentration − α DFV just after dosing time n = 1. If they take the drug at dosing time n = 2, then the concentration in their body will be lower than − α DFV since α − α + 1 < − α . Hence, suppose they miss taking the drug at dosing time n = 2. If they takethe drug at dosing time n = 3, then they will take a double dose and theconcentration in their body will return to − α DFV . If they miss taking thedrug at dosing time n = 3, then the concentration will be even lower. Therefore, λ double ≤ α , which upon combining with (71) yields λ double = α .The proof that λ boost = max { , b α } is almost identical to the proof that λ double = α . The proof that λ triple = α + α is also almost identical, uponnoting that this value of λ triple is attained by a patient who takes medicationat every third dosing time.The proof for the “all dose” protocol follows from noting that if the patientmisses k consecutive doses and then takes the next dose, then the drug concen-tration in their body just after that dose is at least ( k + 1) DFV . Since this is truefor every positive integer k , the result λ all = ∞ follows. Acknowledgments
SDL was supported by the National Science Foundation (Grant Nos. DMS-1944574 and DMS-1814832). SDL thanks Jennifer Babin and Colt Schisler for31elpful discussions.
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