The effect of adherence on antihypertensive therapy plans in patients with diabetes

Abstract

Patient adherence to a medication plan may significantly impact the benefits received by therapy. Thus, considering patient adherence to antihypertensive medication therapy in patients with diabetes, we investigate the impacts of adherence levels on patient health outcomes using a finite horizon, discounted Markov decision process. Health states are based on varying systolic blood pressure levels, cardiovascular complications, adherence levels, and the patient’s current hypertension medications. We model patient transitions through these health states by combining various models from the literature. The model maximizes the expected quality-adjusted life years (QALY). Experimentation on varying levels of patient adherence to medication plans emphasizes the importance of adherence to medication plans with respect to a quality of life metric. Furthermore, sensitivity analysis of model factors finds that smoking as an internal factor, and model parameters as external factors highly influence health outcomes. Furthermore, when patients maintain a higher than average adherence level, they will be able to noticeably increase their expected QALYs.

Keywords:

• Markov decision processes
• diabetes
• hypertension
• adherence

Disclosure statement

No potential conflict of interest was reported by the author(s).

Appendix A: Converting risk probabilities

According to Gidwani and Russell (2020), probability values cannot be manipulated easily. For example, a 100% probability at 5 years does not mean a 20% probability in 1 year. To convert a probability value, it should be transformed to a rate and then a new probability value again. These conversions can be executed by:

Probability to rate: (8) $$r=\frac{-\hspace{0.17em}\text{ln}\hspace{0.17em}(1-p_{\mathit{original}})}{t}$$(8) and rate to probability: (9) $$p_{\mathit{converted}}=1-\hspace{0.17em}\text{exp}\hspace{0.17em}(-rt)$$(9) where $p=$ probability, $t=$ time, and $r=$ rate. Thus, when converting a 3 year probability of 60% to a 1-year probability, $r=$ 0.3054, and the 1-year probability is 0.26 or 26%.

Appendix B: Transition matrix of the natural history

Since the medication plans are not considered in computing ${\sigma }_t(s_{t+1}|s_t),$ we can ignore the medication part and rewrite it as ${\sigma }_t(l_{t+1}|l_t).$ Welton and Ades (2005) and Craig and Sendi (2002) explain how to estimate the transition matrix of a discrete and continuous time Markov model. We use the idea that Craig and Sendi (2002) offer to generate our transition probability matrices among different SBP and complication levels as well as the transition probabilities among different adherence levels. They explain how to build the transition matrix longitudinal cohort data with observation intervals common to all subjects. Let L_o be the common observation interval and L_d the desired cycle length. According to Craig and Sendi (2002), the maximum likelihood estimate of the observed transition matrix ${\widehat{M}}_o,$ associated with the common observation interval L_o, is obtained using the methods of “observation intervals coincide”. By the invariance property, the maximum likelihood estimate of the transition matrix associated with cycle length L_d is ${\widehat{M}}_d={\widehat{M}}_o^t$ where $t=L_d/L_o.$ To apply this model, we use the data from two periods to form ${\widehat{M}}_o$ matrix of transition probabilities of different SBP and complication levels.

Computation of this matrix is straightforward from the decomposition of ${\widehat{M}}_o$ into its eigenvalues and eigenvectors. Based on this decomposition, the s × s matrix ${\widehat{M}}_o$ can be expressed as ${\widehat{M}}_o=PD{P}^{-1}$ where, (10) $$D=\left[\begin{array}{cccc}{\delta }_1& 0& \dots & 0\\ 0& {\delta }_2& \ddots & ⋮\\ ⋮& \ddots & \ddots & 0\\ 0& \dots & 0& {\delta }_s\end{array}\right]$$(10) and δ_i is the ith eigenvalue and its associated eigenvector is the ith column of P. It then follows that ${\widehat{M}}_o^t=P{D}^t{P}^{-1}$ where, (11) $$D^t=\left[\begin{array}{cccc}{\delta }_1^t& 0& \dots & 0\\ 0& {\delta }_2^t& \ddots & ⋮\\ ⋮& \ddots & \ddots & 0\\ 0& \dots & 0& {\delta }_s^t\end{array}\right]$$(11)

The eigenvalues are raised to the power of t but the eigenvectors do not change. To establish these transition probabilities, we need a data set that shows patient levels of SBP and complication type for at least two different periods and their adherence to the medications.

Notes

1 A class of antihypertensive medications.

2 Basu et al. (2017) provide a risk probability model for Myocardial Infarction (MI) that is a serious result of CHDs, and since these two terms have been used interchangeably in the literature (Kurt et al., 2011; Stevens et al., 2001), we use this risk model to estimate CHD risks.