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For reprint orders, please contact: [email protected] Research Article 2015/12/30 A rolling-horizon pharmacokinetic pharmacodynamic model for warfarin inpatients in transient clinical states Aim: To design a pharmacokinetic pharmacodynamic model to make individualized
Yao Zhao1, Nan Liu2, Yijun and adaptive international normalized ratio (INR) predictions for warfarin inpatients Wang1 & Kathleen T Hickey*,3 in changing clinical status. Methods: We tested a new model on 60 inpatients
1Department of Supply Chain at Columbia University. The model personalizes four submodels and minimizes Management, Rutgers Business School, Rutgers – the State University of New the number of parameters to be estimated. Prediction accuracy was assessed by Jersey, Newark, NJ, USA prediction error, absolute prediction error and percentage absolute prediction error. 2Department of Health Policy Results: The INRs were accurately predicted 5 days into the future. Median prediction
& Management, Mailman School of error: 0.01–0.12; median absolute prediction error: 0.17–0.5 and median percentage Public Health, Columbia University, NY, absolute prediction error: 9.85–26.06%. Conclusion: Patients exhibit interindividual
USA 3Columbia University School of Nursing, and intertemporal variability. The model captures the variability and provides accurate Columbia University Medical Center, and personalized INR predictions.
NY, USA *Author for correspondence: First draft submitted: 19 September 2015; Accepted for publication: 20 October 2015;
Tel.: +1 212 305 4944 Published online: 7 January 2016
Keywords: cardiovascular disease • mathematical model • personalized medicine
• pharmacodynamics • pharmacokinetics • warfarin
Warfarin is the most routinely prescribed mins, or herbal products [17,18]. Thus, war-oral anticoagulant for the treatment and farin has a variety of potentially dangerous prevention of thromboembolic events [1–3]. side effects and is among the top drugs with The drug works by affecting the function of the largest number of serious adverse event the coagulation cascade and decreasing the reports [19,20].
clotting ability of the blood [1–3]. Although To prevent such adverse side effects, warfarin is widely utilized in clinical prac- patients taking warfarin are required to have tice, it can be challenging to manage. For regular INR testing. The INR needs to be example, it negatively interacts with many controlled within a narrow range, typically other common medications, such as anti- between 2.0 and 3.0, to achieve anticoagula- biotics and over-the-counter medications tion [21]. A high INR may predispose some (e.g., NSAIDS) [4–9]. Advancements in individuals to an increased risk of bleed-molecular genetics and technology have ing, while an INR below the therapeutic shown individual genetic variations, where range may be insufficient to protect high-particular polymorphisms in the genes of risk individuals against stroke and other certain enzymes have been associated with thromboembolic events [22,23].
altered sensitivity and metabolism of warfa- Numerous factors can affect individuals' rin [10–16]. The international normalized ratio INR levels, which can make determining (INR) levels, a standardized measure of blood the need for increased or decreased warfarin clotting time, of patients taking warfarin can challenging for the practitioner. These fac-also be significantly altered by changes in the tors can include, mostly notably, variations dietary intake of vitamin K from foods, vita- in anticoagulation dosing and clinical pro- 10.2217/pme.15.41 2016 Future Medicine Ltd Per. Med. (2016) 13(1), 21–32
Research Article Zhao, Liu, Wang & Hickey tocols, individual genetic differences in metabolism provided more flexibility to capture the interindividual of warfarin (which may not be known), significant and intertemporal variability previously mentioned. Our chronic disease burdens (e.g., atrial fibrillation, heart refined model also has fewer parameters to be estimated failure, diabetes) and addition or deletion of other (than previous work that include all the submodels) medications that might influence warfarin metabo- reducing the amount of historical data needed for esti- lism [24–30]. These factors coupled with a hospitaliza- mation. Our model is also estimated on a rolling hori- tion can further alter an INR response to warfarin by zon (with dated history replaced by latest observations) exhibiting strong intertemporal variability, meaning using a Bayesian approach to capture the intertemporal an individual's INR response to warfarin could change variation. The purpose of this study was to develop a considerably over time, and interindividual variability, new PK/PD model to make individualized and adaptive meaning the same dose of warfarin may result in very predictions of INR measurements for cardiac clinical different INRs among different individuals.
inpatients whose health status was in a state of flux.
To ensure effective and safe warfarin dosing in clinical practice, an adaptive personalized medicine Methods
approach to care that will account for individual vari-
ability is needed. Warfarin dosing and INR prediction This was a retrospective cohort study conducted at protocols and algorithms have been studied in the lit- New York Presbyterian Hospital (NYPH, NY, USA)/ erature for initiation and maintenance of proper anti- Columbia University Medical Center (CUMC, NY, coagulation in patients under relatively stable clinical USA). The study protocol was reviewed and approved conditions [19,22,31–38]; however, limited literature exists by the Institutional Review Boards at Columbia Uni-on how best to optimize anticoagulation therapies for versity, (NY, USA) and Rutgers, the State University of inpatients receiving intensive care and those who are in New Jersey (NJ, USA).
a transient clinical state. Regression models have been This study collected data from a convenience sam- used to determine loading and maintenance doses, so ple of 64 inpatients cases (admissions), which were as to improve time in therapeutic range [35–37]. Even seen as part of the cardiac services at CUMC during after accounting for information on patients' specific 2012–2013. These patients were also followed by the genotypes (responders vs nonresponders to warfa- anticoagulation clinic at CUMC upon discharge for rin), these regression-based models are not completely their outpatient warfarin management.
individualized but rather adopt a cross-sectional Patients were enrolled consecutively. Eligible patients design, and make dose prediction based on association were those who were initiated on warfarin therapy observed between the warfarin dose and INR changes for any indication as part of their usual clinical care in a particular cohort [12,38].
by their providers. We screened 346 cases total and Computerized algorithms based on pharmacoki- excluded those patients under 18 years of age and those netic/pharmacodynamic (PK/PD) models have the cases with fewer than six INR measurements. We also potential of making personalized and adaptive INR excluded four cases, which have missing data and/or predictions and dose suggestions for patients in a tran- errors. We did not exclude patients with other multiple sient clinical state [39–43]. Several clinical trials demon- chronic diseases, using potential medications known strate the effectiveness of PK/PD models for warfarin to interfere with warfarin metabolism (as we aimed to dosing in comparison to manual dosing practices [44,45]. create a ‘real world' approach that could be applied to A few retrospective studies report on the accuracy of clinical practice in the future).
these algorithms in predicting INR and maintenance Subjects' clinical data were collected in a deidenti- doses for warfarin patients [43,46]. Most of these studies fied manner via chart review and through the elec-focus on patients in a relatively stable state (e.g., reha- tronic medical record (EMR), including age, gender, bilitation) where patients' initial drug responses can self-reported race/ethnicity, documented indication be used to predict maintenance dose or steady-state for anticoagulation therapy, concomitant medications INR [43,44,46,47]. Only a few studies examined patients (including those known to affect the metabolism of with potentially unstable conditions [42,44].
warfarin, such as amiodarone), cardiac history and To examine inpatients whose clinical status may be cardiac testing preformed as part of their clinical care. quickly changing, we developed a PK/PD model that All INR results measured as a part of routine clini-includes all submodels of Holford [40] and personalized cal care were recorded from the first day of warfarin them by having parameters from each submodel esti- therapy until subjects were discharged from the hospi- mated by individual patients' data. Including all the sub- tal. A therapeutic INR target range for anticoagulation models and fully personalizing them (in comparison to was defined for each subject at admission (between partial personalization using population means [43,46,48]) 2.0–3.0) and any changes in or out of the target range Per. Med. (2016) 13(1)
future science group A rolling-horizon pharmacokinetic pharmacodynamic model for warfarin inpatients in transient clinical states Research Article were recorded, from the electronic medical record In summary, our PK/PD model has five parameters review. The warfarin dosing history for each patient, (β, b, α, a, A), which are patient-specific and must be including the daily dose taken, adjustments made and estimated based on individual patients' historical INR the time of warfarin administration was collected on and doses. Once these parameters are estimated, the all subjects with at least six INR measurements.
model can be used to predict future INRs for any dose regimen starting from any initial state, W(0) and C(0).
The parameters are estimated by a Bayesian method The PK/PD model developed in this paper includes all using weighted least squares regression [43,46]. Specifi-four submodels of [40]. Specifically, we consider a single cally, let θ = {β, b, α, a, A} be the set of parameters, compartmental model and assume 100% bioavailabil- y (ŷ ) be the jth observed (predicted) INR, x be the ity of warfarin and instantaneous absorption [43]. The corresponding time, μp be the mean of the pth param-pharmacokinetics model for warfarin metabolism is eter, and n be the number of INR observations used to standard [40], fit the model, the posterior objective function or Bayes risk is given by, j y (i, j)) p (ip n ) where W(t) is the amount of warfarin in plasma com- partment at time t and β is the warfarin elimination To describe the dose–effect in the pharmacodynamics vy is the variation of prediction error of INR, 2 model, we use a hyperbolic tangent function, the variation of the prediction error of the pth param-eter. The first part in the sum represents prediction F (W (t)) = 1 - Tanh (bw (t)) errors, and the second part measures the deviation of the parameters from the population means. The means rather than the Emax function [40–42,46,48] because of β (0.02 h-1, corresponding to a warfarin half-life of hyperbolic tangent has similar mathematical proper- 36 h) and b (0.07) are obtained in the literature [43]. ties but only requires one parameter, b [43]. In contrast, The mean of α (0.05 h-1, corresponding to a clotting Emax requires two parameters.
factor half-life of 15 h) is given by [40]. The mean of The physiological model for the vitamin K clotting A is set to 1 because a normal person without taking factors is standard [40], warfarin (C(t) = C ) has an INR equal to 1 the mean of α is set to 0.5. σ is chosen as a certain dC (t) =-αC(t) + percentage of the mean for each parameter cF (W (t)) model is implemented in Microsoft Visual Studio® and the Bayes' estimators are found by a nested partitions where C(t) is the amount of clotting factors at time method for global optimization [49]. This method has t normalized to ensure γ = 1 (thus γ is not a parameter been validated and used in many research areas, such to be estimated) and α is first-order elimination rate. To as medical treatment (e.g., radiation therapy), engi-keep the model simple, we assume that warfarin in the neering and management [50]. The algorithm usually plasma is immediately available to inhibit clotting factor takes 1–2 min in computation to estimating one set synthesis and b is the same for all clotting factors [42].
of the parameters (Intel Core i5 CPU at 2.40 GHz, To describe the relationship between clotting factors RAM 3 GB).
and INR, we use the inverse functions, Parameter estimationWe estimate the parameters for our PK/PD model on a rolling-horizon basis, where we updated the model upon each new INR response by estimating the param-eters using the latest 5 INR responses. Figure 1 presents where α and A (α < A) are patient-specific param- an example to demonstrate the procedure of model eters [42], C is 1/α representing the amount of clotting estimation and INR prediction. It shows the actual factors in the steady-state in absence of warfarin. We and predicted INRs over time for a case in our sam-choose inverse function because theoretically, INR can ple. The patient is a 48-year-old female (in 2012) on be arbitrarily high as plasma concentration of clotting warfarin therapy for atrial fibrillation.
factors approaches zero. This trend can be properly For this patient, we first estimated parameters for our modeled by the inverse function.
PK/PD model (the solid curve in Figure 1) using the future science group Research Article Zhao, Liu, Wang & Hickey Warfarin dose (mg) 3.0
Figure 1. A case study to demonstrate model estimation and to compare predicted versus actual international normalized ratios. The
horizontal axis represents time (in h), the right vertical axis is INR and the left vertical axis is warfarin dose. The squares are actual INR
responses, and the diamonds are warfarin doses. The solid line represents INR calculated by the model, which is estimated by the first
five INR responses and all warfarin doses prior to the fifth INR response. The dashed line represents the predicted INR ahead of the
latest INR (i.e., the fifth INR).
INR: International normalized ratio.
initial five consecutive INR responses and all warfarin is the sixth INR measure, then the second through sixth doses prior to the fifth INR response. Then we set the INR values are used for model estimation but the first fifth INR as the starting INR (initial condition), and INR is dropped. To make INR predictions using the used the estimated PK/PD model to predict INR for the updated model, we reset the starting INR to be the latest patient once every half-hour (the dash curve in Figure 1) INR.
in the next few days under the dose regimen recorded in the medical record. The dash curve does not connect Study designto the solid curve because there is no guarantee that the For each patient case, we calculated three types of errors: model fits the fifth INR response perfectly. The dash the prediction errors ([PEs] defined as the differences curve shows that our predictions for INRs are suffi- between the predicted INRs and the observed INRs), the ciently accurate for this patient up to about 200 h ahead absolute prediction errors ([APEs] defined as the absolute of the fifth INR. The prediction effectively captures value of PEs), and the percentage absolute prediction INR changes, in particular the quick rising INR values, errors ([PAPEs] defined as the ratios between the APEs on this patient over the course of in-hospital clinical care.
and the observed INRs) for at most 5 days ahead of the At each newly observed INR (the latest INR) beyond latest observed INR. We used PEs to measure the accu- the first five, we estimated the model parameters again racy (bias) of the model, and APEs and PAPEs to evaluate by using the latest five INR responses and all warfarin the prediction precision [51]. We calculated the median doses prior to the latest INR. INRs observed earlier than and 95% CIs for the median of each type of errors, for the latest five are dropped. For instance, if the latest INR each day up to 5 days into the future. These statistics Per. Med. (2016) 13(1)
future science group A rolling-horizon pharmacokinetic pharmacodynamic model for warfarin inpatients in transient clinical states Research Article allowed us to draw inference on the performance of the Assessment of prediction accuracymodel in terms of accuracy and precision. In addition, Table 2 shows the median and CI for PE, the APE and we statistically compared PEs, APEs and PAPEs among the PAPE on future INR predictions for the PK/PD different days into the future using the Kruskal–Wallis model on a fixed-horizon. Figure 2 shows the box-plots analysis of variance (ANOVA) test for medians.
of PE. We made predictions for a maximum of 5 days To assess the importance of individualized treatments into the future. The first column of the table and the on these patients and the necessity of a fully personalized caption of x-axis for Figure 2, ‘days into the future,' model, we studied the variation of the estimated values of indicate the day ahead of the latest observed INR all parameters for the entire cohort. For each parameter, response; that is, Day 1 means 0–24th h (including a case may have multiple estimated values over time. If 24th h) ahead of the latest INR response; Day 2 means all patients are the same in their INR response to warfa- 24th h (excluding 24th h) to 48th h (including 48th h) rin, we would expect that the mean estimated values for ahead of the latest response and so on.
all cases will be close to the corresponding population Table 2 shows that the predictions made by the mean of this parameter. To formally test this hypoth- PK/PD model on a rolling-horizon is unbiased within esis, (i.e., there is no interindividual variability) we cal- the first 3 days into the future because the median PE culated the estimated mean value for each parameter in each patient case, and then conducted a t-test to see if Table 1. Demographics of the cases (n = 60) in the estimated mean value is significantly different from the corresponding population mean for that parameter.
To measure intertemporal variability among patients, Variables
we calculated the coefficients of variation ([CoV] stan- Age (at admission): dard deviation of the sample over sample mean) for each parameter estimated over time for each eligible case if – 18–65 years the case has multiple estimated values. If each individual has the same INR response to warfarin over time, that is, there is no intertemporal variability, then we expect that these CoVs for each parameter have a mean zero. To formally test the existence of intertemporal variability, – Female we conducted a t-test for each parameter.
Reason for warfarin: – Atrial fibrillation – Valve disease Table 1 provides the detailed demographic information – Vascular disease of the n = 60 inpatient cases in this study.
The average number of INR readings in these cases is 12.7 (standard deviation [SD]: 11; range: 6–82). The average number of warfarin doses received is 10.0 doses Ethnicity (n = 60): (SD: 7.6; range: 4–60). The median age (at admis- – White or white-Hispanic sion) of these patients is 60.5 years, and 40% of sub- jects were males. The reason for warfarin administra- tion includes atrial fibrillation (29 cases; 48%), valve disease (10 cases; 17%), vascular disease (17 cases; 28%), history of stroke and other reasons (four cases; – Others, not described† 7%). A total of 46 cases (77%) have an INR within INR target range (at admission): the therapeutic target range of 2.0–3.0 (at admission); nine cases (15%) have a subtherapeutic INR below 2.0, and five cases (8%) had a high INR level above 3.0.
Our cohort also had multiple co-existing cardio- vascular comorbidities, which further necessitated – 2–2.5 the need for warfarin. Of the 60 cases, 47 (78%) had hypertension, 20 (33%) had diabetes, 16 (27%) had – 2.5–3.5 coronary artery disease and 15 (25%) had heart failure. †Others not described refers to mixed races, those who did not This is not surprising given we acquired our data from self-report their race.
our cardiovascular units.
IQR: Interquartile range; INR: International normalized ratio.
future science group Research Article Zhao, Liu, Wang & Hickey Table 2. Assessment of the prediction accuracy using median prediction error, median absolute prediction error, percentage absolute prediction error†.
Days into
Number of
the future observations Median (%) 95% CI
Median (%) 95% CI
Median (%) 95% CI
†In the column ‘Days into the future', 1 means 0–24th h (including 24th h) ahead of the latest actual INR, 2 means 24th h (excluding 24th h) to 48th h (including 48th h) ahead of the latest actual INR, and so on. The number of observations is greater than the number of inpatient cases because most cases have more than six INRs. The 95% CIs are for the median.
‡PE = Predicted INR - Actual INR§APE = PE¶PAPE = PE/observed INRAPE: Absolute prediction error; PAPE: Percentage absolute prediction error; PE: Prediction error; INR: International normalized ratio.
is not statistically different from zero. In the 4th and radation of clotting factors. (4) A model to link the fifth days, the model slightly overpredicts the INR val- activity of clotting factors to the prothrombin time ues. Table 2 presents the precision of the model using (or equivalently, INR). Earlier studies attempted median APE and median PAPE, which shows that the various mathematical equations for each of these four INR predictions of the model are reasonably accurate submodels [41–43,46,48].
for 5 days into the future. The model is particularly pre- Our PK/PD model is novel in that it uses a new cise for the first 2 days with a median APE of no more combination of the mathematical equations, for simi-than 0.25 (95% CI: 0.21–0.27) and a median PAPE lar mathematical properties but fewer parameters. of no more than 13.45% (95% CI: 12.09–14.82%). Specifically, we use hyperbolic tangent function for The Kruskal–Wallis ANOVA tests on PEs (APEs and submodel (2) and inverse function for submodel (4); PAPEs) show that the model is more accurate for a see section of ‘The PK/PD Model' for further details. shorter term prediction, that is, the prediction of the Our model includes all four submodels but only model becomes more accurate the closer they are made has five parameters, with one for warfarin elimina-from the last value into the future (p-value < 0.001).
tion, one for clotting factor elimination, one for dose sensitivity and two for INR; all to be estimated for Interindividual & intertemporal variability individual patients. Thus, the model is able to cap- Table 3 shows the summary statistics of all estimated ture interindividual and intertemporal variability in model parameters for all cases over time and the t-tests all submodels. In comparison, Vadher and Patterson results for interindividual and intertemporal variabil- proposed a PK/PD model with 11 parameters, nine ity. Except for two parameters, we found strong statis- of which come from the literature and thus are identi- tical evidence on interindividual variability (p < 0.05), cal for all patients, the other two parameters (warfarin supporting that individual patients do present very dif- elimination, dose sensitivity) are estimated for indi- ferent INR responses to warfarin. For all of the five vidual patients [43]. Wright and Duffull considered a model parameters, individual CoVs over the course of PK/PD model with eight parameters and six of them our study are significantly different from zero, indicat- are individualized [46]. Wright and Duffull studied a ing strong intertemporal variability, (i.e., patient INR model with eight parameters [41], and the model of response to warfarin does change over time).
Hamberg et al. has 11 parameters [48].
The results in section of ‘Interindividual and Inter- temporal Variability' imply that the inpatients in our Our PK/PD model versus PK/PD models in the sample not only have interindividual variability but also exhibit a strong intertemporal variability in all A PK/PD model for INR response to warfarin typi- four submodels of Holford NHG [40]. These results cally has four submodels [40]: (1) A pharmacokinetic confirm the importance of an individualized PK/PD model for the absorption, distribution and elimina- model particularly for those who were not in their tion of warfarin. (2) A pharmacodynamic model to stable state of health (as in our cohort). The rolling-describe the impact of warfarin on clotting factors. horizon estimation method updates the model at each (3) A physiological model for the synthesis and deg- new INR observation by adding the latest INR but Per. Med. (2016) 13(1)
future science group A rolling-horizon pharmacokinetic pharmacodynamic model for warfarin inpatients in transient clinical states Research Article Days into the future
Figure 2. Boxplots of prediction error (predicted international normalized ratio - actual international normalized
ratio) for the pharmacokinetic pharmacodynamic model on a rolling horizon. Each box represents the median,
75th and 25th percentile of the prediction error and the whiskers extend to cover 99% of the data. The horizontal
axis represents the number of days into the future.
dropping the oldest data point; as such, it can capture Predicting INR for patients in stable versus the changing patient sensitivity to warfarin over time. transient clinical statesOur PK/PD model uses fewer numbers of parameters Previous studies of computerized algorithms of PK/PD as compared with previous models reported in the lit- models focus on patients in a stable health status. erature. Thus, our PK/PD model not only reduces the For instance, Vadher and Patterson predicted the amount of data required for a reliable estimation but long-term maintenance dose by the initial four INR also helps to keep the model current.
responses [43], and Wright and Duffull predicted the Pharmacogenomics information can be included in long-term steady-state INR by the initial 0–6 INR PK/PD models [48]. However, this may not be neces- responses [46]. Pasterkamp et al. considered a reduced sary for the model proposed in this paper because it is model where one parameter can be updated after each already individualized to each patient [41]. That is, for visit for outpatients where INRs are monitored on a each patient, we estimate a model based on the patient's weekly or monthly basis [42]. In contrast, a majority historical doses and responses, and then use the model of the inpatients in our study had fluctuating INRs to make prediction for that patient. The model is also because of their changing clinical status.
updated as the patient's condition progresses to capture Wright and Duffull tested a variant of the PK/PD the intertemporal variability.
model of Hamberg et al. in a study of 55 warfarin Table 3. Summary statistics for the estimated values of all model parameters.
Warfarin half-life (h) b
Clotting factor half-life (h)
Mean sample CoV (%) †p-value < 0.05 suggests strong statistical evidence of interindividual variability.
‡p-value < 0.05 suggests strong statistical evidence of intertemporal variability.
CoV: Coefficient of Variation.
future science group Research Article Zhao, Liu, Wang & Hickey patients on its predictive performance of the steady- multiple times to the hospital during the course of state INR, defined to be the second of two subsequent the study. Patients who did not self-identify with the INR measurements within 80–120% of the target included race/ethnic groups (Table 1) were classified as (usually 2.5) separated by at least 7 days [46]. Wright ‘other' or did not have information about race/ethnicity and Duffull also found that the model performs the available within their chart. Because of the retrospective best when the initial six consecutive INR responses are nature of data collection, further information regard-included, where 99% of the prediction errors are within ing race/ethnicity cannot be obtained. We acknowledge (-1–1) and 50% of them are within (-0.5–0.5) [46]. this as a limitation, given the known differences in the Rather than the steady-state INR, we made predictions metabolism of warfarin in various populations.
on INRs at any time in the next 5 days because the patients in our study exhibit strong intertemporal vari- ability and rarely reached steady-state during their stay We have provided evidence that inpatients exhibit in the hospital. This likely explains why our predic- strong interindividual and intertemporal variability tion errors are larger (especially on day 5); 95% of the in their response to warfarin doses. Our novel PK/prediction errors (for INRs 5 days ahead) are within PD model combined with the rolling-horizon method (-1.7–4.6), 50% of them are within (-0.2–0.8), and can adapt to patients' changing clinical conditions some extreme outliers have PE > 4 (see also Figure 2).
over time and provide an accurate and individualized Our model may outperform previous ones on the prediction of INR levels 5 days into the future.
prediction accuracy of INRs in the near future (in con- Given the requirement of at least six INR measure- trast to steady-state). For instance, Vadher and Patterson ments over the course of treatment, the patients included made an accurate prediction for the fourth INR based in this study only represent those who had significant on the initial three with the median PAPE of 15.3% and underlying cardiovascular risk factors and were admit-a 95% CI of 10.9–17.7 in a sample of 74 inpatients [43]. ted to the hospital for acute care. We plan to expand Based on the latest five INR responses, our model makes this research from INR prediction to warfarin predic-a more accurate prediction for the next INR (see Table 2) tion dosing in the future. We also plan to study inter-where the PAPE has a median of 9.85% and a 95% CI temporal variations and fully personalized models for of 8.55–10.93. A few factors contribute to the improved outpatients receiving warfarin therapy, in whom INR predictive performance of our model and algorithm. levels are typically monitored less frequently than inpa-Apart from the enhanced flexibility provided by indi- tients. This individualized approach holds the promise vidualizing all four submodels and more historical data to improve anticoagulation management in the future.
used, we use a global optimization algorithm to find the
model parameters. In contrast to the commonly used Future perspective
Excel solver, our algorithm ensures that our fitted model Our computerized model provides a personalized
provides the best possible fit to data [49].
rolling horizon approach that is able to adapt in real INR levels are difficult to predict for warfarin time to individual warfarin PK and PD variability. patients, and it is particularly challenging to dose The application of this model will allow for better and obtain an INR within a target range of 2.0–3.0 prediction of anticoagulation response outcomes and for inpatients given their transient clinical state where better dosing without the need for intensive sam-their sensitivity to warfarin may change over time. Our pling of drug concentrations (INR levels) and the use results provide the evidence that the fully personalized of static ‘one size fits all' dosing algorithms. Future PK/PD model developed in this paper, for which the research needs to focus on developing computerized parameters are estimated on a rolling-horizon, can dosing tools based on our proposed methods, and test-provide reasonably accurate INR prediction for these ing such tools via clinical trials on patients receiving patients up to 5 days into the future. This method is warfarin treatment.
much better than manual protocols used by clinicians that are based on generic response tables, the alternative Financial & competing interests disclosurecurrently available for such unstable patients, because The authors have no relevant affiliations or financial involve-the latter typically can only predict INR 1 day ahead.
ment with any organization or entity with a financial inter-est in or financial conflict with the subject matter or mate- rials discussed in the manuscript. This includes employment, Compliance with warfarin therapy and diet were not consultancies, honoraria, stock ownership or options, expert always available on study participants. While a major- testimony, grants or patents received or pending, or royalties.
ity of patients have only one case in this study, a few No writing assistance was utilized in the production of this patients have multiple cases as they were admitted manuscript.
Per. Med. (2016) 13(1)
future science group A rolling-horizon pharmacokinetic pharmacodynamic model for warfarin inpatients in transient clinical states Research Article Ethical conduct of researchThe authors state that they have obtained appropriate institu- animal experimental investigations. In addition, for investiga- tional review board approval or have fol owed the principles tions involving human subjects, informed consent has been outlined in the Declaration of Helsinki for all human or obtained from the participants involved.
Executive summary Background
• Warfarin has a long history of clinical use and is the most routinely prescribed oral anticoagulant.
• The international normalized ratio (INR) is a standardized measure of blood clotting time for those taking warfarin and needs to
be controlled within a narrow range typically between 2.0 and 3.0 to achieve anticoagulation.
• Numerous factors (e.g., comorbidities, diet, medications, metabolism, genetic differences) can affect individuals' INR levels, which make warfarin dosing a challenging task for clinicians.
• The same dose of warfarin may result in very different INRs among different individuals (interindividual variability).
• Patients with multiple co-existing comorbidities may be in a transient clinical status and exhibit strong intertemporal variability; meaning that an individual's INR response to warfarin could change considerably over time.
• No protocols for warfarin dosing have been uniformly accepted, resulting in a significant variation in clinical practice of dosing.
• Recent research has shown that computerized algorithms based on pharmacokinetic pharmacodynamic (PK/PD) models may outperform the manual approach and protocols used in practice in terms of efficacy and convenience.
• However, most studies on PK/PD models focus on patients in a relatively stable clinical state; much less information exists on how to predict INR and how to dose warfarin for inpatients with multiple co-existing comorbidities and in a transient clinical status.
• The purpose of this study was to develop a new PK/PD model to make individualized and adaptive predictions of INR measurements for cardiac clinical inpatients whose health status was in a state of flux.
Methods
• This retrospective study used a convenience sample of 60 inpatients receiving warfarin treatment. This sample is collected via
chart reviews.
• Patients had multiple co-existing cardiac conditions and were in a clinically transient state at the time of the first INR • For patients in transient clinical states, we design a new PK/PD model by personalizing all the four submodels of Holford (1986) and minimizing the number of parameters to be estimated.
• The parameters are estimated for individual patients on a rolling horizon such that the dated historical data are replaced by the latest reading upon each new INR observation. We then predict future INRs for the patient once every half-hour under the dose regimen shown in the medical record.
• We evaluate the INR prediction accuracy by our PK/PD model on all patients in our sample, using standard measures like prediction error, absolute prediction error and percentage absolute prediction error.
• We conduct t-tests to test the existence of interindividual and intertemporal variability in these patients.
Results
• Our sample of patients had multiple co-existing cardiovascular comorbidities such as hypertension, diabetes, coronary artery
disease and heart failure.
• The PK/PD model on a rolling-horizon provides accurate predictions of INR up to 5 days into the future.
• The predictions is unbiased within the first 3 days into the future. It slightly overpredicts the INR values in the fourth and fifth • The predictions are particularly precise for the first 2 days, and statistically, the model is more accurate for a shorter term • We found strong statistical evidence on interindividual and intertemporal variability; that is, INR response to warfarin changed across patients and over time.
Discussion & conclusion
• The novelty of our PK/PD model lies in its use of a new combination of the mathematical equations, for similar mathematical
properties but fewer parameters.
• Thus, it not only reduces the amount of data required for a reliable estimation but also helps to keep the model current.
• Because all submodels are individualized, the model is able to capture individual warfarin PK and PD variability.
• Our fully personalized PK/PD model, estimated on a rolling-horizon, can provide reasonably accurate INR prediction for patients in changing clinical status up to 5 days into the future.
• The commonly-used method for predicting INRs in such unstable patients can only provide predictions 1 day ahead. Our method offers much more flexibility.
• Our model may not outperform previous ones in steady-state INR predictions due to the patients' transient clinical status, but it is likely to outperform previous methods in short-term predictions.
• Our PK/PD modeling approach holds the promise of improving warfarin treatment and management in clinical practice.
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