Prediction of the international normalized ratio and maintenance dose during the initiation of warfarin therapy

Abstract

Aims

A pharmacokinetic/pharmacodynamic model, with Bayesian parameter estimation, was used to retrospectively predict the daily International Normalized Ratios (INRs) and the maintenance doses during the initiation of warfarin therapy in 74 inpatients.

Methods

INRs and maintenance doses predicted by the model were compared with the actual INRs and the eventual maintenance dose. Cases with drugs or medical conditions interacting with warfarin or receiving concurrent heparin therapy were not excluded. As the study was retrospective, model predictions of the maintenance dose were not those that were administered. Mean prediction error (MPE) and percentage absolute prediction errors (PAPE) were used to assess the model predictions.

Results

INR MPE ranged from −0.07 to 0.06 and median PAPE from 10% to 20%. Dose MPE ranged from −0.7 to 0.17 mg and median PAPE from 16.7% to 37.5%. Accurate and precise dose predictions were obtained after 3 or more INR feedback’s.

Conclusions

This study shows that the model can accurately predict daily INRs and the maintenance dose in this sample of cases. The model can be incorporated into computer decision-support systems for warfarin therapy and may lead to improvement in the initiation of warfarin therapy.

Introduction

The initiation of warfarin treatment has generally been poor with only 47% to 50% of the International Normalized Ratios (INR) being within the therapeutic range for inpatients[1, 2]. The time taken to initiate therapy and determine a stable dose by traditional empirical methods may take a week or longer[3]. Also most haemorrhagic complications of oral anticoagulant therapy occur during the first month of therapy [4]. Therefore poor initiation and control of warfarin treatment often leads to increased morbidity and mortality, increased hospital stay and therefore increased health care costs. The main reason for the poor quality of warfarin treatment is its complex pharmacology with wide interindividual and intraindividual variation in the dose–response relationship [5, 6]. An optimal method for the initiation and control of warfarin treatment is therefore needed.

Traditional empirical (trial and error) or regression methods such as that of Fennerty et al. [7] can be used to determine the maintenance dose of warfarin after loading doses. However, computerized pharmacokinetic/pharmacodynamic (PKPD) models of the time course of warfarin action with Bayesian parameter estimation are considered to be theoretically superior although this is unproven [3, 8]. These adaptive methods allow individualized dosing and are not restricted to prediction of certain dose ranges, therapeutic ranges or just for the first few days of therapy. However, four or more feedback Prothrombin time (PT) measurements are still required for accurate maintenance dose prediction [9, 10]. Possible problems of these methods include: (i) use of inaccurate pharmacodynamic models [9]; (ii) use of a hypothetical prothrombin complex in the model with an arbitrary model describing its relationship to the prothrombin time without taking into consideration the thromboplastin sensitivity or the method of prothrombin time determination [3, 11]; (iii) inability to account for dose–response variation due to excess heparin or inhibitors of warfarin metabolism [10].

The objective of this study was to evaluate the ability of a computerized pharmacokinetic/pharmacodynamic (PKPD) model to predict the INR and maintenance dose during the initiation of warfarin therapy. The model uses a hyperbolic tangent relationship to describe the dose response relationship, an empirical model which describes the relationship between the vitamin K dependent clotting factors and the INR, and the Bayesian method for parameter estimation. Although the INR system is unreliable during the initiation of warfarin therapy, it does provide an advantage over the reporting of the results as PT ratios [12]. A retrospective comparison of the maintenance dose prediction ability was also made with the modified regression method of Fennerty et al. [13]. Empirical methods have been more readily accepted by clinicians for routine use mainly because of their simplicity and ease of use [13, 14]. A Bayesian method with an INR model may be more readily accepted than those with Prothrombin complex models if it can be shown to be more effective then simple empirical methods. Our ultimate aim therefore is to develop and implement a simple and practical Bayesian method which is acceptable to clinicians for routine use.

Method

The model

Pharmacokinetic model A single compartmental model, which adequately describes the warfarin dose concentration relationship, was used [3]. The racemic mixture of warfarin was assumed to be a single drug. The bioavailability of warfarin was assumed to be 100% [15] and its absorption was assumed to be instantaneous (because absorption is much faster compared with elimination and can therefore be ignored). The amount of warfarin, D_t, in the plasma compartment at a time point t during a dosing interval being described by:

(1)

where D_o is the initial post ‘bolus’ body warfarin at time t = 0, and K_w is the first-order elimination constant of warfarin.

Physiological model

A similar single compartmental model was also assumed to describe the plasma time course of each of the vitamin K dependent clotting factors. The change in clotting factor with time being described by:

(2)

where F_syn is the rate of functional clotting factor synthesis and F_deg is the rate of clotting factor degradation. First-order elimination of the clotting factors was assumed, therefore:

(3)

where K_f is the first order elimination constant of the vitamin K dependent clotting factor. The normal steady state level of clotting factors was assumed to be 100%, therefore:

(4)

when there is no inhibition of synthesis by warfarin. The values of K_f for the clotting factors were estimated from the literature [16]. Values used were 0.116 h⁻¹ for factor VII (half-life 6 h); 0.017 h⁻¹ for factor X (half-life 40 h) and 0.0115 h⁻¹ for factor II (half-life 60 h).

Pharmacodynamic model

The dose-effect relationship of warfarin was assumed to be described by a function commonly used for biological systems, based on the hyperbolic tangent [17]:

(5)

where m is a non-dimensional multiplier, b a parameter describing the dose sensitivity relationship and D_t the amount of drug in the plasma compartment at time t. The value of m is 1 when D = 0 and m decreases as D increases. The dose response relationship is then governed by the equation:

(6)

The model assumes that all the warfarin in the plasma is immediately available to inhibit factor synthesis and does not account for plasma albumin binding. It also assumes that b, and therefore m, is the same for all the clotting factors, i.e. the percentage inhibition of synthesis of all the vitamin K dependent clotting factors is the same for a given dose of warfarin. Justification for this is that the clotting factors are equally reduced at steady state [18] although this is controversial [19, 20]. The assumptions were made to keep the model simple and of practical value. This pharmacodynamic model is similar to the sigmoid E_max model which is thought to be theoretically more accurate than the log-linear model [3, 21].

Relationship between the clotting factors and the INR

An empirical model was postulated by observation of plots of INR vs clotting factor level assuming 100% as baseline level:

(7)

where the product runs over the three clotting factors, F_i is the percentage of each clotting factor, a_i and s_i are parameters specific to each clotting factor. F_i was modelled as the integral of eqn 6 thus linking the physiological, pharmacodynamic and INR models. Data from various papers describing the relationship between the vitamin K dependent clotting factors, expressed as percentage of their baseline value, and the INR at initiation and steady state were used to determine the factor specific parameters, a_i and s_i [19, 20, 22–25]. The parameter values were determined by using nonlinear least squares regression. Seventy-five INR values were used and the variance of the residuals from the individual fits was 0.15. The parameter values used were a = 0.61 and s = 1.29 for factor II; a = 1.1 and s = 3.95 for factor VII; a = 0.99 and s = 1.17 for factor X. Factor IX was not used in the model as its deficiency does not affect the INR [25]. The INRs of the data used were in the range 1–4.5, so the ability of the model to predict the INR would be restricted to this range.

Parameter estimation

The Bayesian method uses weighted least squares regression for parameter estimation [26]. The parameters K_w and b were assumed to account for most of the interindividual and intraindividual variability in the dose–response to warfarin [27]. K_w and b were assumed to have normal distributions in the population with mean values of K and B, and variance V_k and V_b, respectively. For the first INR prediction mean population parameters were used. After each feedback INR measurement, from 1 to n, new parameters were estimated by minimising the value of the function:

(8)

where INR_obs = observed INR, INR_pr = predicted INR using the new estimated parameters, V_inr = variance of error in the INR measurements (includes error due to random intraindividual variability, model misspecification and laboratory measurement). The value of K (0.019 h⁻¹) was obtained from the literature [3]. No literature value for B was available and therefore it was estimated (value 0.07) by using patient data (mean warfarin dose of patients in the anticoagulant clinic who were maintained at a steady state INR of around 2.5) and the model, with K = 0.019 h⁻¹ for all the patients. The coefficient of variation (CV) of K and B was assumed to be 50% and of the INR measurement 10% [26]. V_k, V_b and V_inr could thus be calculated by the equation:

(9)

where P is a typical value for the parameter (i.e. K for V_k, B for V_b and 3.0 for V_inr). The model was implemented in Microsoft Excel® and Solver was used for parameter estimation which usually took 2 min (maximum time 100 s, iterations 25, precision 0.01, tolerance 0.05) [28].

Study population

Retrospective data from 74 out of 78 consecutive cases being initiated on warfarin therapy as inpatients at the Whittington Hospital, London were used. Five cases were not used as they did not have their eventual maintenance dose, which was defined as the dose which maintained the INR within the therapeutic range (2–3) for 3 successive days, determined satisfactorily. Patients with factors (hazards) affecting the dose–response to warfarin or receiving concurrent heparin therapy were not excluded. Table 1 shows the clinical characteristics of the patients. Sixty-three cases had INRs done daily over the first 5 days of treatment. All the cases had the INR measured around 09.00–11.00 h and received the warfarin dose in the evening. A Sysmex CA1000 optical density coagulometer was used with a low opacity Manchester thromboplastin reagent (ISI 1–1.2) for INR measurement. On two occasions INRs >4.5 were recorded.

Table 1.

The clinical characteristics of the cases (n = 74) used in the study. IQR = interquartile range.

Study conduct

For the first INR prediction mean population parameters and the induction warfarin doses given were used. The observed (measured) INR response and the predicted INR were then used to estimate new patient-specific parameters. These new parameters were then used to predict the first maintenance dose (dose which predicted an INR close to 2.5 on day 14 of therapy in the model) and, together with the actual dose administered, the next INR, i.e. after one feedback INR measurement. As the study was retrospective the predicted maintenance dose was not the actual dose administered. INR prediction for the next observed INR and the maintenance dose prediction were then determined after each INR feedback and new parameter estimates. Altogether INR predictions using the mean population parameters and three feedback INRs and maintenance dose predictions after four feedback INRs were determined. The predicted INRs were compared with the observed INRs and the predicted maintenance doses were compared with the eventual maintenance dose. It was assumed that the warfarin was administered at 17.00 h on each day and the INR measured at 09.00 h the next day.

Data analysis

The accuracy or bias of the predictions (consistent over-or under-prediction) was analysed by computing the mean prediction error (MPE) which is the mean of the predicted minus actual values on each day. The error of prediction (‘spread’ of predictions around the actual values) was evaluated by computing the median of the percentage absolute prediction error (PAPE) which is the absolute prediction error divided by the actual value expressed as a percentage. Absolute prediction errors were not used as an error (e.g. 1) in a small value (e.g. 2) may be clinically more significant than the same error in a large value (e.g. 10). Comparison of the mean prediction errors was by the Student’s t-test and of the percentage absolute prediction errors by the Mann–Whitney test and Wilcoxon test with 2P≤0.05 being significant. The day 4 maintenance dose prediction by the Bayesian method was also compared with the day 4 maintenance dose prediction by the modified method of Fennerty et al. [13].

Results

The model INR prediction was accurate in that mean prediction errors were less than 0.1 (Table 2). The INR mean prediction error using the population parameters was biased due to under-prediction. The percentage absolute prediction error (PAPE) was worse after one INR feedback, compared with the use of population parameters, with improvement after two feedback INRs. The larger percentage prediction errors tended to be for the lower INRs, i.e. INRs less then 1.4 (8 out of 9 times) for PAPE of 50% or more and INRs less than 1.5 (36 out of 74 times) for PAPE of 25% or more. The mean prediction errors were not statistically different in cases with and without hazards except after two feedback INRs (Figure 1). However, the median percentage absolute prediction errors were better in cases without hazards especially after two INR feedbacks (13.5%v 24.3%, P = 0.003) and three INR feedbacks (12.3%v 22.7%, P = 0.0004). On 31 out of the 38 occasions, when the PAPE was ≥25% and INR was >1.5, the relevant cases had one or more hazards to anticoagulation. Figure 2 shows a scatter plot of predicted INR vs the actual measured INR after three feedback INRs.

Table 2.

INR mean prediction errors and median percentage absolute prediction errors, CI = Confidence interval, PE (Prediction error), PAPE (Percentage absolute prediction error).

Figure 1.

Box-plots of the INR prediction error for cases with and without hazards after using the population parameters () and each feedback INR measurement ( 1, 2, 3). The horizontal line inside the box represents the median and the height of the box corresponds to the interquartile range. Outlying cases with values more than 3 box-lengths from the upper or lower edge of the box are designated with an asterisk (*) and cases that are between 1.5 and 3 box-lengths from the upper or lower edge of the box are designated with a circle (o). The whiskers show the largest and smallest observed values which are not outliers.

Figure 2.

Scatter plot of the next observed actual INR vs the INR predicted by the model after three feedback INRs. The solid line is the regression line. The dotted lines refer to the boundaries of the therapeutic range.

The mean prediction error for maintenance dose prediction was accurate without bias and improved with each INR feedback measurement (Table 3). The mean prediction error was statistically significantly different after three feedback INRs compared with four feedback INRs (2P = 0.03, paired t-test). However, this may be clinically insignificant. The percentage absolute prediction errors (PAPE) of the dose prediction also improved with each INR feedback measurement (P<0.001, anova). However, there was no difference in PAPE after three feedback INRs compared with four feedback INRs (2P = 0.21, paired t-test). The larger percentage absolute prediction errors tended to be for the smaller doses (less than 3 mg), i.e. 27 out of 27 times for errors of 100% or more, and both smaller (less than 3 mg) and larger doses (greater than 10 mg), i.e. 66 out of 77 times for errors of 50% or more. The mean prediction errors were not statistically significantly different in cases with and without hazards (Figure 3). Generally median percentage absolute prediction errors were better especially after three feedback INRs (12.5%v 20%) in the cases without hazards although these were not statistically significant. Figure 4 shows a scatter plot of the eventual maintenance dose vs the predicted maintenance dose after four feedback INRs.

Table 3.

Maintenance dose mean prediction errors and median percentage absolute prediction errors, CI = Confidence interval, PE (Prediction error), PAPE (Percentage absolute prediction error).

Figure 3.

Box-plots of the dose prediction error for cases with and without hazards after each feedback INR measurement ( 1, 2, 3, 4). The horizontal line inside the box represents the median and the height of the box corresponds to the interquartile range. Outlying cases with values more than 3 box-lengths from the upper or lower edge of the box are designated with an asterisk (*) and cases that are between 1.5 and 3 box-lengths from the upper or lower edge of the box are designated with a circle (o). The whiskers show the largest and smallest observed values which are not outliers.

Figure 4.

Scatter plot of the eventual maintenance dose (dose which maintained the INR within the therapeutic range of 2–3 for 3 successive days in the study cases) vs the maintenance dose predicted by the model after four feedback INRs. The solid line is the regression line.

Table 4 shows the comparison of maintenance dose prediction by the modified Fennerty method with that by the Bayesian method on day 4. Fifty-eight cases were used for the comparison because the Fennerty method could not predict a maintenance dose in eight cases (INR less than 1.4 on day 4). In the other cases no daily INRs were done or doses according to the Fennerty et al. schedule were not given. Neither method showed bias but the Bayesian method was significantly more accurate. The PAPE of the methods were not statistically different although it was generally better in the Bayesian method.

Table 4.

Comparison of maintenance dose prediction by Bayesian method with that by the Fennerty method of regression on day 4 (n = 58). CI = confidence interval; PE = prediction error; PAPE = percentage absolute prediction error.

Discussion

The model INR and maintenance dose predictions were accurate with no statistically significant bias except for INR prediction using the mean population parameters. The population parameters may not be accurate especially in cases with hazards. Improved parameter estimates may correct the early bias, however, this early bias was very small and may be clinically insignificant. Also patients would normally be closely monitored during this early phase. The accuracy of the model INR and dose predictions were not statistically significantly different in cases with and without hazards except for INR mean prediction error after two INR feedback’s.

The percentage absolute prediction errors (PAPE) of the INRs worsened after one INR feedback compared to using population parameter estimates. The INR responses to the first dose of warfarin will have not fully developed by the second day and because the range of possible INR responses is narrow the PAPE will also be smaller. Model misspecification may also have accounted for this early worsening of the PAPE. However, the percentage prediction errors of the maintenance dose improved with each INR feedback. The percentage absolute prediction errors were worse in the cases with hazards although these were not statistically significant. INR and Dose percentage absolute prediction errors were much smaller in cases without hazards after two and three feedback INRs, respectively. Overall, considering that the eventual maintenance dose was not determined in 19 patients until after 14 days (range 15–99 days), the maintenance dose predictions aimed for an INR of around 2.5 whereas dose stability was defined by control of INR between 2 and 3 and patients with hazards were not excluded, a median PAPE of 16.7% after four feedback INRs is acceptable. Three or more feedback INRs provided the best maintenance dose prediction.

Similar pharmacological models with Bayesian and non-Bayesian parameter estimation have been evaluated [9, 31]. These methods have excluded the use of cases with hazards. Both studies also found that the spread of prothrombin ratio (PTR) prediction errors worsened before they improved with the number of feedback PTR measurements. However, Svec et al. [9] found that the PTR mean prediction error improved progressively with more feedback measurements in contrast with our study and that of Carter et al. [31]. Direct comparisons of the errors in these methods with those in our method cannot be made as the other methods have used the prothrombin ratio instead of the INR. A recent study by Sun & Chang [32] using a log-linear model and the INR showed that the INR prediction error worsens before it improves with the root mean square error (rmse) ranging from 1.04 after four feedbacks to 3.24 after two feedbacks. In the our study the rmse were 0.19, 0.46, 0.57, 0.58 after using population parameters, one, two, and three feedbacks, respectively. The method of Svec et al. has been evaluated in a randomised prospective trial [33]. A stable therapeutic dose was achieved 3.7 days earlier (P = 0.002) by the Bayesian method, with fewer patients over anticoagulated, compared with interns using traditional empirical methods. No direct comparison of the method of Svec et al. has yet been made with a regression method such as that of Fennerty et al. [13].

The Bayesian method, in our study, is more accurate at predicting the maintenance dose compared with the modified regression method of Fennerty et al. [13] although the percentage absolute prediction errors were similar. However, the regression method is restricted in that induction doses are limited to 10 mg on the first two days; maintenance doses greater than 8 mg cannot be predicted; the method is non-adaptive and it cannot be used after 4 days. Also the Bayesian method does not require daily INR measurements and can be used for any target INR.

The main disadvantages of the Bayesian approach are its complexity, the limitations of the INR model, inaccuracy of population parameter estimates, model misspecification especially in patients with hazards and the relatively long computation times required. The INR model is only valid for the INR range of 1–4.5 due to the range of limits of the data used to establish the relationship between the clotting factor levels and the INR. The INR range of the model may be improved by using data with wider INR ranges to determine new values of a_i and s_i specific for each of the clotting factors. The problem of using the INR at initiation is inherent in the Prothrombin time measuring system. Thromboplastins differ in their sensitivity to depletion of the various vitamin K-dependent clotting factors which are depressed at different rates during induction therapy [34, 35]. However, the INR is still less variable than the PT ratio and is a more useful measure of excessive depression of factor VII and hence risk of haemorrhage [12]. Whether our INR model remains valid for all thromboplastins and different laboratory methods of measuring the INR remains to be determined. Because the effect of heparin on the INR is negligible, unless the activated partial thromboplastin time (APTT) is markedly prolonged [36], the effect of over-heparinization was not taken into account in the INR model. Worsening of PAPE for INR prediction after one feedback may have been due to the effects of over-heparinization and also due to the base-line INR for the model being 1.0 compared with 1.0–1.3 in patients. The value of the mean population dose-sensitivity factor, B, was determined using patient data and keeping the value of K_w constant. A better estimate of B and K_w and their variance obtained by using methods suggested by Sheiner et al. [37] may improve the predictive ability of the model further but whether this is clinically significant needs further investigation. The change in warfarin clearance with age, as found by Mungall et al. [38], was not used as it was felt that this would increase model complexity with probably little gain. Model misspecification may account for the higher percentage absolute prediction errors for dose prediction especially in the cases with hazards. However, although the model did not take into account the dose–response variation due to heparin or drug interactions, its overall performance in patients with hazards was not significantly different form those without hazards. Use of a more complex model in an attempt to take into account these factors may lead to little improvement. Also knowledge of how these factors affect the dose–response is limited [39]. Computation times may be improved with the use of more efficient programming algorithms and faster processors. Although not shown in our study the method can also be used for out-patient maintenance dose adjustments. Our method still requires a clinician to accept the dose prediction and determine the optimal monitoring interval. Combination of the method with an interval optimising method may give a more complete decision-aid [40].

In conclusion, our PKPD model accurately predicts the INR and maintenance dose during the initiation of warfarin therapy without significant deterioration in patients with hazards. A promising method of warfarin initiation and maintenance therapy has been developed and validated. However, the method still needs to be evaluated prospectively in randomised controlled trials comparing it with the other methods. Other uses of the model include teaching, training and development of more simple dosing methods [41]. Our approach may lead to an improvement in the initiation and control of warfarin therapy and thereby reduce morbidity and mortality.