### Abstract

- Top of page
- Abstract
- Introduction
- Methods
- Results
- Discussion
- References
- Appendix

**Summary. ** It has been shown that computerized algorithms for the prescription of coumarin derivates can improve the quality of long-term anticoagulation treatment. These algorithms are usually based on an empiric relationship between dosage and International Normalized Ratio and do not quantify the delaying effect of the drug's pharmacokinetics or the effect of alternating doses that are used to approximate a certain average dosage. Our objective was to develop a mathematical model that takes into account these effects and to develop a new algorithm based on this model that can be used to further optimize the quality of long-term anticoagulation treatment. We simplified a general model structure that was proposed by Holford in 1986 so that the parameters can be estimated using data that are available during long-term anticoagulation treatment. The constant parameters in the model were estimated separately for phenprocoumon and acenocoumarol using data from 1279 treatment courses from three different anticoagulation clinics in the Netherlands. The only variable parameter in the model is the sensitivity of the patient, which is estimated during the course of each treatment. A total of 194 dosage and appointment intervals that were proposed by the new algorithm were scored as ‘good’, ‘acceptable’, or ‘bad’ by two dosing experts. One hundred and seventy-eight (91.8%) proposals were considered good by at least one expert and bad by none. In 39 cases the experts disagreed. We believe that this algorithm will allow further improvement of anticoagulation treatments.

### Introduction

- Top of page
- Abstract
- Introduction
- Methods
- Results
- Discussion
- References
- Appendix

Oral anticoagulant therapy with coumarin derivatives is used for a variety of indications such as mechanical heart valve prostheses, deep vein thrombosis and atrial fibrillation [1]. Inadequate treatment with these drugs increases the risk of thromboembolic (under-anticoagulation) and bleeding (over-anticoagulation) complications [2]. Therefore, the intensity of anticoagulation is regularly monitored by assessment of the International Normalized Ratio (INR) and the INR is kept as much as possible within a predefined target range. In the Netherlands the monitoring of the INR is handled by approximately 70 anticoagulation clinics. Patients on anticoagulation therapy visit these clinics every 1–6 weeks to have their INR measured and to receive a new calendar with daily dose prescriptions by mail 1 day later (dose calendar). Acenocoumarol (Sintrom mitis^{®}) and phenprocoumon (Marcoumar^{®}) are the licensed coumarin derivates.

If the measured INR is outside the predefined target range the dosage needs to be adapted. Determining the new dosage is complicated by several factors. First, the sensitivity of patients to the drug is subject to a pronounced inter-individual and, in the course of time, intra-individual variability. Second, because of the influence of delaying processes such as pharmacokinetics, and the complex relationship between dose and INR, it is difficult to make accurate dose adjustments. Third, the INR shows fluctuations that are unpredictable and may be the result of differential sensitivity of the thromboplastin reagent used [3], and unknown variations in diet, medication, compliance or physical condition of the patient.

To facilitate anticoagulation dosing, several computerized algorithms have been developed in the past and have been shown to improve the quality of control [4–8]. In the Netherlands several anticoagulation clinics use an algorithm called TRODIS that was first developed in 1973 [9]. TRODIS uses a complex empiric decision-tree that determines whether the dosage has to be adapted and, if it has, whether a new dosage can be calculated by the algorithm (dosage proposal) or whether it has to be determined by the physician. In practice, TRODIS only proposes a dosage in approximately 60% of all cases, 25% of which are subsequently overruled by the physicians upon review. In the case that a new dosage has to be calculated, TRODIS uses a simple dose–INR relationship that calculates the new dosage based on the INR and previous dosage only. This approach has several weaknesses. First, it does not take into account the pharmacokinetics of the drug or the effect of previous dosage adjustments. Second, TRODIS does not use the actual daily doses that are prescribed on the last visit's dose calendar but the average. On the calendar, for instance, a patient receives instructions to alternately take two and one tablets per day, which TRODIS uses as a dosage of 1.5 tablets per day. Especially for acenocoumarol with its short half-life of 24 h [10] it is relevant to know how many (in this instance either one or two) pills were prescribed to the patient the day before the visit to the anticoagulation clinic. Finally, after the previous visit the dosing physician may have decided to either prescribe an extra loading dose if the INR was too far below target, or stop the dosing for one or more days if the INR was too far above target (stop-dose). These actions are not taken into account if the algorithm only uses the average dosage instead of using the actual prescriptions on the dose calendar.

In conclusion, TRODIS only makes use of a subset of all the information that is available to adjust the dosage. The anticoagulation process is simply too complex to derive an empiric relationship between all the available data and the right dosage that will yield an INR near the target at the next visit.

A common approach in engineering to control complicated systems is to develop a mathematical model that describes the effect of the input on the output. An algorithm that is based on such a model controls the output (INR) by choosing the input (dosage) that yields the desired output (target INR), as predicted by the mathematical model. In the case of computerized anticoagulation dosing it may be expected that an algorithm that uses such a mathematical model will lead to faster and more accurate corrections of the dosage. This in turn will lead to a higher percentage of time-in-range, and fewer visits to the anticoagulation clinic per year. A few mathematical models with this purpose have already been developed, but they either use input that is not available during the therapy, such as individual levels of clotting factors, or they only operate during the initiation of therapy [11–14].

This article describes three steps that were taken to develop an algorithm for the prescription of oral anticoagulants. First a model structure for the above mentioned mathematical model is determined. Second, the parameters of the model are optimized to data from three anticoagulation clinics in the Netherlands. Finally the algorithm is evaluated by two physicians who are expert at prescribing anticoagulation dosages at the anticoagulation clinic in Leiden.

### Discussion

- Top of page
- Abstract
- Introduction
- Methods
- Results
- Discussion
- References
- Appendix

A simplified model with a reduced set of parameters was derived from Holford's theoretical model structure, and this model could be used to calculate dosage proposals for the physician. Using data from three anticoagulation clinics we optimized the parameters of the model for both acenocoumarol and phenprocoumon, while a similar procedure can be used to estimate parameters for other coumarin derivatives, such as warfarin. Finally, evaluations by two experts showed a high acceptance of the dosage and appointment period proposals of the algorithm.

Several earlier studies have modeled pharmacodynamics using a sigmoid *E-max* model. Although these studies use warfarin as the anticoagulant agent, the estimation of the *E-Max* parameter *γ* largely complies with our findings [19,20]. The first ICAD submodel describes the influence of all dynamic processes that determine the time-dependent relationship between dose and INR. Because pharmacokinetics is known to be of major influence, we expected the effective half-life to be at least as long as the half-life of the anticoagulant agent. The half-lives of acenocoumarol and phenprocoumon are, respectively, 24 h and 160 h, whereas we found longer effective half-lives of 50 and 200 h. It is likely that the difference of approximately 1.6 days can be attributed to the remaining processes that have a time-dependent and delaying influence.

Based on data from six anticoagulation clinics, it has recently been shown that the variability in control of treatments with acenocoumarol is higher compared to that with phenprocoumon [21,22]. This could explain why with phenprocoumon a longer mean visit interval is found compared to treatment courses with acenocoumarol (13.3 days vs. 16.4 days), although the mean INR is closer to the center of the INR target range.

A combination of several factors makes it difficult for the dosing physician to determine a reliable new dosage. These are variability of the patient's sensitivity, the delay in the effect of the anticoagulant, the variability in the prescribed doses and the unpredictable variations in the measured INR. TRODIS uses an empiric dose–INR relationship that does not take into account the above-mentioned factors and the dose–INR relationship is chosen to be steep so that proposed dosage corrections are relatively small. As a result, TRODIS reacts slowly to changes in the sensitivity to the drug which leads to lower time-in-range. TRODIS resembles other algorithms that are described in literature like Coventry, Hillingdon and DAWN AC in a sense that they combine a decision tree or table with an empiric dose–INR relationship [5,7,10,23,24].

One drawback of the model-based strategy that we followed is that the quality of the dosage proposals is limited by the accuracy of the underlying mathematical model. In this study, the parameters were fitted using a large amount of historical data, but the equations themselves remain theoretical. A better model could be derived if not only the prescribed doses and measured INR were known, but also intermediate variables like the plasma level of the anticoagulant and the level of individual clotting factors. However, these measurements were not present in the available data of the anticoagulation clinics and are not used in daily practice, in which clotting factor measurements are not routinely carried out.

Another important drawback of the model-based approach is that the model is fitted on historic data and there is no guarantee that the inverse path, calculating the necessary dosage from the target INR, yields accurate dose prescriptions. Although the expert evaluation gives an indication of the quality of the dosage proposals, it cannot be used as a gold standard, because we expect that the new algorithm will yield better controlled anticoagulation therapies than those that are currently achieved by the physicians and TRODIS.

An important advantage is that improvements to the mathematical model can easily be incorporated into the algorithm. The model could, for example, be enhanced with more input variables like individual clotting factors or the sensitivity of the thromboplastin reagent.

The next step will therefore be to start a randomized controlled trial that compares the new algorithm against TRODIS. The expert evaluation shows that the dosage and appointment periods proposed by the algorithm were highly acceptable by the two experts, which makes such a randomized trial feasible and justified. We hope that a model-based and more quantitative approach to the prescription of oral anticoagulants, as proposed in this study, will lead to safer and more effective anticoagulation therapies.