7 minute read
Ibrahim Arogundade
Models for the assignment of dosages to Oncology patients: Bias coin design model and Dosage Stability Model Ibrahim Arogundade
Aliakbar Montazer Haghighi Department of Chemical Engineering and Department of Mathematics
Introduction: Our study in this project is based on Ivanova et al. (2003) with some revisions. This project aims to determine if the bias coin design with all the factors we put into consideration is an effective model for the distribution of dosage levels; this will lead us to find the optimal dosage level that is best suitable for the patient. It should be noted that Dixon and Mood (1948) started the idea of biased coin design. Durham and Flournoy (1994) begun elaborating on biased coin design based on the up and down design. The theory of exact treatment was done the following year by Durham et al. (1993 and 1995).
A patient (or a subject) is to go under repeated treatments for a disease that requires possible repeated use of medicine with different dose levels. Prescribing each dose level needs a period of time to act. We refer to this period as a unit of time. The subject is to return to the physician after each treatment for reassessment. Dose levels are under certain restrictions. We suppose there are ten possible dose levels, from 1 to 10, level 1 being the weakest and level 10 the strongest. Typically, levels three or lower may not be much effective but safe. On the other hand, levels eight or higher are highly toxic and dangerous, yet needed sometimes to take a chance. Hence, based upon the first assessment, if the disease is very mild, level 3 or lower is prescribed. However, if the disease has progressed too much, the level 8 or higher may be prescribed. Thus, we assume that there are three different options to determine the first dose level. (i)If the features show a mild case, dose level k is assigned. The level k = 1, 2, or 3 will be selected randomly. (ii) If the subject shows a high progress case, dose level K is assigned. The level K = 8, 9, or 10 will be selected randomly. (iii) If the first assessment is not indicative as too weak or highly progressive, a dose level d is chosen randomly, where d = 4, 5, 6, 7.
Now, the patient returns, after a unit of time, for the next visit. Required tests will be applied to assess the result of the dose level prescribed previously. There are two cases for the result: (i) The dose was influential in the sense that it affected the patient’s health negatively. This case is referred to as toxicity, and the subject’s case is called toxic. (ii) The dose was effective in the sense that it affected the patient’s health positively. This case is referred to as non-toxicity, and the subject’s case is called non-toxic. At this point, the physician has to decide about a new dose level based on the two cases of toxicity and nontoxicity. There are three possible cases to prescribe the next dose level: (i) Move down one level, (ii) stay put, or (iii) move up one level. The process of determining the next dose level is referred to as the up-and-down method. A factor to take into consideration before determining the dosage level is the probability Γ of toxicity which is used to determine the probability of heads π = Γ / (1- Γ) in the toss of a coin. Now, the question is how to determine the dose level in each of the three cases.
Before selecting the next dose level, the physician will have to include features of the patient such as age, gender, some health conditions such as diabetes, being overweight, having asthma, being allergic to some medicine, etc. That is, several additional factors are being considered before the next dose level is prescribed. Methods and Materials: Among methods to choose the next dose level is the so-called bias coin design, see Ivanova et al. (2003). The method is represented as follows: A patient is assigned a dosage level, dj, j =1, 2, ..., K. The next subject will be assigned as followed: (i) Dose level dj-1, if the dosage level was toxic; and (ii) Dose level dj+1, if the dose level was non-toxic and the coin toss results in a head. Otherwise, dosage level dj will be assigned. We will continue to do this over time until we find a dosage level repeatedly being assigned to the patient; the repetitive assignment of this dosage proves that it is the stable dosage level for the patient. When we have found this stable dosage level for the patient, we will use it to assign a dosage level to the next patient, which will continue to happen for every patient in the group. This model aims to find a dosage level that is stable for the patient and use this dosage level to establish a basis for assigning a dosage level for the next patient. We will use the lines of code from the programming language MATLAB to simulate the bias coin design.
The code directly follows the methods for the biased coin design. The steps for the code and the algorithm are outlined as follows: (i) The first step of the code starts with assigning a number to the patients or subjects, dose levels, and trials. Finally, we create a menu that asks us to select the design used, which will ultimately be the bias coin design. (ii) The second step is the initiation of the trial due to the selection of the bias coin design. A dose level is randomly selected from dose 1 to 3 for the first patient, a coin toss is simulated, and the toxicity is determined. The biased coin design conditions are then applied, and the dose level that was generated is then assigned to the next patient or subject. Two arrays are created; the first stores the dose
level for every subject in a group, and the second stores all the dose levels that were assigned to all the subjects in all the groups. The flowchart displays the methods taken to execute this step. The next step is calculating the proportion of trials of each dose level and plot them on a histogram. Results and Discussion: The simulation resulted the following: (i) Dosage level 1 being the dose level that was most distributed. (ii) Higher dosage levels were less likely to be distributed with the highest dosage level not even assigned at all. (iii) This was the result of the simulation regardless of the change in number of patients, trials and Γ. The results show that the distribution is skewed towards the lower dosage levels, particularly to dosage level 1. This result is because the doses assigned are likely to be toxic due to the probabilities, as two conditions determine if a higher dose level is assigned compared to only one condition in which a lower dose level is assigned. All these effectively lowers the probability of nontoxicdoses and higher dose levels being assigned or distributed. These reasons effectively lead to a higher proportion of lower dose levels regardless of the Γ, the number of patients, and trials. MATLAB simulation has been used and is available.
Conclusion: This version of bias coin design is a much simpler one as it is devoid of several factors and given several assumptions. In this version, we assumed that toxicity does not change with the dosage level. Therefore, in the future, we will consider several factors and assumptions we omitted, such as: (i) A function Q(dj ) that determines the toxicity of each dosage level. (ii) Eliminate the assumption the probability of toxicity is constant with each dosage level.
References:
Ivanova, Anastasia, Haghighi, Aliakbar Montazer, Mohanty, Sri Gopal and Durham, Stephen D. (2003). Improved up-anddown designs for phase 1 trials. Statistics in medicine; 22:69-82. Dixon W. J., and Mood, A.M. (1948). A method for obtaining and analyzing sensi- tivity data. Journal of the American Statistical Association 43, 109-126. Durham, S.D. and Flournoy, N. (1994). Considered two biased coin designs, which are improvements of Dermans (1957) design. Durham, S.D. and Flournoy, N., and Haghighi, A.M. (1993). Up-and-Down Designs. Computing and Statistics: Interface, Vol. 25, pp. 375-384. Durham, Stephen D., Flournoy, Nancy and Haghighi, Ali A. Montazer (1995). Up- and-Down Designs II.: Exact Treatment Moments. Adaptive De- signs (South Hadley, MA, 1992), Institute of Mathematical Statistics Lecture Notes Monograph Series, 25, pp. 158-178.
