15 minute read
Ibrahim Arogundade
Analysis of a Biased Coin Design Model Ibrahim Arogundade
Mentor: Aliakbar Montazer Haghighi Mathematics Department
Goal
Using simulation, this project analyzes the prescription of dose levels to a patient whose disease requires a period of repeated use of the same medicine with different dose levels, such as an oncology patient.
Basic Terminologies
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 to act. We refer to this period as a unit of time. The subject is to return to the physician after each treatment for assessment. Dose levels are under certain restrictions. We suppose there are ten possible dose levels 1,2, ,10 L , level 1 being the weakest and level 10 being the strongest. Usually, levels 3 or lower may not be much effective but safe. On the other hand, levels 8 or higher are highly toxic and dangerous, yet needed to take a chance on emergency cases. 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, level 8 or higher may be specified. Thus, we assume that there are three different options to determine the first dose level. (i) If the features of the patient show a mild case, dose level k is assigned. The level k = 1, 2, or 3, is randomly selected. (ii) If the subject shows a progressed case, dose level K is assigned. The level K = 8, 9, or 10 is randomly determined. (iii) If the first assessment is not indicative of too weak or highly progressed, dose level d is assigned. The level d = 4, 5, 6, and 7 is randomly selected. After a unit of time, the patient returns for a follow-up visit. Required tests will be done to assess the result of the dose level prescribed previously. There are two possible cases: (i) The dose was not effective but harmful because it negatively affected the patient's health. This case is referred to as toxicity, and the subject is referred to as toxic. (ii) The dose was not harmful; that is, it either positively affected the patient's health, or it did not affect it at all. Either case is referred to as non-toxicity, and the subject is referred to as non-toxic. At this point, the physician has to decide about a new dose level based on the two cases of toxicity and non-toxicity cases. 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.
Now, the question is how to determine the dose level in each of the three cases. The process of determining the next dose level is referred to as the up-and-down method. Before choosing the dose level, the physician will have to include features of the patient such as age, gender, some health conditions such as diabetes, being overweight, asthma, allergies to some medicine, etc. We choose the so-called bias coin design (BCD), see Anastasia et al. (2003) among the variations of upand-down methods. For our case, the method is as follows: A patient was assigned a dosage level dj, j=1, 2,...., K, at the previous visit. At the next visit, the subject will be assigned (i) dose level dj-1, the dosage level was toxic, (ii) dose level dj+1, if the dosage level was non-toxic and the coin toss results in ahead, or (iii) otherwise, the dosage level dj will be assigned. We are continuing the assignments at successive visits until we find a repeated dose level is assigned to the patient. This dosage's repetitive assignment suggests that we have reached a stable dosage level for the patient when taking the stability level to assign the dose level for the next patient. Now, if we have a group of patients and follow that method, we will reach a stability lever for each group member. The average of these stability levels can be used as the level for the first visit of a patient rather than the random selection discussed above.
The Algorithm
To right the necessary simulation programming, we are to develop a code that uses a biased coin design to assign and distribute the dose levels among the oncology patients, takes the proportion of trials for each dose level, and visualizes this data with a histogram. A select number of patients are introduced for the initial trial and overtime. A new group of this select number of patients undergoes the same trial until the number of trials is reached while using the bias coin design based on distribution. The biased coin design is a system where you use the toxicity and result of a coin toss to determine the next dose level to assign to the next patient. This can be demonstrated with the following scenario if a patient is assigned dose level 3, a coin is tossed, and the toxicity is determined; from this demonstration, three events are likely to occur. In the first event, the coin toss resulted in head and dose level 3 is not toxic, so the next patient is assigned the next dose level, which is 4. In the second one the dose level 3 is toxic so the next patient is assigned a dose level that is a level before the assigned one, in this case it is dose level 2. In the third event the same dose level is assigned to the next patient as a result of the dose level being non-toxic and the coin toss resulting in tails.
Step 1
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 which asks us to select the design that will be used which will ultimately be the bias coin design
Step 2
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, Figure 1, below displays the methods taken to execute this step.
Assign a Subject with a dose level from dose level 1 to 3
Y Is the subject
N
H What is the result of the coin toss?
T
Assign the next subject with the dose level dj-1.
Assign the next subject with the next dose level dj+1. Assign the next subject with the same dose level dj-1.
Step 3
Figure 1. Code Algorithm Flowchart
The next step is calculating the proportion of trials of each dose level and plot them on a histogram
Graphs
The following graphs are illustration of the simulations. Program is found in the Appendix.
Figure 2: Γ=0.3, Subjects/Patients=10, Trials=1,000
Figure 3: Γ=0.2, Subjects/Patients=5, Trials=1,000 The graphs show that the distribution is skewed towards the lower dose level particularly to dose level 1. The reason for this result is because that the doses assigned are likely to be toxic due to the probabilities, as there are two conditions that determine if a higher dose level is assigned compared to only one condition in which a lower dose level is assigned. In addition, the probability of a dose level being toxic is initially divided evenly. A dose level that is toxic leads to a lower dose level; but this is likely not the case for a non-toxic dose as it has to go through the coin toss test before a higher dose level can be assigned, all of these effectively lowers the probability of non-toxic doses and higher dose levels being assigned or distributed. These reasons effectively lead to a higher proportion of distribution of lower dose level regardless of the Γ, the number of patients and trials.
REFERENCES
Anastasia Ivanova, Aliakbar Montazer Haghighi, Sri Gopal Mohanty, Stephen D. Durham 92003). Improved up-and-down designs for phase 1 trials. Statistics in medicine; 22:69-82.
APPENDIX
%% Research Project- Design
%---------------------------------------------------% STEP 1 % Assign the number of trials, subjects and dosage levels, also select the dosage level that will be used to start the trial. Select % the design that is to be used.
%---------------------------------------------------no_subject=10; % The number of subjects dose_lvl=[1:10]; % The dosage levels Design_select= menu('Which design do you want', 'BCD','K in a row','average up and down','Narayana','CRM','Start-up'); if Design_select strcmp(Design_select,'BCD') trials=0;% The trials are being initiated,we are using it to control the while loop last_trial=1000; dosage_group=[]; % stores the overall dosage trials dosage_array=[];
%-------------------------------------------------------------%STEP 2(I) % TRIAL HAS STARTED AND BIAS COIN DESIGN HAS BEEN SELECTED % Proceed with the trial of each dosage level for each patient in a % group of a certain number(in this case 10) and do this trial 1,000 times % for different groups. If a dosage is toxic asssign the next patient the % dosage level before the assigned one, if a dosage level is not toxic and % the while trials < last_trial % dosage trials for 1,000 groups of 10 individuals=0; %the pateint has just walked in dosages=[]; % stores the dosages that was tried for each group
if trials==0 dj=randi(dose_lvl(3)); % Randomly selects the dosage level, random selection dj is only for the first group of % the initial trial. end while individuals < no_subject % dosage trials for each indvidual in the group gamma=0.3; % this is gamma bias= gamma/(1-gamma); % line 19-26 simulates the cointoss results, % CORRECT COINTOSS_PI TO BIAS pi= rand(1); % Randomly selects a value between 0 and 1 toss_result=['']; if pi <= bias toss_result=['head']; elseif pi > bias toss_result=['tail']; end Toxicity=randi([0,1]); % randomly determines the toxicity
%STEP 2(II) % If a dosage is toxic asssign the next patient the % dosage level before the assigned one, if a dosage level is not toxic and % the coin lands on head, assing the next patient the next dosage.If the dosage level is not toxic and the coin lands on tails, assign the next patient the same dosage level % Store all the dosage level in an array.
if Toxicity== 1 dj= dj-1; % if toxic the next dosage assigned is a dosage lower elseif Toxicity==0 && strcmp(toss_result,'head') dj=dj+1; % if non toxic the next dosage assign is a dosage lower else dj=dj;
end if dj < 1 % line 47-50, makes sure the dosage level doesnt go lower than 1 or higher than 10 dj=1; elseif dj > 10 dj=10; end dosages=[dosages,dj]; % stores the dosages that was tried for each group individuals=individuals+1; %responsible for the iteration of dosage assigned for each person in the group end dosage_group=[dosage_group,dosages];% stores the overall dosage trials trials=trials+1; %responsible for the iteration of dosage trials, used to control the while loop end end % STEP 3
%The next step is claculating the proportion of trials for each dosage %level and plot them on a histogram % Calculation of the proportion of trials for dosage level 1, calculate dosage1=sum(dosage_group==dose_lvl(1)); pro_dos1= dosage1/(no_subject*trials); % Calculation of the proportion of trials for dosage level 2, calculate dosage2=sum(dosage_group==dose_lvl(2)); pro_dos2= dosage2/(no_subject*trials); % Calculation of the proportion of trials for dosage level 3, calculate dosage3=sum(dosage_group==dose_lvl(3)); pro_dos3= dosage3/(no_subject*trials); % Calculation of the proportion of trials for dosage level 4, calculate dosage4=sum(dosage_group==dose_lvl(4)); pro_dos4= dosage4/(no_subject*trials);
% Calculation of the proportion of trials for dosage level 5, calculate dosage5=sum(dosage_group==dose_lvl(5)); pro_dos5= dosage5/(no_subject*trials); % Calculation of the proportion of trials for dosage level 6, dosage6=sum(dosage_group==dose_lvl(6)); pro_dos6= dosage6/(no_subject*trials); % Calculation of the proportion of trials for dosage level 7, calculate dosage7=sum(dosage_group==dose_lvl(7)); pro_dos7= dosage7/(no_subject*trials); % Calculation of the proportion of trials for dosage level 8, calculate dosage8=sum(dosage_group==dose_lvl(8)); pro_dos8= dosage8/(no_subject*trials); % Calculation of the probability of trials for dosage level 9, calculate dosage9=sum(dosage_group==dose_lvl(9)); pro_dos9= dosage9/(no_subject*trials); % Calculation of the probability of trials for dosage level 10, calculate dosage10=sum(dosage_group==dose_lvl(10)); pro_dos10= dosage10/(no_subject*trials); probabilities=[pro_dos1,pro_dos2,pro_dos3,pro_dos4,pro_dos5,pro_dos6,pro_dos7,pro_dos8,pro_dos9,pr o_dos10]; x=[1,2,3,5]; % %histogram(x) %histogram(probabilities) bar(dose_lvl,probabilities,'hist') ylim([0.0 1]) ylabel('Proportion of trials') xlabel('dosage level')
Vanadium Toxicology in Crustaceans Tristan Roland
Mentor: Andrea Ashley-Oyewole, Ph.D. Chemistry Department
Introduction: Crabs are eatable arthropods in the subphylum Crustacea and the order Decapoda (Ruethers et al., 2018). They and other shellfish are rich in dietary Vanadium (Jakimska, Konieczka, & Namieśnik, 2011). Research shows that the edible portion and internal organs of crustaceans bioaccumulate trace metals, and very little research is available on crustaceans' vanadium behavior (Shalini et al., 2020 (Miramand, Guary, & Fowler, 1981)). The research hypothesis is that there is a strong correlation between Shellfish poisoning/allergies and dangerous vanadium concentrations. There is a likelihood that shellfish allergies and shellfish poisoning may result from Vanadium stored in shellfish muscles (Miramand et al., 1981; Ruethers et al., 2018).
Shellfish Allergy and Vanadium toxicity Background:
According to the Cleveland clinic shellfish allergies are more common in places where fish consumption is high Invalid source specified.. 60% of Shellfish allergies occur in adulthood (Orford & Wilson, 1985); Ruethers et al., 2018). The symptoms for shellfish allergies include: Hives, itching, or eczema (atopic dermatitis), swelling of the lips, face, tongue, and throat, or other parts of the body, wheezing, nasal congestion, or trouble breathing, abdominal pain, diarrhea, nausea, or vomiting, dizziness, lightheadedness, or fainting. Many studies and report on Vanadium used in implants have failed due to Vanadium ionic species may be released and people being allergic to them (Costa et al., 2019). So, the Vanadium in the implant causes erythematous eczema around the incision site, widespread systemic skin manifestations, persistent synovitis leading to prolonged pain, and aseptic loosening. The connection between shellfish allergies and vanadium allergies in metal implants is that they both cause eczema. The thought process here is that if Vanadium is high in shellfish and one primary allergic symptom of shellfish is the same major allergic symptom of Vanadium. Subsequently, humans may not be allergic to shellfish, but they are allergic to Vanadium in shellfish. We are investigating the amount of Vanadium stored in crabs so that once we find that out, we can begin to see how this has affected human allergies in the long run. Materials and Methods: The research plan involves obtaining the live crab from a commercial vendor and weighed after the acquisition. Then we will treat the specimens with a vanadium-rich diet of algae for approximately two weeks. For comparison, two control crabs fed with a diet consisting of naturally occurring concentrations containing Vanadium. We will record the specimens' weight after an interval of three weeks based on previous exposure experiments (Miramand et al., 1981). We will prepare triplicate samples of the muscle tissue for direct analysis. Representative samples must be thoroughly dried to remove all traces of water. X-ray photoelectron spectrometry (XPS) has become a gradually more accessible and powerful tool for characterizing different surfaces, including biological samples. XPS and Scanning Electron Microscopy (SEM) instruments are used in metals analysis and can only operate effectively under ultra-high vacuum conditions. For this reason, all samples must be dehydrated. XPS and SEM analysis will be necessary to detect and quantify Vanadium levels in the dried muscle tissue without destroying the samples (Baer & Engelhard, 2010). Little or no sample preparation is needed for these techniques. X-ray fluorescence (XRF) is a another very quick and usually non-destructive analytical technique for measuring the levels of elements in samples (Byers, McHenry, & Grundl, 2019). Upon completion, data should provide information on an estimate of vanadium concentration available. We will
attempt to make inferences about whether those concentrations may contribute to shellfish allergies after comparing established contractions in the literature. Conclusion(s) or Summary: The hypothesis of my research so far is that there is a strong correlation between Shellfish poisoning/allergies and dangerous vanadium concentrations. This project should provide more clarity on the amounts of Vanadium that may be ingested by consumers. After comparing with National Health standards and recommendations for safe levels of ingestible Vanadium, we will make further inferences. Since there is no concentration limit set by the United States Environmental Protection Agency, or we did not find any available information. The Food and Drug Administration has a Toxic Elements Working Group (TEWG) which intends to decrease contact with to toxic metals in food, cosmetics, and dietary supplements. The 2006-2013 Toxic and Nutritional Elements study did not include Vanadium content of food Invalid source specified.. This information will help fill some gaps in the knowledge about total vanadium levels ingested by the public and Vanadium concentrations in drinking water.
References:
Baer, D. R., & Engelhard, M. H. (2010). XPS analysis of nanostructured materials and biological surfaces. Journal of Electron Spectroscopy and Related Phenomena, 178-179, 415-432. doi:https://doi.org/10.1016/j.elspec.2009.09.003 Byers, H. L., McHenry, L. J., & Grundl, T. J. (2019). XRF techniques to quantify heavy metals in vegetables at low detection limits. Food Chemistry: X, 1, 100001. doi:https://doi.org/10.1016/j.fochx.2018.100001 Costa, B. C., Tokuhara, C. K., Rocha, L. A., Oliveira, R. C., Lisboa-Filho, P. N., & Costa Pessoa, J. (2019). Vanadium ionic species from degradation of ti-6Al-4V metallic implants: In vitro cytotoxicity and speciation evaluation. Materials Science and Engineering: C, 96, 730-739. doi:https://doi.org/10.1016/j.msec.2018.11.090 Jakimska, A., Konieczka, P. S., & Namieśnik, N. (2011). Bioaccumulation of Metals in Tissues of Marine Animals, Part I: the Role and Impact of Heavy Metals on Organisms. Polish Journal of Environmental Studies, 20(5), 1117-1125. Retrieved December 29, 2020 Kurowski, K., & Boxer, R. W. (2008). Food Allergies: Detection and Management. American Family Physician, 77(12), 1678–1686. Retrieved from MayoClinic.org Miramand, P., Guary, J. C., & Fowler, S. W. (1981). Uptake, assimilation, and excretion of vanadium in the shrimp, lysmata seticaudata (risso), and the crab, carcinus maenas (L.). Journal of Experimental Marine Biology and Ecology, 49(2), 267-287. doi:https://doi.org/10.1016/0022-0981(81)90076-9 Orford, R. R., & Wilson, J. T. (1985). Epidemiologic and immunologic studies in processors of the king crab. American Journal of Industrial Medicine, 7(2), 155-169. doi:https://doi.org/10.1002/ajim.4700070207 Ruethers, T., Taki, A. C., Johnston, E. B., Nugraha, R., Le, T. T. K., Kalic, T., . . . Lopata, A. L. (2018). Seafood allergy: A comprehensive review of fish and shellfish allergens. Molecular Immunology, 100, 28-57. doi:https://doi.org/10.1016/j.molimm.2018.04.008 United States Food and Drug Ad,omostratopm. (2020, November 10). Analytoca; Results of the Total Diet Study. Retrieved December 30, 2020, from USFDA Science & Research (Food): https://www.fda.gov/food/total-diet-study/analytical-results-total-diet study.
