There is a growing medical interest in combining several agents and optimizing their dosing schedules in a single trial in order to optimize the treatment for patients. Evaluating at doses of several drugs and their scheduling in a single Phase I trial simultaneously possess a number of statistical challenges, and specialized methods to tackle these have been proposed in the literature. However, the uptake of these methods is slow and implementation examples of such advanced methods are still sparse to date. In this work, we share our experience of proposing a model‐based partial ordering continual reassessment method (POCRM) design for three‐dimensional dose‐finding in an oncology trial. In the trial, doses of two agents and the dosing schedule of one of them can be escalated/de‐escalated. We provide a step‐by‐step summary on how the POCRM design was implemented and communicated to the trial team. We proposed an approach to specify toxicity orderings and their a‐priori probabilities, and developed a number of visualization tools to communicate the statistical properties of the design. The design evaluation included both a comprehensive simulation study and considerations of the individual trial behavior. The study is now enrolling patients. We hope that sharing our experience of the successful implementation of an advanced design in practice that went through evaluations of several health authorities will facilitate a better uptake of more efficient methods in practice. The precision of the dose selection in the initial Phase I study has been shown to be decisive for the success of the complete development process.Another emerging field in early phase drug development is dose‐schedule clinical trials in which the safety and tolerability of a monotherapy administered using various schedules is studied. These also require specialized approaches to ensure that the target dose‐schedule regimen is identified accurately.Building on the ideas of combining various treatments and administering them under different dosing schedules, there is also an interest in combination‐schedule clinical trials.R‐packages,One of the reasons for the slow uptake of more efficient clinical trial designs in early phase trials was found to be the absence of guidance in the literature on how novel methods can be implemented in actual clinical trial settings.In this work, we share our recent experience of extending and implementing an advanced model‐based dose‐finding design in a three‐dimensional combination‐schedule oncology clinical trial with many possible combination‐schedules. We have extended the partial ordering continual reassessment methods (POCRMs) proposed by Wages et al2N), and a novel experimental agent M1774 (denoted by M), both administered orally once daily. There are two doses of niraparib, 200 mg (approved monotherapy dose) and 100 mg. There are five doses of M1774 that can be potentially investigated, 30, 60, 90, 130, and 180 mg. In addition to the five doses of M1774, two its administration schedules, S1 and S2, were defined. Schedule S1 refers to continuous once daily dosing of M1774, whereas S2 incorporates breaks from treatment and is approximately half as intensive as S1 over the dose‐limiting toxicity (DLT) evaluation window of 28 days. Therefore, the underlying dose‐finding problem is three‐dimensional: 5 doses of M1774, 2 schedules of M1774, 2 doses of niraparib, resulting in 20 combination‐schedules (to which we will refer to as to regimens) that could potentially be tried in the trial. The primary objective is to find the regimen that corresponds to a DLT risk in the target range of , and recommend it for further studies. The study has been opened for enrolment, with the first cohort of patients in the study being dosed at 200 mg of niraparib, 90 mg of M1774 using schedule S1, based on prior knowledge and assumptions about the expected toxicity.This work is based on an actual oncology clinical trial  conducted by Merck KGaA and for which the authors serve as statistical and clinical collaborators. The clinical trial will study the combination of an already approved agent niraparib .It is known prior to the trial that the toxicity of a regimen increases with the dose of each agent, and with the use of a more intensive schedule. However, the order of regimens that correspond to an increase (decrease) in doses and less (more) intensive dosing is unknown. For example, it is unknown prior to the trial whether a two‐fold increase in a single dose of M1774 for a half as intensive schedule of M1774 (with the dose of niraparib being unchanged) will result in an increase or decrease in the probability of a DLT; or whether a decrease in the dose of N while increasing the single dose of M1774 and using more intensive schedule will increase or decrease the risk of DLT. This results in a number of plausible ways of how the regimens could be ordered with respect to their monotonically increasing toxicity. We will refer to these as As a result, a model‐based design for monotherapies that relies on the monotonicity assumption of the dose‐toxicity curve could not be applied in this setting.We present a modified POCRM design below and then outline the specific arguments why it has been chosen for the considered trial.33.1R feasible orderings of the regimens satisfying the monotonicity assumption within each agent are considered. Let i be the index of the regimen, i=1,…,20, r be the index of ordering, r=1,…,R, πir be the standardized regimen level at regimen i under the ordering r, and let pir be the corresponding probability of a DLT. Then, the regimen‐toxicity model takes the formαr is the  model parameter under the ordering r that has a normal prior distribution, 𝒩, with the density function denoted by f0. The working models πir are constructed from standardized values, π˜i  by reordering them according to the order r.Assume that A and B, with two doses of each  resulting in four combinations. There are two possible orderings of these combinations if one assumes monotonicity within each agent. Assuming the skeleton π=, the two working models are given in Table As an illustration of reordering, consider a simple dual‐agent study of agents of each A,A2,B1,B2r, a working model is fitted using the Bayesian framework. Let nj(i) be the number of patients received regimen i after j cohorts have been enrolled in the study, and let yj(i) be the number of DLTs observed in these nj(i) patients. Here, the cohort is a small group of patients . Then, under ordering r′ after j cohorts have been enrolled in the trial, the likelihood is given byαr′ takes the formα^r′. Then, assuming that each of r=1,…,R considered orderings had a prior probability of being the true one equal to q01,…,q0r,…,q0R, the posterior probability of ordering r′ being the correct one after j cohorts have been enrolled in the trial is given byUnder each ordering r=1,2,…,R, and the ordering corresponding to the maximum probability of being the correct one, r⋆, is found. This most likely model is used to make dose‐escalation decisions for the next cohort of patients using a prespecified criterion. Using the parametric model and by using all available DLT information for the decision‐making, the model supports finding of the optimal dose of administration for the two treatments to trigger further development. We provide the proposed design based on the POCRM inference in Section The above inference is performed for the working model under each ordering 3.2A number of advanced dose‐escalation designs for the dual‐agent combination trials within the framework of a Bayesian logistic regression model (BLRM) approach have been proposed and applied in practice over the past decade.To deal with the problem of various schedules that allow relaxation of this assumption, a number of model‐based designs have been proposed recently.A natural extension of these approaches would be to use a model with as many parameters as the factors contributing to the toxicity and (possibly) their interaction(s), that is, modeling schedules and total average dose through additional covariates. While in the dual‐agent combination setting, this would typically result in 4 to 5 parameters,r=1,…,R but accounts for the uncertainty in the toxicity through the prespecified orderings. This flexibility allows it to be applied in both a two‐dimensional setting of dual‐agent combination, as originally proposed by Wages et alThe fundamental idea of the POCRM is that it benefits from the one‐parameter model within each of the considered orderings Furthermore, the clinician knowledge about the agents based on monotherapy experience with both niraparib and M1774 resulted in a decision for the first cohort of patients to receive a combination in the middle of the grid of regimens. The trial team expects that if a tolerable toxicity profile is observed with the starting regimen, many of the considered regimens and dose/schedule levels might not be tried in the trial at all . These de‐escalating regimens serve as backup if unexpected toxicities occurred. If certain schedules or doses will not be tested, then it would not be feasible to accurately estimate the corresponding model parameters. However, it is unknown prior to the start of the trial which doses/schedule levels will actually be tried. In other words, the modeling of the dose‐schedule combination toxicity relationship with respect to all potential drivers of toxicity was not the primary objective of the trial that the BLRM approaches are aimed at. Rather, the goal is to define a set of admissible dose‐schedule combinations that the SRC can select from for the next cohortWhile this manuscript addresses the specific trial setting, more generally, early phase clinical trials with more than one (monotherapy) or two  factors contributing to the overall toxicity become more widespread. Depending on the number of factors contributing to the overall toxicity, starting combinations and assumptions about how these factors relate to each other, each of subsequent trial may require setting up a new parametric model. Utilizing a flexible POCRM approach in this trial allows setting up an infrastructure, as a strategic resource, for application beyond this trial.These reasons together have served as the main motivation for opting for the POCRM for the considered trial rather than alternative approaches mentioned above. We specify below how this model was used in the trial design.3.3i is considered to be safe ifcoverdose is a probability constant controlling overdosing ;The total average dose per week or single dose of M1774 cannot be more than doubled compared to the current dose;If at least one DLT in the current cohort of patients is observed, the total average dose of M1774 or dose of niraparib cannot be increased.The POCRM model defined above will be used to drive escalation/de‐escalation subject to several restrictions that were not originally defined in the POCRM designThe first cohort is allocated to the starting regimen defined by the trial team.R orderings and finds the posterior distribution of αr under each of them. Then, the ordering corresponding to the highest posterior probability of the ordering being the true one, r⋆, is selected.After the DLT outcomes for the previous cohort of participants are evaluated, the POCRM fits a model under each of the r⋆ satisfying the escalation restrictions above are found.The set of admissible regimens under the most likely ordering i corresponding to the minimum ofγ is the middle of the target toxicity range, pir⋆ denotes the toxicity at regimen i under the most likely ordering r⋆, b is an asymmetry parameter, and the expectation is taken with respect to the posterior probability fj(αr⋆).Among the admissible regimens, the regimen Steps 2 to 4 are repeated until the maximum number of patients is reached or the trial is stopped for safety .The proposed design takes the following form:b allows to tackle this balance explicitly if there are some concerns in the escalation being too aggressive . At the same time, it is known that for a=2×γ, the criterion  when recommending the next regimen. Therefore, the primary role of the proposed design is to define the set of admissible regimens to which the next cohort of patients can be assigned, and to summarize and quantify the DLT risks at the prespecified regimens to the Safety Monitoring Committee (SMC). The final decision on de/escalation will be taken by the SMC and will also take into account the cause of DLT. This implies that the recommendation of the model is not binding and can be overruled by the SMC. This was explicitly mentioned in all discussions with the trial's team, SMC's members, protocol and statistical analysis plan. However, at the planning stage, it is crucial to understand how this can affect the properties of the design. We will investigate how the deviation of the recommended escalation/de‐escalation affects the operating characteristics in Section 4An integral part of relaxing the monotonicity assumption in the POCRM design is through prespecifying a set of orderings reflecting various assumptions on potential main drivers of toxicity in the regimens. Throughout the discussions with the clinical team, it was crucial to communicate that these orderings are the main mean to mitigate the risk of overly strong assumptions on the order of drivers of toxicity, and allow for the uncertainty in defining these drivers a‐priori. At the same time, it stresses that their choice is important. The prespecification of these toxicity orderings can have a noticeable impact on the performance of the POCRM design.4.1The first step of specifying the toxicity orderings is to understand all potential drivers of toxicity that can influence the toxicity outcomes in patients. The clinical team formulate that these are (i) dose of niraparib, (ii) total average dose of M1774, (iii) schedule of M1774, and (iv) the single dose of M1774. The second step was to present all possible regimens as a grid to support the visualization of all of the combination/schedules that could be tried in the trial to ensure aligned thinking between the clinical and statistical teams, and to formalize the monotonicity assumptions within each treatment and treatment schedule that are known to the trial team a‐priori. The toxicity assumptions can also incorporate any additional information on the relationships between the regimens that do not bear the uncertainty in the toxicity ordering according to the clinical judgment. It is important that all such assumptions of the certainty in the toxicity orderings are incorporated in trials relaxing monotonicity assumptions as it will results in gains in the accuracy and safety of the trial.In Table together are assumed to result in a larger increase in the toxicity than a decrease in a single dose of M1774. The same line of arguments were extrapolated for the rest of the grid.The latter is designed to reflect the clinicians' knowledge of how much each of the factors  contribute to the overall toxicity of the regimen. For example, comparing (14) and (18), the clinicians were confident that (18) is more toxic than (14) despite a decrease in the single dose of M1774 from 130 to 90 mg due to the increase in the total average dose of M1774 (from 455 to 630 mg) and change to a more intensive schedule. The latter two effects In Table 4.2The combination‐schedule grid in Table Dose of niraparib has the highest effect, followed by the schedule of M1774 , then the total average amount of M1774.Dose of niraparib has the highest effect, followed by the total average amount, followed by the schedule of M1774.Dose of niraparib has the highest effect up to some total average dose of M1774, followed by schedule of M1774 , followed by the total average amount of M1774.Following the discussion with clinicians, the assumption on the order of effect of each of the four factors, and following the guidance on the ordering specification by Wages and Conaway,N having the highest effect, followed by schedule of M1774 , then the total average amount of M1774.Additionally, to communicate different orderings according to various toxicity assumptions between regimens, each of them was presented as a figure displaying the increasing risk of toxicity under the made assumptions. An example of such visualization of the toxicity ordering is illustrated in Figure Within each of the assumptions, following the suggestion by Wages and Conaway,In the first iteration, fourteen orderings were provided to the clinicians, of which 10 were agreed as being plausible for the considered trial. For example, one of the orderings excluded from the consideration by the team is one of the “default” orderings proposed by Wages and Conaway4.3After the orderings were agreed, the POCRM requires to reflect a‐priori toxicity assumptions on how likely each of the prespecified orderings to occur during the conduct of the study. The default specification of the POCRM considered in the literature focuses on the equal a‐priori probability. However, given the clinicians knowledge about the two agents, it was agreed that not all of the considered orderings are equally likely and this should be reflected in the model specification.The specification of the POCRM requires to assign the probability to each of the ordering directly. However, eliciting the prior probability for each of the 10 orderings can be challenging given that each of the ordering consists of 20 regimens and some of them might differ only partly and it can be difficult to grasp all of the differences between various orderings,To tackle this, we have developed an approach of eliciting the probability ordering through the prior probabilities for pairs of anti‐diagonal regimens with uncertainty in the toxicity ordering. For example, regimen (12) and (16) that has the same total average dose of M and the same dose of N but less intensive schedule yet higher single dose of M1774. The pairs of regimens for which the prior probability of ordering was elicited and the prior probabilities provided by the clinicians are given in Table M. In contrast, comparing, (14) and (17), it is thought that there is an equal chance of one of these being more toxic than the other given that the total average dose of M1774 is slightly higher for (14) together with the single dose of M1774 but the schedule is less intensive. Similar prior probabilities were assumed to hold for 100 mg of niraparib, for example, regimen (2), (6) and (4), (7). Additionally, the clinical team argued that orderings 9 to 10 should have lower a‐priori probabilities compared to orderings 1 to 8, and the sum of the first 5 orderings (corresponding to the ordering where agent N is the dominant factor in toxicity) should be between 67% and 75% as they were quite confident that agent niraparib will be the main driver in toxicity. Note that the remaining probability of 25% to 33% safeguards from relying overly strongly on this a‐priori toxicity assumptions and enhances the identification of such orderings even if these are thought to be less unlikely.For example, considering (12) and (16), it is believed that there is 90% probability that (16) is more toxic given that schedule is thought to have a larger effect on toxicity than the single dose of Given these constraints, the prior probability of each complete ordering implying these prior probabilities on ordering on pair of regimens was found by grid search. The prior probabilities of ordering resulting in the lowest sum of squares of the elicited and computed prior probabilities were found and are provided in the Supplementary Materials. The prior probabilities on the pairs of anti‐diagonal regimens implied by these orderings' probabilities are given in the last column of Table 5In this section, we finalize the design parameters required for the considered trial and evaluate the performance of the proposed design in individual escalation paths.5.1The proposed POCRM design requires a number of parameters to be specified additional to the prespecified orderings. Some of these parameters are chosen through discussions with clinicians and others through statistical considerations and tuning via simulations. We consider the parameter that were specified from discussions with the clinicians below and the parameter obtained via tuning in Section γ=0.275 was used for the criterion b=2γ approximately results in the standard squared distance criterion, a slightly higher value of b=0.60 (but still below the double of the upper toxicity bound 2×0.35=0.70) was used to explore the combination‐schedule grid more thoroughly as higher values of b were found to result in higher accuracy.c=3 patients is used, and the sample size is to be defined by simulations but maximum sample sizes between 42 and 54 are to be considered. The trial will start at the regimen (18).The target toxicity range was specified as 20% to 35%, and the target toxicity level lying in the middle of this interval 5.2π˜i, prior mean μ, prior variance σ2, and the overdosing threshold coverdose. After the prior clinical knowledge about the agents was included via the specified probabilities of each ordering, these were selected via the grid search by conducting extensive simulations for various combinations of values of these parameters and searching for the one yielding high accuracy.ν, between the neighboring combinations π˜i=0.01+ν×i where i=0,…,17 and the last two values  and (20)) were tuned separately from the escalation conditions below. The following grids were used: μ={0,0.50,0.55,0.80,1,1.25,1.40,0.15,1.75}, σ2={0.40,0.45,0.50,0.75,1.0,1.25,1.34,2.0}, coverdose={0.25,0.30,0.50}, ν={0.025,0.03,0.035,0.04,0.0425}.The proposed design requires specification of several model‐specific parameters including the skeleton While this was agreed that the design parameters could be chosen from the statistical considerations, that is, from the accuracy and safety in the simulation study, it was essential that the resulting parameters yield intuitive dose‐escalation/de‐escalation decisions for the first three cohorts of patients. Specifically, using the search over the grid specified above, any combination of values within this grid is excluded if they do not lead to the following escalation/de‐escalation decisions for the first cohort of 3 patients. Specifically, after 0 DLTs, it was required that (19) is the next safe regimen and (20) is yet inadmissible, and after 1 DLT on the starting regimen, regimens 14) and (17) are in the admissible set while the starting regimen (18) is not. This additional step was taken to overcome the “black‐box” concern. Then, for these combinations of parameter value, a simulation study across various scenarios was conducted using 1000 simulations for each scenario. Then, the design parameters resulting in the highest geometric mean in the proportion of correct regimen selection is taken forward for further evaluation. The list of the design parameters tried are given in Table  and (17)To choose scenarios for calibration, one can opt for a limited number (around four) of qualitatively different scenarios with the optimal regimens being located in various parts of the grid.μ=1.50, prior variance, σ2=1.50, overdose threshold coverdose=0.25, and the skeleton values The set 4) in Table  in Table5.3An important aspect of a model‐based design is that its escalation and de‐escalation recommendations are aligned with the clinician's knowledge and expectations. To communicate the model's recommendations, Yap et alWhile the original DTP and its implementation⋆), the later version of the tree has also included the admissible set to illustrate the choice the SRC will have at the given step of the trial. At the same time, as within the given path, there might be several admissible regimens, plotting a tree for each case will make the decision‐tree infeasible to read and communicate. Therefore, within each path, the regimen recommended using the model's criterion Importantly, the first version of the decision‐tree included only one regimen per box as in the original DTP proposal, that is, the regimen recommended by the model according to the criterion 6) and subject to the safety constraint. However, the primary use of the proposed model is not only to recommend the single regimen for the next cohort but to define the set of admissible regimens that the SRC can select from for the next cohort. Therefore, such a tree was agreed not to convey the required message. Additionally, to the recommended regimen (marked with a  and subjThe decision‐tree starts from the regimen (18). If no DLT is observed in the first cohort of 3 patients, the set of admissible regimens is (15) and (19) that correspond to an increase in the dose of M. Among these two, the regimen (19) would be recommended by the model. Note that, as required, the regimen (20) is not in the admissible set. We see that it is only possible to escalate to regimen (20) if no DLTs are also observed for the second cohort of patients.If 1 DLT is observed for the first cohort, the regimens (14) and (17) are in the admissible set of regimens together with lower regimens, and the model recommends de‐escalating to regimen (14). If no DLT is observed at this regimen, the model will recommend to escalate back to the starting regimen (18).If 2 DLTs is observed in the first cohort, there is only one regimen with 200 mg of the agent niraparib which is safe to administer. Finally, if all 3 patients in the first cohort experience the DLT, there are no regimens with 200 mg of niraparib that are deemed safe and it is recommended to de‐escalate the total average dose of M1774.This tree was presented to the clinical team and it was agreed to satisfy the required conditions for the first cohort and lead to intuitive escalation/de‐escalation decisions for the subsequent cohorts.Additionally to the decision‐tree that presents only the recommended and admissible set of regimens, an example of the output with the characteristics of all 20 regimens was presented for some of the branches of the tree. This included (i) a table  and (ii) a figure illustrating the probabilities of overdosing, underdosing, and being in the target range. These served as one more check that the model's recommendations are aligned with the clinicians beliefs and knowledge. This, again, would resolve the “black box” concerns about the properties of the proposed model. Additionally, this also served as a tool to initiate the process of refining the details that should be included in the actual analysis during running of the trial. We refer the reader to the Supplementary Materials for the example output and the description of characteristics of the regimens that were provided.6Once it was agreed that the proposed design results in intuitive escalation/de‐escalation decisions for the first cohorts in the trial, the proposed design was extensively evaluated via a simulation study to ensure that it results has good operating characteristics in a variety of possible scenarios throughout the whole trial.6.1n=42, 48, 54, were investigated to make the decision on the sample size .The simulation study has used the setting of the considered trial targeting the toxicity interval of  with the cohort size of 3 patients, and using the calibrated design parameters from above. Three maximum sample sizes, As mentioned above, based on the observed data, the model recommends a subset of admissible regimens with acceptable toxicity and provides a non‐binding suggestion for the next regimen for the next cohort from the estimated set of acceptable regimens. However, the clinical team may choose a different regimen than suggested by the model. This, however, will clearly affect the operating characteristics. Therefore, we consider two options of the allocation in our simulations.1/δi. This mimics that, on average, the regimens with the target toxicity closer to the target of 27.5% are more likely to be allocated . We will refer to it as the “within admissible set.”The design under the first considered allocation criterion strictly follows criterion . We refeoptimal regimen selection—the regimen corresponding to the toxicity risk closest to the middle of the target interval, that is, 27.5%.Probability of correct regimen selection corresponding to 20% to 35% DLT risk.Probability of >35% DLT risk) regimen selection.Probability of overly toxic  is(are) located and the first figure in the scenario number corresponds to the diagonal the optimal regimen(s) is (are). For example, under scenario 1, the optimal regimen is the highest regimen. Under scenarios 2, the optimal regimen(s) is on the second to the right anti‐diagonal. Specifically, under scenario 2.1 there are two target regimens—(19) and (15), under scenarios 2.2—only regimen (19) is the target, and under scenario 2.3—only regimen (15) is the target. Following the same structure, other scenarios were constructed by moving the diagonal with the optimal doses across the grid up to scenario 8, in which the optimal dose is the lowest regimen. While the scenarios were initially constructed by the statisticians, they were then reviewed by the clinical team to ensure that they indeed reflect possible regimen‐toxicity relationships.6.3We will compare the performance of the proposed POCRM design to two competing approaches. The first one is a conventional rule‐based 3+3 design . Although, there is a long record of studies showing the inferiority of the 3+3 design compared to more statistically sound approaches,The proposed study is a dual‐agent combination trial while the 3+3 design is a method for single‐agent studies. To apply 3+3 design to the proposed trial, one needs to assume a single monotonic dose‐toxicity ordering to mimic the setting of a single‐agent trial. In the comparison below, we assume the monotonic ordering with respect to the combination numbers.…, 20 due to computational purposes. The operating characteristics of the 3+3 design are found analytically (rather than by simulations) by computing all possible paths the trial can take. With more than 9 combinations, this computation can become infeasible. Note that this assumption favors the 3+3 design in the comparison as it considers fewer combinations in the trial than the proposed design. These nine combinations are selected such that they correspond to the 200 mg of the N compound that is a more clinically interesting (and the approved) dosage of the agent. The 3+3 design is implemented using the bcrm‐package.Furthermore, for the 3+3 design, we will limit consideration of combinations to 9 combination/schedules: 11, 12, The second competing approach, as suggested by a reviewer, is a variant of the proposed POCRM that instead of choosing one, the most likely model as originally proposed by Wages et al,It worth noting that this approach was not originally chosen for the motivating trial mainly for two reasons. First, it is more computationally expensive to fine‐tune and evaluate in a timely manner (that was an important consideration at the time of the planning of the study): given the 10 orderings, the computational costs are 10‐fold higher for this design. Second, while under the proposed approach one might select an ordering that is slightly more likely than others, the main objective of the design is to provide the SRC with the set of admissible combinations rather than a single recommendation. Hence, it was thought that the effect of choosing different model will have a modest effect in practice.6.4n=48 is given in Table n=42, 54 being reported in the Supplementary Materials).The summary of the performance of the proposed design in the 20 considered scenarios based on 2000 simulations under the sample size M is the optimal one but for 100 mg of N. This happens due to the selected prior probabilities of orderings—the orderings that are close to this scenarios (orderings 9 and 10 where the toxicity ordering goes throughout the grid) have low a‐priori probability, therefore, the considered sample size might not be always enough to overrule it. The same explains the low proportion of the optimal selections under this scenario. Hence, the likelihood of these scenarios should be considered when interpreting the results.Overall, the proposed design under the escalation criterion for the allocation of patients (“next best”) has good accuracy properties—the average proportion of the optimal regimen selections is 39%, and the average proportion of acceptable selections is 68%. At the same time, the proportion of overly toxic selections is small—21%, on average—with proportion being below 25% in the half of all scenarios. Higher probabilities of making an overtoxic recommendations correspond to more toxic scenarios 4.3 to 8. At the same time, a particularly high proportion of overly toxic selections can be observed under scenarios 2.3 for which the highest dose of The average proportion of DLTs is below 33% under 18 out of 20 scenarios with the average proportion of DLTS being 28%. This corresponds to approximately middle of the target toxicity range, and is hence acceptable.The allocation rule within the admissible set decreases the proportion of overly toxic selections for the same sample size due to a better exploration of the grid . Specifically, for the considered sample size, it allows to decrease the mean proportion of overly toxic selection from 21% to 18% and to increase the proportion of scenarios with overly toxic selection below 25% from 50% to 75%. Furthermore, the proportion of overly toxic selections under scenarios 2.3 drops from 33% to 19%. Importantly, the proportion of acceptable selections remains unchanged—68% with the price for not selecting the next best after each cohort being in the lower proportion of optimal selections from 39% to 36%. Similar patterns can be observed for other sample sizes .Comparing the proposed design to the 3+3 design, as expected, the 3+3 design results in noticeably lower proportion of optimal and correct regimens identifications. The probability to select the optimal combination is 15% lower, and the probability to select the correct combination is 21% lower compared to the proposed design under the “next best” allocation rule. The proportion of scenarios under which the optimal/correct selection is above 40%/60% drops to 15%/35% compared to 55% and 75%, respectively, under the “next best” allocation. Furthermore, the 3+3 design results, on average, in noticeably higher probability of recommending unsafe combination, 31% vs 21% for the POCRM, and 9% higher average proportion of patients experiencing a DLT. In 60% of scenarios this proportion is above 25%. This means that the 3+3 design results is less accurate and does not safeguard the patients to the same extent as the proposed design. However, it is important to mention that the 3+3 design results in noticeably lower expected number of patients than the proposed model‐based design due the early stopping rule—the average sample sizes ranges between 12 and 21. However, more toxic scenarios correspond to higher expected sample sizes. Furthermore, it worth reiterating that the maximum sample size for the 3+3 design is 54 patients, higher than for the proposed one, and this is the number of patients that a trial should be costed for were one to follow the 3+3 design.Comparing the proposed design to the POCRM with model averaging, it is found that the model averaging approach tends to result in lower proportion of optimal selections (by 4%‐5%), similar proportion of correct selection (within 2%), and smaller proportion of overtoxic selections (6%‐7%). These conclusions are consistent across both allocation rules. This makes the POCRM with model averaging a viable alternative to the proposed design. Importantly, one could attempt to improve the operating characteristics of the model averaging approach as the parameters calibrated for the original proposal are used. However, this will be computationally expensive.n=48 was made given that the advantage of the six extra patients  were limited in terms of the accuracy.Overall, the proposed design was concluded to be able to identify the optimal and correct regimens with high probability. Given the trade‐off in the accuracy, safety, and costs, the decision to fix the sample sizes at 7Accurate identification of the optimal doses and schedules in early phase trials that balance safety and efficacy is of paramount importance for the whole drug development path. To find the optimal combinations of doses of several drugs and how these should be administered to the patients, various dosing and schedules parameters should be studied simultaneously as assuming one of them being fixed can result in suboptimal recommendations. However, this poses a number of challenges, both statistical and clinical. We have proposed an extension of the POCRM design and provided a step‐by‐step summary of how it was implemented in practice. The proposed design supports the decision‐making process during dose‐escalation and relies on a parametric model that allows borrowing of information across all regimens. As the result, it increases the likelihood of finding the dose with optimal benefit while safeguarding the patients by mitigating the risk of allocating patients to ineffective or overly toxic doses.We have also proposed a novel approach to defining orderings and their prior probabilities to inform the POCRM design, utilizing the a‐priori clinical knowledge. This approach is more straightforward to communicate to the trial team than the alternative of specifying the probability of each complete ordering directly and ensures that the design reflects clinically plausible scenarios.The proposed POCRM can incorporate prior clinical knowledge through the probability of each ordering. This approach is different to how the prior information is leveraged in the BLRM‐type approaches, which typically formally constructs a prior distribution on the model parameters themselves. At the same, being a model‐based approach too, the POCRM design can use an informative prior distribution on the model parameter. However, the one‐parameter working model used in the POCRM might not be flexible enough to achieve a good approximation of the prior toxicity estimates and sufficiently different level of uncertainty at various dose‐schedule combination. Moreover, under the proposed design of choosing the most likely ordering, this could be challenging as well, as changing the order will change how the prior information is included. The model‐average POCRM alternative might be a more natural way of achieving this if one would consider inclusion of in a similar to the BLRM‐type approaches way.A number of tools, the DTP, probability grids, and complete output in several escalation paths that facilitate the communication of the properties of the designs have been developed. While simulations are an integral tool for understanding the statistical properties of the design, we stress that they should be considered together with the individual trial behavior and the investigation of the design should not be limited to simulations only. Giving more consideration to individual trial behavior is thought to resolve one of the concerns associated with model‐based designs, them being a “black box.”Importantly, an implementation of such an advanced design in practice requires not only support by experienced statisticians but also a close collaboration with the whole trial team. For example, it took multiple iterations and meetings to agree on the plausible toxicity orderings, to define the corresponding prior probabilities, to agree on the scenarios for the simulations study, and define the operating characteristics that are of interest to the team. The developed supporting visualizations were the key in these meetings.Finally, there is an increasing acceptance of more efficient statistical approaches in early phase clinical trials by pharmaceutical companies and academic institutions. The proposed design is currently being implemented in a clinical trial enrolment to which has commenced and has been reviewed by the FDA and MHRA. While implementing such a design does take resources, the use of more efficient methods earlier in the drug development path is expected to pay off later in development. Sharing such examples as presented in this work will contribute to improving the uptake of more efficient methods for advanced dose‐escalation trials.Thomas Goddemeier and Anja Victor are employees of Merck Healthcare KGaA, Darmstadt, Germany. Marianna Grinberg was an employee of Merck Healthcare KGaA, Darmstadt, Germany, at the time the reported work was conducted, and Marianna Grinberg is currently employee of UCB, Statistical Sciences and Innovation Division, Monheim, Germany. Marianna Grinberg and Anja Victor hold stocks of Merck Healthcare KGaA. Ioannis Gounaris is an employee of Merck Serono Ltd., Feltham, UK, an affiliate of Merck KGaA. Pavel Mozgunov and Thomas Jaki served as statistical consultants for Merck Healthcare KGaA.Table S1 Operating characteristics of the proposed design under 3 sample sizes, n=42, 48, 54, and under two allocation rulesTables S2 to S4 Example of the output provided to the SRC after 0, 1, and 2 DLTs were observed in the first cohort of 3 patients on the starting regimenFigure S1 Example of the visualization of the probability of risk of toxicity for each combination‐schedule being in the underdosing, target, and overdosing interval used the grid representation of the regimensClick here for additional data file.


Highly reactive thermosets are currently expanding the processability of high-performance structures for transportation industry. The short polymerization time makes it a suitable process to replace metallic structures with polymer matrix-based composite materials. The resin characterization is a fundamental step to obtain the properties and the associated constitutive models, which are required to design and optimize the manufacturing process parameters of composite materials. However, the short time on polymerization requires to use the characterization equipment at their performance capability limits. This work presents a comprehensive methodology to characterize the thermo-chemical properties of highly reactive resin systems, which are relevant for resin impregnation into the preform for liquid injection processes. Four different commercial resin systems are analyzed in this study. Experimental methodologies are analyzed and adapted for best data acquisition at high temperature isothermals. Based on the experimental data, Cure kinetics and viscosity equation-based models are used to describe the behaviour of these complex resin systems. Processing maps are developed based on the cure kinetics and viscosity models to predict the processability time for specific process conditions than can be used on liquid injection moulding processes. One of these areas is the vehicle weight reduction, which is an important improvement toward higher fuel economy and electric battery range. A lower vehicle mass requires less energy during acceleration due to lower inertial forces. A general rule is that for every 10% reduction in vehicle weight, the fuel consumption of vehicle is reduced by 5%–7 %.The current global environmental situation requires a greater effort to reduce greenhouse emissions. New challenges for engineers are related to the efficient use of energy and materials. Cheah analyzed the different areas of enhancement to the vehicle design in order to reach these fuel economy targets. The main factors restricting the development of polymer composites in vehicles today are the long production cycle and the cost of fibres.Composites are extensively known as efficient materials due their high specific strength and rigidity. Plastics and polymers composites currently make up about 8% of a vehicle by weight and 50% by volume, and these numbers are expected to increase over time. Highly reactive thermosets are good candidates to shorten the production cycle and accomplish the vehicle production requirement. Manufacturing processes for composite structures in transportation applications should be economic, versatile, and fast. One process with high production volume rate and that is affordable is Compression Resin Transfer Moulding (CRTM). Khoun and Trudeau (2019) describes the implementation of highly reactive thermosets for CRTM. The process consists of five stages, with an optimal duration of 2 min as seen on Innovative resin systems require shorter cure cycles (2–5 min) compared with traditional resins (1–2 h).8 Transportatioin industry requires a high rate production, thus resin systems that cures in seconds to minutes are the most suitable.The overall quality of the part made from liquid injection moulding process depends on the impregnation of the preform during the injection. The viscosity of the resin is a fundamental property that drives the impregnation of the preform. The viscosity is strongly dependent to the degree of cure and temperature. As the processing time is considerably shorten, the characterization techniques need to be adapted for the acquisition of the data in a very short time. A deep comprehensive understanding of the resin processing properties, their constitutive models and the interaction involved is required to define and optimize the parameters of the manufacturing process. Moreover, the high reactivity of the thermosets requires a promptly acquisition of the data from the beginning of the test, which can be challenging because the reaction can occur during the preparation of the sample and setting of the equipment .10 that is used for further research and process simulation implementation.12 Different semi-empirial models have been developed to describe the behaviour of cure kinetics18 and viscosity.22 The material models used on this manuscript are intended to be simple and accurate. This would facilitate the implementation for process modelling. Four different systems were studied with different techniques and material models from literature.The current characterization techniques are discussed in the methodology section. Improvements on the settings for the differential scanning calorimeter (DSC) and rheometer are discussed to measure the properties of the resin during the highest reaction rates. The implementation of existing material models for the characterization of thermosets has been a source of valuable material data,The key processing properties during the impregnation of the preform is the viscosity and the cure kinetics. A sequence of characterization steps was performed to obtain these properties. First, a thermogravimetric analyser was used to determine the temperature of degradation and the thermal stability of the resin systems. The curing behaviour was then investigated with a differential scanning calorimeter (DSC) to obtain the evolution of the degree of cure, α, as a function of time and temperature. Finally, the viscosity, µ, was measured with a rheometer, and was described as a function of degree of cure, the degree of cure at the gel point, and the temperature.Based on the high reactivity and processability time, three different commercial thermosets were analyzed . One sysThe thermal stability tests were carried out on a thermal gravimetric analyser Q500 from TA Instruments. A temperature ramp at 20°C/min from room temperature to 700°C was applied under air condition. A weight percentage reduction of 5% is observed on Airstone system on the range of the processing temperature of 100°C–140°C as seen on Two different equipment were used with different methodology for the resin systems. The first equipment was a modulated differential scanning calorimeter (MDSC) Q100 from TA Instruments for the Hexion and Airstone resin systems. Dynamic scans were performed at a heating rate of 10°C/min, from −50°C to 250°C. Isothermal scans were performed at 80°C, 90°C, 100°C, and 110°C for both systems. The second equipment was a double-furnace differential scanning calorimeter (DSC) 8500 from Perkin Elmer that was used for the Gurit Standard and Fast resin systems. Dynamic scans were performed for these resins under the same conditions of a heating rate of 10°C/min, from −50°C to 250°C. Isothermal scans were performed at 70°C, 80°C, and 90°C for Gurit Standard. Gurit Fast was tested with isothermals scans of 50°C, 60°C, 70°C, and 80°C.Highly reactive thermosets present a challenge when characterizing at high isothermal temperatures. The first methodology was implemented on the Q100 DSC, which consist of placing the pan with the uncured resin into the DSC cell at a desired initial temperature. The cell is opened, the sample pan is placed in the cell, then the cell is closed, and the DSC starts acquiring the data. However, the reaction starts while placing the pan and closing the cell. Initial reaction is not monitored, and the data acquire is not fully reliable. The data lost rises as the temperature is higher.The second methodology was implemented on the Perkin Elmer DSC, where the pan is placed at room temperature. Then the cell is closed, and no reaction is expected during this process. The cell is then heated at the maximum rate of the DSC or 300°C/min. In this method the data is recorded at the beginning of the ramp. In this case there is not lost on data, but the data acquired during the ramp to reach the isothermal temperature is not properly measured. Due to rapid temperature fluctuations, there is a possibility of a time lag or thermal inertia causing a disparity between the furnace temperature and the sample temperature. This leads to uncertainty and errors that increase as the temperature is higher. and the strains used are shown in The resin rheological behaviour was analyzed using two types of rheometer. The Hexion and the Airstone resin systems were characterized using AR2000 rheometer from TA instruments. The Anton Paar CTD 600 was used to characterize the Gurit standard and fast resin systems. Both machines used an environmental test chamber and the complex viscosity of the resin was measured in both dynamic and isothermal conditions.  Large oscillation strains are required at the start of the tests to obtain good signals at low viscosity. However, with the cure progression of the resin, there is an increase in the storage (G′) and the loss moduli (G″). With this increase, only a small oscillation strain should be used to maintain torque within the capabilities of the instrument. Therefore, to obtain a noise free data over the course of the test, a good oscillation strain control is necessary.The resin systems studied in this work reach minimum viscosity very fast followed by a steep rise is viscosity which is typical of a fast curing epoxy resin. This technique advices user not to exceed strain beyond LVR. This is ideal for slow and medium cure resin systems. But the resins tested here cure extremely fast and the properties of the resin (G′ and G″) evolve rapidly. As a result, this technique produces noisy data at low viscosity. However, if we use high strains (beyond LVR) at initial stages until the viscosity starts increasing and reduce the strain as the time progresses to get within the LVR range, we can eliminate or minimize noisy data and get a fine curve. G′, G″ and the gel time data for both tests.TA instruments uses a method called “Auto-strain” which controls the oscillation strain throughout the cure of the sample.Ht is the total specific heat of reaction of the resin. T is 410 ± 3 J/g for a replicated experiment.The results from the dynamic and isothermal tests are converted into cure rate as the rate of the reaction, equation (1)dαdtThe degree of cure, equation .(2)α=114 or bi-reactive, where two reactions are assumed to occur, equation  versus 1/T at a low degree of cure range . An average of 61.29 kJ/mol was found for the activation energy of the Gurit Fast resin system, as seen in The first step is to determination of the activation energy n, were calculated using a least squares nonlinear regression between the cure rate and the degree of cure for all the isothermal temperatures. An R2 fitting value is expected to be higher than 0.95 to have a good agreement with the experimental data. equation (1/K)n=n0+nmT21 Equation  from from21 EEquation  can be eAµ and Eµ from equation  solution method. Constants calculated for all the resin systems are tabulated in Linear regression was used to calculate the constants equation , using d2 value greater than 0.80 for all the resin systems. However, capturing data at high temperatures becomes extremely difficult due to the fast curing nature of the resin systems. The resin would have undergone significant curing reaction by the time the resin is placed between the preheated parallel plates and the sample trimmed to the desired gap. The maximum temperature up to which the isothermals where performed was 120°C. The models developed can also be extended to the dynamic data as shown in Large scale production of composite materials for the transportation industry is mainly constrained by the lack of knowledge on the material processing. CRTM is one of the most suitable processes for an affordable and quick processing of composite materials for large production. Impregnation plays an important role during the processing of composite materials. The overall quality of the final composite part strongly depends on the impregnation of the preform. The resin viscosity plays an important role during impregnation, and the viscosity is directly linked to the cure kinetics evolution of the resin. The work performed on this manuscript focuses on the resin properties and behaviour during CRTM process. The characterization planning consists of 3 stages: I) Thermal stability, II) Cure kinetics, III) and Rheological behaviour.Arrhenius based cure kinetics models were implemented accounting for diffusion. It was observed that the parameters of these models remain constant below a critical temperature. Isothermal tests were performed above the critical temperature, which shown that the parameters increase linearly. A logistic function was defined to account for this behaviour.Castro-Macosko model was used to capture the viscosity behavior of all the fast curing resins studied in this work. The model is simple and can be used to predict the processing window necessary for CRTM applications.The cure kinetics and viscosity models can be used to produce time versus temperature graphs as seen on The material models developed here can be implemented in state-of-the-art finite element tools like COMSOL Multiphysics, PAM RTM or COMPRO, RAVEN. These models can be used to solve coupled flow, heat transfer and cure processing problems, to optimize the CRTM process parameters. These models can also be used to solve thermochemical-stress processing problems encountered while making a composite part using CRTM. The detailed procedure and methods explained in this paper can be applied for characterization of wide range of fast curing resin systems.


Although the cost-reference particle filter (CRPF) has a good advantage in solving the state estimation problem with unknown noise statistical characteristics, its estimation accuracy is still affected by the lack of particle diversity and sensitivity to the particles’ initial value. In order to solve these problems of the CRPF, this paper proposed an intelligent cost-reference particle filter algorithm based on multi-population cooperation. A multi-population cooperative resampling strategy based on ring structure was designed. The particles were divided into multiple independent populations upon initialization, and each population generated particles with a different initial distribution. The particles in each population were divided into three different particle sets with high, medium and low weights by the golden section ratio according to the weight. The particle sets with high and medium weights were retained. Then, a cooperative strategy based on Gaussian mutation was designed to resample the low-weight particle set of each population. The high-weight particles of the previous population in the ring structure were randomly selected for Gaussian mutation to replace the low-weight particles in the current population. The low-weight particles of all populations were resampled in turn. The simulation results show that the intelligent CRPF based on multi-population cooperation proposed in this paper can reduce the sensitivity of the CRPF to the particles’ initial value and improve the particle diversity in resampling. Compared with the general CRPF and intelligent CRPF with adaptive MH resampling (MH-CRPF), the RMSE and MAE of the proposed method are lower. State estimation has interested scholars around the world for a long time. It is an important problem in the fields of parameter detection, automatic control, fault diagnosis and navigation guidance ,2,3. A pAt present, many scholars have conducted a significant amount of research and proposed some improved resampling algorithms to solve the problem of lack of particle diversity. The paper  proposedTherefore, this paper proposes an intelligent resampling method based on multi-population cooperation, which improves the previous resampling method and further improves the particle diversity. The new resampling method is applied in the cost-reference particle filter (CRPF) to solve the problem of lack of particle diversity and sensitivity to the particles’ initial value in the CRPF, which is good at estimating the system state when the statistical characteristics of noise are unknown. Firstly, the particles are divided into several independent populations, and each population performs importance sampling using the distribution of different initial values. Then, the particles of each population are sorted according to the weight from large to small, and the golden section ratio is used to divide the particle set into three parts: high, medium and low weight; while retaining the set of particles with high and medium weights, Gaussian mutation is performed on the particles with high weights with a certain probability. By using the cooperative strategy, the low weight particles in the current population are replaced by the high weight particles after mutation in the previous population, so as to realize the resampling of the low weight particle set in turn. In this paper, a new method is proposed to realize information sharing among populations through the ring coordination mechanism, which makes the particles in the population closer to the posterior probability density of the state. The superimposed Gaussian mutation effectively increases the diversity of the particles in the resampling process. The final state estimate is obtained according to all particles and their corresponding weights. Finally, the effectiveness of the proposed method was verified by common one-dimensional and multidimensional models.k. Suppose the nonlinear discrete system model is described as follows:i-th particle k is defined as follows:Unlike the traditional particle filter, the CRPF does not need to know the statistical characteristics of noise when estimating the state. Instead, it proposes a cost function and a risk function, and finally calculates the weight of the particle by them. The cost function of the i-th particle at time k is defined as follows:In the formula, According to Equations –5), whe, whe5), According to the above steps of the CRPF algorithm, the process noise and measurement noise of the system are not used in the whole process of system state estimation. Instead, it is achieved through the cost function and risk function defined by the CRPF algorithm. Therefore, compared with the conventional particle filter, the CRPF has a significant advantage in solving the state estimation problem with unknown noise statistical properties.Algorithm 1 The algorithm steps for the standard CRPF.(1)k = 0(1)N particles are extracted from the initial distribution (2)The initial particle set is Initialize (2)(1)The risk function value of the particle is calculated by Formula ;(2)The weight of the particle is calculated from the risk function value according to the following formula:(3)Polynomial resampling is performed on the particle according to the weight value, and the particle set after resampling is Resample(3)(1)Carry out particle update according to the following formula:(2)Update particle cost by Formula  and 4);;4);(3)Calculate the updated particle weight according to the following formula:Update(4)State estimationThe estimated system status isAlthough the CRPF has a good filtering effect when the statistical characteristics of noise are unknown, it still uses the traditional resampling method. It copies high-weight particles and discards low-weight particles, so the problem of lack of particle diversity also exists in the CRPF. When the particles with high weight are copied and most particles with small weight are discarded, this leads to the result of state estimation deviating from the real state, due to the entire set of particles being unable to completely cover the posterior distribution, as shown in In order to solve the problem of low estimation accuracy due to the lack of particle diversity in the CRPF when the initial value is not accurate, an intelligent cooperative resampling mechanism based on ring structure is proposed in this paper. The structure of the intelligent cooperative resampling mechanism is shown in Different from the traditional resampling methods used in the CRPF, new resampling mechanisms are proposed in this paper. Instead of simply copying the high-weight particles to replace the low-weight ones, this method adopts an intelligent cooperative strategy to improve the diversity of the particles. Compared with the previous intelligent resampling methods, the intelligent cooperative resampling mechanism adopts a multi-population ring cooperative structure. Each population uses the importance distribution of different initial values to generate particles, which reduces the sensitivity of the particle filter to the initial values. In addition, the collaborative strategy was used to exchange high-weight particle information between different populations, which improved the information sharing ability of the particles and effectively improved the diversity of the particles. At the same time, the traditional resampling step is canceled to reduce the complexity of the algorithm.Since the particles in the high weight and medium weight particle sets have large weights and are all effective particles, these sets must be retained when resampling the particles. Only the low weight particles in each population are resampled by the collaborative strategy. This can not only improve the efficiency of resampling, but also increase the diversity of the particles. However, the traditional resampling method can no longer be used for cooperative operation. Copying particles with high weight and directly replacing particles with low weight will cause the same particles to be copied in large numbers, resulting in the serious loss of particle diversity and the reduction in particle filtering accuracy.In this paper, a collaborative strategy based on Gaussian mutation is designed to resample low-weight particles in each population, avoiding blindly copying high-weight particles to improve the diversity of the particles. This strategy is based on the ring structure of Gaussian mutation is calculated as follows:q are the mutation rate.In Formula , xksls iq is set appropriately, the particle with Gaussian mutation must be located near the particle with high weight. Different from the existing high-weight particles, this method can effectively improve the diversity of the particles and ensure the quality of the whole set of particles. At the same time, q should not be too large; otherwise, the newly generated particles will be far away from the high probability region, and then reduce the accuracy of filtering. The organic integration of cooperative strategy and ring structure ensures that the information of high-weight particles generated by different importance distributions of each population is shared. After several iterations, the particles generated by the importance distribution far away from the true posterior distribution are gradually abandoned, and more and more particles close to the posterior distribution are retained. This strategy effectively improves the diversity of the particles, thereby reducing the impact of inaccurate initial value settings on the state estimation results.According to the above equation, when Algorithm 2 Steps of Intelligent CRPF Algorithm Based on Multi-Population Cooperation.(1)(1)Set the population number M, the particle number N of each population and the uniform distribution parameter (2)Randomly extract the initial particle of each population from the uniform distribution Initialize(2)(1)The risk function  Formula , and the Formula ;(2)Arrange the particles in each population according to the weight from large to small, and retain (3)Resampling the particle  Formula  and Sect Formula ;(4)The high-weight particle set Resample(3)(1)The resampled particles are updated according to Formula , the updFormulas  and 4)  Ckis is  Formula ;(2)Calculate the estimated value  Formula ;(3)Finally, calculate the final state estimate of the system according to the estimated value UpdateThe multi-population cooperative intelligent resampling proposed above is introduced into the CRPF algorithm to obtain the intelligent CRPF based on multi-population cooperation. The algorithm steps are as Algorithm 2.K represents the number of repeated experiments, L is the length of time series, k in the i-th simulation.In order to verify the effectiveness of the proposed method, three simulations were designed. The standard CRPF and the adaptive MH resampling-CRPF  are used as comparison algorithms, and each experiment was independently repeated 100 times. The three simulation models are the mathematical model of one-dimensional non-stationary economic growth ,18, the The one-dimensional non-stationary economic growth mathematical model has the characteristics of being strong nonlinear and bimodal, and it is difficult to estimate its system state, so the model is widely used to verify the effectiveness of the particle filter. Its model is described as follows:In this simulation, The one-dimensional nonlinear univariate time series model is also widely used to verify the effectiveness of the particle filter due to its strong nonlinearity. The model is described as follows:In this simulation, k is the number of cycles of a lithium–ion battery, Due to the high-dimensional, nonlinear and non-Gaussian characteristics of the remaining useful life prediction model of a lithium–ion battery, the nonlinear filtering algorithm is usually used to predict the remaining life of a lithium–ion battery. Therefore, this model can also be used to verify the effectiveness of the filtering algorithm. The model is described below.This simulation selects the data of the B0007 lithium–ion battery, which is widely used in the verification experiment of lithium battery remaining life prediction . These dAccording to the above three simulation results, each population uses different initial state value to generate particles, and the multi-population cooperative intelligent resampling mechanism based on ring structure and Gaussian mutation strategy is proposed in this paper. It can effectively reduce the dependence of state estimation results on the initial state value. In addition, the high-weight particle information of each population is shared to other populations based on the ring structure. The high-weight and medium-weight particles of each population are retained, and the low-weight particles are resampled, which can effectively improve the diversity of the particles, so as to improve the estimation accuracy of the CRPF.When the statistical characteristics of noise are unknown, the CRPF shows good filtering performance. An intelligent CRPF method based on multi-population cooperation is proposed to solve the problem of low estimation accuracy caused by the lack of particle diversity in the CRPF when the initial value of the particles is not set accurately.The main contribution of this method is that the multi-population cooperative intelligent resampling method based on ring structure and the cooperative strategy based on Gaussian mutation are introduced into the CRPF. The particles are divided into several independent populations with different initial state values, and after the importance sampling is completed, the resampling process of low-weight particles is improved based on the ring structure. This method promotes the cooperation of particles among various populations and enhances the interaction ability of particle information. At the same time, Gaussian mutation is introduced to improve the particle diversity. The results of three simulation experiments show that, compared with the standard CRPF and MH-CRPF, the proposed method can effectively improve the estimation accuracy of the CRPF when the initial value of the state is not set accurately.Although the proposed method has good estimation results, how to better set the particle initial value of each population and the mutation rate of Gaussian mutation, and how to enrich the cooperative strategy between multiple populations, so as to achieve more accurate estimation results, are the future research directions and research goals.


Despite the fact that multicore processors have a better instruction execution speed and lower power consumption, they also encounter a set of design challenges. The appearance of multicore and many core architectures has raised the problem of managing shared hierarchical memory systems. The main focus of this paper is to evaluate the behavior of shared hierarchical memory systems by modeling their response time analytically. Since the gap between the memory and processor speed increases rapidly, it gets more crucial to find an analytical model that includes the significant factors that affect the performance of hierarchical memory systems. The proposed model considers the interdependence between different memory layers and differentiates between the memory response time and memory system time. Moreover, the model evaluates the effect of memory hierarchy on the variance of the memory access time. The existence of a large variance can lead to extremely long wait queues which can dramatically affect the performance of multicore processors Nevertheless, simulations may become very time consuming due to the complexities of architectural design spaces and workloads. In modern technology, there is a growing need for high performance computing. As Moore’s Law continues to hold, we will be able to pack even more transistors on a single die. But what happens if this trend continues? On the positive side, we have the advantage to enhance the performance of multicore processors by increasing the number of cores. Multicore processors consume less power and generate less heat per core than the same number of single-core processors.Performance modeling plays an essential role in processor design. It can help in determining the architectural parameters that are crucial for optimal performance. Earlier researchers used simulations to evaluate their designs2. Although there are many improvements in memory system performance but still there is a significant gap between memory system speed and processor speed. Even if the new multicore processors use a faster memory, there will still be a chance for contention as long as the memory system is shared among the cores. As the number of cores increases, the performance of more applications will be significantly affected by the memory hierarchy contention4.Multicore processors use shared memory hierarchies to achieve a high-speed memory system. One important characteristic of multicore processors is that they have a different degree of memory system sharing at different levels of the memory hierarchy. Most multicore processors' have cores with a private L1 cache. According to the architecture, an L2 cache may be shared by two or more cores; and an L3 cache is shared as well. The main memory level is shared among all cores of the processor. The memory sharing may vary from one multicore processor to another. While the performance of memory hierarchies is essential to single core processors, it is even more critical to multicore processors. Due to this memory sharing, there is a possibility that for memory intensive applications to occupy the shared memory system leading to a degraded performance. Contention for shared memory hierarchies may be so critical that the only way to run a task may leave many cores idleIn this paper, we propose an analytical model for memory hierarchy systems that takes into account the essential parameters that affect the performance of memory systems. These parameters include the number of layers, the hit ratio of each layer, the access time of each layer, the search time in each layer, the arrival distribution of memory requests and the service distribution of memory requests. We use Markov chains and the M/G/1 queuing model to estimate the average and the variance of the response time for hierarchical memory systems. We use the Linear Algebraic queueing Theory (LAQT) to be able to model deep memory hierarchies. The proposed model can be used to estimate the impact of the number of levels of the memory hierarchy on the variance of the response time. Large variance can produce long wait queues for shared memory systems which can significantly degrade the performance of multicore processors. The rest of the paper is organized as follows, in "5 approximated the level one cache performance analytically using the random replacement policy. Another model that derives stack distance (the number of unique memory objects accessed during a reuse epoch) histogram from reuse distance (the number of unique data elements accessed between any two accesses to the same element) histogram to predict level one LRU cache behavior is proposed6. Pan et al.7 used Markov chains to estimate the cache memory performance under three different replacement algorithms. Artificial neural networks are used to evaluate the impact of out-of-order executions8. The second category focused on those models of hierarchal memory systems in a single processor. Ji et al.10 created an analytical model to estimate the cache misses' ratios using the level one cache stack distance histogram. In11, another analytical model was introduced to estimate the level two cache behavior based on the effect of cache inclusion/exclusion policies.The previous work in modeling hierarchal memory systems can be organized into three categories. The first one is focused on those models of one memory level in single processor machines. Berg et al.12 presented an analytical model for a bus-based shared memory with only private caches. The model intended to capture the whole range of invalidated cache coherence protocols. The model assumes that the memory hierarchy is not shared and the only sharing exists in the main memory. This model focuses on the influence of cache-coherence protocols on the memory system performance but did not take into account other significant factors such as sharing the memory hierarchy. Taecheol et al.13, proposed an analytical model to study the impact of memory size and off-chip bandwidth on the performance of multicore processors. The model assumes that the sharing exists on the last level of cache only. This assumption is also used in15. Jin et al.16 presented an analytical model to estimate the performance of two-level memory systems but we see in their presented model a big inconsistency between the forecasted and measured memory access times. Eklov et al.17 presented a model that approximates the shared memory miss rates of co-scheduled tasks on memory hierarchal systems. They considered the locality features of the memory requests for each processor core. Though, they did not take into account the impact of data sharing among different processes, which may lead to significant errors when estimating the effect of data sharing on multi-threaded applications. Jasmine et al.18 employed Markov chains to model the stack distance histogram of multi-threaded programs. Wu et al.19 presented a model to examine the fluctuation of reuse distance by obtaining the profiling information of concurrent reuse distance (the reuse distance of a data/memory reference when the thread is interfered by references from other threads) in Loop-based parallelism. Balasubramonian et al.20 indicated that the memory hierarchy systems that do not meet the applications demands will results in a degradation in the performance of most applications. In21, the authors have proposed a data-sharing aware analytical model for estimating the miss rates of multi-level cache hierarchies for multi-core processors. The proposed model can also evaluate coherence misses. The authors claim that the overall average absolute error is 5% in four hardware cores configurations. The authors did not take into account the queueing overhead or the utilization of the memory system. The authors in22 introduced an experimental study to the impact of multi-level cache hierarchies on the performance of different architectures. For this study, they performed experiments in the Broadwell CPU and Pascal GPU, using applications from the Rodinia benchmark suite. The experiments showed that the main performance limit is the accesses to main memory. In Pascal architecture, the overall memory utilization rate is directly linked to application performance. In the Broadwell architecture, results showed that it is more important to have a total high hit ratio in the memory hierarchy than simply having a high cache hit rate in upper levels such as L1.The third category focused on those models of hierarchal memory systems in multicore processors. NikolovOur motivation can be explained as follows. The hierarchical memory systems were considered by several work but all the previous studies to analyze the performance of memory hierarchal systems were limited. These limitations were due to the dependency of the models on the application type or from the analytical model that cannot model deep hierarchal memory systems. In some of the previous studies, we noticed inconsistency between the values produced by analytically models and the measured values due to ignoring the queueing delay for shared memory systems. Also, previous work did not consider the variance of memory access times and its effect on the performance of shared memory systems. In this paper, we propose an analytical model based on Markov chains and the M/G/1 queueing model. The proposed analytical model can be used to model deep memory hierarchies. Also, the proposed model differentiates between the memory response time and memory system time. Moreover, the model evaluates the effect of memory hierarchy on the variance of the memory access time.i. The average access time of layer i is ti which is divided into two-time parts. The first part is the time taken to search the memory layer for a specific address. The second part is the time taken to read or write the data if found in this layer. We assume that the model uses the write-back protocol. A memory request will be delivered to the cache memory first level. The process of accessing any level of the memory hierarchy starts with searching that level for the required data. The data will be found by probability h then the read/write operation will be performed on that data. The memory request will not be found with probability (1 − h). In this case, the memory request will be forwarded to the next memory level of the hierarchy until we reach the final level of the hierarchy. The required data will be found in the last level with probability one.We assume that the hierarchical memory system consists of N layers as shown in Fig. We present two cases of memory use. The first case is the exclusive use of the memory where only one core can access the memory system. The second case is the concurrent access where any number of cores can access the memory system at the same time. If the memory system is idle the request will be handled immediately. Otherwise the request will wait in a FIFO queue until it gets its turn.0 represents the multicore processor and the last state in the state diagram M is the main memory in the hierarchy. With probability θ the instruction will not need a memory access. We focus on the memory hierarchy only and we will ignore the CPU time and include more details about the memory system to our model. The state diagram in Fig. 23:The memory hierarchy system in Fig. T is a random variable corresponding to the memory hierarchy response time.P is the transition probability matrix.p is a vector whose component i is the probability that the memory request starts in state i when it begins using the memory system. The size of row vector the row vector p is n = 2 × N + 1, where N is the number of intermediate levels.€ is a unit column vector of size n = 2 × N + l.i = 1/ti,M is the rate matrix; M is a diagonal matrix where Mij is the probability of the system is being in state i and moves to state j.The rows in the matrix P represent the different states in the state diagram. The probability pi at any memory layer can be either exponentially or non-exponentially distributed. For example, for shared memory systems that are using a bus, we can use exponential distribution because it is difficult to predict how many other memory requests are waiting to use the memory system. In the proposed model, we assume that at each state i would be an exponential server with average service time ti. we consider X to be a column vector such that each component xi is the average time a memory request takes to finish using the memory system, assuming it started using the memory system at state i. First the memory request is served by state i, and on the average spends a time of xi = (M−1e′)i. After that it either leaves memory systems or enters state j with probability pij. It will spend on average time xj to finally leave the memory system. Mathematically we have, in vector form,The memory system consists of a set of states as shown in Fig. Now, the random variable T denoting the time a memory request spends in memory system can be calculated as follows,23 as,The rate matrix B is definedThe average visit time matrix is calculated as,ij) represent the average time a memory request takes at state j from the time it first enters state i until it finishes using the memory system.The elements of the V matrix  is given by,From23,Which confirms the result in Eq. . The varThis model can be used as a baseline model since we assume there is no memory sharing. We can use this model for a specific system to calculate the values of the V matrix then from its values we will be able to indicate which level of the memory hierarchy represents the performance bottleneck. Hence, we can find for a specific application which level of memory needs to be enhanced to improve the system performance. Also, it can be used to measure the effect of sharing on the memory system performance. Finally, we can calculate the effect of the hit ratios and look up algorithms on the performance of existing computer systems. This model can be used for light loaded systems or for computer systems with a single customer.If concurrent access is allowed that means we can except more than one request to the memory hierarchy at the same time. If a request from one of the cores reaches the memory hierarchy when it is idle, it will be serviced immediately. If a request from one of the cores reaches the memory hierarchy when it is busy, it will wait in a FIFO queue until it gets its turn as shown in Fig. s) spent by a memory request in the M/G/1 queue is given by:2 is the coefficient of variation of the memory access time, The P–K formula is combined with Little’s theorem to show that the average memory system time E and Variance (T) can be calculated as we showed in the exclusive access case.s)). The difference between the memory response time and memory system time represents the queueing delay for the memory system that has been ignored by previous models.Previous work did not consider the variance of memory access times and its effect on the performance of shared memory systems. In Eq. , we estiWe begin our parametric study by examining the effect of the system state on the performance of the memory system in exclusive memory access case. We use the proposed model to study the behavior of the memory hierarchy system of three levels. We will consider a memory system that consists of two levels of cache memory and one level of main memory as shown in Fig. C1 = the first level search process.C2 = the second level search process.M = accessing the main memory.D1 = the first level read/write operation.D2 = the second level read/write operation.The stochastic Matrix P can be constructed as follows,The rate matrix M will be constructed as follows,D1 as a one unit of time and we will use the rest of the time values according to this value as follows,Here we will use the access time for the first level of cache TThe two parameters γ and α are used to reflect the cost and speed of the memory system. Small values of γ and α means a fast but more expensive system. The visit rate matrix is calculated as,The average service time is calculated as,2) memory level on the total memory access time for different values of γ  when the hit ratio of the first memory level (h1) is constant. We can see clearly in these figures that the effect of h2 becomes more significant on the total memory hierarchy access time for large values of γ and small values of h1. For large values of h1 the effect of h2 on improving the total memory hierarchy access time increases for large values of γ. If we compare Figs. 1 equals 0.95, we see that the improvement of the total memory hierarchy access time approaches 20% when γ equals 5 and increases to 45% when γ equals 20.In Figs. Now we consider examining the effect of the system state on the performance of the memory system in the concurrent access case as in Fig. 1. When we compare Fig. 1 and h2). When we increase the values of the hit ratio over 0.9, the effect of the system sharing drops to 70% increase on average.As we have seen in Fig. In Fig. When we compare Figs. 26. Even if we have a small value for memory access time, we could have a large variance if the distribution of memory access times has a long tail (coefficient of variation > 1). Memory access time large variance can produce long wait queues for the use of the shared memory systems that can significantly degrade the performance of multicore processors. A memory access time larger than a pipelined CPU depth can significantly influence the performance of pipelined processors because of the pipeline hazards and pipeline stalling27. The proposed model can take into account the dependency that may occur among the different memory hierarchy levels that influences the performance of the memory system.None of the previous work studied the effect of the variance of the access time on the performance the memory hierarchy. It is well known that long tail distributions exist in many areas of computer systems such as FTP data connections, traffic arrivals in local and wide area networks, file sizes, and CPU burst times1) on the coefficient of variation of memory access time. We use γ equals 20. As we can see, the coefficient of variation increases as we increase h1 but with different behavior for the different values of the hit ratios. For small values of the hit ratios, the coefficient of variation increases almost linearly. When the hit ratio h1 approaches 0.8, the coefficient of variation blows up.Next, we are going to study the effect of the memory hierarchy system on the coefficient of variation of memory access time. In Fig. 1) on coefficient of variation of memory access time when γ equals 50. We can notice the same behavior as in Fig. 1 (greater than 0.8). This is a very important characteristic to consider when designing memory hierarchy systems. Since we always try to use memory levels with high hit ratios to improve the performance and this is true for single core processors. But for the memory hierarchy systems of multicore processors, we should consider the drawback of the queuing time that a memory request may face. The proposed model indicates that choosing memory levels with high hit ratios will improve the performance of memory hierarchy systems of multicore processors but with certain conditions.One of these conditions is to have a lightly loaded system. We should try to avoid building up the queue of the shared memory system. Otherwise the performance of the memory hierarchy system will degrade significantly.The other condition, is to have the relative speed between the memory levels (γ) as small as possible otherwise the variance of the memory response time may increase significantly.In Fig. In this paper, we have proposed an analytical model based on Markov chains and the M/G/1 queueing model. The proposed analytical model can be used to model deep memory hierarchies for multicore processors. The main objective of the model is to evaluate the behavior of shared hierarchical memory systems by modeling their response time analytically. We have shown the impact of the number of hierarchical memory levels on the variance of the memory hierarchy response time. By using the proposed model, we were able to show that increasing the number of hierarchical levels of the memory increases the variance of the memory hierarchy response time. The large variance can produce long wait queues for the use of the shared memory systems which can significantly degrade the performance of multicore processors. The model can identify the performance bottlenecks in the hierarchal memory system by showing which level or levels of the memory hierarchy degrade the performance significantly. The model also can be used to estimate the waiting delay of the use of the shared memory systems. This delay explains the inconsistency between the values predicted by analytically models and the measured values published by previous researchers. We have shown that the waiting delay increases significantly when the relative speeds between memory levels increases.