DETERMINING AN OPTIMUM BIOLOGICAL DOSE OF A METRONOMIC CHEMOTHERAPY

The surrogate markers(SM) are the important factor for angiogenesis in cancer patients.In Metronomic Chemotherapy (MC) , physicians administer subtoxic doses of chemotherapy (without break) for long periods, to the target tumor angiogenesis. We propose a semiparametric approach, predictive risk modeling and time to control the level of surrogate marker to detect the perfect dose level of MC. It is based on the controlled level of surrogate marker, and the aim is to detect an Optimum Biological Dose (OBD) finding rather than a traditional Maximum Tolerated Dose (MTD) approach. The methods are illustrated with MC trial dataset to determine the best OBD and we investigate the performance of the model through simulation studies.


Introduction
The doses of chemotherapy selected in clinical practice are routinely derived by the MTD approaches. It is believed that an increase in dose will lead to an increase in tumor response. The dose selected for phase 2 and further studies is one dose level below the MTD level in Phase I studies [Skipper 1970, Devita 2008, Briasoulis 2009]. However, this conventional dosing of chemotherapy has routinely compromised the QOL of patients and may lead to selection of chemo-resistant clones [Schmid 2005, Golfinopoulos 2007, Saltz 2008]. The MC has been suggested as an alternative strategy to overcome such effects. The strategy in MC is to administer sub-toxic doses of same chemotherapy drugs for long periods to target angiogenesis [Bhattacharjee 2016, Patil 2015,Kerbel 2002, Kerbel 2004,Scharovsky 2009]. The effect on angiogenesis is achieved by targeting CEC. The effect on CEC is restricted to an antiangiogenic window in tumor cell line studies [skipper 1970, Hobson 1984, Vacca 1999]. In tumor cell line studies a drug would start exerting an antiangiogenic effect at a low dose level and it would continue to exert it till an upper dose level. Drug levels below the lower dose level and above the upper dose level won't exert an antiangiogenic effect ( Figure 1). The challenge is to establish the optimal biological dose (OBD) of MC which would have maximized inhibiting effect on CEC. The aims of any phase I traditional clinical trial is to detect the MTD of chemotherapy. As when conventional chemotherapy is administered toxicity is a concern.
However, in case of MC the toxicity is not a concern. As the Metronomic doses are nearly 1 10 ℎ of MTD of conventional chemotherapy. Therefore the aims of phase 1 MC clinical trial is to obtain OBD of a drug [Marshall2012].
The simple phase I trial design is to obtain MTD of a cytotoxic agent through standard '3+3' design accrues patients in cohorts of three and escalates until a pre-defined DLT is observed [Rosemarie1993]. The Continual Reassessment Method (CRM), is an another dose-response finding model and recently gained popularity [O'Quigley2006]. But both the models directly failed to detect the aim of any MC i.e. OBD . This paper is aimed to illustrate the application of OBD in MC. The OBD of MC is determined based on the performance of desired level of Surrogate Marker(SM). The serum creatinine (Scr) is considered as a choice of SM in this study. The OBD for dose-finding designs to search the perfect dose in MC has been proposed in this article.

MC trial data set
It is a partial data set considered from ongoing 7-years duration of a longitudinal study. This study is continuing under the supervision of the authors as principal investigator and is not matured enough to state more about it. This work is dedicated only on dose-response modeling section. It is expected that the full data will be matured enough by the end of the year 2020 in terms of completed follow-up of the patients with strong enough statistical power to serve the primary objective as the event of interest i.e. death. In this manuscript, we focused only on initial dose-response modeling of MC with cancer patients. A total of 110 patients MC therapy data has been utilized in this paper. In MC the main goal is to find out the OBD among all different low dose levels. The serum creatinine (SCr) is considered as response of interest for OBD. A total of six doses-levels coded as 10mg/m2,15 mg/m2,20 mg/m2, 25 mg/m2,30 mg/m2 and 35 mg/m2 of IV docetaxel are considered in this study. The data set is observed with individual-specific information for patient 's with doses and corresponding marker levels. The Scr measurements are retrospectively observed for six different cycles of MC therapy. This study is focused on Scr data observed till end of first few weeks of MC Trial.

Data Exploration with Mixture Distribution
The parametric model is assumed for dose-response curve preparation. The challenge is to specify the dose-response for the optimum biological response modeling. The literature for dose-response modeling is well established through different types of MLE assumptions [islam2016,Aghamohammadi2010,Schmoyer1984], and Kernel estimator [Muller1988].The non-parametric setup has been found suitable for dose-response through Dirichlet Process Prior [Mukhopadhyay 2000] and quantile response [Dette 2010].The application of semi-parametric through mixture of parametric and non-parametric explored for continuous variable [Einsporn 1987]. This mixture approach for density estimation [Olkin 1987], hazard function [Kouassi 1997], robust regression [Mays2001] and the linear mixed model [Waterman 2007] are already established for repeated [Pickle 2008] and non-repeated measurement [Robinson 2010]. This work is extended with the application of semi-parametric estimates of the dose-response relation for the exploratory data analysis. The weight is used to reduce the bias balanced the adaptively borrowed across the two method, i.e. parametric and non-parametric The scenario of parametric model correctly specified, the weights towards the parametric model will be more otherwise non-parametric portion will gain more weight. Let the response of interest is denoted as Y following the Bernoulli distribution with the probability f(x). The dose-response f(x) is estimated through the semi-parametric modeling. The effective dose-response is denoted as = −1 ( )for 0 < < 1$. The inverse functions f(x) is −1 ( )$. The aim of any toxicology studies is to estimate for smaller values of . The dose-response relation is estimated through parametric and non-parametric modeling. The parametric model is assumed as ( , ) for the dose -response curve and is the unknown parameter. Suppose the nonparametric and parametric estimates are ̅ ( ) and ( ,̂) respectively. The estimated value of the parameter of interest is ̂ obtained by MLE .The dose is denoted as x and corresponding response as Y. The dose-response relation is observed through the mixture of parametric and non-parametric modeling.
The equation is defined as The weight ( ∈ [0,1]) factor is used to make balance between parametric and nonparametric estimates. The equation (3.1) provides the option to incorporate the parametric and non-parametric both the options simultaneously. The estimates (either parametric or nonparametric) fits more appropriately will gain more weight and complementary value of i.e 1will be gained by another one. The dose range is denoted as ( , ). The estimated value of $\omega$ is obtained from the minimum estimated value of the mean integrated squared error(MISE) (Yuan2011) The dose-response relation is explored through semi-parametric estimation. The function values of the above equation are defined as The optimum dose level is selected with =0.5.
Further, x is defined as = −1 5 for d=1,2,….6.The Probit model in parametric portion and Kernel estimation for non-parametric portion assumed to perform the analysis. A total of 1000 boot strapping has been carried to obtain the estimates of weight ω. The value of has been observed through 0.05,0.1,0.15,0.20,0.25,0.30 to avoid the complicated estimation of Bias and MSE estimates of observed simulations of parametric and non-parametric estimates are detailed in Table 1 and Table 2 respectively.

Marker (SM)
The event is defined as the presence of a Surrogate Marker (SM) within the controlled limit. Let SM under consideration is Y. The objective is to reduce or induce the level of response Y through specific dose i.e. x. The target level of Y to be controlled by x is decided based on supportive literature of trial. Suppose patients i is having response value and if = , patients has experienced the controlled level of SM in the jth interval of time, −1 < < for j=1,2,....C and the sequence of time denoted as 0 < 0 < 1 < ⋯ … < < ∞ where The conditional probability is defined as ( , , ) = ( ≤ − 1| ≥ , , )for = 1,2, … − 1 and m = 1, … . M (3.8) Further, ( , , 0 ) gives the probability that a patients who successes in 0 − 1 without event will expereinced by * = − 1 at dose .
The likelihood is expressed as The extended function through consideration of latent variable is Patients safety also been considered to obtain the OBD controlled surrogate markers. It is assumed that when a group of new patients entered in a trial and treated with specific dose but not fully followed then two scenario may occurs to them either all of them are having controlled surrogate markers or none of them have controlled surrogate markers at time * . Since, none of the dose is having problem for occurrence of high toxicity level so six different desirable doses are administered simultaneously to the randomly selected patient and level of surrogate markers observed in the follow-up visits. The basic intention is to develop probability measures of either tiny or large surrogate markers levels observed at the specific dose $d_{k}$ to be fixed and OBD dose or move their dose to near at dose (3.13) Further, suppose dose is mentioned as $m$, the total number of patients received the dose denoted as , and partially evaluated patients for dose m is . Now Further ̅ 1 ( ) = 1 if all all patients fully evaluated, and ̅ 2 ( ) =

Probability Model for Surrogate Marker(SM) levels Distribution
Let the duration between initiation of trial and evaluation of surrogate marker is $"t^{*}"$ about the decision of * , the number of patients enrolled up to " * ", the study duration when the patients enrolled is and i=1,2,... * . Suppose the duration and possibly unobserved duration when patient 'i' successes the controlled level of surrogate marker. The amount of time patients $i$ is observed that It is defined as Δ i = 1 if = otherwise Δ i = 0 . Let = { ,1 , … . , } gives the successive patient times at which the ith patient receives the MC agent. This notation is useful to capture the actual patients administration time to deviate from his/her scheduled times. It is the provision that if the investigator wish to study m treatment i.e. (1) , (2) , … . . ( ) where ( ) = ( 1 , 2 , … … . , ) and that the jth dose has a total of ( ) administration. Further, $q_{i}$ gives the index of the last administration received by patient 'i' at the interim study time * . However, ( ) administration are scheduled for any patient to schedule ( ) at * it may be < ( ) due to administrative reason. Let shows the maximum duration of follow-up of each patients decided by oncologist. A fixed target probability is decided from the medical oncologist to define the targeted threshold changes of the surrogate marker for any time from enrollment to .

Monitoring
Let the total number of patients are N, and each patient assigned to a treatment dose. Each patient is assumed to follow days. Let the desired threshold value of surrogate marker is for ( | , ( ) .Here, two criteria's are selected It follows that 1 ( ) ≤ 2 ( ) ≤ ⋯ . ≤ ( ).. Further the best dose is defined as ( ), ̅ .

Surrogate Marker Distribution
The patient index is i. It is assumed that same dose of MC has been administered in different visits. Let ℎ( | ) is the event of high CEC level (i.e. more than 166 cells/ml) attributes to a single administration and parameter of interest is v. The event at time t for a patient treated with schedule s is defined as

Specifying the Single-Administration to Control Surrogate Marker
The single-administration's treatment failure ℎ( | ) can be quite general, provided that it reasonably reflects the risk of toxicity for the agent under study and is sufficiently tractable to facilitate the necessary computations. In general, because certain toxicities may be hard to identify before the trial, one can include in the definition of toxicity any adverse event sufficiently severe that it precludes further administration of the agent. We assume that the hazard of toxicity from a single administration has a finite duration and vanishes to zero within Thus, ( 1 , 2 , 3 ) with 1 the time at which ℎ( | ) reaches its maximum, 2 , and 3 the time when h(.) vanishes to zero. Figure 4 illustrates this function. Initially, we assumed that ℎ(. )had only the two parameters 1 and 2 , with 3 fixed and assumed known. However, we found that fixing 3 severely hindered the method's ability to locate the optimal schedule when the actual duration of ℎ.was much longer than the assumed value of 3 .
Other forms for ℎ( | )are possible, depending on the particular application. For example, the Weibull hazard ℎ 1 ( | ) = 1 2 ( 1 ) 2 −1 allows the risk of toxicity to continue indefinitely, with the shape parameter 2 determining whether the risk increases, decreases, or remains constant over time. One also could vary the hazard of toxicity for each administration.

Likelihood and Posterior
The most recent data at study time * collected on patient i for = 1,2, … * , are = ( , , ).. The optimal treatment sequence assigned to patient * + 1$ who enters the trial at * is based on the posterior of v given the data available at * , which we denote by * = ( * , 1 , 2 , … . * ).
The likelihood at * is

Application to MC trial data set
The idea is to study the level of Scr to a restricted cycles of $m=6$ for duration of administration 1,3,5,7,9 and 11th weeks. The maximum days to monitor the controlled level of SM is specified as = 80 days. However, during the MC therapy the schedule has been finalized for each patient. With individual specific prognostic function. The physician assumed that the event of controlled level of SM for a single administration will be achieved by 10 days, with a range of 7 to 80 days. The therapy is assigned to administer with different dose level with 110 patients with parameter * = 0.30, = 0.05, = 0.30 and ̅ = 0.90. It is assumed that patients will attend the clinic in 7 days intervals. It is assumed that the parameter 1 will be having the maximum value at the end of 7 days after the first administration. The derivation of 3 , 3 , for ( 3 ), and 1 and 1 for ( 1 / 3 ) are computed. The probabilities of event for 80days for each are detailed in Table6.
The proposed method has been illustrated on MC therapy. The results are detailed in Table  6 and corresponding data explored in Figure 4. It gives the dose-response estimation through parametric, non-parametric and semi-parametric methods. The estimation of dose-increases rapidly from 0.05 to 0.15, and thereafter slowly increases with dose. For doses, 0.20 and above response started to decrease. Parametric, non-parametric and semi-parametric fits are more or less identical with different doses. The weight assigned through semi-parametric method is 0.3 and parametric close to 0.4 and non-parametric with 0.05. Criteria 1 and 2 are selected with 0.2< ̅ <0. 6. It has been found that ̅ =0.60 performed best over others. The table is formulated for $\bar{p}=0.6$ for both the criteria. The parameter 1 is assumed to be occurring at 7 days and 2 as 10 days to 40 days. The designs performed are observed for criteria 1 and criteria 2. In different scenarios, the value of 2 is varied to reflect which schedule is optimal for different values in Table1. The decision is obtained from the Table of detailed value of , 1 ( ), ̅ 1 ( ), and ̅ 2 +1 ( ). The numerical values of the different states are also detailed in Table 3. The Operating Characteristics of PRMOBD design based on the choice of six different doses choice for the maximum duration of 80 days are detailed in Table 4. The corresponding simulation result for the OBD design for the maximum duration of 80 on 1 ( ), ̅ 2 ( ), and ̅ 2 +1 ( ) with the space of 5,10,20,40,60 and 80 days are given in table 5.The PRMOBD model suggests deciding the OBD with $0.20mg$ for 20days. It is expected that the trial with dose 0.35mg because it provides = 0.20. It's the value of $\eta=1$, then there would be a fixed OBD decided based on particular observed all cases. Similarly, = 0 may influence to observe a less number of patients towards control the CEC and the decision about OBD.

Discussion
The trinomial continual reassessment method is found suitable for OBD through highest probability of but failed to consider the toxic effect suitably [Zhang2006]. The two stage design is proposed on interim analysis for detecting the optimal dose [Polley 2008, Zang 2014 proposed dose-finding designs to search for the OBD by three adaptive dose-finding designs on molecularly targeted agents. Low-dose chemotherapy drugs are more effective to suppress tumors by restraining tumor vessel growth and preventing the repair of damaged vascular endothelial cells [Shen 2010]. High dose chemotherapy drugs like Cisplatin contributes to serious side effects [Shen 2010]. The target of MC therapy is the vascular endothelial cells [Shaked 2006,Wu 2007. The growth of new vessels for a long run survival time treated with traditional maximum tolerated dose (MTD) through high dose chemotherapy has been confirmed [Lam 2007]. The anti-angiogenic plays the important role as clinical potential [Lu 2007]. The metastasis and the growth of tumor cells depends on neovascularization [Folkman 2003]. The anti-tumor drugs could cause inhibition of tumor neovascularity [Moreira 2007]. The Scr have emerged as a promising candidate surrogate marker to assess the efficacy of antiangiogenic therapies [Malka 2011,Jubb 2006,Bhatt 2007,Bertolini 2006].The low-dose chemotherapy drugs, as one-tenth of the MTD, administered continuously and frequently, could selectively suppress vessel growth in tumor tissues and prevent the repair of damaged vascular endothelial cells (VECs) [Shen 2010].This model is appropriate in dose-response modeling having the avoidable level of toxicity in any clinical trial. Based on our knowledge this is the first statistical methodological attempt that has been considered to deal with OBD in MC trial. It is expected that this above mentioned methods will be useful for OBD detection in MC trials in future as well.