How to Analyze Cancer Progression in COVID-19 Pandemic?

sharvareeshukla@gmail.com Abstract The constant news about the corona virus is scary. It is not possible to separate treatment for Cancer due to COVID-19. An effective treatment comparison strategy is needed. We need to have a handy tool to understand cancer progression in this unprecedented scenario. Linking different events of cancer progression is the need of the hour. It is a methodological challenge. We provide the solutions to overcome the issue with interval between two consecutive events in motivating head and neck cancer (HNC)


Introduction
Cancer patients were more likely to develop COVID-19 because they are immunocompromised1. The individual risk of COVID-19 infection varies from patient to patient. It is required to assess the risk of both COVID-19 and tumor control on a case-bycase basis with the patient. The treatment effect of head and neck cancer (HNC) is obtained in the presence of multiple events like locoregional relapse (LRC), progression (PFS), and death (OS). These events are analyzed separately by Kaplan-Meier and Cox PH. COVID-19 makes these events related. Time lag/interval between different types of event are to be explored. This article is dedicated to explore the time lag effect and make statistical 2 inference about the best experimental arm using Accelerated failure time model and regression methods. The work is presented as the occurrence of other events as a hazard rate after the first event (relapse). The time lag effect is linked. It is true that local relapse biologically triggers cancer progression and death. But we have never measured. Now all events are likely to be influenced by COVID-19 infection. These cannot be isolated.
Further, two-time points are generated as the duration between relapse to progression and progression to death. These are transition periods. The Cox proportional hazard model and show the efficacy of our study. An accelerated failure time model is also applied with the transition states and the dependency structure between the gap times are explained using auto-regression. The effects of Arms are compared using the coefficient of auto-regression and AFT models. The complete analysis using Bayesian techniques is executed with R opensource software and OpenBUGS.

Dependency modelling
It is difficult to stop the spread of the infection of COVID-19 from vulnerable cancer patients. We cannot deny treatment to several thousand cancer patients. It is really challenging to handle several thousand cancer and COVID-19 patients separately. There is a very minimal chance that cancer patients will not be infected by COVID-19 in the long run.
There is no doubt we have to run several clinical trials in the presence of COIVD-19 infection. Suppose we are having n number of cancer individuals. Events may occur as locoregional relapse, progression, and death. The events are marked as 1, 2 and 3 respectively. The events are ordered which implies that the Loco-regional event cannot happen post-progression or death, and death as a terminal event. Let, be the total number of events that happened to a patient? Thus, an individual I corresponds to a cluster of the 3 size of those events. Here, our interest is to measure the event occurrence rate at each of the interval or gap time between two events. Let, be the true event time for individual i.
The j could be 1, 2 or 3. Also, we consider that all the individuals have experienced at least one event. The intervals between two subsequent events are defined as follows: (1) In our study, the gap times are assumed to be dependent. Events are ordered. To model the dependency structure, we assume that the 1st event corresponds to , the duration from beginning of the study to occurrence of 2nd event, 2nd event corresponds to and so on.
So, the dependency structure is considered among , etc.
We assume that a simple linear regression model between and . The regression We fit two separate linear regression model for two different Arms. defines the change in for unit change in for Arm 0. The same inference can be drawn for So, ignoring the intercept term in the regression model, the difference between the coefficients -signifies the change in dependent gap time due to change in the Arm.
We fit AFT models for and and obtain the corresponding coefficients of Arm to measure the change in survival due to change in treatment.

AFT Model with Gap Time
The accelerated failure time (AFT) model is a popular alternative of proportional hazard model to analyse survival data. It is also applicable in the current COVID-19 scenario. To observe the dependency pattern between observed times, it is more efficient to model the survival time rather than hazard rate. In AFT model, it is assumed that the effect of covariate is to accelerate or decelerate the survival duration by some constants. The AFT model can be expressed as, where, .
As the number of independent parameters in a Bayesian hierarchical model is not clearly defined, DIC estimates the effective number of parameters by difference of the posterior mean of the deviance and deviance of posterior means.

Bayesian Cox PH regression separately for each event
The Cox proportional hazards model is applied in time-to-event data analysis [7][8][9] . It is defined as (9) or, .
The baseline hazard and hazard at time t is defined by and for i th patient.

Results
Dataset was presented to resemble a motivating example of head and neck cancer (HNC). A total of 148 patients treated with two chemotherapeutic arms were illustrated. The duration between treatment initiation to the time of progression or the last followup visit for patients who had not progressed was considered. The progression is defined as RECIST criteria version 1.1. Disease-free survival was considered as the duration while the person experienced a complete remission. We considered the duration between LRC and progression as and between progression and death as .
One of the aims of the trial was to investigate the best effective arm to prolong the PFS.
The trial was followed by the loco-regional recurrence and overall survival. In this example, we measured the LRC as the duration between dates of registration to the date of first locoregional relapse. Similarly, the date of registration to date of progression is defined as PFS.
The OS was defined as the last date of follow-up or date of death from the date of The Cox proportional hazard model applied in this dataset is defined as, The 3 covariates considered for the modeling are Arm, Age, and Gender. The results are illustrated in Table 1. The survival curves corresponding to LRC and PFS are shown in Figure   1, Figure 2. The Kolmogorov-type supremum test is performed to obtain the p-value. The AFT models are computed considering Arm as the only covariate. The model is, The posterior mean and standard deviation of the Arm effect is obtained from both the AFT and regression model and from the density plots of the difference of Arm effect from both the model shown in figure3. We can draw this inference that the dependency of gap times is translated through the regression structure. So, adding the Arm effect from the AFT survival model for the first gap time and the Arm effect obtained from the regression model, we will obtain the same output for the AFT survival model for the second gap time. Thus, given the information of time between LRC and PFS and dependency structure between gap times, the survival duration between PFS and OS can be predicted. The details of the posterior means obtained using the Bayesian AFT model are given in table 2. The analysis was done using R opensource software10 and Bayesian computing was performed using OpenBUGS.

Discussion
The novel corona virus that causes COVID-19 appeared more than twice as high among individuals with cancer than the general population 11 . In survival analysis of disease-related to oncology, the patients commonly experience multiple events like locoregional relapse, progression, death across the follow-up period. The interest lies in the prediction of survival duration for a particular event and evaluating effective treatments. The analysis is carried out by assuming the independence of the events. However, due to a missed visit of the patients, information regarding the complete follow-ups of the patient is often unknown.
So, their survival duration cannot be predicted based on the analysis carried out on the previously occurred events. The dependency modeling of the durations between consecutive events will assist to predict the occurrence of the next event.
The generalized version of the multi-state model is well-documented [12][13] . The simplest form is defined as mortality model having two states, 'alive without disease' and 'dead' and linked transition between these two states. The competing risk model can be defined as a provision where individual may die due to another causes [14][15][16]

Conclusion
The constant news about the corona virus is scary. It is not possible to separate treatment for Cancer due to COVID-19. An effective treatment comparison strategy is needed. We presented a handy tool to understand cancer progression in this unprecedented scenario. Linking different events of cancer progression is the need of the hour. It is a methodological challenge. We provide the solutions to overcome the issue with interval between two consecutive events in motivating head and neck cancer (HNC) data.
Ethics approval and consent to participate: Not Applicable.