Parametrization of Survival Measures (Part III) Clinical Evidences in Single Arm Studies with Endpoint of Overall Survival

Many clinical trials have prospective or retrospective data-sets without comparison to the control-group formed by the same cohort as the active one. The measured single arm naturally contains the relevant information, however, in most of the cases, it is impossible to obtain it from the complex survival curve without a reference. In our previous articles [1] [2], we had shown that the self-similar Weibull distribution fits the self-organized biological mechanisms well, and so it is the best option to study the single-arm survival curves, where self-organizing process is actively present. With the Weibull decomposition of the survival curve, we can fit at least two subgroups of patients. The weighted sum of the decomposed fractions could be optimized analytically and determining the best parameters of the components and the best composition ratio of the weighted sum is also possible. In this part of our series of articles, we will show how the method works in a real clinical envi-ronment through modulated electro-hyperthermia (mEHT) as a complementary method, applied curatively when no other conventional curative therapies are available. The decomposed function of the non-responding group provides an excellent agreement with the historical controls in pancreatic cancer and non-small-cell-lung-cancer studies. In the case of glioblastoma multiform,


Introduction
Overall survival (OS) is the most reliable parameter which characterizes the efficacy of a clinical trial. Evidence-based medicine (EBM) needs statistical evidences by clinical trials [3]. The clinical evidences are categorized into phases from basic preclinical (Phase 0) to the postmarket surveillance (Phase IV) [4].
The next steps are clinical, showing safety (Phase I study), efficacy (Phase II study), and an extensive, stable applicability (Phase III study), post-sale surveillance (Phase IV study) [5]. The usual basis of statistical evaluation is the randomized separation of a well-chosen cohort to control and active arms, while being as objective and as double-blinded as possible. Extracting strong reliable evidence from single-arm survival studies is rather challenging due to the missing control in the cohort. Due to the missing control-arm, the hypothesis check is unavailable. The information about the success of the treatment is of course somehow well-embedded in the results of the active arm, but without a reference set of values, its proper selection from the data is highly difficult and, in many cases, even impossible. Double blind categorizing is when neither the patient nor the therapist has any information about the actual treatment is impossible in cases of many medical equipment approvals, because the equipment usage cannot be hidden if no other treatment is possible.
In systemic diseases like malignancies, local response is not relevant in trials, because the local responding tumor refers to the local advantage, while no data is collected for the systematic behavior of the malignancy. Late stages probably have micro or macrometastases that essentially modify the survival. Local success does not give reliable information about the survival of the patient. The decreased overall survival among others for breast carcinoma [6] [7], for non-small-cell lung cancer [8] [9], for uterine cervix [10] [11], and even for the easily "heatable" surface tumors [12] were measured together with anyway significant local response (shrinking) of the tumor. Consequently, relevant information about the success of the treatment can only be obtained, if the endpoint is OS.
The effect of active therapy that changes the patient's survival is the hidden information in the measured OS. Looking for the embedded data is hard, and the success is doubtful when only a standard data-mining is used. The actually unavailable comparison to reference could lead to misinterpretations in a single-arm study [13]. Furthermore, the enormously massive bio-variability of the A. Szasz et al. International Journal of Clinical Medicine participating individuals creates a stumbling-block for objective evaluation even in a well-chosen cohort; and covers the useful data. The life conditions (lifestyle, diet, social position, etc.) of the studied individuals are also very different; these may modify the results [14], which gets even worse when the patient uses additional supportive therapies like "home medicine", that can be picked up easily from the widely available uncontrolled internet sources.
Naturally, when the well-controlled single arm study offers obviously much better results than expected from the historical data, we tend to regard it as a breakthrough, however significant heterogeneity is observed in these comparisons [15]. When the survival is not obviously much better than the historical data, the evaluation of a single arm study is complicated and in most of the cases impossible.
A commonly applied possibility to evaluate single-arm therapy information is when researchers use a historical control from the same clinic/hospital, choose retrospectively the same conditions of the cohort-selection. The evidence of the retrospective data-collection from historical archives is, of course, weaker than the randomization. The propensity scores method offers an increased reliability of the obtained results [16] [17] by adding a database construction of the control arm to single-arm results [18]. Data mining in large and representative databases selects a comparative group of patients, with relevant and characteristic properties of the disease and the conditions of the patients, supposing that these (directly independent parameters from the actual therapy) do not change during the complete curative or palliative process. The method can be verified statistically if the confounding variables are chosen well [19]. Advanced cancerous cases limit the applicability of the propensity scores method, because the patients might have had a large variety of previously failed treatments and could develop various metastatic lesions.
For improving the statistical relevance, another method has been developed: the sequential trial [20]. It is a method during which we continue the study until the number of the patients reaches a level where it can be regarded as statistically significant. In such sequenced study, cumulative data is analyzed interim after the treatment of the chosen group of patients and a decision is made to continue or stop the given treatment at every step [21]. The sequenced trial is commonly applied in small studies [22], as a tool for evaluating the interim data for statistically significant values [23] [24] [25]. It is a useful tool for studies of advanced cases when other ways do not exist [8] [26]. This way, multiple survival endpoints could be evaluated [27]. Just like the propensity score method, the sequential method also has complications, for example when the patients are in late metastatic stages with multiple pretreatments and possible comorbidities (like organ failure or unsatisfactory laboratory results).
Oncological hyperthermia is one of the therapies that cannot be blinded, due to its machinery application, the sham treatment is usually well-sensed by the patient, so it is not possible to be blinded. Moreover, the medical staff who takes care of the patient, must also know that the treatment is a sham or not for safety International Journal of Clinical Medicine issues. These conditions challenge the evidence, affect the reliability of these trials and make them less comparable to the evidences of conventional applications.
The chosen end-points of oncological local/regional hyperthermia clinical studies are often connected to local responses (local remission rate, local remission free survival, local progression-free survival). This choice is a logical consequence of the local treatments-however, the problem of malignancy is far beyond the local response. Malignant diseases have the possibility of forming micro-and macro-metastases by systemic dissemination far from the original tumor. The development of metastases is more life-threatening than local tumor development [28], and the invisible micro-metastases worsen the life-prognosis further [29] significantly. Unfortunately, there are multiple studies with effective and significant local control, but at the same time a decreased OS is shown in well-conducted studies among others for breast carcinoma [6] [7], for non-small-cell lung cancer [8] [9], for uterine cervix [10] [11], and even for the easily heatable surface tumors [12]. An important fact is that the inclusion criteria was "locally advanced", so no metastases were observed at inclusion. This raises doubts [30], that could block the application of hyperthermia in oncology, [31].
One of the categories of oncological hyperthermia methods is the modulated electro-hyperthermia (mEHT, trade name: oncothermia) [32]. The mEHT method is usually applied in the stages when conventional curative methods fail, and conventionally only palliation would be applicable. The method of mEHT is able to resensitize the previous refractory treatments, and usually, it is applied for late-stage patients. In most of the cases, the quality of life (QoL) is in the focus of the trials in palliative care [33]. These studies provide evidence of the palliation being mostly irrespective of the tumor-type and the selection is usually only based on the unavailability of curative approaches [34] [35].
The direct rationale of mEHT is that it attacks the malignancy in its systemic conditions, so instead of the local responses of the actual tumor, the complex issue of the overall survival with the QoL together is the usual endpoint of its studies. The basic idea behind mEHT is the selective heating of the malignant cells in a highly heterogeneous tumor. The bioelectromagnetic interactions [36] with the physiology differences of the malignant and non-malignant cells [37], allow the attack and induce the apoptosis [38], in malignant cells, while no change has been made in healthy neighboring ones. The process produces a damage-associated molecular pattern and immunogenic cell-death [39], which has a crucial role in the abscopal effect of the mEHT method [40] [41]. The immune-effect is so strong that after the treatment the re-challenging of the same tumor was unsuccessful [39]. Significant differences can be shown in a comparison of mEHT to conventional water-bath heating [42] or with other bioelectromagnetic heating methods [43].
Considering the possible controversial endpoint response-related parameters of clinical studies, the appropriately combined endpoint with QoL should be the  [45], which is viewed as the future of hyperthermia in oncology [46].
Clinical results prove that the improvement of the survival is induced by mEHT [47]. There are studies for multiple localizations, like pancreas carcinoma [48] [49] [50]; small-cell [51] and non-small-cell lung cancer [

Method
First, we define the inclusion criteria for unifying the mEHT cohort. Selecting late-stage patients for mEHT in curative approach, when conventionally only palliation is available makes mEHT studies complex, and obtaining evidence difficult. The before-mentioned problems have aroused because the actual cohort contains only those late-stage patients, for whom conventional therapies are unavailable due to their refractory cases, organ failure, inadequate hematology measures, multiple relapses, or simply, there are no curative possibilities in that concrete stage of the disease. In this meaning, mEHT starts as a definitive palliation, but its intent is curative. Specialized medical facilities like hospitals, university clinics, and private services use mEHT treatment for a broadly heterogeneous group of patients who are not treatable by conventional therapies anymore. The long years of mEHT usage in oncological practice shows that the actual stage of the patient determines the time of the first, so the point of the inclusion of the patients to mEHT process is based on identical criteria: conventional curative possibilities (chemotherapy, radiotherapy, gene therapy, etc.) no longer available for this group of patients. The blind process in a clinical study is obviously impossible in the case of mEHT. In many mEHT clinical studies, even the simple, non-blinded randomization is impossible because late-stage patients need the only applicable curative possibility, therefore, the option of exclusion from the mEHT by randomization would be unethical.
The point, when the patient leaves conventional therapy to start complementary mEHT can be regarded as the end of an independent trial. This stage is usually grouped by late palliative intent, and the aim is to provide the best supportive care. The time when previous treatments fail forms a reference point for a cohort of patients, for whom conventional curative protocols alone do not work anymore. At this point, mEHT treatment can be started and/or promoted to a complementary, but curative therapy in order to be able to use conventional approaches again. The failure of conventional therapies as the only inclusion criteria of the study unifies the mEHT cohort. In consequence, the starting point International Journal of Clinical Medicine of mEHT is defined by the cohort-forming condition, therefore, the time between the diagnosis and the start of mEHT treatment ( s T ) has importance. All the information about the efficacy of the mEHT treatment is included (but hidden) in the single arm process as well. The information describes the obtained OS, however without the reference arm, information cannot be seen. By adding quasi control-arms, the accuracy of the estimation can be improved, and the double-checking of the subgroup division becomes available, that can be compared to the historical control arm of the group with the same (but retrospective) inclusion criteria. The simplest way to create the control arm in late-stage treatments is by choosing the patients for whom conventional treatment was ineffective, or those, who were censored or deceased earlier than the end of the protocol. Note, that local response is not relevant information in trials with OS as an endpoint, because the locally responding tumor excludes the systematic behavior of the malignancy. Late-stages probably involve micro or macrometastases, which essentially modify the survival. Local success does not give reliable information about the survival of the patient.
As it was shown earlier [59] [60], the survival is expected to fulfill some universal rules originated from the self-organizing and self-similarity of the bio-structures. In consequence, the parametric Weibull function (WF) fits to the non-parametric KM plot with high accuracy. The regression curve has simple information, considering, that all the individuals in the cohort have identical fate because of the development of the malignant disease. This universality gives the possibility to extract the outliner changes from the data-coherency in the parametric curve. So, when the observed KM survival plot does not fit with appropriate accuracy by WF, the weighted sum of two or more WFs with different parameters gives a satisfactory solution [59]: where M subgroups exist in the complete cohort of N patients, and in every Other reference groups may be compared to the historical arm or make a decomposition of KM with the process of WF-fit [59]. For the usual facilities of the trial, we may group the patients roughly into two groups: responding (r) and non-responding (nr). In this grouping only two subgroups of KM in (1): This bi-grouping is not always possible. The measured accuracy of the obtained ( ) ( ) KM W t decides the necessity of further subgroups. For detailed investigation we, had chosen two single arm trials performed by mEHT: inoperable, advanced pancreas [36] and advanced non-small-cell lung cancer (NSCLC) [40], as well as for glioblastoma multiform (GBM) [42].

Inoperable Pancreas Carcinoma, a Palliative Stage with Curative Intent
A study for the mEHT treatment of inoperable advanced pancreas carcinoma [36] involves 99 patients in the active arm from two centers (73 and 26 patients) and a historical control with 34 patients. The overall survival is shown in Figure  1.
The measured KM of historical control compared to the KM of OS of mEHT treatment is shown in Figure 2. WF can be well fitted to the historical control, but to the OS plot it is far from accurate.  The WF decomposition fits significantly well to the OS (regression by minimizing the sum of deviations points by points), which is shown in Figure 3.
The comparison of the historical control and the non-responding subgroup obtained from the WF decomposition of OS in Figure 3 shows that mEHT has slightly effected even the non-responding group, but its statistical difference from the historical control is not significant ( 0.23 p = ) (Figure 4). The information contents, that are measurable by the Shannon entropy of the Weibull probability distribution [72]; are remarkably equal in this case:  This equivalence well verificates the decomposition concept by identifying the responding and non-responding patients, and so forming references to the single-arm study. This is not only a simple reference, but at the same time, it shows the percentages of the patients, whom the active treatment helps.
Despite the late stages and the conventionally palliative phase of the inoperable pancreatic patients, mEHT had shown curative features. The survival time from the first mEHT application has 6.1 m median ( Figure 5).
Studying the KM of the time from the first mEHT treatment gives another opportunity for controlling the WF approach. Decomposing the KM by WFs, the sum properly fits the measurement ( Figure 6).  it is over 30%, support further the previously observed accuracy of the WF decomposing fit.

Non-Small-Cell Lung Cancer (NSCLC) Palliative Phase Curative Intent
Another clinical trial was conducted for the advanced NSCLC by mEHT treatment [40], where patients were selected based on their finished conventional therapies without curative possibilities. The study involves 258 patients from two centers (197 and 61 patients, respectively) and a historical control from another hospital including 53 patients. Figure 7 shows the overall survivals for the two centers and the completely unified cohort. The measured KM of the historical control (in the case of patients who were treated with only palliative treatments, due to failed curative possibilities) compared to the KM of the OS of mEHT treatment is shown in Figure 8. WF can be well fitted to the historical control, but fitting to the OS plot it is far no accurate.
The WF decomposition, a regression by minimizing the sum of deviations data by data, fits to the OS of NSCLC is shown in Figure 9 significantly well.
The comparison of the historical control and the non-responding subgroup obtained from WF decomposition of OS in Figure 8 shows that mEHT has slightly affected even the non-responding group, but its statistical difference from the historical control is not significant (Figure 10). The information coin-

Glioblastoma Multiform
A clinical trial for the advanced GBM was performed by mEHT treatment after the conventional therapies had no more curative possibilities [42]. The study involves 94 patients. The overall survival is shown in Figure 12. International Journal of Clinical Medicine The overall survival curve in case of GBM cannot be fitted by two regression curves (responders and non-responders). The non-responders are unfortunately large (67.5%), but the responders' group has two subgroups, superior response (17.5%) and response (15%). The survival from the first mEHT treatment (finalizing the complete therapy set) shows the same behavior ( Figure 13), but much fewer non-responders (14.7%) in this part of the treatment.
It is remarkable that the non-responding subgroup in the period of mEHT treatment is only 14.7% (compared to the complete OS, where it was 67.5%).

Discussion
By the proper decomposition of KM to WF sub-groups, we were able to unhide the well-buried information in the single-arm study, and we were able to see the percentage of responding and non-responding patients when fitting WF-curves by best regression possibilities. The pancreas and NSCLC analyses had an accurate decomposition of KM into two parts, but GBM needed three subgroups for an accurate regression. The analysis of the elapsed time until the first mEHT treatment compared to the time when the mEHT was active shows huge differences between the groups, Figure 14. While the two KM plots are well-distinguishable in the pancreas and NSCLC cases. However, the elapsed time to the first mEHT and the survival from that have similar curves in GBM plots.
The comparison of OS KM curves to the KM of mEHT involvement ( Figure  15). It is obvious (that can be seen from the Figure 14 where ( ) e KM t is the probability of the KM-plot of elapsed time from the first diagnosis to the first mEHT, μ is the mean and σ is the standard deviation (σ 2 is the variance) of the distribution; and α is a normalizing factor. The parameters are fixed by physical assumptions (Figure 18): 1) The 0 µ = was chosen to fix the strict monotony of the plot.
2) The α was chosen to have the function value 1 at the mean of the distribution.
3) For the σ we use the percentage of the non-responding patients, as a dividing parameter for groups of patients in weak and strong condition. The group which is higher than σ, has a lower extrapolated survival. They start the mEHT earlier than the limit made by non-responders, and other patients are late starters, due to their defense systems, that can be regarded stronger.
The modified HJP (HJPm) and the elapsed time from the first diagnosis to the first mEHT show significant differences ( Figure 19); the elapsed time bases the approximated expected survival well after the failure of the conventional curative therapies.
The comparison of the HJPm survival curve, the historical control and the non-responding fraction from the WF decomposition, the three control arms are practically equivalent (Figure 20).
In the case of GMB, no historical control exists, therefore we may compare the OS and the HJPm approximation. The WF decomposition, in this case, has three A. Szasz et al. Figure 18. The HJP (HJPm) principle modified by the differences of the patients leaving the conventional curative treatment period. For simplicity, we used the median (50% -50% of patients in weak and strong conditions). Below the median shows patients who have left early due to insufficient improvement or personal weakness; while the patients over the median have appropriate defense and/or benefited from the treatments for a long time.    where the QoL is taken into consideration too. The quality-adjusted survival (QAS) [78] [79], which considers the QAS without symptoms and toxicity (Q-TWIST) [80] would be an important extension to the single-arm study.

Conclusion
The WF regression fit to KM non-parametric estimate works precisely in real clinical studies of advanced pancreatic cancer, NSCLC and GBM trials where the mEHT method was applied as a complementary treatment when no more conventional curative possibilities were available. The WF decomposition method creates an estimated reference-arm in a chosen homogeneous cohort. The control arm is correct, if we assume that patients start their mEHT treatment when the conventional therapies fail, so their overall status (relative to the lines of the conventional therapies) groups them into groups with similar conditions. The mEHT method has no harm for the patients (no adherent effects to make tumor-progress by the treatment alone), so the possibility of the treatment results has two categories only: effective or ineffective, which fits the decomposition concept well. Regression is accurate, and the control-arm from the decomposed WF corresponds well with the modified Hardin-Jones-Pauling statistical estimation too, when the number of patients is high enough (>30) for statistical evaluation.