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In some clinical applications in oncology randomized, double armed, and double-blind trials are not possible. In case of device applications, double-blinded conditions are nonrealistic, and with many times the randomization also has complications due to the high-line treatments where the reference cohort is not available; the active “arm” has mainly palliative initiative. Sometimes highly personalized therapies block the collection of the homogeneous group and limit its double-arm randomization. Our objective is to discuss the situations of the single arm evaluation and to give methods for the mining of information from this to increase the level of evidence of the measured dataset. The basic idea of the data-separation is the appropriate parameterization of the non-parametric Kaplan-Meier survival pattern by the poly-Weibull fit.

Survival studies most frequently use the Kaplan-Meier (KM) non-parametric estimate. The KM estimator is fixed by the duration of participation in the observation. Both the start of the observation time and the end of the observation of the individual by events (censored due to death or dropped out from the cohort) are not absolute and have inexplicit values. The precariousness flows from the differences between real lifetime to observational time. We summarize the characteristic points of the life of a cancer-patient in

could be zero in any actual case. One point (start of the disease) is elusive because the real symptoms of any disease could be later than the starting point of the disorder. This situation usually happens because the observation facilities of malignant diseases are technically limited. We could have only guessed the latency period, which starts with an avascular situation, forming a dormant microscopic cluster [

The real survival period in this content is blurry, so the definitions of the real survival points are the sum of the observation and the post-observation period. The evaluation may concentrate on disease-related death or any deaths in the observational period, irrespective of the cause. The observation period may only contain careful watch, then the treatment, and at the end, a long follow-up too. The observation period usually is evaluated statistically by the Kaplan-Meier non-parametric estimates (

The start of the observation could be after the routine screening when patients complain (about symptoms) and the statistically valuable period starts at the first diagnosis. The latent period can be long, even years before the discovery of cancer [

Measuring the effect of the treatment has various approaches, since having complications of the bio-variability and personal sensitivity of the treated individuals as well as the variation of the results depends on the social background and lifestyle of the patients. Randomized clinical trial (RCT) is a commonly used study design to measure lifetime. In an RCT, the active (investigated) arm can be statistically compared to the well-randomized control group in a carefully chosen, unified cohort. To evaluate a clinical intervention with the optimal possibility of RCT has ethical issues [

appropriate cohort is a complex issue. Cohort forming sometimes uses forced conditions by reaching a definite toxicity predefined by the protocol (like in high-dose chemotherapy [

Sometimes, in cancer treatment, a misleading (or at least not complete) evaluation is practiced by measuring the local control of the tumor, instead of the systemic development of the malignancy in the whole body. The problem of the overall control of the system is complicated and not even possible with imaging because of micro-metastases and such adverse effects which cause comorbidities for the patient. Therefore, parametrization would only be effective if the end-point of the study is the overall survival and the quality of life combined.

Before deciding on the RCT, both sides of the balance of measured efficacy and the adverse effects must be taken into account. In case of serious diseases or terminal cases, no curative treatment is available, or further curative therapy is simply not possible because of comorbidities like organ-failure, low-blood-count, etc. Note that some conditions limit the RCT evaluation even in the double arm construction: the false inclusion and exclusion criteria (sometime “cherry picking”); the missing normal distributions; or the changing time series that have the same statistical momentums but their time-fluctuations differ. The data-set in the last case is out of the applicability of the usual analysis of variance (ANOVA). Furthermore, the ethical selection issues oppose the randomization, so the trial must be solved in a simple non-randomized design of single arm.

Due to the possible problems of RCT, some prospective clinical trials register the data in the single arm only. The most frequent reason is the targeted far advanced disease where the conventional curative therapies have failed, and no other treatment is available except for the newly tried one. In these cases, the best supportive care (BSC) could be applied [

The single arm design is popular in the Phase I process when safety data is collected. The goal in this phase of the study is to determine the toxicity, the side effects and the dose with dose-escalation process. The investigation of efficacy is not included in Phase I trials. The Phase II studies concentrate on efficacy of the applied safe process [

Lifetime studies have a surprising universality by the self-organizing [

Due to the self-similarity, most of the biological structures and processes can be described by a simple-power function (like P ( x ) = a x α ), where a and α are constants, and so the form of P ( x ) remains only multiplicated by the constant during any m magnification of x: P ( m x ) = a ( m x ) α = m α a x α = m α P ( x ) . This magnification process (scaling [

In consequence of the widely applicable universality behavior, the general ontogenic growth [

The real challenge is how we can reveal the hidden data in the single active arm in case of the missing randomization that forms reference in double arms. We have limited possibilities for mining the available information without a reference set, even though we know it well, that the information is in the data. The general self-similar behavior of the various tumors has different parametrization and so can be distinguished from each other. Consequently, the fitting to survival curves gives hints on how to extract information from the single arm alone.

Experimental data fit well to the empirical data in biology as well as it has been widely investigated and proven in solid-state reactions (precipitations, phase-transitions, aggregations, nucleation, growth, and others) [

The mortality can be approached by the fitting of different distributions [

In such advanced situations, when the malignancy is double refractory, the WF provides the best fit to the KM [

The approximation with a simple WF function in real cases of the KM non-parametric survival curve is not precise enough. The missing preciosity apparently contradicts the WF self-organized basis. When the survival is self-organized in the same way as we observed in all the biological processes, the fitting to the non-parametric KM has to show the self-similarity, because it is entirely rigorous due to the universality of the lifetime of the living systems and the growth dynamics of the tumors. The contradiction is due to the fact that the self-similar WF only fits to strictly homogeneous patients’ cohorts. WF parameters characterize the group of generally equal participating individuals, which is of course not acceptable. The KM represents a cohort group of patients with the equipoise of individuals made as ideal as possible, choosing explicit inclusion and exclusion criteria. Nevertheless, the choosing criteria in the situation when we are not able to apply RCT cannot be fixed well. The only inclusion is the failure of conventional curative treatments and the only exclusion is when the patient is in such terminal stage when any extra intervention could be fatal.

Due to the enormous variability of the living conditions (like social, diet, habits, etc.) and bio-variability of the individuals (like genetic variability, immune-variability, sensing-variability, etc.), any chosen cohort has inhomogeneities. However, it is possible to divide the cohort into more homogeneous subgroups than the full set of individuals, expecting that the fitting of the self-similar WF will be better by the growing homogeneity of the subgroup to which it is applied.

Usually, the groups of local responses (complete response (CR), partial response (PR), no change (NC), or progression of the disease (PD)) come into the center of the attention automatically at the finishing of the study. We could make similar subgrouping in systemic (lifetime, survival) measurements, and WF fit them individually. The measured data is the summary of the complete cohort with overlapping data in the experimental non-parametric KM estimates, containing the data of all the subgroups. For simplicity, using the same subgrouping as in local response, the subgroup of those patients who could be regarded is introduced as “cured” (CP), the subgroup for those whom the treatments helped (they as responding patients (RP), and the patients who had no benefit from the therapy as non-responding patients (NP). The KM in the real experiment measures is only the sum of these (in the same way as in the analysis of the local response). Fit WF for subgroups and sum it for fitting to complete KM:

W ( K M ) ( t ) = n C P N e − ( t t 0 ( C P ) ) n ( C P ) + n R P N e − ( t t 0 ( R P ) ) n ( R P ) + n N P N e − ( t t 0 ( N P ) ) n ( N P ) and n C P + n R P + n N P = N (1)

where n C P , n R P , n N P are the number of patients in CP, RP and NP groups, and N is the number of patients in the complete cohort. Note, that the difference between the CP and RP groups is only in the definition, just like in the local response between the CR and PR categories. Usually CP can be defined to the lifetime of the healthy group of patients in an age-normalized comparison. Consequently, for easy categorizing, usually the CP is the long, RP is the medium and NP is the short survival.

Simpler and more roboust WF regression received, when the fitting is divided into only two different functions [

W ( K M ) ( t ) = c R P e − ( t t 0 ( R P ) ) n ( R P ) + c N P e − ( t t 0 ( N P ) ) n ( N P ) (2)

where the Weibull parameters denoted by (RP) and (NP) superscripts, according to their sub-cohorts. Due to the complete set of patients, c R P + c N P = 1 , so (2) is:

W ( K M ) ( t ) = c R P e − ( t t 0 ( R P ) ) n ( R P ) + ( 1 − c R P ) e − ( t t 0 ( N P ) ) n ( N P ) (3)

Using the regression with division into only two subgroups by temperature development criteria was used by others [

The two-subgroup division has five parameters to fit. Looking for the only concentration parameter ( c = c R P ), some examples look like it is shown in

In that special case when the RP subgroup is cured, meaning no disease-specific death happen in the whole observation period (including the available follow-up time too), the e − ( t t 0 ( R P ) ) n ( R P ) ≅ 1 , so the WF-like curve will have the following form:

W ( K M ) ( t ) = c c u r e + ( 1 − c c u r e ) e − ( t t 0 ) n (4)

According to our general knowledge in oncology, the size of the malignant tumor certainly affects the lifespan of the cancerous individuals. The ratio of the actual basal metabolic rate (basal energy consumption) of the malignant lesion E ( t ) to the healthy one E 0 with the same volume modifies the survival distribution ( P S ( t ) ) which modifies the simple Weibull-related distribution as follows [

W S ( t ) = exp ( − E ( t ) E 0 ( t t 0 ) n ) . (5)

The modification of (5) can be interpreted as the change of the t 0 , and the scale factor of the Weibull function:

t ′ 0 = t 0 ( E ( t ) E 0 ) − 1 / n ⇒ W ′ ( t ) = exp ( − ( t t ′ 0 ) n ) (6)

Consequently, the scale-factor of WF (the time-factor of survival fit) contains the information about the tumor-growth in the way it was shown in (6). The original Weibull-based parametric approach of KM survival curve from the 0th stage gives a reference to the E 0 value.

On this basis we study the changes of the two Weibull-parameters by fitting the cumulative distribution curve to the hypothetical choice of the survival studies in different stages of the disease, which is directly connected to the inclusion criteria of the study. Also, we follow the change of parameters by the endpoint of the studies fitting to the finishing conditions. The mathematical fit of the curves uses the least square method by digital stepping of the functions in large number (n > 1000) steps and optimizing the square of Pearson parameter (maximize) and also the sum of squares of deviations (minimize). We used two software supports: the Excel (Microsoft 365) and the MathCad 15.

Using the hypothesis, that the self-similar WF follows the real bioprocesses in survival, the effect of the malignancy staging at the first diagnosis could be followed with the Weibull fitting method, hypothesizing, that the staging strongly correlates with the time of the first actual diagnosis in the same cohort of patients. Diseases discovered earlier have lower stages than the ones diagnosed later. First, we are dealing with the survival curves of the patients in the control arm (reference arm, which in principle could be placebo as well), so the treatment modification will be considered later.

The start of the treatment is not immediate. Even the most accurate and modern detection methods do not allow the diagnosis in a latent state. The earliest time when the first diagnosis can be made is only after the dormant (untraceable) period of the disease. The traces of the disease cannot be detectable by imaging (due to its lower sensitivity), but some blood-test could detect the signal of disseminated circulation cancer cells or its parts. Overall Stage Grouping uses stages 0, I, II, III, and IV to characterize the progression of cancer [^{−2} m range, which is about 1 cm^{3} volume, having already billions of tumor-cells. Supposing a cluster contains 30 cells (~3 cells in a diameter) and supposing it takes 100 days to double its size, the tumor will be in the preclinical (latent) state for approx. 8 years, without the existing malignant tumor being observable, but we assume the self-organized growth during this time-period too.

Considering the basic survival curve from the start of the malignant behavior even from a single “renegade cell” [

W b ( t ) = e − ( t t 0 ( b ) ) n b (7)

Following the staging of the tumor status with WF when the diagnosis is based on the development of the malignant lesion related to (5):

W S ( i ) ( t ) = exp ( − E i ( t ) E 0 ( t t 0 ( i ) ) n i ) ( i = I , II , III , IV stages ) (8)

Hence, according to (6), the measured t 0 ( i ) in subsequent stages from

t ′ 0 ( i ) = t 0 ( E i ( t ) E 0 ) − 1 / n i ⇒ W i ( t ) = exp ( − ( t t ′ 0 ( i ) ) n i ) ( i = I , II , III , IV stages ) (9)

Let us denote the time when the tumor is observed like in carcinoma in situ, by T 0 . Due to the supposed continuity of the tumor-growth from the latent to the observable stage, the WF fit could follow triple parametrization to the KM non-parametric estimate. In this case a location parameter is added to the shape and scale parameters:

W 0 ( t ) = e − ( t + T 0 t 0 ( 0 ) ) n 0 (10)

This gives a “truncation” possibility of this basic (Equation (7), hypothetical) overall survival plot (

Following the complete survival until the last event (or censoring) in the studied group of patients, the start of the study will be at the shifted time, which determines the truncations of the basic WF to its parts (

The survival studies of different stages could be regarded as studies in shifted time ( T i ), starting the observation of the patients (first diagnosis) a certain time later than the guessed start (stage 0) of the malignant process. The new start is of course regarded as a new study, considering again 100% of the patients who are involved in this stage, with a probability of 1. The truncated curves (

Screening could be misleading for survival evaluations because sometimes the elongation of overall survival with a certain time is an addition to the differences between the first diagnosis [

of the tumor, reorganizes the complete structure in the lesion, so the shape parameter also changes. Note, that normally different tumors can be detected in different stages. For example, most of the breast and cervical cancers are detected in the stages 0 or I, while lung cancer is usually detected in stage III or IV, depending on the observed symptoms or the accident screening without indicated complaints of the patient. Due to the developing technical conditions, the complete process depends on the historical time of the screening.

Considering T i , the shift for the studies in subsequent stages, we get:

W i ( t ) = e − ( t + T i t 0 ( i ) ) n i (11)

The T 0 is the start of the observational period: optimally the immediate treatment, or at least the watchful waiting (watch and wait, WAW period); when the treatment cannot be decided yet. For simplicity we consider the studies as time-to-event (TTE) data, where time is denoted from a starting point to a certain event, such as death. When the end of the study fixed differently, we must use the fit shown in (2). All studies start as new one, of course, there is no knowledge about the unmeasured early treatments; consequently, survival probability at the start of the treatment is 1, irrespective of when it started. We show the later starting points in the time-line of the disease in

We start counting the elapsing time from T i , by time-shift in (12). The complete time-scale is shifted by T i value. The number of patients at the starting of the trial is considered 100% for KM, consequently, the truncated “remains” must be normalized to 1 to be able to fit with WF fitted. Usually the cancer in T 0 does not cause symptoms for the patients. When the symptoms appear, and a

patient recognizes the problem, it is usually in a later stage, when a higher number of cancer cells are already present, or even when they have already been disseminated from the local site. The WF fittings to the truncated “remains” (not showing the carcinoma in situ 0th stage), are shown in

The curves in

logarithmic dependence on the T i late start time in

In reality, the real KM curve could be decomposed to at least two components like it is shown in (2). An example is shown in

The form of

The late (at a more serious stage) start of the treatment is not the only challenge in the evaluation. Another common challenge at the KM evaluation is the

end-time of the study. Most of the clinical studies have limited time for follow-up, so they are usually finished before all involved patients are deceased or censored, and they do not force the TTE condition. At the end of the study, a certain group of patients remains (patients at further risk, PFR), or patients are completely cured (PCC). Identifying the PCC group in the practical applications is very unprecise, and by definition, the PFR at five years point regarded as PCC. However, there are doubts about this strict limit [

P F R i N = exp ( − ( F i t 0 ) n ) (12)

where the P F R i values are patients that are alive (they are at risk, belonging to the actual PFR) at the early finish time when the actual study ends. When the study finishes before all events happen at F i , the patients at risk is P F R i , and the number of events (loss of patients due to death or censored) until this point will be: ( N − P F R i ). The finish of the study ( F l a s t ) is when a single patient remains at risk ( P F R = 1 ), and censored from the initial set of N individuals,

F l a s t = t 0 ( − ln ( 1 N ) ) 1 n (13)

According to the Hardin-Jones-Pauling’s (HJP) biostatistical theory [

F e n d = t 0 [ ( − ln ( 1 N ) ) 1 n + Γ ( 1 + 1 n ) ] (14)

The early finished studies, when a certain number of patients remain in risk are shown by an example in

The studies finishing early have a slight shift in t 0 when elongating them and the number of patients at risk decrease (

Both the two independent Weibull parameters change by inclusion criterial of staging. Both the shape and the scale factors are decreased when treatment starts later, which is natural. In case of an unchanged n shape-character, the decrease of the scale factor is less than in case of a changing n.

Using (9) we get:

t 0 ( i ) = t 0 ( E i ( t ) E 0 ) − 1 / n i ⇒ E i ( t ) = E 0 ( t 0 ( i ) t 0 ) − n i ( i = I , II , III , IV stages ) (15)

Expression (16) allows an approximating of the metabolic rate from the change of t 0 ( i ) by WF fit to various KM non-parametric estimates. Metabolic activity could be measured approximately by positron emission tomography (PET), evaluating the standardized uptake value (SUV) of the radiolabeled tracer

2-deoxy-2-[18F] fluoro-D-glucose (FDG) uptake in tumors in various stages at the start of the trial ( S U V i ), so:

( E i ( t ) E 0 ) = ( t 0 ( i ) t 0 ) − n i ⇒ ( S U V i ( t ) S U V 0 ) ≈ ( t 0 ( i ) t 0 ) − n i ( i = I , II , III , IV stages ) (16)

where S U V 0 is the FDG uptake of the neighboring healthy tissue. The metabolic ratio, calculated by ( t 0 ( i ) t 0 ) n i at the late start process above gives a quite accurate linear dependence from the T i late start time (

In this way we could also approximate the basic survival curve, when the PET is actually sensitive enough to measure cancer in situ lesions, supposing the time when the tumor starts to form in a microscopical region and its clusters are still undetectable with our present diagnostic methods.

The treatment of the chosen patient cohort is expected to change the KM of the active arm compared to the control arm, which is untreated with the same protocol, and formed from the same cohort. The changes of KM in active arm will modify the WF fit, too. The measured change of metabolic rate by SUV indicates the effect of the actual treatment. When the malignant tissue shows a lower metabolic rate (lower SUV ratio) the treatment regarded effective. The lower SUV has a longer scale parameter ( t 0 ) according to (17). In case of a successful treatment, the shape-parameter (n) decreases, “smooths” the probability of event with a longer, heavier tail.

The question is: how the situation changes by treatments in the study? The WF changes of course and the evaluation use this change to compare it to the reference (control arm) WF. There are different parametric estimations for the result. The first attempt is always the median survival, which looks undecided about the efficacy of the treatment in the measuring process. However, this single parameter is not nearly enough to see the complete picture. It is possible that the treatment is effective without the change of the median of the KM, while the distribution has a long tail; patients over the median lifetime live longer. for example

For the decision of the efficacy we must use an information parameter from the WF, an important parameter of a probability distribution: the Shannon-entropy ( S S h ) [

S S h ( n , t 0 ) = γ ( 1 − 1 n ) + ln ( t 0 n ) + 1 = S S h 1 ( n ) + S S h 2 ( t 0 ) (17)

where γ is the Euler-Mascheroni constant: γ ≅ 0.5772 , and

S S h 1 ( n ) = γ ( 1 − 1 n ) − ln ( n ) + 1 ; S S h 2 ( t 0 ) = ln ( t 0 ) (18)

The information source of S S h is produced by a stochastic data-source, like the probability distribution of the survival time. In the simple formulation, it refers to the amount of uncertainty about an event associated with a given probability distribution. At the probability of the survival, this directly means, that the decreasing entropy shows the increasing probability of death. The easiest way to decide the advantage of a treatment which changes the parameters of the WF, is with this parameter, because the survival is better when S S h is higher. It is due to the meaning of the entropy: a larger entropy means less information and a higher uncertainty of death. Visualizing it on the image of the pdf, it has more located peak when n grows, and its width is shrinking by t 0 , therefore both make death more definite. The growing n and decreasing t 0 both decrease the entropy, making the certainty of death higher. In the case of

The entropy evaluation in the case shown in

Interestingly, despite the more moderate decrease of the scale factor when the shape factor decreases in optimal fit, the Shannon entropy shows an advantage for these optimal WF sets, compared to the constantly fixed shape. The reason is that the patients with longer survival time are fit for the later start of the treatment and were selected by their other, less hazardous conditions than the others.

The Shannon entropy can be evaluated for late-start treatments (treatments in various stages of the tumor) like that it is shown in

for non-responding patients (group A), and for responding ones (group B) is shown in

The Shannon-entropy decreases the number of patients at risk linearly, due to the increasing certainty of death (

We assume, that no extra comorbidity developed (or at least it is controlled) over the elapsed time, consequently, we kept the original two parameters (shape and scale) unchanged, regarding the same cohort of patients participated; only their study started in different F i times. When we calculate with the developing comorbidities, then both parameters of WF will be changed in a direction that S S h decreases, indicating a higher certainty of the event.

We discussed a method of data mining from the single-arm clinical study without a reference group. We studied the possibility to open the hidden information in the measured Kaplan-Meier non-parametric estimate by the composition of proper parametrization of cumulative Weibull functions. We had shown the

changes of the two independent parameters of the Weibull cumulative distribution by the study design, namely their dependence on the inclusion criteria (staging) and the intended end-point (finishing). We had shown that the various studies with different inclusion and exclusion criteria and different endpoints could be well described by the decomposition method. The fit of these results to real studies in clinical applications will be shown in the next part of this series of articles.

This research was supported by the Hungarian Competitiveness and Excellence Program grant (NVKP_16-1-2016-0042).

The authors declare no conflicts of interest regarding the publication of this paper.

Szasz, A., Szigeti, G.P. and Szasz, M.A. (2020) Parametrization of Survival Measures (Part II): Single Arm Studies. International Journal of Clinical Medicine, 11, 348-373. https://doi.org/10.4236/ijcm.2020.115032