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ISSN: 2155-6180
Journal of Biometrics & Biostatistics
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Parametric Cancer Survival Analysis Based on the Boag Model: A Clinician?s View

Shunzo Maetani1* and John W Gamel2

1Tenri Institute of Medical Research, Tenri, Japan

2University of Louisville School of Medicine, Louisville KY, 40202, USA

*Corresponding Author:
Shunzo Maetani
Tenri Institute of Medical Research
200 Mishima-cho, Tenri
632-8552, Japan
Tel: +81-743-63-5611
Fax: +81-743-63-1530
Email: [email protected]

Received date: March 20, 2012; Accepted date: April 25, 2012; Published date: April 26, 2012

Citation: Maetani S, Gamel JW (2012) Parametric Cancer Survival Analysis Based on the Boag Model: A Clinician’s View. J Biomet Biostat S7-017. doi: 10.4172/2155-6180.S7-017

Copyright: © 2012 Maetani S, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract

As many cancer patients have recently been cured, it has become necessary in cancer survival analysis to distinguish between cure and delayed death, which make a great difference in survival benefit and quality of life. Also, cancer patients must be provided with relevant and comprehensible information to make optimal decisions. For this purpose, the Boag parametric analysis with a cured fraction has emerged as a relevant model. The authors evaluated this model compared with the Cox model using life-long follow-up data. The parameters of the Boag model provided the comprehensible information patients wish to obtain; particularly, the cure rate served as a useful measure of survival benefit. In contrast, the hazard ratio, a parameter of the Cox model, failed to distinguish cure from delayed death. The Boag model could be extended to regression analysis to evaluate the long-term effects of various factors, including cancer treatment. Also, it could be extended to predict the overall survival

curve and mean survival time using limited follow-up data. In conclusion, the Boag model offered a more relevant measure of the long-term benefit of cancer treatment and other factors than conventional methods, although an ideal model has yet to be developed

Keywords

Survival analysis; Boag model; Cox model; cure, Mean survival time; Patient-centered medicine

List of non-standard abbreviations

OS: overall survival curve

ALL: acute lymphoblastic leukemia

6-MP: 6-mercaptopurine

Introduction

When the majority of cancers were incurable, the statistical analysis of cancer survival data did not differ from that of other kinds of data except when it came to dealing with censored observations. This situation has changed since Cox [1] developed the proportional hazards model in 1972. His method is based on the assumption that at any point in time the hazard (instantaneous mortality) for one group remains proportional to the hazard for the other group. Thus, the hazard ratio between two groups and the closely associated log-rank statistic have been widely used to compare survival between groups [2]. Furthermore, using the hazard ratio as the dependent variable, Cox extended his model to regression analysis. This allows statisticians and researchers to evaluate the effects of cancer therapy, demographic and clinicopathological findings, etc., on the hazard ratio.

Recently, with progress in the diagnosis and treatment of cancer, an increasing number of cancer patients have been cured. It has now become necessary to distinguish between the two primary outcomes, cure and delayed death. Another recent advance is patient-centered medicine, in which patients play a major role in selecting their optimal treatment. To achieve this goal, we must make the relevant information clear to patients and physicians alike. Thus, the cure model pioneered by Boag [3] has emerged as a relevant survival model. In this paper, using life-long follow-up data, we evaluate the Boag model and its extensions compared with the Cox model. The authors’ goal is to present findings from the clinician’s point of view.

Cure and Delayed Death

When diagnosed with cancer, patients often ask, “How great is my chance of being cured?” and “If I am not cured, how long will I live?” Whether or not cure is achieved makes a great difference in both the quality of life of patients and their survival benefit, since, if cured, patients are saved from the physical and mental suffering due to relapse and can gain additional decades of life. For children, the average gain may exceed 50 years. Thus, in survival analysis one has to distinguish between therapies that cure patients and those that only delay death from cancer. Unfortunately, this basic question has not been addressed by conventional survival methods based on the Cox model, which has prevailed for the past four decades. This issue was brought up by Gamel et al. [4,5] and other investigators [6-8]. Cox himself [9], commenting on the authors’editorial [10] in Surgical Oncology, acknowledges the limitations of his model in the study of long-term survival.

Cancer Progression

Cancer is a chronic disease that progresses step by step in a predictable manner. It differs from acute illnesses such as cerebral infarction or aneurysmal rupture, in which death (event) may occur at any time. In contrast, cancer patients usually have event-free intervals near the beginning of a study. This is also true for patients with advanced cancers, since imminently fatal cases usually are not included in the studies (Figure 1). This difference between cancer and acute illnesses is important in formulating an appropriate survival model.

biometrics-biostatistics-event-time-in-cancer

Figure 1: Distribution of event time in cancer and acute illness.

If a patient has a longer time to event (failure time) than another patient, the former is generally considered to fare better than the latter. However, unless patients are cured, they have to experience a variety of disagreeable symptoms during this period, including pain, dyspnea, nausea, vomiting, general weakness, etc., all of which worsen with time. Some of these symptoms result from the therapy. Except for pain management, these symptoms are difficult to control and, in some cases, patients may even wish to die early. Unfortunately, their wish is not realized until after event-free interval.

In a study published by Silvestri et al. [11], the coauthors interviewed 81 patients with stage III to IV non-small cell lung cancer who previously received chemotherapy. Given that a hypothetical subject has metastatic non-small cell lung cancer, patients are asked how they value the trade-off between the life-prolonging effect of chemotherapy and its toxicities. One of the questions is whether the subject would receive cisplatin-based chemotherapy if it prolongs his or her life by 3 months. Only 18 (22%) choose chemotherapy. When asked whether the subject would choose chemotherapy if it reduces symptoms without prolonging life, 55 patients (68%) are willing to receive chemotherapy. Thus, the prolongation of survival time from cancer therapy does not necessarily mean a benefit to the patients. To solve this dilemma, we need a better instrument for measuring the long-term impact of various treatments.

Boag Parametric Cure Model

In 1949 Boag proposed a parametric survival model that provides desirable information needed for patients to select the best treatment. He assumes that in a group of patients with a specific cancer, a fraction c is permanently cured of their disease. In addition, he assumes that the log failure time of the uncured patients follows a normal distribution with mean m and variance s2. By using the maximum likelihood method, he estimates the three parameters (c,m,s) manually. Thus, for the above patient, the chance of cure and median survival time are estimated as c and exp (m), respectively. The parametric survival curve based on the Boag parameters (disease-specific survival curve) differs from the usual survival curve (overall survival curve, or OS) in that only death from the disease is counted as an event (failure), while deaths from other causes are coded as censored at the time of death. This curve is crucial in measuring the long-term effect of cancer therapy, since it can distinguish therapies that cure the disease from those that merely prolong the time to death from cancer.

It must be noted, however, that a high cure rate does not always assure a prolonged survival. Patients, even if cured of the original cancer, will eventually succumb to death from other causes, as is the fate of all living things. Such deaths are expected to be more common in older patients or those living in countries where the life expectancy of the general population is relatively short. Thus, in determining the long-term impact of cancer treatment it is important to consider deaths from all causes, not just the cancer under study. For this purpose, the OS is useful.

Extensions of the Boag Model to Estimation of Overall Survival Curve and Mean Survival

The OS is estimated from two curves based on the assumption that disease-specific death is statistically independent of death from all other disease-unrelated causes (competing independent risk model [12]). Under this assumption, the survival rate of OS at a given point in time is the product of the survival rates of the disease-specific and disease-unrelated survival curves at the same point in time. The Boag parametric curve is substituted for the disease-specific curve, and the disease-unrelated survival curve is approximated by the survival curve of the contemporaries who match the patients for age and sex [13,14]. The latter curve is obtained from the life tables of the appropriate nation [15]. If a substantial portion of patients die from surgery or early chemotherapy, the product of the two survival rates must be further multiplied by (1- therapy-related mortality) in order to obtain the overall survival rate. Once the entire length of OS is obtained, the mean survival time is calculated as the area under the curve and may be used for cost-effectiveness analysis and other economic studies.

Extension of the Boag Model to Regression Analysis

In survival analysis, it is important to evaluate the effects of therapy and other prognostic factors on the Boag parameters, particularly their effects on cure rate. This is done by extending the Boag model to three regressions, whose dependent variables are cure rate (c ), and mean (m) and SD (s) of log failure time [16].

Thus,

Where the ci, mi, and si represent the parameters of the model (regression constant and coefficients) and xi represents the covariates of a given patient (i = 1,2,...p). In contrast, note that in the Cox regression, the dependent variable is the hazard ratio.

To evaluate the effect of a treatment, the regression of cure rate is important. On the other hand, for the reasons explained later, it is difficult to evaluate its direct effect on failure time unless the treatment has no effect on the cure rate.

Validation of the Boag Model and its Extensions

The parametric survival model has an advantage of extrapolating survival beyond available follow-up. In other words, it can predict the future survival curve. However, its validity needs to be confirmed in advance using life-long follow-up data. Fortunately, such data were provided by the Cancer Institute Hospital in Tokyo. The authors used only early data consisting of 3,597 gastric cancer patients who were consecutively operated on from 1950 through 1969 and were followed up for 30 to 50 years; death was confirmed in 97% of patients. To predict OS, the outcome of each patient was checked at 5 years after surgery, and only death from the disease occurring at 5 years or earlier was counted as an event, while survival times longer than 5 years were censored at 5 years. Then, the Boag model was applied to these limited follow-up data and extrapolated beyond 5 years; the mean survival was predicted using the competing independent risk model. The validity of the model was assessed by fitting the predicted OS to the observed OS (Kaplan-Meier survival) and by comparing the predicted mean survival with the observed one, which was derived from the full follow-up data.

In Figure 2, the observed and predicted OSs of the whole group are compared. The two curves show an adequate overlap although the predicted survival rate was slightly higher than the observed one after 15 years. In Figure 3, the same patients are classified into three subgroups by age. The reason why age was used for classification is that with increasing year it has a greater impact on OS than other prognostic factors, so that the difference between the predicted and observed curves is more clearly seen. To predict OS at 5 years extensions of the two models were used, the Kalbfleish & Prentice method [17] based on the Cox model (A) and the competing independent risk model based on the Boag model (B). Although the Cox model uses the full-follow-up data to derive the base-line survival curve, the predicted OSs based on this model deviate more from the corresponding Kaplan-Meier curves than those based on the Boag model.

biometrics-biostatistics-overall-survival-curve

Figure 2: Estimation of the overall survival curve based on the Boag model and competing independent risk model.
OS: overall survival

biometrics-biostatistics-overall-survival-curve-models

Figure 3: Estimation of the overall survival curve based on the two models.

In Figure 4, the patients were classified into 50 subgroups by demographic and prognostic factors. Their mean survival times were predicted at 5 years by the competing independent risk model, and correlated with the observed mean survival [14]. Although the model slightly overestimated the mean survival, the predicted survival was reasonably close to the diagonal equality line irrespective of the length of mean survival and the prognostic factors by which the patients were classified into subgroups.

biometrics-biostatistics-predicted-mean-survival

Figure 4: Regression of the predicted mean survival on observed mean survival.

Evaluation of Chemotherapy in Acute Lymphoblastic Leukemia

The fact that different chemotherapeutics have different effects on survival curves is demonstrated in children with acute lymphoblastic leukemia (ALL). Figure 5 shows the OSs of ALL patients treated at different years with different regimens [18,19]. Before 1960, the survival curves gradually shift rightwards with time, indicating prolonged failure time, but all curves fall almost to zero. After 1960, the tails of the curves progressively move upwards in parallel with the time axis, indicating increased cure rates.

biometrics-biostatistics-chemotherapy-in-childhood-acute

Figure 5: Results of chemotherapy in childhood acute lymphoblastic leukemia in different calendar years (reproduced by permission from Pedia Clin N Amer18 and N Engl J Med19).

We compared the Cox and Boag models using the data of Freireich et al. [20], who performed a randomized clinical trial evaluating the effect of 6-mercaptopurine (6-MP) versus a placebo on the steroid-induced remission of ALL patients. The results showed that using the Cox model, the hazard ratio of 6MP versus the placebo was 0.22 (95%CI: 0.10 to 0.49), which is commonly interpreted as indicating that 6MP would prevent relapse in 78% of placebo-treated patients (Figure 6). This result does not agree with the clinical observation that before 1960 the disease is almost always fatal [21]. The Boag model also showed that 6-MP failed to prevent relapse, but extended the median remission period about 3.8 times, (95% CI: 2.07 to 7.15), or from 1.43 to 5.67 months (Figure 7).

biometrics-biostatistics-effect-of-6-mercaptopurine

Figure 6: Effect of 6-mercaptopurine on steroid induced remission in acute lymphoblastic leukemia20 (Kaplan-Meier curve).

biometrics-biostatistics-steroid-nduced-remission

Figure 7: Effect of 6-MP on steroid-nduced remission in acute lymphoblastic leukemia20 (Boag parametric curve).

Effects of D1 versus D2 lymphadenectomy in resection of gastric cancer

What follows is another example of a randomized clinical trial which the authors studied using the conventional method and the Boag model. From 1989 to 1993, the Dutch Gastric Cancer Group compared the results of limited lymphnode dissection (D1) with extended dissection in a total of 711 patients with adenocarcinoma of the stomach who underwent curative gastrectomy. The operations were performed under the instruction of a Japanese surgeon [22]. The results show that the D2 resection was associated with a significantly higher mortality (P=0.004) and morbidity (P<0.001) and a significantly longer hospital stay (P<0.001), but there was no significant difference in 5-year survival; the hazard ratio was not significantly lower than 1. The cumulative risk of relapse was not significant, either. The authors conclude that their results do not support the routine use of D2 resection advocated by Japanese surgeons.

In 1999, the authors were given the opportunity to analyze the same Dutch group data using the Boag model. The results showed that the cure rate after D2 dissection was significantly higher than after D1 dissection with a difference of 11.5% (Figure 8). In 2010, after a median follow-up of 15 years, the Dutch group reached a different conclusion, stating that the D2 procedure was associated with significant locoregional recurrence and higher gastric-cancer-related death rates than D1 surgery [23]. The death rates were 48% and 37%, respectively, with a difference of 11%, which is close to what the authors estimated 10 years ago.

biometrics-biostatistics-risk-cumulative-relapse

Figure 8: Observed and Predicted Risk of Cumulative Relapse after D1 AND D2. Smooth black curves are predicted by the Boag model (Data provided by the Dutch Gastric Cancer Group).

It is noteworthy that after D2 dissection, the mean log failure time of the non-cured fraction was shorter than after the D1, although the difference (0.201) was not statistically significant (95% CI:-0.027 to 0.428). This result is contrary to the authors’ intuitive expectation that treatment which increases the size of the cured fraction should also prolong the lives of non-cured patients. However, this conflicting result may be explained by a report of radical re-excision of recurrent rectal cancer [24]; long-term cure is achieved only in patients with slow-growing tumors whose failure time is longer. If such prognostically favorable patients move from the non-cured fraction to the cured fraction, the mean failure time of the remaining non-cured fraction is shorter (Figure 9).

biometrics-biostatistics-migration-of-patients

Figure 9: Migration of patients from the non-cured fraction to the cured fraction after D2.

Discussion

Cancer has two distinct effects on its victims: lowering the quality of their lives, and shortening the length of their survival. Many fear the former more than the latter. In either case, curative treatment offers much greater survival benefit than palliative treatment, although curative treatments may sometimes cause serious complications. If the Boag model is valid, its parameters allow us to estimate the percent of patients cured in a given group. In theory, the parameter of the Cox model (hazard ratio) is also related to the curarbility of the disease. It is generally believed that if the hazard ratio for a new treatment group versus a control group is 0.7, then 30% (1-0.7) of the deaths occurring in the control group are prevented by the new treatment. (For example, see the Glossary of Clinical Evidence published yearly by the British Medical Journal). Romond et al. [25] states in the New England Journal of Medicine thatTrastuzumab therapy for breast cancer is associated with a 33 % reduction in the risk of death when the hazard ratio is 0.67. Giann, et al. [26] also describe their results in the Lancet as a 41% reduction in the risk of recurrence, progression or death when the hazard ratio is 0.59 with the addition of trastuzumab. Such results, however, have not been verified by long-term observations. Contrary to this common interpretation, the trial of Freireich et al. showed that virtually all patients with or without 6-MP therapy die, although the hazard ratio of 6-MP to the placebo is 0.22.

Peto et al. [2] derives a somewhat different interpretation of the hazard ratio which they call the average death rate ratio. According to them, a hazard ratio of 0.6 very crudely suggests that the new therapy (Busulphan) prevents or delays about 40% of the deaths that would occur with radiotherapy alone. If this is indeed a more precise interpretation of the hazard ratio, it should not be used as a measure of cure, and the editors of the top journals should encourage the authors to use it properly. Kay [27] states that the hazard ratio, like the odds ratio, is not only difficult to explain, but its use and interpretation continue to confuse both statisticians and nonstatisticians.

The Boag model has also been the target of much criticism. For one thing, the validity of its parameters, especially the cure rate, has seldom been confirmed by life-long follow-up data. Other arguments against the Boag model are: 1) its lognormal distribution is only one of the possible distributions and others such as exponential, Weibull, gamma, etc. may give different estimates of the cure rate, and 2) the survival curve should not be extrapolated beyond the available data. The authors have selected the Boag model because its hazard curve follows the logical pattern that clinicians expect; it starts at 0, rises to a peak, and declines slowly towards 0. Such a hazard pattern is seen only with lognormal or loglogistic distribution; which are very similar and may be used interchangeably. Should early deaths from disease occur, preventing an adequate fit of the lognormal model, then therapy-related deaths may be misclassified as such, and should be classified as censored cases. It should be noted that the lognormal model sustains its advantage over nonparametric methods even when applied to simulated data that follow other distributions such as the log-logistic or Weibull [5].

As for extrapolation of the survival curve, the authors combined the Boag model with the competing independent risk model, and could roughly predict the entire pattern of OS (up to 50 years) from 5-year follow-up data. Obviously, not all disease-unrelated deaths are independent of death from disease. For example, gastric cancer patients receiving blood transfusion show much higher disease-unrelated mortality than sex- and age-matched contemporaries [28]. Since blood transfusion (particularly transfusion of HV-positive blood) was overused in the past, it may be the major reason why the observed OS was slightly lower than the predicted curve. If such a reason is found and characterized, the results can be adjusted. The authors suggest that in cancer survival analysis, extrapolation beyond the available data can be based on sound logic, just as weather experts can predict the paths of hurricanes.

In conclusion, despite the increasing demand for patient-centered medicine, survival information conveyed to patients is still difficult to understand, and is seldom based on life-long observations. Although an ideal model has yet to be developed, the Boag parametric cure model may provide better long-term information than the conventional models.

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