Original Article

Probabilistic modelling of pancreatic cancer survival: can Markov chains predict survival in stage IV pancreatic cancer?

Josie Currie¹, Geoffrey Currie^2,3 , Eric Rohren^2,3

¹Rural Medical School, University of New South Wales, Wagga Wagga, Australia; ²School of Dentistry & Medical Sciences, Charles Sturt University, Wagga Wagga, Australia; ³Department of Radiology, Baylor College of Medicine, Houston, TX, USA

Contributions: (I) Conception and design: J Currie, G Currie; (II) Administrative support: J Currie, G Currie; (III) Provision of study materials or patients: None; (IV) Collection and assembly of data: J Currie; (V) Data analysis and interpretation: All authors; (VI) Manuscript writing: All authors; (VII) Final approval of manuscript: All authors.

Correspondence to: Geoffrey Currie, BPharm, MMedRadSc(NuclMed), MAppMngt(health), MBA, PhD. School of Dentistry and Medical Sciences, Charles Sturt University, Locked Bag 588, Wagga Wagga 2678, Australia; Department of Radiology, Baylor College of Medicine, Houston, TX, USA. Email: gcurrie@csu.edu.au.

Background: Pancreatic cancer staging is used to predict prognosis accurately in early stages of disease, however, stage IV with fewer treatment options, is harder to define an accurate life expectancy. Various machine learning methods have been used to improve predictive accuracy of survival. Markov chains are another way to mathematically model a sequence of probability vectors or eigenvalues and could provide a simple yet accurate method for predicting pancreatic cancer survival. The aim of this investigation was to use matrices, eigenvalues and Markov chains to predict survival rates in pancreatic cancer patients based on stage, particularly stage IV.

Methods: Matrices and eigenvalues/eigenvectors were used to create transition coefficients that were subsequently feed into the Markov chain and modelling. Outcomes were compared to literature values and decision tree analysis.

Results: For all pancreatic patients, 4-week survival is 85% at the time of diagnosis using Markov modelling. The Markov modelling revealed that, for those with advanced disease (stage IV) at presentation, 4-week mortality is 13.3% for those where treatment is undertaken and 21.3% where treatment is not an option. Matched pairs t-test revealed that Markov modelling had a 0.798 correlation coefficient compared to decision tree analysis (R²=0.637) and similarly 0.804 with the published literature (R²=0.647).

Conclusions: The decision tree analysis provided modelling of survival at a more granular level and as a result, would be more suitable than Markov modelling or current models based on regression analysis for predicting survival for patients and their families.

Keywords: Markov chain; decision tree; survival; pancreatic cancer

Received: 17 April 2024; Accepted: 07 June 2024; Published online: 05 August 2024.

doi: 10.21037/apc-24-8

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Key findings

• Markov modelling is based on a number of assumptions that may not stand at the more granular sub-stage level of analysis. Despite accurate modelling across pancreatic cancer stages, Markov chains do not appear to provide additional insights at the sub-stage level of stage IV.

What is known and what is new?

• Markov chains have been used to model patient progression and predict survival in cancer patients. Modelling at a more granular level within disease stage IV has not been reported previously and has the potential to enhance patient care. The investigation revealed limitations in extending Markov modelling into the sub-stage level of patient analysis and confirmed superiority of the less complex decision tree analysis for the same.

What is the implication and what should change now?

• Oncology patients present to medical imaging with heightened vulnerability and varying degrees of uncertainty with respect to the path ahead. The ability to provide patients with more refined timelines does not change their outcomes but does afford them the opportunity for better informed decision-making. This approach is consistent with person-centred care but requires exploration of alternative modelling approaches to enhance insights at the more granular level of stage IV.

Introduction

Pancreatic cancer accounts for only 2% of all cancers but is responsible for 5% of cancer deaths (1-3). Symptoms are not obvious early which makes early diagnosis difficult and leaves many patients presenting with advanced metastatic disease (3-5). Surgery is not an option for the 80% of patients presenting with locally advanced or distant metastatic spread (6). Even with treatment, relapse is common for this typically aggressive cancer and 5-year survival is only 2–9% (2-4,6). While radiographic imaging plays an important role in the assessment of the pancreatic cancer patient, novel radiopharmaceuticals targeting the hallmarks of cancer have changed the diagnostic and therapeutic landscape in nuclear medicine. Indeed, neuroendocrine tumours represent the original prototype for theranostics.

Staging employs the TNM (tumour, nodes, metastases) approach and is used to predict prognosis (6). Treatment options also depend on the stage and grade of pancreatic cancer at presentation and include combinations of surgery, chemotherapy, radiation therapy, immunotherapy (including radioimmunotherapy), hormone therapy, peptide receptor radionuclide therapy and palliative care (1,3,7). While survival prediction of patients in early stages of disease is accurate within a wide window depending on therapies, stage IV in particular, with fewer treatment options, is harder to define an accurate life expectancy. For example, stage IV simply means the patient has distant metastases and has 3–7 months median life expectancy (6,8-10). Within that group, metastatic spread to specific organs or tissues may alter life expectancy and it is in these patients that more accurate prediction of life expectancy will be of most benefit.

Staging alone is not the only predictor of survival in pancreatic cancer. Older patients have poorer survival and treatments options also vary survival outcomes. Pancreatic cancer survival is also influenced, to varying degrees, by genetics, gender, ethnicity, marital status, histology, site of primary and sites of metastatic spread (8,10). The relationship between these variables and survival outcomes can be evaluated using conventional inferential statistical approaches and these multivariate approaches are the foundation of current survival predictions (7,11). The increasing digital footprint of patients provides richer data for computational analyses of survival. Various machine learning methods have been applied to understand the weighted and scaled combination of different variables to enhance survival prediction (7,11); quantifying relationships among variables not evident or intuitive to the human observer. For example, Wang et al. (12) used 86 different signatures (variables) and 76 artificial intelligence machine learning algorithms to predict 1-, 2-, and 3-year survival in pancreatic cancer patients. Predictive accuracy, despite complexity, was in the order of 60–75% depending on the subgroup.

Markov chains are another way to mathematically model a sequence of probability vectors or eigenvalues (13-15). If pancreatic cancer survival is considered from the perspective of a dynamic system comprised of variables that change over time, then the probability of a variable (survival) being in any given state (vector) can be predicted for a point in time (13,16). This stochastic process predicts a state based on the previously known state. Markov modelling of random process sequences (Markov chain) takes into account the dependencies among variables (14). In the case of pancreatic cancer, stage II is preceded by stage I, stage III is always preceded by stage II, and stage IV comes after stage III. Within stages, treatment options change from optimal to less optimal with disease progression and with that, a decline in probability of survival. Markov chains, named after Russian mathematician Andrei Markov (16), could provide a simple yet accurate method for predicting pancreatic cancer survival. Limited predictive insights associated with conventional inferential statistical approaches and Kaplan-Meier survival analysis mean that a probabilistic model using Markov chains may provide more accurate information on survival which could then allow better end of life planning for patients and their loved ones, particularly in stage IV; consistent with person centred healthcare.

Methods

The aim of this investigation was to use Markov chains to predict survival rates in pancreatic cancer patients based on stage, particularly stage IV. Matrices and eigenvalues/eigenvectors were used to create transition coefficients that were subsequently feed into the Markov chain and modelling. Outcomes were compared to literature values and decision tree analysis. Ethical considerations were taken into account but this literature-based enquiry was exempt from institutional ethics approval because no human data was collected.

Principal component analysis

When data is high dimensionality, like the variables that may influence cancer survival, there could be hundreds of variables with varying degrees of influence on the outcome of interest. Principal component analysis was used to identify the few variables that account for the greatest variability in survival (17) by assessing the recent literature for published survival predictors. Four recent publications were identified based on reliability of data and were used for analysis (6,9,10,18). In each of these published works, a large patient database was used to evaluate the hazard ratio (HR) of variables to determine which variables most strongly predicted overall survival (Table 1). A number of sociodemographic variables were omitted from principal components because reports varied, lacked statistical power or were made redundant by other included variables (e.g., ethnicity, gender and marital status). Primary site and grade were also excluded because both are made redundant by inclusion of disease stage as a principal component. The principal components for pancreatic cancer survival relate to stage of disease (Table 1) (6,9,10,18). Importantly, representativeness of data was assessed by inclusion of large multi-center analysis of databases [e.g., the Surveillance, Epidemiology and End Results (SEER) registry in the USA] and calibration and rationalization of that data against smaller single centre datasets. Large multi-center registries provide rich data but can suffer the effect of data variability. In this case, the principal component analysis and modelling data contained in Tables 1,2 and consistent and representative across large-multi-center registries and region specific single centre datasets (Table 3).

Table 1

Summary results of principal component analysis identified using previously published HR for OS

Sources	Component	Class	HR for OS	P value
Shi et al., 2022 and Li et al., 2022	Age	≥65 years	1.49	<0.01
	Stage	II	1.58	<0.01
		III	2.48	<0.01
		IV over III	1.31	<0.01
		IV	3.44	<0.01
Yao et al., 2022 and Liu et al. 2019	Metastases	Solitary of liver, lung, lymph nodes or peritoneum	1.51	<0.01
		Multiple including bone	1.36	0.03
		Multiple including brain	1.21	0.64
Shi et al., 2022, Li et al., 2022 and Yao et al., 2022	Radiation therapy	Yes	0.74	<0.01
	Radiation therapy	No	1.08	0.08
	Surgery	Yes	0.56	<0.01
	Surgery	No	2.15	<0.01
	Chemotherapy	Yes	0.27	<0.01
	Chemotherapy	No	1.82	<0.01

The principal components for further modelling relate to stage, sub-stage and treatment approaches. HR, hazard ratio; OS, overall survival.

Table 2

Summary of key data for the Markov chain and decision tree analysis (6,9,10,18-23)

Items	Stage I	Stage II	Stage III	Stage IV
HR for death	Reference	1.58	2.48	3.44
Incremental HR for death	–	–	Reference	1.31
Proportion of patients	0.08	0.21	0.16	0.55
No therapy (proportion)	0.32	0.08	0.19	0.32
No chemotherapy proportion (HR)	–	–	0.35 (1.82)	0.35 (1.82)
No surgery proportion (HR)	–	–	0.75 (2.15)	0.75 (2.15)
No radiation therapy proportion (HR)	–	–	0.87 (1.27)	0.87 (1.27)
1-month survival with resection (proportion)	0.96	0.96	0.90	0.90
1-month survival without resection (proportion)	0.86	0.86	0.63	0.63
Solitary metastases proportion (HR)	–	–	–	0.50 (1.51)
Multiple metastases proportion (HR)	–	–	–	0.20 (1.36)
Brain metastases proportion (HR)	–	–	–	0.0033 (1.21)

HR, hazard ratio.

Table 3

Breakdown of key data attributes of sources used to inform analysis

Studies	Study type	Source	Subject number	Collection period
Zhang et al., 2024	Retrospective	SEER national registry in USA	14,406	2010–2015
Zhang et al., 2023	Retrospective	SEER national registry in USA	2,709	2010–2015
Liu et al., 2019	Retrospective	Multi-center [4] registry in Taiwan	746	2010–2016
Yao et al., 2022	Retrospective	SEER national registry in USA	11,287	2010–2016
Shi et al., 2022	Retrospective	SEER national registry in USA	1,256	2000–2018
Li et al., 2022	Retrospective	Single-center study in China	625	2013–2017
Yu et al., 2015	Retrospective	SEER national registry in USA	13,131	2004–2011
Lee et al., 2018	Retrospective	Single-centre study in Korea	1,646	2007–2013
Huang et al., 2018	Retrospective	Multi-national investigation using multiple European and USA registries	125,183	2003–2014
Shakeel et al., 2020	Retrospective	Ontario (Canada) multi-center registry	6,437	2007–2015
Zhang et al., 2021	Retrospective	SEER national registry in USA	18,832	2010–2016

SEER, Surveillance, Epidemiology, and End Results.

Markov model

Mathematically, the challenge associated with pancreatic cancer survival modelling is that information about disease progression is generally expressed as a disease state (e.g., stage) at a specific point in time (13). Markov modelling is useful to estimate survival based on disease state. Consider the model outlined in Figure 1 using the data in Table 2. The model assumes stage IV in the sequence must pass through previous stages (even if early stages occur prior to presentation) and that spontaneous regression is not possible (14,24). At time n the state of the system S can be expressed as a state matrix S_n which represents the proportion of patients in a given state starting at time zero (S₀) and the state matrix takes the form (13):

$S_{n} = [\begin{matrix} - (μ_{1} + λ_{12}) \\ λ_{11} \\ 0 \\ 0 \end{matrix} \begin{matrix} λ_{12} \\ - (μ_{2} + λ_{23}) \\ λ_{22} \\ 0 \end{matrix} \begin{matrix} 0 \\ λ_{23} \\ - (μ_{3} + λ_{34}) \\ λ_{33} \end{matrix} \begin{matrix} 0 \\ 0 \\ λ_{34} \\ - (μ_{4}) + λ_{44} \end{matrix}]$ [1]

Figure 1 Schematic model of the disease states (stages) and transition rates for the Markov model simplified to have a single absorbing state rate (μ) for each state. The absorbing state refers to death. As with the previously cited matrix, the notation convention for the transition rate (λ) from stage I to stage II is λ₁₂ and thus, λ₁₁ refers to those commencing the step in state 1 (stage I) who remain in state 1 after the designated time period (step). In this case, a step is 4 weeks.

where, μ is the single absorbing state (death) rate for each state and λ is the transition rate.

Decision tree analysis

Decision tree analysis is a technique used to provide evidence-based problem solving (25,26). Decision tree analysis allows complex medical scenarios to be mathematically modelled and displayed in a simplified manner while removing subjectivity (25,26). There are other evidence-based approaches to decision making analysis (e.g., Bayes’ theorem and receiver operator characteristic curves) but decision tree analysis is used here due to its versatility to help populate the Markov chain (Figure 1). The data sources for the decision tree analysis are summarised in Tables 1,2.

The initial decision tree analysis (Figure 2) highlights a number of favourable utilities. Good stage I survival is influenced numerically by a very low proportion (0.08) of patients presenting with stage I pancreatic cancer. Good stage II survival is influenced numerically by a high proportion (0.92) of patients suitable for therapy. Poorer survival for stage IV is a product of the nature of late-stage pancreatic cancer, the high proportion of patients who first present at late stage (0.55) and the high proportion of patients not suitable for therapy (0.32). This decision tree analysis does, however, pool sub-stages within stage IV and given the purpose of this modelling was to improve survival prediction among the difficult stage IV cohort, further decision tree analysis was undertaken in isolation for stage IV (Figure 3).

Figure 2 Decision tree analysis representing conditional probabilities for the disease state today and for the disease state (probability of death) in 4 weeks. The overall probability of death within the 4-week window of 0.17 (17%) can be calculated as the sum of total deaths as a fraction of the 100,000 starting population. As demonstrated in Table 2, the shared conditional probabilities among sub-divisions of stage I and stage II justify pooling at a stage level but not at an outcome level.

Figure 3 Stage IV only decision tree analysis representing conditional probabilities for the disease state today and for the disease state in 4 weeks. The overall probability of death within the 4-week window of 0.24 (24%) can be calculated as the sum of total deaths as a fraction of the 100,000 starting population.

For those pancreatic cancer patients first presenting with stage I or stage II disease, if resection of tumour and therapy are an option, there is a 4-week mortality of just 4% for those under 65 years of age and 6% for those 65 years or older. For those where treatment is not an option (e.g., not available or unsuitable), 4-week mortality is 14% for those under 65 years and 21% for those 65 years or older. This observation is consistent with therapy reducing the tumour doubling rate by 70% (20). Understandably, these values are higher for stage III and stage IV pancreatic cancer. For those where therapy is suitable when patients first present with stage III or stage IV disease, 10% of those under 65 years and 15% of those 65 years and older will die in the 4 weeks after diagnosis. This increases to 37% for under 65 years and 55% for 65 years or older when therapy is not available or unsuitable. This will result in many more actual deaths because there is a predominance of patients who present with advanced disease.

Statistical analysis

Correlation was determined using a match pairs t-test and a P value less than 0.05 was considered significant.

Results

Markov chain calculations

Median survival for stage I, II and III pancreatic cancer are sufficiently long to allow planning, even with some variability in survival predictions. Stage IV pancreatic cancer has a typically short median survival with variability producing significant inaccuracies that, in effect, negatively impact the patient and their families (7,27). Stage I, II and III survival predictions are robust to variations in key variables, within an acceptable margin of error. A more detailed Markov model for stage I can be used to demonstrate the complexity of the process (Figure 4). Stage IV has significant compounding of variables shortening survival. For example, by definition stage IV pancreatic cancer patients have metastases but the extent and the specific organs involved substantially vary survival. While the main goal of this investigation was to delineate stage IV survival using Markov chains (advanced disease), each stage (I, II, III and IV) has been modelled.

Figure 4 Markov modelling for the first disease state (stage I pancreatic cancer) with transition rates demonstrating the potential complexity of the model.

For any given stage, the starting status is alive and thus the probability of a patient starting a stage of pancreatic cancer dying in the subsequent 4-week window can be expressed in the transition matrix (T) form where T = t_ij (the probability of state i moving to state j) or eigenvalue (13). Each column in the transition matrix represents the current state and each row represents the next state and the values (P) are the eigenvectors (13).

$\begin{array}{l} \begin{matrix} I & II & III & IV & Death \end{matrix} \\ T = [\begin{matrix} 0.928 & 0 & 0 & 0 & 0.072 \\ 0 & 0.952 & 0 & 0 & 0.048 \\ 0 & 0 & 0.849 & 0 & 0.151 \\ 0 & 0 & 0 & 0.814 & 0.186 \\ 0.072 & 0.048 & 0.151 & 0.186 & 0 \end{matrix}] \begin{matrix} I \\ II \\ III \\ IV \\ Death \end{matrix} \end{array}$ [2]

From Table 2;

$\begin{matrix} (0.32 \times 0.86) + (0.68 \times 0.96) = 0.928, 1 - 0.928 = 0.072 \\ (0.08 \times 0.86) + (0.92 \times 0.96) = 0.952, 1 - 0.952 = 0.048 \\ (0.19 \times 0.63) + (0.81 \times 0.90) = 0.849, 1 - 0.849 = 0.151 \\ (0.32 \times 0.63) + (0.68 \times 0.90) = 0.814, 1 - 0.814 = 0.186 \end{matrix}$ [3]

At time n the state of the system S can be expressed as a state matrix S_n which represents the proportion of patients in a given state starting at time zero (S₀), where μ is the single absorbing state (death) rate for each state and λ is the transition rate, with 4-week steps the matrix taking the form (13):

$S_{1} = [\begin{matrix} - (μ_{a} + μ_{b} + λ_{12}) & λ_{12} \\ λ_{11} + λ_{21} & - (μ_{a} + μ_{b} + λ_{23}) \end{matrix}]$ [4]

$S_{2} = [\begin{matrix} - (μ_{a} + μ_{b} + λ_{23}) & λ_{23} \\ λ_{22} + λ_{32} & - (μ_{a} + μ_{b} + λ_{34}) \end{matrix}]$ [5]

$S_{3} = [\begin{matrix} - (μ_{a} + μ_{b} + λ_{34}) & λ_{34} \\ λ_{33} + λ_{43} & - (μ_{a} + μ_{b}) + λ_{44} \end{matrix}]$ [6]

Pooling all stage IV pancreatic cancer patients does not provide the insights expected nor does it demonstrate a full understanding of the disease process. Stage IV alone represents its own Markov chain and can be modelled as such (Figure 5). A simple view using the data tabulated in Table 3 would generate the following matrices:

$T_{4} = [\begin{matrix} 1.0 & 0.9 \\ 0 & 0.1 \end{matrix}] for those with resection$ [7]

$T_{4} = [\begin{matrix} 1.0 & 0.63 \\ 0 & 0.37 \end{matrix}] for those without resection$ [8]

Figure 5 Markov modelling of stage IV pancreatic cancer as a Markov chain of sub-stages for those receiving treatment. For those not having resection μ₁ is 0.37.

Stage 4 is differentiated from earlier stages by the presence of distant metastases (mets) and disease progression is characterised by involvement of different organs. The most commonly reported solitary metastases are liver and lung while multiple metastases almost always involved liver or lung plus bone (8,9,18-21). The presence of bone metastases (multiple metastases) can be viewed as meeting the Markov properties of a new sub-stage. Other less common sites of metastases like brain, muscle and nerves can be classified as a subsequent sub-stage. This allows Markov modelling to predict 1 month survival. The state matrix takes the form (13):

$S_{4} = [\begin{matrix} - (μ_{1} + λ_{2}) & λ_{12} & 0 & 0 \\ λ_{11} & - (μ_{2} + λ_{23}) & λ_{23} & 0 \\ 0 & λ_{22} & - (μ_{3} + λ_{34}) & λ_{34} \\ 0 & 0 & λ_{33} & - (μ_{4}) + λ_{44} \end{matrix}]$ [9]

$S_{4} = [\begin{matrix} - 0.6 & 0.5 & 0 & 0 \\ 0.4 & - 0.55 & 0.4 & 0 \\ 0 & 0.45 & - 0.15 & 0.01 \\ 0 & 0 & 0.85 & 0.76 \end{matrix}]$ [10]

For any given stage, the starting status is alive and thus the probability of a patient starting a stage of pancreatic cancer dying in the subsequent month can be expressed in the transition matrix (T) form where T = t_ij (the probability of state j moving to state i). Each column in the transition matrix represents the current state and each row represents the next state and the values (P) are the eigenvectors. Mathematical resolution of such Markov chains is complex but matrices allow simpler calculations.

$\begin{array}{l} IV Solitary Multiple Multiple Death \\ mets plus bone other \\ mets mets \\ T = [\begin{matrix} - λ_{1} & λ_{1} & 0 & 0 & 0 \\ 0 & - λ_{2} & λ_{2} & 0 & 0 \\ 0 & 0 & - λ_{3} & λ_{3} & 0 \\ 0 & 0 & 0 & - λ_{4} & λ_{4} \\ 0 & 0 & 0 & 0 & - λ_{5} \end{matrix}] \begin{matrix} IV \\ Solitary mets \\ Multiple plus bone mets \\ Multiple other mets \\ Death \end{matrix} \end{array}$ [11]

$\begin{array}{l} IV Solitary Multiple Multiple Death \\ mets plus bone other \\ mets mets \\ \begin{matrix} T = \\ resected \end{matrix} [\begin{matrix} 0.50 & 0 & 0 & 0 \\ 0.40 & 0.40 & 0 & 0 \\ 0 & 0.45 & 0.01 & 0 \\ 0 & 0 & 0.85 & 0.88 \\ 0.10 & 0.15 & 0.14 & 0.12 \end{matrix}] \begin{matrix} IV \\ Solitary mets \\ Multiple plus bone mets \\ Multiple other mets \\ Death \end{matrix} \end{array}$ [12]

$\begin{array}{l} IV Solitary Multiple Multiple Death \\ mets plus bone other \\ mets mets \\ \begin{matrix} T = \\ unresected \end{matrix} [\begin{matrix} 0.50 & 0 & 0 & 0 \\ 0.13 & 0.40 & 0 & 0 \\ 0 & 0.45 & 0.01 & 0 \\ 0 & 0 & 0.85 & 0.88 \\ 0.37 & 0.15 & 0.14 & 0.12 \end{matrix}] \begin{matrix} IV \\ Solitary mets \\ Multiple plus bone mets \\ Multiple other mets \\ Death \end{matrix} \end{array}$ [13]

Matrix calculations and probability determination

The following Markov chain probability vectors have been calculated on the basis that the Markov property has been satisfied. For the model depicted in Figure 1 for stages I to IV for pancreatic cancer with death being the absorbing state, the matrix calculations are outlined below. The probability vector for the Markov model indicates that the probability of death in any given 4-week window for pancreatic patients is 15%.

$\begin{array}{l} Stochastic state Initial state \\ [\begin{matrix} 0.928 & 0 & 0 & 0 & 0 \\ 0 & 0.952 & 0 & 0 & 0 \\ 0 & 0 & 0.849 & 0 & 0 \\ 0 & 0 & 0 & 0.8 & 0 \\ 0.072 & 0.048 & 0.151 & 0.2 & 1 \end{matrix}] \times [\begin{matrix} 8 \\ 21 \\ 16 \\ 55 \\ 0 \end{matrix}] = \end{array}$ [14]