Modelling and performance analysis of clinical pathways using the stochastic process algebra PEPA
- Xian Yang†1,
- Rui Han†1,
- Yike Guo1Email author,
- Jeremy Bradley†1,
- Benita Cox†2,
- Robert Dickinson†3 and
- Richard Kitney†3
© Yang etal.; licensee BioMed Central Ltd. 2012
Published: 7 September 2012
Hospitals nowadays have to serve numerous patients with limited medical staff and equipment while maintaining healthcare quality. Clinical pathway informatics is regarded as an efficient way to solve a series of hospital challenges. To date, conventional research lacks a mathematical model to describe clinical pathways. Existing vague descriptions cannot fully capture the complexities accurately in clinical pathways and hinders the effective management and further optimization of clinical pathways.
Given this motivation, this paper presents a clinical pathway management platform, the Imperial Clinical Pathway Analyzer (ICPA). By extending the stochastic model performance evaluation process algebra (PEPA), ICPA introduces a clinical-pathway-specific model: clinical pathway PEPA (CPP). ICPA can simulate stochastic behaviours of a clinical pathway by extracting information from public clinical databases and other related documents using CPP. Thus, the performance of this clinical pathway, including its throughput, resource utilisation and passage time can be quantitatively analysed.
A typical clinical pathway on stroke extracted from a UK hospital is used to illustrate the effectiveness of ICPA. Three application scenarios are tested using ICPA: 1) redundant resources are identified and removed, thus the number of patients being served is maintained with less cost; 2) the patient passage time is estimated, providing the likelihood that patients can leave hospital within a specific period; 3) the maximum number of input patients are found, helping hospitals to decide whether they can serve more patients with the existing resource allocation.
ICPA is an effective platform for clinical pathway management: 1) ICPA can describe a variety of components (state, activity, resource and constraints) in a clinical pathway, thus facilitating the proper understanding of complexities involved in it; 2) ICPA supports the performance analysis of clinical pathway, thereby assisting hospitals to effectively manage time and resources in clinical pathway.
Today, hospitals are asked to serve more and more patients while maintaining the quality of healthcare with limited medical staff and equipment. This situation causes many serious problems, including overcrowded emergency departments, delayed treatment of urgent patients, long waiting time and decreasing satisfaction of both doctors and patients . Within such a context, it becomes essential to apply information and communications technology (ICT) to achieve more efficient hospital management. Health informatics (also named clinical informatics) applies ICT to healthcare and biomedicine for promoting public health, facilitating hospital management and reducing healthcare cost . Among its various branches, clinical pathway, emerging in the 1980s , is a popular tool to outline the sequence and timing of actions necessary to a desired outcome with optimal efficiency . The sequence of clinical actions performed by a multidisciplinary team moves a patient with a specific diagnosis progressively through a clinical experience to a desired healthcare effect [4, 5].
A clinical pathway management platform, Imperial Clinical Pathway Analyser (ICPA), is introduced for quantitatively analysing clinical pathway. This platform can construct models of clinical pathway using existing clinical data and conducts performance analysis based on these models. The analysis results can benefit hospitals by providing crucial information for clinical pathway management.
Developed from performance evaluation process algebra (PEPA) , a stochastic model clinical pathway PEPA (CPP) is introduced to accurately describe different aspects of a clinical pathway, including its state transitions and treatment activities as well as their associated resources and constraints.
Using CPP, performance analysis of a clinical pathway can be conducted. The analysis results provide a variety of useful information in the clinical pathway. Firstly, resource utilisation can help hospitals to optimise resource allocation. Secondly, the passage time under different patient inputs shows patients' expected residing time in hospitals. Thirdly, the maximum number of input patients reveals the capacity of hospitals.
A stroke clinical pathway obtained from Charing Cross Hospital of Imperial College London is demonstrated. We choose this example for medical and economic reasons. Firstly, stroke is a typical acute disease and the third most common cause of death worldwide. Hence any delay in treatment may result in severe disability [15, 16]. Secondly, according to a recent National Audit Office report, 4-6% of the total NHS expenditure in the UK is spent on stroke treatment. It is predicted that the better management of stroke care can bring £20 m in annual savings, 550 fewer deaths and 1700 fewer cases of disability in the UK [17, 18].
First, an ambulance scheduled by London Ambulance Service (LAS) picks up the patient, where a Fitness and Anthropometric Scoring Template (FAST) test is carried out.
The patient is sent to the accident and emergency (A&E) resuscitation department and referred to the stroke team, if the FAST test shows s/he potentially has a stroke. Otherwise s/he is sent to the normal emergency department.
Next, if the patient is further diagnosed to have a stroke, s/he is sent to the CT department; otherwise s/he is transferred to the normal A&E department.
If a CT scan shows the patient does not have a stroke, s/he will be switched to the non-stroke treatment department. If the patient has a haemorrhagic stroke, s/he is transferred to the Hyper-Acute Stroke Unit (HASU). If the patient has an ischemic stroke, clinicians need to further determine whether the patient needs thrombolysis therapy.
Afterwards, the patient is sent to the HASU to be cared by occupational therapists for two or three days.
Finally, the patient is discharged from hospital, and the community therapy team takes over.
There are many uncertainties in the clinical pathway, so that a model may need to simulate pathways in a stochastic manner. For example, the stroke clinical pathway in Figure 1 has a changing number of incoming patients, and the time spent in each department for different patients may vary significantly. Patients with different diagnoses will be transferred to different departments, taking different routes through the pathway. Therefore a modelling method, which can cope with these complexities, is required.
Performance analysis for different scenarios should be supported. For example, patients would be interested in the expected time they would spend in hospital, while hospitals need to estimate the maximum number of patients they can treat every day and manage the clinical pathway in a cost-effective way. Thus a clinical pathway management platform, which can explicitly evaluate the clinical pathway performance, such as passage time and throughput, would be of interest.
The general impact of information technology on the clinical pathway like the influence of IT support on the satisfaction of both patients and medical staff is studied [8–10]. The clinical pathway can be implemented manually on paper  or electronically . With a good understanding of the clinical pathway, a set of evidence-based recommendations forming guidelines for clinical practice can be developed to both formalize and optimize the care process . Thus IT techniques embedded into clinical pathways can efficiently decrease undesired practice variability and improve clinician performance.
Some researchers further propose methods of modelling the clinical pathway. By using quantitative models, the influence of individual resources on overall pathway performance can be evaluated directly. However, little work has been done in the area of modelling clinical pathways with the aim of quantitatively improving system performance. Although in , a pathway model to facilitate the stroke care planning process is introduced, without a mathematical representation, the model performance cannot be shown explicitly. The work has been done in  applies the workflow graph to model the clinical pathway. Using this model, a similarity function is proposed to evaluate the temporal equivalence of two clinical pathways in order to reduce a complex pathway scenario into a simple one. In , ontology is used to describe the clinical pathway. This model allows the hierarchical representation of the clinical pathway, e.g., a clinical pathway can be described as a combination of a high-level outcome flow and a detailed workflow with care time constraints. However, the work done in  and  cannot build a model to explicitly measure the performance of the clinical pathway, such as throughput and passage time. Some researchers focus on analysing time and resource information in clinical pathway using stochastic Petri net. In , Rui et al. introduce the Probabilistic Time Constraint WorkFlow Nets (PTCWF-nets) to model a process. A static analysis method is then proposed to analyse each activity's probability of meeting its time constraints (e.g., deadline) in a process before the process is actually executed. In addition, they develop a dynamic method during the execution of the process . This method can update remaining activities' probabilities of successful execution whenever some activities are completed and their actual durations are known. Time schedulability is only one aspect of performance analysis in clinical pathway and it is closely related to another aspect: resource analysis. In , they further apply PTCWF-nets to manage resources. Their approach can schedule a clinical pathway among multiple available resources and allocate each activity to an optimal resource, i.e., the resource that has the highest probability to finish the activity in time. Authors in  also attempt to model clinical pathway using Petri net. They introduce performance trees to provide a standard unifying framework for expressing performance measures and performance requirements. Benefit from performance trees, the Petri net model for clinical pathway can provide estimation of steady state distribution and passage time.
Besides the work has been done to model the clinical pathway using Petri net, process algebra is also introduced to formally specify the interactions of different entities in the clinical pathway . Process algebra is a mathematical framework used to describe a complex parallel system. In this framework, both the behaviour and properties of the system are described in the form of algebra, facilitating accurate definition and rigorous reasoning about the system in mathematics. PEPA is an enhanced process algebra mainly used to describe and analyse the performance of concurrent systems . It inherits most characteristics of process calculus while incorporating features to specify a stochastic model, which potentially behaves as a continuous time Markov process. Comparing PEPA with other frequently used modelling tools, a queuing network offers compositionality but lacks formal definition, while a Petri net has formal definition without good compositionality. Thus in paper , PEPA is used to model the healthcare system. Based on PEPA, the execution duration and throughput of the clinical pathway can be evaluated. However, the role of resources, which constrain the activities and further limit pathway outcome, is not explicitly shown in this model. Therefore, this paper proposes a general clinical pathway model based on the CPP to analyse the performance of system and optimize pathway output.
The remainder of this paper is organised as follows: the Method Section introduces the architecture of ICPA, defines the CPP model, and presents key theories in conducting performance analysis on clinical pathway; the Results Section reports our experimental tests of the CPP model's effectiveness; finally, the Conclusion Section summarises our work.
The core element of ICPA is the CPP modelling method. In ICPA the constructed model, whose modelling language should follow the rules of PEPA, is first checked by the CPP Model Checker. By applying the CPP modelling method, the clinical pathway can be modelled in a stochastic manner where the time duration of each activity in the disease treatment process is a variable. Furthermore, the CPP method supports multiple parallel patients, enabling competition among patients for the same treatment resources. As the underlying stochastic model of CPP is a continuous time Markov Chain, the steady state distribution and an estimation of passage time can be produced. In the following subsections, technical details of the CPP modelling method will be discussed.
Definition of CPP
Inheriting and developed from PEPA, the basic elements of CPP are state components, resource components and activities. A state component represents the status of a patient by showing in which department the patient is being treated. A resource component is used to specify the state of each resource, which can be either busy or idle. The third component is the activity, which decides transitions between different state components. As each activity relies on one or more resources, the state transition can only take place when the associated resource components are in their idle states. Therefore, the number of resources constrains the frequency of state transitions and limits the system throughput. The model defined by CPP which represents states and resources separately is suitable for resource optimization and status monitoring of patients. Definitions of CPP are as follows.
Definition 1. In CPP, the clinical pathway is represented by a five-tuple, < S, R, Act, C, F C >, where:
S is a finite set of states s ∈ S, showing places that patients are being treated;
R is a finite set of resources, r ∈ R, required during treatment;
Act is a finite set of activities a ∈ Act; Each activity, a, is represented by a two-tuple (α, rate), where α is the action type and rate is one over the mean value of execution duration which is an exponentially distributed random variable;
C is a set of constraints c ∈ C;
F C is a set of functions that determine action rate: rate = f (cr 1, cr 2, . . . , c rn ) where f ∈ F C , c ri ∈ C and 1 ≤ i ≤ n.
For example, means that the component P becomes Q with the completion of the activity (α, rate). The expression P + Q represents that the system can behave either as P or Q. It enables all the activities of P and Q, and the first completed activity determines how the system behaves. The cooperation operator in the expression forms the basis of composition and can specify two components working cooperatively with shared activities defined in L.
Three key parts of the CPP model
State definition The state definition part shows how patients proceed through healthcare system. It consists of multiple state components representing places that patients are being treated. Based on their corresponding pathology states, patients with different diagnosis results are transferred to various departments. Two related state components are connected by sequential and choice operators. The sequential operator shows the time sequence of states, while the choice operator represents competition between two states. Here is an example of the state component definition:It shows that patient can move from the state Patientplace 1to Patientplace 2or Patientplace 3with probabilities of ρplace 2and ρplace 3after the completion of the activity whose action type is α.(2)
Resource specification The resource specification part defines all resources including equipment and medical staff required by state components. Transition between different state components requires the availability of corresponding resources. Assume the resource Resourceplace 1needed by the state component Patientplace 1is defined as
where Resource_busyplace 1represents that the resource is currently busy in completing the activity whose action type is α, while Resource _idleplace 1shows the resource is currently idle waiting for the completion of activity whose action type is β. The activities with action type of β and α can occur in:where the state Patientplace 0is the previous state of Patientplace 1and Patientplace 2is the next state of Patientplace 1. With completion of the activity whose action type is β, patients can move from Patientplace 0to Patientplace 1. Meanwhile, the resource required by Patientplace 1becomes busy in treating the patient. After the treatment, denoted as the activity with the action type of α, the resource becomes idle again waiting to treat another patient.(3)(4)
System description The last part of the CPP model is the system description part, describing the whole clinical pathway by using the cooperation operator to denote interactions between components. Suppose only one patient defined in Eqn.4 and one resource component defined in Eqn.3 are involved in the CPP model. The cooperation between state component Patientplace 0and the resource component Resource_idleplace 1can be represented as
where β and α are the action types on which two components Patientplace 0and Resource _ idleplace 1synchronise. More specifically, the patient which is currently in the initial state Patientpalce 0can only be scheduled to the following states when successive treatments are carried out.
As the healthcare system usually contains multiple patients and multiple copies of resources, it is necessary to represent parallel patients and resources as follows:
where || is the parallel combinator, equivalent to , showing that two components are in parallel. Therefore, we can describe the system which consists of multiple Patientplace 0and Resource _idleplace1 in the form of
Although definitions in Eqn.3,4,7 can emphasize the influence of resource on patient health care process, it has a limitation. When all copies of the resource are in their busy state Resource _busyplace 1, a new coming patient cannot be transferred from Patientplace 0to Patientplace 1. The patient will be stuck in the state Patientplace 0until at least one copy of resource become idle enabling it transfer to Patientplace 1and then to Patientplace 2. In reality, even if all the copies of resource are busy, we still want the new coming patient to be able to move from Patientplace 0to Patientplace 1and then stay in Patientplace 1waiting for at least one copy of the resource becoming idle. Therefore, we need to revise the resource definition part by inserting an additional resource as follows:
Then the system description part is:
The number of copies of the inserted resource Wait_room 0_idle should be sufficiently large to guarantee that at least one copy is available at any time, that is NumWait_room 0≥ Num Patient . Moreover, the action rate rate γ must be large enough to enable instant transition from Wait _room 0_busy to Wait _room 0_idle whenever at least one copy of resource is in its idle state Resource _idleplace 1.
Key techniques of performance analysis using CPP
Rooted in a continuous time Markov process , CPP model can be used to estimate the performance of clinical pathway including throughput (the number of patients that the healthcare system can serve every day) and resource utilisation (the percentage of time that a resource is in use).
The underlying stochastic process
We can therefore use Eqn.10,16 to obtain steady state distribution.
Calculating the resource utilisation and system throughput
where is the total number of Resource placei copies.
showing that the system throughput is associated with the utilisation of the resource whose state transition depends on the type λ activity.
We can use PEPA eclipse plugin  to simulate the developed CPP model, from which resource utilization and throughput can be directly obtained.
Critical assessments-state explosion problem
When there are many patients and multiple copies of resources involved in the system description part, the state explosion problem occurs. For instance, when there are eight patients, eight waiting rooms and three resources in Figure 4 clinical pathway, the number of states increases from 10 to 75582. Then the dimension of generator matrix Q becomes 75582 × 75582. Hence, calculating steady state distribution by solving Eqn.16 turns out to be computational intensive and requires large storage space. To address these problems, this paper uses two methods which are state aggregation and fluid analysis.
which are similar with Eqn.26.
Calculating passage time
In this section, the CPP modelling method is applied to model the stroke clinical pathway. It begins with the parameter settings of experiment. Then detailed description of the CPP model for stroke clinical pathway is shown. By simulating the pathway using the CPP modelling method, we can optimize the resource allocation, estimate the passage time and find the maximum throughput with the current resource distribution.
HASU activity data from NHS 
Number of beds in HASU
Number of stroke admissions
Number of 'mimic' admissions
Number of Transient Ischaemic Attack (TIA)
Number of patient directly admitted to a HASU
Number of stroke patient thrombolysed
Average door to needle (thombolysis) times (min)
Number of patients receiving a brain scan within 24hrs of admission to HASU
HASU median length of stay (day)
Total number discharged home directly from HASU
The settings of experiment
A&E for stroke
The CPP model of the stroke clinical pathway
- 1.State definition(38)
- 2.Resource specification(39)
In order to get the average patient incoming rate to 4, the parameter Num patient is set to 1000 and r income is 0.004.
Three performance analysis scenarios
Three scenarios are tested in this subsection to show the application of the CPP model in performance analysis. The CPP model can detect the optimal resource allocation, estimate the passage time and determine the maximum throughput of the healthcare system.
Scenario 1: Resource optimization
Figure 10 is used to show how our performance analysis result guides medical staff to find optimal resource allocation, i.e., the amount of resource is kept minimal while still maintaining the maximum throughput in the clinical pathway. The three 3D graphs look similar and they actually demonstrate the process of reducing redundant resources while keeping the throughput unchanged. In these 3D graphs, we give two examples. In the first example, the numbers of beds and scanners are reduced from 20/3 to 8/1, respectively. This example shows that twelve redundant bed resources and two scanner resources can be removed and the clinical pathway's throughput is not influenced. The second example further reduces two redundant stroke team resources and keeps the throughput unchanged.
Comparison of resource utilisation: The utilisations of resources resulting from the model with initial and optimal parameter sets are compared.
Initial parameter set
Optimal parameter set
Scenario 2: Passage time estimation
Scenario 3: Maximum input estimation
Therefore, by simulating the CPP model, we can find the maximum number of input patients that can be supported by the healthcare system. This maximum input estimation is significant for hospital to determine whether it can accept more stroke patients or not. For example, suppose there are already 10 patients on average coming to the hospital from the surrounding area. If the national health community asks whether this hospital can serve patients from larger area, meaning that more than 10 patients will arrive every day, by estimating the maximum input this hospital can determine whether this is possible and whether more resources are required to support the increased number of patients.
CPP model can be extended to incorporate survival analysis by collecting clinical data including patient recovery speed and survival rate. The augmented model can then be used to examine the influence of treatment delay on patients' recovery processes. Therefore, the results from the ICPA can be used to increase patient recovery probability and decrease the recovery time.
Multiple hospitals can be managed concurrently using ICPA. At present, there are seven hospitals in London with HASU departments to treat stroke patients. ICPA can be extended to combine their clinical data to build a uniform model. By analysing this model, patients can be dynamically scheduled to different hospitals in order to optimise their treatment process.
The stroke clinical pathway discussed in this paper is described in the coarse granularity. With clinical data, ICPA can be applied to analyse an element in the clinical pathway in the fine granularity. For example, the HASU department, one element of the stroke clinical pathway, needs to treat patients with multiple therapies. If this type of information can be obtained and modelled by CPP, ICPA can view HASU as a clinical pathway and conducts performance analysis on it.
A user friendly interface can be build to facilitate medical staff's access to ICPA. Currently, ICPA only provides analysis results such as optimal resource allocation to medical staff. By interpreting these performance analysis results, medical staff can optimally re-allocate medical resources and re-configure treatment process. In the future, we plan to develop a user portal to help medical staff construct the clinical pathway model by themselves.
The authors would like to thank Michelle Osmond and Moustafa Ghanem for their helpful remarks on this paper.
This article has been published as part of BMC Bioinformatics Volume 13 Supplement 14, 2012: Selected articles from Research from the Eleventh International Workshop on Network Tools and Applications in Biology (NETTAB 2011). The full contents of the supplement are available online at http://www.biomedcentral.com/1471-2105/13/S14
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