Team:CPU CHINA/Model
Cost-Effectiveness Analysis
Model part
Q: How to determine if a new therapy is interesting for practical application?
Q: Also, how to balance the pay and gain of the patients? How to set up a reasonable price for the patients?
Background
According to the World Health Organization, primary liver cancer is globally the sixth most frequent cancer (6%) and the second leading cause of death from cancer (9%) [1].
China has a particularly high incidence of liver cancer in its population. The number of new cases of liver cancer every year accounts for about half the total number worldwide. This disease takes the lives of about 110,000 people each year, which is also nearly 45% of the number worldwide. Especially in recent years, liver cancer has risen from the third place to the second place in the cause of cancer-related death in China [2].
Hepatocellular carcinoma (HCC) has a huge impact on human health and remains a huge burden to people all over the world. Intense research has been and will be made against this disease, so the number of available medical treatments should be expected to continue growing. For researchers and investors, the current market and competition tend to be obscured by ever-new attractions and tendencies. In such case, mathematical modeling would come in handy as it helps to explain and to study the effects of different factors, and to make predictions about new products. Therefore, efforts have been made by us to set up a model that estimates the efficiency and cost of these medical treatments.
Concepts:
1. Utility
Within economics the concept of utility is used to model worth or value, but its usage has evolved significantly over time. In the field of health care, the utility value associates with a given state of health by the years lived in that state [3]. Preference-based utility can be influenced by a lot of factors such as the patient’s age, income, and education. Usually and as it is in our model, an assumption is made. The assumption is that the utility of death is 0 and the utility of total health is 1 so that utility is always between 0 and 1. However, it also occurs sometime that the value becomes below 0, which indicates something that’s even worse than death, such as lying in bed for a long time with unbearable pain.
2. QALY
When we consider both mortality and disability that some diseases can bring, there appears a new index called QALY (Quality Adjusted Life Years). QALY combines both the length and quality of life, which summaries the influences medical treatments have on patients. This index can be used to inform personal decisions, evaluate programs, and set priorities for future program. To determine QALYs, one multiplies the utility that we mentioned above. For example, a year of life in perfect health is worth 1 QALY (1 year of life × 1 Utility value). A year of life in a state of less than perfect health is therefore worth less than 1 QALY. 1 year of life lived in a situation with utility 0.5 (e.g. bedridden, 1 year × 0.5 Utility) is assigned 0.5 QALYs. Similarly, half a year in perfect health is equivalent to 0.5 QALYs (0.5 years × 1 Utility). Death is assigned a value of 0, and in some circumstances, it is possible to accrue negative QALYs to reflect health states deemed "worse than dead."[3]
3. Cost-utility analysis (CUA) and incremental cost-effectiveness ratio (ICER)
CUA (cost-utility analysis) is well-known in Pharmaco-Economics. It is a type of financial analysis used to guide procurement decisions. Cost is measured in monetary units while the benefits often are expressed in QALYs.
The incremental cost-effectiveness ratio (ICER) is a statistic for summarizing the cost-effectiveness of a health care intervention. It is defined by the difference in cost between two possible interventions, divided by the difference in their effect [5]. It represents the average incremental cost associated with 1 additional unit of the measure of effect.
ICER =
C1 and C0
E1 and E0
C1 and E1 are the cost and effect in the intervention group while C0 and E0 are the cost and effect in the control group. Costs are usually described in monetary units, while effects can be measured in terms of health status or another outcome of interest. A common application of the ICER is in CUA, where the ICER gets synonymous with the cost per quality-adjusted life year (QALY) gained.
Figure1. Clinical stages and treatment pipeline of liver cancer[6]
PS (performance status): It is an attempt to quantify cancer patients' general well-being and activities of daily life. This measure is used to determine whether they can receive chemotherapy, whether dose adjustment is necessary, and as a measure for the required intensity of palliative care. It is also used in oncological randomized controlled trials as a measure of quality of life.
TACE (Transarterial chemoembolization): It is a minimally invasive procedure performed in interventional radiology to restrict a tumor's blood supply. Small embolic particles coated with chemotherapeutic drugs are injected selectively through a catheter into an artery directly supplying the tumor. These particles both block the blood supply and induce cytotoxicity, attacking the tumor in several ways.
Great efforts have been taken to cure liver cancer and different therapies in different stage are shown in Figure.1. We performed CUA of some of the therapies and set up a mathematical model to figure out which one costs less and benefits the patients more. This model is expected to be capable of stimulating a situation that helps us to determine if a new therapy is interesting enough to be introduced into the market, on both the medical level and the financial level.
Assumption
1. All patients can be divided into three stages: if the patient is dead, we call it the Death stage; if the tumor does not grow or the health state of the patients doesn’t deteriorate, we call it a Progression-Free Survival (PFS) stage; else it’s a Progression Disease (PD) stage.
2. Patients cannot switch from Death stage to any other stage.
Our Model
Introduction
A Markov chain is a stochastic process with the Markov property. The term "Markov chain" refers to the sequence of random variables such a process moves through, with the Markov property defining serial dependence only between adjacent periods (as in a "chain"). It can thus be used for describing systems that follow a chain of linked events, where what happens next depends only on the current state of the system [7,8].
The construction of the model
Here we discuss a discrete-time Markov chain. When an actual problem is being described by a Markov chain, the first thing to determine is its state space and parameter set, and then its one-step transition probability. Since we assume that the patients have only three statuses, the state space would be E={0,1,2} (0 is PFS, 1 is PD, 2 is death) and patients are distributed into the three states. Every month the state of the patients (εn) changes and we can record these information in the form of a month sequence {εn,n=1,2,...} εn=j,j∈E
Figure.2 The state space and the transfer relationship in our model
Transfer probabilities (P) can be obtained from the inherent laws of the problem, experience, or estimation from observational data. In our model, we use the monthly transition probabilities derived from the survival data and a formula described by previous studies. The transfer matrix is shown below:
Table.1 The transition probability of the Sorafenib and combination therapy [9]
| Sorafenib | Combination therapy | |
| PPFS-PFS | 0.6639 | 0.7826 |
| PFS-PD | 0.2264 | 0.1399 |
| PPFS-death | 0.1097 | 0.0739 |
| PPD-PD | 0.8088 | 0.8553 |
| PPD-death | 0.1912 | 0.1447 |
Table.2 The transition probability of the Sorafenib and SBRT[10]
| Sorafenib | SBRT | |
| PPFS-PFS | 0.784 | 0.805 |
| PFS-PD | 0.063 | 0.109 |
| PPFS-death | 0.153 | 0.086 |
| PPD-PD | 0.882 | 0.960 |
| PPD-death | 0.118 | 0.040 |
Table.3 The transition probability of the FOLFOX4 and Sorafenib[11]
| FOLFOX4 | Sorafenib | |
| PPFS-PFS | 0.686 | 0.680 |
| PFS-PD | 0.211 | 0.219 |
| PPFS-death | 0.103 | 0.101 |
| PPD-PD | 0.819 | 0.829 |
| PPD-death | 0.181 | 0.171 |
Transfer matrix example of FOLFOX verses Sorafenib
We simulated the life course of 1,000 patients treated with the above drugs in silico. At the beginning, each patient's state is determined by a random function to make our simulation close to the practical situation. Also, the switch between states of every patient in every month is random but the probability complies with transfer probabilities. The simulation ended when all the patients die and the output is the number of death in every month.
a. FOLFOX4 versus Sorafenib
b. SBRE versus Sorafenib
c.Sorafenib combination versus monotherapy
Figure3. The comparison of the survival curves in different cases. We have found that Sorafenib monotherapy is not better than the SBRT therapy or the Sorafenib combination therapy, while it shows little advantage over FOLFOX4.
During the clinical trials, researchers can’t really follow and record the patients until they die. However, in this model, we can define several states of the patients in advance. After every month’s medical intervention and collection of the data, we can put patients into the corresponding states and calculate the transfer probabilities. By running the program that we have written based on this model, we can predict the survival curve of each medical treatment.
Then the utility can be estimated by doing some questionnaire survey and lab tests in specific disease contexts. As we mentioned before, with the life span and the utility, the quality-adjusted life years then can be calculated.
Since we can get the efficacy of different medical intervention in the model, the next step is to estimate a reasonable price (Figure 4). Companies can calculate the ICER of their products with their prospect price and compare it to other drugs of the same type contemporarily on the market.
