Some economics on personalized and predictive medicine

Abstract

Objective

To contribute to the theoretical literature on personalized medicine, analyzing and integrating in an economic model, the decision a health authority faces when it must decide on the implementation of personalized medicine in a context of uncertainty.

Methods

We carry out a stylized model to analyze the decision health authorities face when they do not have perfect information about the best treatment for a population of patients with a given disease. The health authorities decide whether to use a test to match patients with treatments (personalized medicine) to maximize health outcomes. Our model characterizes the situations under which personalized medicine dominates the alternative option of business-as-usual (treatment without previous test). We apply the model to the KRAS test for colorectal cancer, the PCA3 test for prostate cancer and the PCR test for the X-fragile syndrome, to illustrate how the parameters and variables of the model interact.

Results

Implementation of personalized medicine requires, as a necessary condition, having some tests with high discriminatory power. This is not a sufficient condition and expected health outcomes must be taken into account to make a decision. When the specificity and the sensitivity of the test are low, the health authority prefers to apply a treatment to all patients without using the test. When both characteristic of the test are high, the health authorities prefer to personalize the treatments when expected health outcomes are better than those under the standard treatment. When we applied the model to the three aforementioned tests, the results illustrate how decisions are adopted in real world.

Conclusions

Although promising, the use of personalized medicine is still under scrutiny as there are important issues demanding a response. Personalized medicine may have an impact in the drug development processes, and contribute to the efficiency and effectiveness of health care delivery. Nevertheless, more accurate statistical and economic information related to tests results and treatment costs as well as additional medical information on the efficacy of the treatments are needed to adopt decisions that incorporate economic rationality.

Appendix

Let us assume that the test classifies a patient as being of type 1. Then, she receives treatment A and her expected net benefit is

$$- C_{\text{A}} + { \Pr }\left( {1 |t1} \right)b_{1} + { \Pr }\left( {2 |t1} \right)\beta b_{2} - h$$

(7)

where ${ \Pr }\left( {j |t1} \right)$ is the posterior probability that the patient is of type j, j = 1, 2, given that the test identified the patient as being of type 1.

$$\begin{gathered} {\text{Pr }}\left( {1 |t1} \right) = \frac{{\Pr \left( 1 \right){ \Pr }\left( {t1 |1} \right)}}{{\Pr \left( 1 \right){ \Pr }\left( {t1 |1} \right) + \Pr \left( 2 \right){ \Pr }\left( {t1 |2} \right)}} = \frac{\pi s}{{\pi s + \left( {1 - \pi } \right)\left( {1 - e} \right)}} \hfill \\ \Pr \left( {2 |t1} \right) = \frac{{\Pr \left( 2 \right){ \Pr }\left( {t1 |2} \right)}}{{{ \Pr }\left( 2 \right){ \Pr }\left( {t1 |2} \right) + \Pr \left( { 1} \right){ \Pr }\left( {t1 |1} \right)}} = \frac{{\left( {1 - \pi } \right)(1 - e)}}{{\pi s + \left( {1 - \pi } \right)\left( {1 - e} \right)}} \hfill \\ \end{gathered}$$

where ${ \Pr }\left( {t1 |j} \right)$ is the probability that the test identifies a patient as being of type 1 when she is of type j, j = 1, 2.

By plugging these expressions into (7), we have that the net expected benefit of a patient identified as being of type 1 is

$$\Pi \left( {t1} \right) = \frac{{\pi sb_{1} + \left( {1 - \pi } \right)\left( {1 - e} \right)\beta b_{2} }}{\pi s + (1 - \pi )(1 - e)} - \left( {C_{\text{A}} + h} \right)$$

The probability that the test identifies a patient as being of type 1 is result of the test is $\Pr \left( {t1} \right) = \pi s \, + \, (1 - \pi )(1 - e)$.

Let us assume that the test classifies a patient as being of type 2. Then, she receives treatment B and her expected net benefit is

$$- C_{\text{B}} + { \Pr }\left( {2 |t2} \right)b_{2} + { \Pr }\left( {1 |t2} \right)\alpha b_{1} - h$$

(8)

where ${ \Pr }\left( {j |t2} \right)$ is the probability that the test identifies a patient as being of type 2 when she is of type j, j = 1, 2. Following Bayes, we have

$$\begin{gathered} {\text{Pr }}\left( {1 |t2} \right) = \frac{{\Pr \left( 1 \right){ \Pr }\left( {t2 |1} \right)}}{{\Pr \left( 1 \right){ \Pr }\left( {t2 |1} \right) + \Pr \left( 2 \right){ \Pr }\left( {t2 |2} \right)}} = \frac{\pi (1 - s)}{{\pi (1 - s) + \left( {1 - \pi } \right)e}} \hfill \\ \Pr \left( {2 |t2} \right) = \frac{{\Pr \left( 2 \right){ \Pr }\left( {t2 |2} \right)}}{{{ \Pr }\left( 2 \right){ \Pr }\left( {t2 |2} \right) + \Pr \left( { 1} \right){ \Pr }\left( {t2 |1} \right)}} = \frac{{\left( {1 - \pi } \right)e}}{{\pi (1 - s) + \left( {1 - \pi } \right)e}} \hfill \\ \end{gathered}$$

where ${ \Pr }\left( {t2 |j} \right)$ is the probability that the test identifies a patient as being of type 2 when she is of type j, j = 1, 2.

By plugging these expressions into (8), we have that the net expected benefit of a patient identified as being of type 2 is:

$$\Pi \left( {t2} \right) = \frac{{\pi \left( {1 - s} \right)\alpha b_{1} + \left( {1 - \pi } \right)eb_{2} }}{\pi (1 - s) + (1 - \pi )e} - \left( {C_{\text{B}} + h} \right)$$

The probability that the test identifies a patient as being of type 1 is result of the test is $\Pr \left( {t2} \right) = \pi \left( {1 - s} \right) + \,\left( {1 - \pi } \right)e$.

Therefore, if the test is administered, the expected net benefit $\Pi (T,h)$ is:

$$\begin{aligned} \Pi \left( {T,h} \right) & = \Pr \left( {t1} \right)\Pi \left( {t1} \right) + \Pr \left( {t2} \right)\Pi \left( {t2} \right) \\ & = \left[ {\pi s + \left( {1 - \pi } \right)\left( {1 - e} \right)} \right]\left[ {\frac{{\pi sb_{1} + \left( {1 - \pi } \right)\left( {1 - e} \right)\beta b_{2} }}{\pi s + (1 - \pi )(1 - e)} - \left( {C_{\text{A}} + h} \right)} \right] \\ & \quad + \left[ {\pi \left( {1 - s} \right) + \left( {1 - \pi } \right)e} \right]\left[ {\frac{{\pi \left( {1 - s} \right)\alpha b_{1} + \left( {1 - \pi } \right)eb_{2} }}{\pi (1 - s) + (1 - \pi )e} - \left( {C_{\text{B}} + h} \right)} \right] \\ & = \pi b_{1} \left[ {s + \alpha (1 - s)} \right] + (1 - \pi )b_{2} \left[ {e + \beta (1 - e)} \right] - C_{\text{A}} + \Delta C\left[ {\pi \left( {1 - s} \right) + \left( {1 - \pi } \right)e} \right] - h \\ \end{aligned}$$