## General Strategy For Developing And Using Dynamic Biomarkers

To summarize, there are many types of information, only two of which were described here. In general, dynamic biomarkers will have more information than static biomarkers; multivariate biomarkers will have more information than univariate biomarkers. These ideas are foreign to most biologists and will take some time to spread among them. The standard entities used to evaluate biomarkers, such as sensitivity, specificity, and ROC curves, have questionable or limited utility but can easily be modified to fit the Shannon information framework. Some steps for biomarker development and implementation are given here as a guideline and probably contain significant gaps in knowledge, requiring further study:

1. Choose the signals (S) or the surrogates (D) that are relevant (biology).

2. Choose the best models (P) for the given S (mathematics/statistics) and all available biomarkers (Y) (biology).

3. Choose the best decision rules (R) given P (mathematics/statistics).

4. Choose the best subset, or subspace, of Y (statistics).

The mathematical entities P, R, and Y determine the diagnosis of S. Currently, most of the effort is focused on Y.

### MODELING APPROACHES Signal Types

This section deals with some very specific aspects of the models for P(Y Most people think of signals as being electrical. This is probably because most of the terminology and use comes from electrical engineering. However, the mathematics is completely general. Signals can be static or dynamic, meaning something that is measurable with a constant value or a time-varying value, respectively. Biomarkers are just biological signals. Signals are classified as analog (continuous) or digital (discrete). Birth and death are discrete signals; blood pressure and serum glucose levels are continuous signals. Dynamic signals vary over time. Time can also be classified as continuous or discrete, as described above.

Anything that changes in time and space is a dynamic system. This generally implies that space has more than one dimension, but it does not have to be physical space. The space of biomarkers would be an example where each univariate biomarker (signal) defines a spatial dimension. A continuous system is modeled with a set of differential equations where the variables defining the space usually appear in more than one equation. A discrete system is modeled with a set of difference equations.

### Mathematical Models of Dynamic Signals

Mathematical models are usually hypothesized prior to the experiment and then verified by the experimental data. Historically, these models have been deterministic differential equations. Pharmacokinetics is an example of this. Following are some typical mathematical models (ordinary differential equations) and their solutions:

Linear model:

Quadratic model:

Exponential model:

Sigmoidal model:

Sine model:

dx _ P dt = Pl dx dx _Pi+2p 21 dt dx dt=Pi x f x ^- x >

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