Introduction

The purpose of this chapter is to introduce some approaches to thinking about biological dynamics. Pathodynamics is a term used by the author to describe a quantitative approach to disease that includes how the biological system changes over time. In many ways it is analogous to thermodynamics [1,2] in that it deals with the macroscopic, measurable phenotypic aspects of a biological system rather than the microscopic aspects such as those modeled in mathematical physiology [ 3]. For the purposes of this chapter, macroscopic will refer to measurements that do not involve the destruction of the biological system being studied but may vary in scale (i.e., cell, tissue, organ, and body). For example, clinical measurements would be macroscopic whether invasive or not, but histopathology, cell lysis, or genotype would be microscopic. Another way to view this is that when the dynamics of the system being studied result from something greater than the sum of the parts, usually from networks, it is macroscopic; otherwise, studying the parts of the system is microscopic. One of the problems in macroscopic biology is that many of the underlying system characteristics are immeasurable. This is why the biological

Biomarkers in Drug Development: A Handbook of Practice, Application, and Strategy, Edited by Michael R. Bleavins, Claudio Carini, Malle Jurima-Romet, and Ramin Rahbari Copyright © 2010 John Wiley & Sons, Inc.

variation is just as important, if not more important than the mean behavior of a system because it represents the system changes that cannot be measured directly.

The term parameter is used rather loosely in biology and other sciences and has a very specific meaning in mathematics. For the purposes here, it will be defined in a statistical manner as a quantity which is a characteristic of the system that is not directly measurable but can be estimated from experimental data and will be represented by Greek symbols. Quantities being measured will be referred to as variables. The letter y will designate a random, or stochastic, variable, one that has a probability distribution, and the letter x will represent a deterministic, or controlled, variable, one that is measured "exactly," such as gender, temperature, and pH. The letters s, t. and T will refer to time and will be either deterministic or random, depending on the experimental context. Biomarkers are really mathematical functions of parameters. In modeling, parameters appear as additive coefficients, multiplicative coefficients, and exponents. Sometimes parameters are deterministic, sometimes random, and sometimes functions of time or variables. Probably the most common parameter in biology is the population mean of a random variable. The sample mean is an estimate of this parameter; it is also a statistic, a quantity that is a function of the data. However, under the definitions above, some statistics do not estimate parameters and are known as nonparametric statistics. Nonparametric statistics are generally used for hypothesis testing and contain little or no mechanistic biological information [4-7].

Analogy to Thermodynamics

Thermodynamics is a macroscopic view of physics that describes the flow of energy and the disorder of matter (i.e., entropy) [1,2]. The former is reflected in the first law of thermodynamics,which says that

Aenergy = Awork + Aheat which is basically a law of the conservation of energy. The second law has various forms but generally states that entropy changes of a system and its exterior never decrease. Aging and disease (at least some) may be examples of increasing entropy.

In chemistry, classical thermodynamics describes the behavior of molecules (particles) and changes in the states of matter. Equilibrium occurs at the state of maximum entropy [i.e., when the temperature is uniform throughout the (closed) system]. This requires constant energy and constant volume. In a system with constant entropy and constant volume, equilibrium occurs at a state of minimum energy. For example, if constant entropy occurs at a constant temperature, equilibrium occurs when the Helmholtz free energy is at its minimum. In all cases, the particles continue to move; this is a dynamics part. For nonequilibrium states, matter and energy will flow to reach an equi librium state. This is also a dynamics part. Modern thermodynamics is a generalization of classical thermodynamics that relates state variables for any system in equilibrium [2] - such as electromagnetism and fluid dynamics. The open question is whether or not these concepts apply to biological systems.

For this chapter, the pathodynamics concept is that the states of a biological system can be measured and related in a manner similar to thermodynamics. At this time only the simplest relationships are being proposed. The best analogy seems to be viewing a biological system: in particular, a warm-blooded (constant-temperature) mammal as a particle in a high-dimensional space whose states are measured via (clinical) laboratory tests. The dimension of this space will be defined by the information contained in these biomarkers (see below). This particle is in constant motion as long as the system is alive and the microscopic behavior of the biology is generally unobservable, but the probabilitistic microscopic behavior of this particle is observable through its clinical states. The probabilistic macroscopic properties are described by the probability distributions of the particle. These distributions are inherently defined by a single biological system, and how they are related to population distributions for the same species is unknown. Hopefully, there will be some properties that are invariant among individuals so that the system behavior can be studied with large enough sample sizes to get sufficient information about the population of systems. One way to visualize this concept of patho-dynamics is to imagine a single molecule moving in a homogeneous compressible fluid [8] - A drop of this fluid represents the probability distribution and is suspended in another fluid medium. The fluids are slightly miscible, so that there is no surface on the drop, but there is a cohesive (homeostasic) force that pulls the particles toward the center of the drop while the heat in the system tends to diffuse the particles out into the medium. Dynamic equilibrium occurs when the drop is at its smallest volume, as measured by levels of constant probability density. The conservation of mass is related to the total probability for the particle (mass = 1), which is analogous to the mass of the drop. The drop becomes distorted when external forces on it cause distortion of the shape and may even fragment the drop into smaller drops or change the drop so that holes appear. These external forces are due to factors such as external environment, disease, and therapies. A goal of pathodynamics is to infer the presence of an external force by observing the motion of the particle and finding the correspondence between the force and known causes.

The Concept of Time

Since dynamics relates to the changes over time, some mention of time is appropriate. Everyone has a general concept of time, but in mathematical and physical thinking, time is a little more complicated [9,10]. There are two main classes of time: continuous (analog) and discrete (digital). Although physicists use the reversibility of time, which at least makes the mathematics easier, Prigogine [10] argues that at the microscopic level, the world is probabilistic and that in probabilistic (stochastic) systems, time can only go forward. Furthermore, he argues that "dynamics is at the root of complexity that [is] essential for self-organization and the emergence of life."

There are many kinds of time. In the continuous category are astronomical time (ordinary time), biological time (aging), psychological time, thermody-namic time, and information time [11]. Discrete time is more difficult to imagine. Whenever digital data are collected, the observations occur only at discrete times, usually equally spaced. Even then, the process being observed runs in continuous time. This is probably the most common case. However, discrete time is really just a counting process and occurs naturally in biology. Cell divisions, heartbeats, and respirations are a few of the commonly observed discrete biological clocks. In this chapter only continuous-time processes are discussed. An example of discrete -time pathodynamics can be found elsewhere [12].

BROWNIAN MOTION Diffusion

One of the most basic continuous stochastic processes is Brownian motion. The concept of Brownian motion was born when a biologist named Robert Brown observed pollen under a microscope vibrating due to interactions with water molecules [13]. This concept should be familiar to biologists. Brownian motion has been modeled extensively [14,15]. Standard Brownian motion (B' is a Gaussian stochastic process with mean and variance

^ = 0 o,2 = t respectively. In the laboratory, Brownian motion is a good model for diffusion (e.g., immunodiffusion), where the concentration of the protein is related to the time of observation and the diffusion coefficient. In an open system where time goes forever or when the particles have no boundary, the diffusion of a particle is unbounded because the variance goes to infinity. To make it a useful concept in biology, the particle needs to be either in a closed container or stabilized by an opposing force.

Homeostasis: Equilibrium Pathodynamics

In biological systems, diffusion occurs only on a microscopic level and is not usually measurable macroscopically. However, in a probabilistic model of pathodynamics, the particle representing person's clinical health state can be thought of as a microscopic object (e.g., a molecule in a fluid drop), while the probability distribution is the macroscopic view. The reader should note that the concepts of microscopic biology as defined in the Introduction and this imaginary microscopic particle of pathodynamics are different uses of the microscopic/macroscopic dichotomy. The Ornstein-Uhlenbeck (OU) stochastic process [16] is a stationary Gaussian process with conditional mean and variance given y0:

respectively, where y0 is the baseline value, p is the equilibrium point (mean of y t), and o2 is the variance of yt. The autocorrelation between two measurements of y is

In thermodynamic terms, the average fluctuation of a particle is dpt , x

Read "dx" as a small change in x. This ordinary differential equation (ODE) is analogous to the stochastic differential equation (SDE) [14,15] that generates the OU process, dyt = -a (yt -p)dt W2a a dBt which has a biological variation term that is driven by Brownian motion. In statistical physics, this is called the Langevin equation , or the equation of motion for a Brownian particle [17].

Here a link between thermodynamics and pathodynamics will be attempted. Using the Einstein theory of fluctuations [2,17], the change in entropy is

Since this quantity is always zero or negative, this says that in the equilibrium state the entropy is decreasing with time, which suggests that there is some organizing force acting on the system. This is the homeostatic force

In statistical terms under near-equilibrium conditions, the drag in the system increases as the autocorrelation increases or as the variance decreases.

When the time rate of change of AS is equal to the product -JF, where J is the current or flow of the particles in the system, the system is in an equilibrium state. Solving this equation gives which is the entropy current. In homeostasis the average force and the average flow are zero and the average of the change in entropy is -A. This can all be generalized to multivariate biomarkers by making the measurements and their means into vectors y and || respectively, and by making a and o2 into symmetric positive definite matrices A and X. respectively. The usual thermody-namic parameters are embedded in these statistical parameters and can be determined as needed for specific biological uses.

To illustrate this physical analog a little further, suppose that a drop of particles were placed in a well in the center of a gel, the external medium, and allowed to diffuse for a fixed time tH. If Fick's law applies, there is a diffusion coefficient D and the Stokes-Einstein relation holds such that where kB is Boltzmann's constant, T is the absolute temperature of the system, and y is the viscosity (friction) coefficient. Now suppose that the particles are charged and an electrical field is applied radially inward at tH so that the field strength is proportional to the distance from the center of the sample and is equal to the force of the diffusion. This means that the distribution of the particles is in steady state and that the particles experience no acceleration. This leads to the Langevin equation with the friction coefficient in the system at y = 1/ao2. It turns out that at this equilibrium state o2 = 2DtH and then by substitution

As long as T is constant, a is constant and inversely proportional to T, which may represent physical temperature or some biological analog but is assumed to be constant as well; and ao2 is proportional to D/T.

Signals of Change: Nonequilibrium Pathodynamics

Changes from equilibrium are the signals of interest in pathodynamics. In the simplest case, the signal can be an observation that occurs outside the dynamic reference interval ("normal limits"). This interval can be constructed by estimating its endpoints at time t using a =

where s is the time of a previous measurement and would be zero if it is the baseline measurement. In either case, two measurements are required to identify a dynamic signal when the parameters do not change with time. Some care needs to be exercised when estimating this interval . 18] . The interval clearly will be shorter than the usual one, j ± 1.96a, will be different for each person, and will be in motion unless there is no autocorrelation. A value outside this interval would indicate a statistically significant deviation from homeostasis with a probability of a false positive being 0.05 for each pair of time points. Simultaneous control of the type I error requires multivariate methods.

Nonequilibrium states might be modeled by allowing one or more parameters to change with time. If ( is changing with time, this is called convection, meaning that the center of gravity of the system is changing, resulting in a flow, or trajectory. If a or a is changing, the temperature or the diffusion properties are changing. When the particle is accelerating, an acceleration term needs to be added to the Langevin equation. The Fokker-Planck equation [17] provides a way to construct the steady-state probability distributions of the particle along with the transition probability states for the general Langevin equation. It is possible for a new equilibrium distribution to occur after the occurrence of a disease or after a therapeutic intervention, which would indicate a permanent residual effect. In this paradigm, a "cure" would occur only if the particle distribution returned to the healthy normal homeo-stasis state. The observation of the dynamics of the particle may suggest that an external (pathological) force is acting on the system. Any changes in "thermodynamic" variables may form patterns that lead to diagnostic criteria. The construction of optimal criteria and the selection and measurement of biomarkers are discussed in the next section.

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