Models and Control of Biological System

Model and control of biological systems

A biological system (or organ system or body system) is a group of organs that work together to perform a certain task: for example, for the glucose homeostasis are involved pancreas, liver, skeletal muscle, adipose tissue, brain, and gut. These systems are regulated at scale of many order of magnitude in space and time. Modelling biological processes often requires accounting for action and feedback involving a wide range of spatial and temporal scales.

Applications of biological system modelling

Biological/physiological system modelling finds application in:

• Medical research: The use of models can yield quantitative insights into the manner in which physiological systems are controlled. Principal component of the type 1 diabetes simulator: a model of the glucose-insulin system, a model of the sensor, a model of the insulin pump and subcutaneous insulin kinetics, and the controller to be tested.
• Educational settings: Medical students can use computer model simulation to explore the dynamic effects of pathophysiological processes or of drug therapy.
• Supporting clinical practice: Models (mathematical) can enable estimates to be made of physiological parameters that are not directly measurable; useful for example in diagnosis, as well as enabling predictions to be made as to how changes in drug therapy will impact variables of clinical importance such as blood pressure or blood glucose concentration.

Understanding biological system models

A biological system model is a representation of reality, better is an approximation of that reality since not all the reality can be incorporated into any model. The form of the model developed is approximate for its purpose, but can have a wide range of possible purposes for modelling. Example: the form of a model adopted for the purpose of understanding some complexities of the control of breathing might well be quite different from that adopted as an aid to weaning the patient in an intensive care unit off a ventilator. This is the case even though in both examples the physiological focus is the respiratory system.

Developing a model

The way in which we develop a model will be dependent on our knowledge of the relevant physiology and the availability of relevant experimental data. So, in essence, the process of building a model can be regarded as a mapping of physiological knowledge and experimental data into the model. In the case of a model that is essentially a representation of the experimental data available, it is those data that dominate in this mapping process (data-driven model). On the other hand, if the model is one designed to provide a representation of the physiology more explicitly (system model), then it will be the physical and chemical knowledge of that physiological system which dominates in the building of the model.

Physiological complexity

In various ways, all physiological systems are characterized by their complexity. The human organism can be seen as a complex multi-input, multi-output system, with linkages involving an array of physico-chemical processes. Finally, it includes many of the standard functions to be found in any complex control system: sensing, decision making and control, actuating or effecting, feeding back of information. Complexity manifests itself in several ways:

• Components: The greater the number of neurons in a central nervous system or the larger the number of intermediate substances in a metabolic pathway, the greater will be the complexity.
• Interconnectivity: In the case of the central nervous system this would correspond to the number of interconnections between neurons.
• Nonlinearity: Occurs when at least one element in the system relates to and varies in a nonlinear way with another. Almost all physiological systems are nonlinear.
• Asymmetry: Occurs when symmetry in a system’s relationships no longer holds. Example: differential cells growth resulting in cell specialization.
• Nonholonomic constraints: Relate to the integrity of systems, so that holonomic constraints are constraints that relate to laws affecting an entire organism. The behavior and control of the parts cannot easily be predicted just on the basis of knowledge of the overall system characteristics.

Yates (1978) suggested that complexity arises when one or more among these five attributes are found. Complexity also arises as a consequence of stochastic and time-varying dynamic effects.

Feedback in physiological systems

Feedback is a fundamental feature of all physiological systems. It ensures physiological regulation and control. It is an “ingredient” of the complexity that characterizes physiological systems. Feedback can be regarded as a mutual causality, whereby variable X has an effect on variable Y, and in turn variable Y has an effect on variable X.

Types of feedback

• Negative feedback: An increase in variable X brings about an increase in variable Y, and that increase in variable Y brings about a decrease in variable X. Example: glucose regulation. Negative feedback mechanisms are initiated to maintain or regulate conditions within a set and narrow range.
• Positive feedback: Variable X causes an increase in variable Y, where in turn, that increase in Y brings about a further increase in X. This is clearly a destabilizing phenomenon. Example: oxytocin during labor. During labor, a hormone called oxytocin is released that intensifies and speeds up contractions. The increase in contractions causes more oxytocin to be released and the cycle goes on until the baby is born. The birth ends the release of oxytocin and ends the positive feedback mechanism. Positive feedback mechanisms are designed to accelerate or enhance ongoing output that has been activated by a stimulus (“push” levels beyond normal ranges).

The feedback effect can be directly proportional to the variable that contributes to this effect. However, other forms of feedback are possible:

• Derivative feedback: Sensing not only a change in variable X but also its rate of change (dX/dt); brings about a more speedy response within the control loop and a more rapid achievement of the desired regulation. Example: insulin is being secreted not only in response to elevation of glucose concentration but also to the positive rate of change of glucose concentration.
• Integral feedback: Signal proportional to the integral between the desired value of the variable being controlled and its actual value. One of the problems with control based on proportional and derivative feedback is that there can be an error in the final steady state that is achieved once the correcting feedback has taken its effect. Example: regulation of the thyroid hormones is achieved through a combination of proportional and integral feedback.

Changes occurring with feedback processes are also closely associated with the transition from the healthy state to that of disease. Feedback is intimately related not only to the manner in which physiological systems are regulated, but also the patterns of dynamic behavior that they exhibit. Feedback can affect stability and speed of response.

Control in physiological systems

The overall control of the chemical processes of physiology involves the nervous system as well as hormones. Many of these are still not very well understood, though considerable advances in understanding have taken place in some areas. At a simple level, there is strong local control action. This enables many of the chemical processes to be regulated without recourse to higher levels of control. There is a hierarchy of regulatory mechanisms in terms of intrinsic chemical, enzymic, hormonal, and neural control. This multitude of controllers ensures that the organ systems and the organism as a whole are able to withstand disturbances. There is also redundancy. Example: a number of hormonal control loops are involved in the maintenance of blood glucose levels. Hierarchy and redundancy are further features of complexity!

N.B. Physiology is complex and the availability of measurements to access the dynamics of this complexity is limited!

Models and the modelling process

A model is a representation of reality involving some degree of approximation. Models can take many forms but we will focus on mathematical models, a representation of physiological reality that are expressed in the form of mathematical equations. Most of reality is so complex that the mathematical formulation is close to impossible. Simulation models/computer models/computational models can be used to describe and study phenomena even when traditional mathematical approaches fail. An example is the agent-based models, where the model represents components of the real system explicitly and keeps track of the behavior of individuals over if-then statements.

Example: If a lion is in hunting mode and there is a prey animal closer than 10 meters, then attack.

Purpose of a model (why)

The way in which we formulate a model, and the degree of detail that is incorporated into it, are determined principally by its intended purpose. Four general types of purpose for which models are developed:

• Descriptive: The descriptive use of mathematical models is the expression of quantitative relationships in terms of equations. These equations provide a concise and economic description of the system under consideration and facilitate analysis and handling of data. For example, if a variable in a system is directly proportional to another, then a linear equation relating the two is more concise and easy to handle than a graphical or verbal description.
• Interpretive: Models can also be used for interpreting experimental results. For example, a single exponential decay, as a mathematical expression, provides a compact representation of data that approximate to a first-order process. By a first-order process, we mean one in which, for instance, the rate of loss of a substance (e.g., drug) is directly proportional to the quantity or concentration of that substance in the pool or compartment from which it is being lost. This could apply to the rate of clearance of a drug from the bloodstream.
• Predictive: Models can be used to address the question as to how a system would respond to a stimulus or to a change in the system. An example would be predicting how the human organism, or a specific physiological organ system, might respond to the injection or infusion of a drug. Suppose that a mathematical model has been formulated which includes the processes that are affected by a particular drug. The model can then be used, in simulation mode, by applying a stimulus to it corresponding to the drug injection or infusion. In the model, one can observe how the concentration of the drug in the body changes with time, or how blood pressure changes over time following the administration of a drug that is designed to reduce blood pressure.
• Explanatory: Models can be used to help provide a physiological explanation for observed dynamic effects. In this way, a model can, for instance, be used to help understand how changes in physiological parameters can cause changes in the uptake of substances, including drugs, by various organs of the body. For example, if the parameters of the model correspond to explicit physiological processes or effects, then changes in observed behavior can be interpreted in terms of changing parameter values.

Purpose of a model (how)

In developing a mathematical model, two fundamentally distinct approaches are possible:

• Based on experimental data, it is essentially a data-driven or black-box modelling approach. It seeks quantitative descriptions of physiological systems based on input/output descriptions that are derived from experimental data collected on the system. These are mathematical descriptions of data that only correspond implicitly to the underlying physiology. It is a particularly appropriate approach where there is a lack of knowledge concerning the underlying physiology and an overall input/output representation of the system’s dynamics is all that is needed; that is, there is not the need to know specifically how the physiological mechanisms gave rise to such input/output behavior.
• Based on the explicit representation of the underlying physiology, it aims to model the system. It seeks to provide an explicit representation of the underlying physiology and offers the advantage that features of dynamic behavior that are observed can be directly related to physiological parameters and variables that are explicitly incorporated into the model. The Windkessel is a lumped model. In other words, this lumped model describes the whole arterial system, in terms of a pressure-flow relation at its entrance, by two parameters that have a physiological meaning. The two-element Windkessel model tells us that the load on the heart consists of peripheral resistance and total arterial compliance.

Modelling process: system approach

Since any model is an approximation of the underlying reality a number of simplifying assumptions will need to be made. These can typically be categorized under the headings of:

• Aggregation – Treating the kidney as a single lumped entity as opposed to providing distinct model representation of the different types of nephron in the kidney.
• Abstraction: Degree to which only certain aspects of a system are considered in a model. For example, in modeling blood glucose regulation, the bloodstream is regarded as containing only glucose and the hormones involved in its regulation. In such abstraction, other features that are important but do not relate directly to glucose regulation are neglected.
• Idealization: Structures or behavior that are difficult to describe or treat can be approximated by simple idealized ones. Example: In a metabolic system, the injection of a metabolite can be regarded as being instantaneously distributed throughout the system although, in fact, the distribution takes a finite time.

Modelling process

It is helpful to break a big problem into smaller, manageable pieces.

Model formulation

• Creating a qualitative model: Qualitative model formulation is the conversion of an objective statement into an informal conceptual model and a set of hypotheses and assumptions. Qualitative models can take any form (except mathematical), but diagrams are the usual representations. Its purpose is to provide enough details and structure so that a consistent set of equations can be written.
• Converting this into a quantitative model: Writing model equations. Different types of equations lead to different types of models: linear or non-linear, deterministic or stochastic.

Model solution

Solve the model: obtain explicit relationships between variables and/or parameters in the model. Within the model, the relevant variables are commonly connected through complex mathematical relations such as differential equations. Obtaining the required explicit relations is commonly done by computer implementation of the model.

• Case 1: The structure and parameter values of the model are known a priori. The model can therefore be solved and its validity further assessed.
• Case 2: Uncertainty in the structure of the model and/or its parameters. In this situation, the solution is not possible directly. → Model identification.

Model identification

The model may not be complete because some of the parameter values are unknown. This may be the case regardless of whether our model is data-driven or an explicit model of the underlying physiology. Solving this identification problem requires data. Experiments usually need to be designed.

These experiments involve applying some stimulus to the system and observing the dynamic response of one or more of the variables. The application of a trace quantity of a metabolite is a typical input stimulus. The experiment will involve applying some type of test signal to our system and measuring the response of one or more variables. The selection of appropriate test signals is of essential importance to the identification process.

• The test signals should be convenient to generate. Examples would be the administration of a drug or metabolic compound in the form of an injection, or applying a step change in the concentration of oxygen being inhaled by a subject in an investigation of the dynamics of the respiratory system.
• The signals should be as large as possible so as to produce a high output signal-to-noise ratio. However, the magnitude of the test signal is limited by practical constraints. First, the extent of perturbation of the system under test must not be excessive. Furthermore, the extent of the disturbance to the system posed by the test signal must not move it outside the region for which the assumed model is valid.
• The test signal should, as far as possible, minimize the time required for the identification process.
• The test signal should, when taken together with the model to be identified, result in a convenient and accurate identification procedure.

The most commonly applied test signals in studies on physiological organ systems are those that result in a transient response of one or more system variables. Impulse (e.g., injection) or step (e.g., infusion) inputs are the usual form of input in this context. The impulse response or the step response is obtained!

Determine whether the experimental data are sufficiently rich to enable unique estimates to be made of all the unknown parameters: A-PRIORI IDENTIFIABILITY. Problems of identifiability arise where there is a mismatch between the complexity of the model and the richness of the experimental data.