Introduction to Biological Systems Modeling

Conceptual Definitions of Modeling in Common and Scientific Language

A model can be defined differently depending on whether it is viewed through the lens of common language or a rigorous mathematical and scientific framework. In common language, a model is often viewed as an object or a behavior meant to be copied. Examples include a painting model or societal models of lifestyle, such as showgirls (veline), soccer players (calciatore), or even extreme examples like a kamikaze. The properties of such models involve completeness and an ideal perfection; they are typically richer and more complex than reality itself.

In the mathematical and scientific language, however, a model is seen as a theoretical scheme, an abstract structure, or a site for simulation. Examples include a architectural model of a building or a Bohr model of the atom. Scientific models represent an impoverished image of the real world. They rely on abstraction, focusing only on aspects deemed relevant from a specific point of view. Consequently, they are poorer and more schematic than the actual systems they represent.

Scientific Models and Biological Complexity

In the scientific field, a model is a simplified representation of an object, such as a phenomenon, a mechanism, or a system. This representation is based on a code of correspondence between the elements of the model and the elements of the object. Essentially, a model is an abstraction or a simplified representation of reality tailored toward physical, technological, biological, or physiological systems. In biological and physiological contexts, modeling is even more necessary than in physical systems due to inherent biological complexity and significant limitations in direct measurement.

Biomedical Modeling and Bioengineering

Defining and using models, particularly mathematical ones, is a primary sector of bioengineering. These models allow for the quantitative description of physiological systems at multiple levels: cellular, tissue, organ, and the entire organism. This activity often involves modeling artificial systems that function similarly to their physiological counterparts. While mathematical modeling in biomedicine has ancient roots, its widespread adoption and sophisticated application date back to the 1900s.

Objectives of Modeling in Biological Systems

The primary goals for developing models in this field are multi-faceted. First, they serve to understand the phenomena under study by providing a synthetic and formal description. They allow for the examination of alternative hypotheses with the specific aim of demonstrating the falsehood of some. Furthermore, models enable the measurement or estimation of variables and parameters that are not physically accessible, such as non-observable physiological compartments or the resolution of deconvolution problems.

Modeling is crucial for the accurate design of experiments, facilitating better characterization of phenomena through the planning of dynamic response experiments with optimized inputs and sampling schemes. As simulation tools, they allow researchers to study the effects of external signal modifications or changes to parts of the model, helping to distinguish physiological states from pathological ones. They function as diagnostic tools to differentiate between diseases, varying severities of the same condition, or healthy vs. affected subjects. Finally, they assist in determining correct therapeutic protocols regarding administration schedules and dosages, while serving as instruments for teaching and knowledge transfer.

Methodological Principles and the Utility of Models

Constructing a model is often described more as an art than a rigid technical procedure, though it utilizes specific analytical tools. The methodology is dictated by the relationship between the model, the system, and the intended purpose. A model cannot exist until a scope and specific objectives have been defined. There is no universal model that applies to every context.

A foundational creed in modeling states that all models are wrong, but some are useful. The challenge is not to define a "correct" model of a system, but rather the most useful model for a specific goal. Models can be classified by their scope:

1. Descriptive Models: These represent a synthesis of available knowledge to facilitate understanding without necessarily explaining underlying causes.

2. Interpretative Models: These attempt to provide an interpretation of observed phenomena based on general laws or internal properties of the object.

3. Predictive Models: These are designed to provide forecasts regarding the possible evolution of a given phenomenon.

Classifications and Types of Models

Models differ based on their level of abstraction and the formalisms they use:

1. Verbal Models: Qualitative descriptions using words, used when precise formulations are impossible. Darwin’s original theory of natural selection is a prime example.

2. Iconic or Physical Models: Scaled reproductions used for physical analysis, such as wind tunnels for fluid dynamics or anechoic chambers for studying radar reflectivity in aircraft.

3. Analogical Models: These replace system components with different physical components that are equivalent in terms of interaction, such as using electrical circuits to study hydraulic systems.

4. Conceptual Models: These represent relationships through diagrams or schemes. They are highly important in complex domains like biology for metabolic pathways or gene regulatory networks.

5. Mathematical or Symbolic Models: These represent the system through formal equations relating characterizing variables, offering the highest level of abstraction.

Advantages and Properties of Mathematical Models

Mathematical models are powerful because they allow for a synthetic, rigorous, and well-structured representation of empirical knowledge. They provide the ability to rigorously deduce the consequences of hypotheses and predict the results of experiments that have not yet been performed. They also allow for the evaluation of hypothetical modifications to a system to determine potential advantages or disadvantages.

Modeling Approaches: External vs. Internal

External modeling, known as Black Box modeling, treats the system as an opaque entity. Researchers apply stimuli (inputs) and observe responses (outputs). The mathematical laws derived only link inputs and outputs without making hypotheses about internal structure. This is useful when internal knowledge is limited, focused on interactions with the external world. However, parameters in a black box model rarely have immediate physical or functional meaning.

Internal modeling, involving Transparent or Opaque Box models, explicitly accounts for internal structure and the physical nature of components. Known as structural modeling, there is a close correspondence between model elements and system components. These are typically interpretative. The level of detail depends on a compromise between simplicity and the required level of detail for the specific purpose.

Structural Classification of Mathematical Models

Mathematical models can be categorized into various types, and any combination of these properties is possible:

• Static vs. Dynamic

• Deterministic vs. Stocastic

• Time-Invariant vs. Time-Variant

• Lumped Parameters (Parametri Concentrati) vs. Distributed Parameters (Parametri Distribuiti)

• Linear vs. Non-linear

• Continuous Time vs. Discrete Time

Mathematical Examples and Formulas

For a general data model relating inputs and outputs without structural assumptions, a sum of exponentials might be used: $y(t) = \sum_{i=1}^{2} A_i e^{-\alpha_{i} t}$ In this case, the unknown parameters are $p = [A_1, A_2, \alpha_1, \alpha_2]^T$.

In compartmental modeling, a single-compartment model representing a physiological system might be defined as: $\dot{x}_1(t) = -k_{01} x_1(t) + u(t)$ $x_1(0) = 0$ $y_1(t) = \frac{x_1(t)}{V_1}$ Parameters for this model include $p = [k_{01}, V_1]^T$. A two-compartment model increases complexity: $\dot{x}_1(t) = -(k_{01} + k_{21})x_1(t) + k_{12}x_2(t) + u(t)$ $\dot{x}_2(t) = k_{21} x_1(t) - k_{12} x_2(t)$ $x_1(0) = 0$ $x_2(0) = 0$ $y_1(t) = \frac{x_1(t)}{V_1}$ Parameters here are $p = [k_{01}, k_{12}, k_{21}, V_1]^T$.

For distributed parameters, such as transcapillary substance exchange, partial differential equations are employed: $\frac{\partial C_p(t, x)}{\partial t} = -v_{LF, g} \frac{\partial C_p(t, x)}{\partial x} - \frac{PS}{V_p} [C_p(t, x) - C_{isf}(t, x)]$ $\frac{\partial C_{isf}(t, x)}{\partial t} = \frac{PS}{V_{isf}} [C_p(t, x) - C_{isf}(t, x)]$ Parameters involve factors such as $p = [RDlv, PSg, V_p, V'_{isf}]^T$.

Simulation, Identification, and Validation

Simulation is the calculation process that examines system behavior in response to stimuli. It involves performing numerical experiments "in silico" on the model rather than analytical resolution. This is preferred when physical experimentation is inappropriate, inconvenient, or dangerous. A "complete" model—where all parameters and initial conditions are specified—is required for simulation. If incomplete, parameter estimation techniques are used.

Model Identification is a process involving experiment design, physiological system observation, model structure definition (based on a priori knowledge and assumptions), and data analysis. If results are poor, it may be due to the "Garbage In - Garbage Out" paradigm, where either the model is poor or the data itself is flawed.

Validation concludes the cycle, ensuring the model accurately represents the intended system before it is accepted as a final model.

Golden Rules for Modelers

Practitioners follow several "golden rules" to ensure effective modeling. First, acknowledge that there is no perfect model; all are approximations. Always define the domain of validity and avoid using the model outside those bounds. Predictions should be compared with independent data not used during the identification phase.

Models should be designed in an "open" manner to allow for the inclusion of new knowledge. They must also be "transparent" to allow for easy falsification. One should remain critical, avoiding over-attachment to a single model, and ideally maintain both a simpler and a more complex alternative to the proposed model for comparison.