# Mathematical models of biological systems

• Method

Why does Ki use mathematical models of biological systems? In line with Ki goal to explain the relationship between nutrition, growth faltering and neurocognitive outcomes, mathematical models of biological systems are employed by Ki to model growth physiology. Three models have been developed to hypothesize pathways and quantitatively describe the interaction of nutrients, gut function, infectious and noninfectious microbes, and other risk factors that impact birth and growth outcomes.

### WHAT ARE MATHEMATICAL MODELS?

Mathematical models are used in many domains to understand and quantify complex system dynamics and test mechanistic and causal hypotheses. In the health sciences, applications of mechanistic models include epidemic forecasting, intra-host models of pathogen and immune response, and modeling cost effectiveness and intervention outcomes.[2,3]

Mechanistic mathematical models differ from statistical models. Statistical models, such as regression methods, generally aim to fit a function or curve to model data. In contrast, mechanistic models are comprised of equations or computation rules that explicitly describe how elements in a system relate to and affect one another.

Mathematical models are comprised of states (or compartments) and the flows between them. Often, mechanistic models employ a system of ordinary or partial differential equations (ODEs or PDEs) to explicitly describe the relationship between states and flow.

Specifically, mathematical models of biological systems and physiology explicitly represent underlying biological processes. Figure 1 shows an example of model parameters and differential equations used in Ki’s Gut and Growth Model. Figure 2 is a simplified representation of the key pathways and clinical endpoints of interest for the Gut and Growth Model.

Mathematical models employ data to accurately specify parameters in the equations, to constrain and calibrate the biological pathways represented, as well as to validate models and model predictions. FIGURE 1. Model parameters and system of ordinary differential equations (ODE) describing the gut and growth model. FIGURE 2. Simplified diagram of the gut & growth model.

#### Advantages of mechanistic models

• Allows modeling of complex, dynamic and non-linear biological systems over time.
• Explicitly incorporates knowledge of underlying biology.
• Many models are able to incorporate the effects of interventions or
novel perturbations.

#### Disadvantages of mechanistic models

• Some model parameters rely on mean data values from the literature, with some subsystems calibrated using animal data.
• Relies on making correct assumptions regarding biological mechanisms and causality.

### KI UTILIZATION OF MATHEMATICAL MODELS

#### Gut & growth mechanistic model

Physiological ordinary differential equations (ODE) model of food intake and gut absorption to derive the available energy for maintenance and growth in children age 0 to 2 years. The model incorporates microbiota health, gastrointestinal pathogen infection, immune activation and inflammation.

This model can predict fat mass, fat-free mass, changes in intermediate measures (e.g. fecal energy loss, indicators of microbiota health or inflammation). From fat and fat-free mass output, this model also allows for calculations of body weight and length.

#### Mother-fetus model

Model describes the partition of protein-energy between the pregnant woman and fetus, where protein energy is distributed across anatomical units: brain, adipose tissue, skeletal muscles, and other lean tissues. The mass of each unit is calculated based on its composition of proteins, lipids and water.

The primary outcome of this model is fetal growth (measured as weight, length, and body composition).

#### Infant-child body-brain model

Model describes the partition of protein-energy within children age 0 to 5 years. The model is partitioned by the same anatomical units and mass components as the Mother-Fetus Model.

### References

1. Morimoto M, Powell L, Santhakumar R, Gowins A, Arsenault J, Chow C, et al., editors. Quantitative Physiologic Model Of The Interaction Between Nutrition And Infection To Determine The Energy Available For Growth. Grand Challenges; 2017; Washington D.C.
2. Phair RD. Mechanistic modeling confronts the complexity of molecular cell biology. Molecular biology of the cell. 2014;25(22):3494–6.
3. Boogerd F, Bruggeman F, Richardson R. Mechanistic Explanations and Models in Molecular Systems Biology. Foundations of Science. 2013;18(4):725–44.
4. de Graaf AA, Freidig AP, De Roos B, Jamshidi N, Heinemann M, Rullmann JAC, et al. Nutritional Systems Biology Modeling: From Molecular Mechanisms to Physiology. PLoS Computational Biology. 2009;5(11):e1000554.
5. Hall KD, Butte NF, Swinburn BA, Chow CC. Dynamics of childhood growth and obesity: development and validation of a quantitative mathematical model. The Lancet Diabetes and Endocrinology. 2013;1(2):97–105.
6. Hall KD. Predicting metabolic adaptation, body weight change, and energy intake in humans. American Journal of Physiology – Endocrinology and Metabolism. 2010;298(3):E449-E66.
7. Bajaj JS, Subba Rao G, Subba Rao J, Khardori R. A mathematical model for insulin kinetics and its application to protein-deficient (malnutrition-related) diabetes mellitus (PDDM). Journal of Theoretical Biology. 1987;126(4):491–503.
8. Hove-Musekwa SD, Nyabadza F, Chiyaka C, Das P, Tripathi A, Mukandavire Z. Modelling and analysis of the effects of malnutrition in the spread of cholera. Mathematical and Computer Modelling. 2011;53(9):1583–95.

#### Last Updated

October, 2020 