The BRAINCIRC model: Modelling the Human Brain Circulation

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Basic motivations

The human brain circulation responds in a complex way to a large number of stimuli, and failures of the circulation both locally and globally are implicated in a number of pathologies. Failure of autoregulation can result in ischaemia, (decreased blood flow) or hyperaemia (increased blood flow, usually accompanied by increased capillary pressure). These in turn can result in blood-brain barrier dysfunction, haemorrhage, and ultimately cell death.

There are a great number of factors which can affect the circulation. These can be metabolic, neural or endothelial for example. Both chemicals and physical stimuli (such as shear stress) act as vasodilators and vasoconstrictors by activating different feedback pathways. It is for this reason that in order to understand how the brain circulation functions in vivo, we need to adopt a modelling approach.

A good introduction to the literature on the cerebral circulation is (1).

Introduction to the model

We have constructed a computational model of the cerebral circulation in the hope of uniting in a single framework the complexity of the regulatory mechanisms, and the variety of ways in which things can go wrong. This website is concerned with documenting the model, and to an extent the physiology underlying the model. There are links below to detailed documentation and to the first released version of the model itself, available for download. We ask the user to bear in mind that this is a first version, and that the model is under constant development.

Construction of the model can be viewed at a number of different levels. The first level is it's very broad structure: choosing which groups of physiological processes are represented, where they take place, and how they connect up with each other. This structure can be considered independently of the details of how particular processes are represented.

At this broad level, the sites of interest are the circulatory system itself, the brain tissue, and vascular smooth muscle (VSM) cells in the walls of certain blood vessels which cause the vessels to contract or expand, and hence regulate blood flow. Different sets of processes are important in each of these sites: chemical reactions, ion channel dynamics, electrophysiology, etc. The sites interact with each other via a variety of means - for example by the direct exchange of chemical substances, or via activation of more complex pathways. Figure 1 sketches this broad structure.

Figure 1: The broad structure of interactions in the model.
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1) and 2) Changes in blood flow rate cause changes in delivery and removal of various substances, and hence changes in tissue chemistry; similarly blood chemistry is affected by the transport of substances from the tissue. 3) Changes in transmural pressure and blood flow rate have direct and endothelially mediated effects on vascular smooth muscle; 4) Changes in chemical concentrations in tissue have effects on vascular smooth muscle; 5) The key feedback step - changes in vascular smooth muscle cause changes in the diameter of resistance vessels in the circulation and hence in blood flow rate.

The model can be conceptualised at a second level - still qualitative but more detailed - which involves tracing pathways of cause and effect. These pathways often have a complex topology, involving branches and loops. A single stimulus such as a change in arterial blood pressure or an increase in CO2 concentration typically activates a number of pathways. It is the concurrence of a number of effects which will eventually result in the observed behaviour. An example of this process is presented in Figure 2.

Figure 2: Some feedback loops by which various stimuli cause changes in blood flow.
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gKCa , gKATP , gKIR and gKV are conductivities of calcium sensitive, ATP sensitive, inward rectifier and voltage dependent potassium channels respectively in vascular smooth muscle (VSM) cells. VSM tension acts to control vessel diameter and hence cerebral blood flow. NO is nitric oxide, a potent endogenous vasodilator.

A third level involves looking at individual processes, and how these might be treated mathematically. Essentially this involves choosing functional forms. At this level, there is often incomplete knowledge and poor experimental data. For example, although it is known that adenosine binds to adenosine receptors on VSM cells and ends up activating ATP-sensitive potassium channels in these cells, the full details of this pathway are not, to our knowledge, known. Alternatively the understanding of a process might be comprehensive at a physiological level, but of a complexity which makes its detailed mathematical treatment difficult. Several biochemical processes fall into this category, including the full range of interactions between, haemoglobin, oxygen, CO2 and protons. On other occasions, the situation may already have been modelled mathematically in some detail, but it remains questionable whether the level of complexity involved is one which is worth incorporating into a wider model. As an example, there has been a fair amount of work on the way in which myosin light chain phosphorylation levels affect force development, but it might nevertheless be sufficient to use simple caricatures as a substitute.

A fourth level involves looking not just at how a process is represented mathematically, but at how parameter values are set once functional forms are chosen, and how model behaviour is affected by the precise values of the parameters. The setting of parameters can itself depend on detailed arguments and indirect estimation. Parameter setting in a large computational model is very likely to contain at least some educated guesswork, and part of the process of analysing the model's behaviour must involve some understanding of how robust particular model behaviours are to errors in parameter estimation. Parameter setting is discussed extensively in the detailed model documentation.

Computational structure

The dynamics of a given variable are often determined by a large number of processes (for example reactions and transport processes). For example the differential equation determining the evolution of the concentration of protons in a given compartment can easily contain more than ten terms. However in constructing a model, or attempting to understand it's structure, we generally think in terms of particular reactions, transport processes, biophysical processes, etc. rather than at the level of the evolution of a particular variable in its entirety. In this light, it is useful if the computational methodology used to construct the model is designed to parallel the physiological structure, in the sense that where possible it is physiological processes which are stored rather than simply the mathematical equations derived from them. A more detailed account of the philosophy underlying the modelling process can be found in (2).

Computationally the description of a chemical reaction consists of a list of its substrates and products, along with their stoichiometries, if necessary the compartments in which they reside, and a rate term, or a means of determining a rate term from some general principle. Similarly, describing a transport process involves knowing which chemical(s) are transported, which compartments are involved, and how the transport rate is determined. Of course in some situations, especially when incorporating parts of existing models, we might also wish to incorporate entire differential equations, or parts of differential equations into the model wholesale.

The way that model construction proceeds at a computational level is shown schematically in Figure 3. This structure has the advantage that when it comes to refining the model, say by altering the mechanism of a particular chemical reaction or by adding a new one, this can be done independently of the rest of the model.

Figure 3: The computational methodology used in constructing the model.
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Detailed documentation and BRAINCIRC model download

Ultimately the model can only be understood by running simulations and by careful exploration of the various processes it contains. To aid in this we have provided two resources: The detailed documentation contains descriptions of all model processes, variables, parameters, and parameter setting procedures. The model itself is being released under a public license, allowing the user to experiment with it as he or she pleases.


L. Edvinsson, D. N. Krause (Eds.), Cerebral Bood Flow and Metabolism, Lippincott Williams and Wilkins, 2002.

M. Banaji, S. Baigent, A flexible, iterative, approach to physiological modelling, in: R. Paton, L.A. McNamara (Eds.), Multidisciplinary approaches to theory in medicine, Elsevier, forthcoming.

Murad Banaji