Early stages of infection

To estimate the influence of gastrointestinal tract infection in the early stages of disease progression, two sets of simulations are performed, with and without local specification for nodes associated with the GI tract. Significant differences appear, due to GI tract inclusion. In the latter configuration, as soon as the GI tract is infected, the virus finds large amounts of potential targets, and viral spread is enhanced. After two weeks, most of the network is infected. On the other hand, if the GI tract is not represented, the virus still spreads rapidly through the lymph chain where the infection was initiated, but mostly remains restricted to these areas, and two weeks are not enough to obtain complete infection of the overall system. Another difference is in the proportion of infected cells, once the tract is included: ten days after local infection, twice as many cells are infected in several nodes of the GI tract, amounting to 70% of overall node cell count. This value is in agreement with biological studies, which found peak infection occurred at day 10-11, with a subsequent loss of 60 to 80% of memory cells [8].

GI tract inclusion clearly affects the overall viral spread, but does not alter the variability of this spread. Our hypothesis is that this variability is primarily a consequence of inter-node cell mobility, which is not altered by including local properties in the GI tract.

Similar patterns are observed for simulations on various sizes of networks. Thus the proposed model successfully implements gastrointestinal tract properties. Since such an attempt has not been previously reported, and given the importance of the tract in early disease progression, this is a useful complement to existing models.

Long-term progression

With respect to the length of the latency period, values obtained from the simulations and shown in Table 1 are slightly lower than clinical results, but the difference is not significant. Overall, GI tract inclusion reduces the latency period. It is also interesting to note that inclusion also improves the model behaviour, which tends to slightly over-estimate the latency period when scaled-up without GI tract inclusion. This is confirmed by looking at ratio between the peak of acute phase, (t), and the end of the latency period, (T). The ratio between these two time points, (which are the easiest to observe and quantify in both the simulations and the real system), is just under 70,(~ 69.6). In simulations without GI tract, we obtained T / t = 62.3 while, with the tract included, T / t = 66.7, which is a significant improvement. Future efforts on refining this GI tract implementation will probably further improve model realism, (and, therefore, this ratio). Our understanding of the model is also that the function used to simulate the input of new cells may now be the limiting factor, and that explicit thymus implementation may significantly enhance the model realism.

Translational potential

There are a number of examples where, using high-end computing, it is possible to scale up well-known approaches and, as a result, gain new insights into the complex systems under investigation. There are, however, cases where scaling up alone does not provide any answer. Thus, complementary solutions, such as model hybridisation, must be sought.

The model presented here is a typical example, where refinement to the agent-based model derives from the inclusion of network-based elements, (and where scaling-up alone was shown to over-estimate the latency period). Hybridisation has been used in other immune models, (see e.g. shape-space models [11]), but this particular hybrid framework may also be applied to other biomedical systems, such as epidemics. Dynamics of disease spread within a population are of crucial importance in terms of public health, (e.g. monitoring of existing outbreaks, preemptive evaluation of intervention policies). Here, a model would have to deal with millions of people living in modern urban environments, each with a refined social behaviour. To date, most approaches have focused on addressing one aspect only.

Network-based models have been used to investigate social structures and their impact on disease spread, (see e.g. [12]). This relies on the fact that, for most infectious diseases, contact is required for a new infection to occur, (sexually-transmitted infections being obvious examples). Social structures are represented by a network where nodes correspond to individuals (or groups), and edges correspond to social links between these. Infection spread is then implemented as a stochastic propagation over the network [13]. This approach, however, is limited by the size of the networks it requires: these are difficult to obtain from real data, or to generate ab initio. More crucially, because they are based on social structures, these models can not account for casual contacts between strangers, (e.g. in crowded areas and public transports), which are crucial in common infectious diseases such as influenza.

Conversely, agent-based approaches are suited to model such infections between strangers, which emerge from individual behaviour: an infection between travellers on a bus occurs due to individual choices from each, which lead to them boarding, and not because of any link between them, (as opposed to colleagues who have to be in the same office, because of this work relationship). Such a system can be efficiently parallelised, as described above for the immune model. The main limitation, however, is the lack of formal framework to include social complex structures.

Given the limitations of both paradigms, it becomes apparent that the hybrid framework presented here for immune modelling is the most efficient solution, combining elements of both approaches. In particular, network-based concepts can be used to generate socially-realistic populations, while the agent basis can simulate an epidemic outbreak within these. This hybrid framework was successfully translated into the epidemic context, and optimally parallelised, providing a realistic framework on which to investigate disease outbreaks and related policies [14].

Although a social basis for agent-based epidemics has been considered by a number of authors, (see e.g. [15]), the focus has been on using activity generators: the social network is then an emergent property of this generator, rather than an explicit model component that can be easily customised by the user. Due to this, handling several types of links, as needed in our model, had not previously been considered. Moving forward, advances in both hardware and computational techniques permit the investigation of larger, and increasingly complex, systems. A socio-medical was described as an example here, but similar frameworks can be applied to microscopic systems such as Epigenetics, where multiple phenomena interact over multiple scales, at the genetic level [16]. Model hybridisation offers a powerful method to account for all aspects of these systems.

Agent implementation

In this model, we implement four agent populations which correspond to CD4 and CD8 cells, antigen-presenting cells, and virions, as previously detailed, (see e.g. [19]). Each agent type corresponds to a specific class in C++, and inherits from a common, abstract base class. This class contains attributes and functions needed for management of ageing and of location within matrices. At the start of a simulation, initial agent populations are specified in a parameterisation file. Agent populations then dynamically evolve, following rules summarised in Table 2. The reciprocity between agents and biological entities, (cells and virions), mirrors real-system interactions between the latter. These are summarised in Figure 1. To implement these interactions, each agent is assigned an objective:

Variations	Viral agents	CD4 agents	CD8 agents	APC agents
Increases	Production by infected CD4	Created in thymus, or produced by multiplicating agent	Created in thymus, or produced by multiplicating agent	New agent created
Decreases	Agent ingested by APC, or infecting CD4	Infected agent destroyed, or end of life	End of immune response, or end of life	Presenting agent destroyed, or end of life

At the start of a simulation, initial agent populations are specified in a parameterisation file. Agent populations then dynamically evolve, following specific rules as when agents may be created or destroyed.

Lymph network implementation

In the context of HIV infection progression and the immune response to this, most of the interactions take place within the lymph nodes. As a consequence, our model relies on an explicit implementation of these: the “world' which hosts our agents is a collection of nodes, which are linked through a network whose structure mimics that of the lymph network.

In order to efficiently account for cell interactions and corresponding physical contact, accurate localisation of agents within the lymph node is required, implying division into smaller subunits. A standard approach to spatial representation requirements is to use a 2D or 3D matrix.

Here, matrix elements can be considered to be “physical”, containing several agents of each type, with a limit based on the size of the modelled entity. This implies that all cell interactions will take place within this physical neighbourhood, and considering surrounding matrix elements is not necessary, (contrary to models where matrix elements can only contain one element, where it is necessary to consider Moore or Von Neumann neighbourhoods for interactions). In our model, lymph nodes are, therefore, implemented as 2D matrices where each element represents a physical, 3D, neighbourhood. The typical size for each element is 10⁻¹²m³, (assuming a 10⁻⁸m³ volume for each lymph node).

Our implementation of these structures is based on static memory allocation. Large arrays of agents, (one for each agent type), are allocated when simulation starts, and store the maximum number of agents which can be present in the whole matrix at the same time. Each matrix element then only stores integers, used as offsets, to locate agents currently held in these arrays. This guarantees optimality of operations such as creation of new agents and agent mobility within a node. Agent mobility between multiple nodes must also be implemented, and this is obtained by linking these nodes on a network structure. A first step is the inclusion of entry and exit points to each matrix. Upon arrival in a node (through an entry point), an agent randomly circulates in its new environment, until it reaches an exit point and travels to another node, (as detailed below). This typically occurs after 20 to 30 hours, (time step: one minute), and the transfer between two neighbouring nodes markedly shorter (~minutes).

As key defense units, lymph nodes are distributed throughout the body. Humans have around 500 lymph nodes, and lymphocytes constantly circulate through these lymph nodes. To guarantee this cell circulation, connectivity is an essential property of the lymph network: a cell newly produced in the thymus must be able to reach any lymph, and efficient immune response implies interactions between nodes, (by means of cell exchanges).

The lymph network, however, is not equivalent to a complete graph: between any given pair of nodes, there is a path, but not necessarily a direct connection. In contrast, the lymph network is organised as a set of chains. These “clusters” of nodes can be found in the neck, chest, abdomen, underarm, etc.

The circulation of immune cells between nodes is not trivial. Cell migration is non-random [20], and is, in particular, type-dependent: CD4/CD8 ratio is always higher in the recirculated compared to overall population, (and often more than twice as important). Interestingly, however, transit kinetics are equivalent for both subsets of T cells. Naive and memory cells also have different recirculation pathways [21].

Circulation is also tissue-specific, in the sense that T lymphocytes preferentially recirculate back to the tissues they came from. This is controlled at the molecular-level through specific chemokines [22].

It is important to account for lymph network structure: where this, (as mentioned earlier), is similar to a directed connected graph, but is not complete. It implies that the final destination of an agent leaving a node can theoretically be any node, but its immediate destination is limited to a small subset of nodes. In our model, selection of the final destination is random, but not based on a uniform distribution. To account for preferential recirculation back to tissues of immune cell origin, each lymphocyte agent leaving a node has a 0.5 probability to be assigned this node as its final destination, ensuring circulation through the whole lymph chain corresponding to this node. To respect the higher lymph node to overall population CD4/CD8 ratio in departing agents, a function is also added to guarantee that not all CD8 agents evolving in the neighbourhood of the exit point are selected for departure. More refined implementations exist, (see e.g. [23]), and such an approach may be incorporated in our model at a later stage.

Selection of the immediate destination is based on the lymph network structure modelled. The number of lymph nodes modelled during a given simulation may vary, for instance depending on available computing resources. We implemented an automated technique to generate a network of nodes reproducing the lymphatic chain structures for any number of nodes. This guarantees a anatomically realistic network structure for any simulation size. Figure 2 illustrates this with a 24-node network.

To generate a network with n lymph nodes, this technique works as follows:

Create main-chain of user-defined length (e.g. n/4)

Do

Select a node already affected

If it is not suitable (e.g. degree to high)

then select a neighbor

End

Selected node becomes start of lymph chain

Select length for lymph chain

Create lymph chain using nodes not yet affected

Select a node already affected (same technique as above)

Selected node used to close chain

Until all nodes are affected

Parallelisation and GI tract implementation

The system under investigation has a strong parallel nature, in the sense that each node is largely independent of the others. There is no direct interaction between cells located in separate lymph nodes, since cell-level interactions require physical proximity. The consequence is that, apart from cell transfer, each node is independent of the others. A node influences its neighbours solely through cells exiting their current location and reaching the next node.

We can, therefore, use a spatial parallelisation, based on a reciprocity between the lymph nodes and the computer nodes of a cluster. Each lymph node of the model is assigned to a computer node of the parallel architecture, and the communication network is designed to mimic cell mobility along the lymph network. The tests carried out to evaluate several communication strategies are detailed in [24, 25].

With this parallel implementation, it is possible to run simulations involving several hundred lymph nodes, and more than one billion agents. This scale permits the definition of localised subgroups of nodes with specific features.

In an agent-based model, entities update their internal state based on interactions with other agents, but also with the environment itself. It is, therefore, possible to create compartmental properties which will locally alter agent behaviour.

Explicit implementation of the lymph nodes is, in that sense, crucial. Since each node is modelled separately, it is possible to select a subset of them and add additional properties for agents in those nodes. For instance, a higher probability of activation by some foreign antigen, not related to HIV, can be specified. Here, we do not distinguish between node types at the structural level, but rather in terms of properties of the cells they host. Providing this can be supported by enough data, later model versions may also include physical differences between lymph nodes.

Thus implementation of the gastrointestinal tract is currently obtained through selection of a long lymph chain in the lymph network structure and alteration of local properties: