A patient-specific in silico model of inflammation and healing tested in acute vocal fold injury. PMC 7. Vodovotz, Y. Translational systems biology of inflammation. PLoS Comput. PMC 8. Mi, Q. Nieman, G. A two-compartment mathematical model of endotoxin-induced inflammatory and physiologic alterations in swine. Care Med. Azhar, N. Analysis of serum inflammatory mediators identifies unique dynamic networks associated with death and spontaneous survival in pediatric acute liver failure. To our knowledge, this is the only hierarchical agent approach that to date has been applied to simulate a systemic disease process.
As an alternative to explicitly representing subcellular structures using a hierarchical agent approach, subcellular biological scales may be incorporated phenomenologically into a cellular level model. More recently, a phenomenological representation of the loss of the metastasis-suppressor E-cadherin protein has been used in an agent-based model to explore the interactions between normal and mutated cells within mixed populations D. Walker et al.
Complex Systems and Computational Biology Approaches to Acute Inflammation
This category of model explicitly incorporates representations of processes which change on a faster time scale than cellular scale phenomena most notably proliferation. Examples include i the intercellular diffusion of autocrine or paracrine growth factors, which can typically diffuse a distance roughly equivalent to a cell diameter in less than a second, ii receptor-binding or intracellular protein phosphorylation events, which occur within seconds to minutes at typical concentrations or iii biophysical phenomena such as vascular flow, that may result in the transport of biochemical factors, or even cells.
These approaches are more powerful in explicitly capturing the complex interactions between cells and their biochemical and biophysical microenvironments, as well as the detailed dynamics of intercellular interactions. The modelling of integrated multiplex biological processes with inherent time scale differences e. The two model components are usually run consecutively, with an update of the agent or CA model followed by an update of the submodel representing the faster process.
In the case of intercellular signalling via the diffusion of biochemical factors, changes in cell number, location or factor secretion will influence the boundary conditions or sources of molecular species in the signalling model, whereas the spatial distribution or intracellular concentration of key molecular or protein species will influence cell fate decisions. This scheme is thus well suited for exploration of the concept of intercellular interactions by diffusive or membrane bound signalling, or cellular interaction with the molecular environment, for instance, tumour cell behaviour in response to diffusing oxygen or nutrients.
These extended models explicitly encapsulated the processes of ligand release and diffusion, binding of receptors and activation of an intracellular-signalling cascade that controlled the decision to adopt a migratory or proliferative phenotype, and provided a feedback mechanism by modulating the expression of genes encoding receptor and ligand. Simulations generated using this multi-scale model suggested that the density of receptors on the cell surface could influence the rate of tumour expansion up to a maximum saturation level, and that both migratory and proliferative cell phenotypes were necessary for a high rate of tumour expansion.
A 3D implementation of this model was also applied to investigate the effect of epidermal growth factor EGF signalling in non-small cell lung cancer [ 41 ]. In a recent paper, the ODE-based Tyson-Novak model of the cell cycle [ 42 ] was incorporated within modelled cells, allowing a more in-depth exploration of proliferation control [ 43 ].
Multi-scale methods have also been used in order to study the role of biochemical signalling in the context of growing epithelial cell populations. In one example, differences between predicted and measured growth kinetics obtained respectively from a lattice-free agent model simulation and epithelial cells grown in culture [ 24 ] indicated the existence of a contact-mediated pro-proliferative mechanism that was not present in early versions of the model. These observations prompted the development and integration of model components representing the production, diffusion and binding of autocrine growth factors in growing epithelial cell populations [ 44 ], and juxtacrine-mediated intracellular signalling via EGF receptor EGFR activation [ 45 ].
In the latter case, the size and frequency of intercellular contacts, which provided sites of intercellular signalling, were closely based on observations derived from time-lapse microscopy studies of real cells grown in different culture environments. Again, diffusive intercellular signalling processes were simulated using PDEs, and juxtacrine and intracellular signalling was represented using ODEs, with data passed at each iteration from the slower agent model to the faster signalling model.
Simulations generated by this model indicated that the response of individual cells was influenced by the local microenvironment, leading to population heterogeneity. This heterogeneity was masked when data was averaged over the entire cell population as is the case for many cell biological-based assays, such as western blotting and highlighted the danger of extrapolating population-derived data to individual cells.
Similar results were reported in [ 46 ], where a generic model consisting of a fixed ring configuration of cells, each encapsulating a generic three-step activator-inhibitor signalling pathway with feedback properties and coupled by diffusion of the end product of the pathway, was used to demonstrate the loss of resolution associated with averaging measurements over space and time.
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The role of direct cell—cell contact-mediated signalling in the context of tumour invasion has also been investigated using multi-scale models [ 47 ]. The results suggested that tumour growth could be controlled directly by intercellular interactions. Finally, multi-scale approaches have also been used to integrate individual based descriptions of cells with representations of vascular flow—a biophysical process which, like signalling processes, occurs on a faster time scale than cellular growth and proliferation.
In [ 48 ], a CA-based model of individual tumour cell migration and proliferation was integrated with an explicit PDE-based representation of oxygen diffusion from blood vessels. The topology of the blood vessel network was explicitly represented and was itself the emergent outcome of a separate model that was run prior to the tumour cell CA model. A pressure gradient was applied to an initial configuration of vessels, with the resultant flow rate through the network calculated using Kirchoff's Laws. Vessel radii were updated to reflect remodelling in response to changes in flow, and the new flow rate calculated and fed back into the first model until a steady state was reached.
The final network topology, and hence haematocrit distribution, provided the oxygen source for the tumour CA model described above which, again, was extended to incorporate the Tyson—Novak cell cycle model [ 42 ]. Hence, as well as incorporating subcellular scale phenomena into a cellular level CA model, this work demonstrated the applicability of physics-based models in generating emergent tissue structures in this case, vessel networks that may in turn influence the behaviour of the surrounding cells.
Although the vessel and tumour growth are separated in time, with the final vessel network geometry forming static nutrient sources in the tumour growth model, in reality, the two systems would evolve together. This suggests that full integration of the two models could provide a more realistic representation of the system. A fully integrated approach to combining vascular flow and cellular interactions has been developed in order to study inflammatory processes [ 49 ]. In this case, endothelial cells lining the vessel surface and circulating leukocytes were represented as individual agents, whereas bulk blood transport was modelled separately using a network flow model.
Although the transport of leukocytes by flow was not explicitly modelled, the prevalence of cells available for interaction at each time-step was determined by the computed haemodynamic parameters. Rules for adhesion and migration depended on the concentrations of these factors, as well as wall shear stress computed by the flow model and the density of adhesion molecules on the leukocyte and endothelial agent surfaces. In simulations, this model successfully represented the sequence of leukocyte-endothelial interactions observed in vivo initial contact, rolling, firm adhesion and extravasation.
A potential criticism of agent- and CA-based models of cellular behaviour is that they are intrinsically phenomenological and not capable of encapsulating the complexity of a biological system. As argued above and in [ 14 ], careful comparison of virtual and experimental models provides a key to allaying these criticisms. The development of multi-scale agent models, and in particular those that aim to capture the protein or gene level mechanisms that determine cell phenotype, offers a further opportunity to ensure that simulations capture the real system processes as accurately as possible.
Many of the examples discussed above consist of a cellular and subcellular usually signalling-related component and fit a category of bi-scale models. Relatively few models have encompassed supra-cellular level scales, with the exception of An's work on modelling systemic inflammatory disease [ 34 ]. However, methods of representing multiple spatial scales are non-trivial. Approaches adopted within the materials modelling community include classical multi-grid or domain decomposition methods, or more recently, gap-tooth or Heterogenous Multiscale Methods HMM.
The details of these methodologies are beyond the scope of this review. However, we refer interested readers to [ 51 , 52 ] for a discussion of these techniques. It is worth noting that these methodologies have been developed for continuum type e. More appropriate may be the continuum-atomistic coupling methodologies discussed in [ 53 ]. As yet, there is no universally adopted theoretical or computational framework for the assembly of multi-scale biological models.
A recent description of a process for designing multi-scale, multi-science i. This involves dissecting a complex system into constituent processes and representing each process within logarithmic axes representing time and space. The relationship s between any pair of processes—whether overlapping, in contact, or separated, and in the latter case, the relative direction of the offset, determines the optimal computational methodology for representing the processes. The relative positions on the SSM of the three categories of model identified in this review are shown in Figure 1.
In terms of the categories identified in [ 54 ], it can be seen that cellular steady-state approaches described above correspond to a special uni-directional case of micro-macro coupling case 3. Spatially hierarchical agents represent coarse and fine structures operating on similar temporal scales case 2 and temporally separated processes coupling cellular models with blood flow correspond to case 3. There are other theoretical considerations associated with the development of multi-scale, multi-paradigm models.
For instance, within equation-based models of signalling mechanisms, the process of quantifying uncertainty with approaches such as sensitivity analysis is well established. How errors and uncertainties propagate from one biological scale e. In addition to the scarcity of theoretical frameworks for the development of multi-scale models, there is also a lack of software frameworks the reader is referred to [ 57 ] for a general review of single-scale agent-based modelling platforms.
The option exists to substitute model modules e. The developers have placed emphasis on usability for non-expert programmers. A potential drawback of multi-scale, as opposed to single scale, modelling approaches is the additional computational overhead required to solve these models. Particular agent-based frameworks e. As far as we are aware, none of the models discussed in this review have been implemented on a parallel architecture, and hence all are ultimately limited in the number of agents and the complexity of the subcellular processes represented.
These issues raise particular challenges for computer scientists and software engineers involved in the design of such computational frameworks. The vision for CA and software agent-based models is that they will provide a solution for systems biology to interpret dynamic, biological relationships, most importantly at the cellular level, that can then be integrated into more general multi-scale models.
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However, fulfilment of this vision will depend on future development and amalgamation of bioinformatics and modelling tools, including automated methods to process and encapsulate information from large-omic datasets. As argued by An [ 34 ], the strength of the software agent paradigm is that it offers a conceptually intuitive, non-mathematical framework within which researchers in the biomedical community can represent, exchange and revise dynamic biological knowledge. Individual based models can serve as a substantiation of conceptual models, and provide a means for simulation to test hypotheses.
The inherent capacity of the paradigm to represent non-deterministic and heterogeneous behaviour offers the opportunity to investigate the role of diversity in cellular populations, and highlight potential implications for experimental methodologies [ 45 ]. There are many challenges, both computational and theoretical, facing those involved with the development of multi-scale models.
These are in addition to the challenges relating to the engagement of biologists and extraction of appropriate biological data to both inform and validate models, which are central to all computational biological investigations of any scale or paradigm. The virtual cell is a promising, paradigm for middle-out modelling because, as the basic unit of life, the cell is responsible for processing and integrating information from other cells and from the environment to regulate homeostasis and emergent tissue-level behaviour.
Software agents and cellular automata provide useful paradigms for modelling biological systems as they can be used as direct representations of cells, facilitating model validation against experimental systems. The modelling of complex biological phenomena that extend across temporal and spatial scales may be achieved by combining different modalities e. Existing models are primarily bi-scale, consisting of a cellular level component integrated with a single supra- or subcellular model.
In the majority of cases, the second component represents an inter- or intra-cellular signalling process. Remaining challenges for computational modellers include the development and adoption of a generalized framework and software for multi-scale modelling, alongside the continuing challenges of engaging with biologists and extracting appropriate biological data to inform and validate models.
Yoram Vodovotz, PhD
The future vision for computational modelling is to provide predictive models of complex multi-scale, dynamic biological systems, but to reach this full potential is a grand challenge that will require automated methods to integrate omic datasets. Oxford University Press is a department of the University of Oxford.
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