Mathematical and computational techniques

Sensitivity analysis

Improved methods of sensitivity analysis will allow a more rigorous and in-depth analy-sis of the model. Morris sensitivity analyanaly-sis could be improved by choosing the param-eter distributions more carefully, choosing different functions as the output of interest, and investigating the effect of using different numbers of repeats. Other methods of sensitivity analysis could also be investigated, such as a variance based approach using Monte Carlo methods [262].

Parameter estimation

Parameter estimation could be improved by using different frameworks, such as Bayes-ian methods. This has already begun to carried out, and will therefore be described briefly here.

For a set of model parameters θ and measured data D, Bayes theorem states that P(θ|D)= P(D|θ)P(θ)

P(D) (7.1)

where P(θ) is the prior probability distribution of the parameters, P(θ|D) is the likeli-hood of the measured data given the parameters and P(D) is known as the marginal like-lihood. Bayesian inference aims to determine the posterior distribution P(θ|D). Usually, the relative probabilities are compared and P(D) is not calculated.

For a deterministic model M and a set of measurements at different times D with inde-pendent identically, normally distributed errors with precision τ= 1/σ², the likelihood at each time point is given by

r τ

and the log of the overall likelihood for N time points is

log P(θ|D) = −τ

Therefore, maximising the likelihood is equivalent to minimising the mean square error between modelled and measured data, as is typical in traditional parameter optimisation.

Bayesian methods are superior to these methods in two ways: i) they incorporate prior information about the distribution of the parameters and ii) instead of finding a single point in the parameter space, they generate a posterior parameter distribution, usually by the use of a Markov chain Monte Carlo (MCMC) sampling method. However, the disadvantages of these methods are that they require many more model evaluations, and the assumption of independent, identically distributed errors may not be appropriate, or the precision of the errors may not be known. Bayesian methods have recently begun to be used for parameter estimation in models of dynamic biological systems [269].

If a model is not deterministic, calculating the likelihood is usually not possible within a reasonable time frame. Likelihood-free Bayesian inference methods provide an esti-mate of the posterior distribution without requiring calculation of the likelihood. One of these methods is approximate Bayesian computation (ABC). In its simplest form, ABC can be implemented with a rejection sampler. A trial point is chosen by sampling from the prior distribution. A distance between the simulated data at this trial point and the measured data is then calculated. A typical choice for this distance is the sum of the squared errors at each of the time points. If the distance is less than a chosen tol-erance, the trial point is accepted, otherwise it is rejected. This is then continued until the chosen number of points have been accepted, and these points make up the estimate of the posterior distribution. This method can be very inefficient (i.e. have a very low

sample populations until a final small tolerance is reached. ABC methods have been used in systems biology for both parameter estimation and model selection [270, 271].

ABC-SysBio [272] is a software package that performs parameter estimation and model selection using ABC. It is now being used for parameter estimation with the Brain-Signals model. This involved writing extensions to the package to enable the simula-tions to be carried out using the BRAINCIRC environment. An example of applying this method to hypercapnia data from one of the 14 subjects described in Chapter 4 is described here. To take into account the uncertainty in the measurements used as in-puts, normally distributed random errors with fixed variance were added to the input data at each time point for each simulation. As in Chapter 4, the PaCO₂, SaO₂ and Pa

measurements were used as inputs, but this time with added random errors of standard deviation 1 mmHg, 1 % and 0.5 mmHg respectively. An example is shown in Figure 7.1.

0 200 400 600 800 1,000 30

40 50 60

time (s) PaCO2(mmHg)

0 200 400 600 800 1,000 96

98 100 102

time (s) SaO2(%)

0 200 400 600 800 1,000 70

80

time (s) Pa(mmHg)

Figure 7.1: Measured PaCO2, SaO2 and Paduring a hypercapnia challenge in one sub-ject (black), and model inputs generated by adding normally distributed noise to the measurements (red).

Two parameters were used (the same as those optimised in Chapter 4): [Hbtot] with a normal prior of mean 9.1 mM and standard deviation of 5.0 mM and RCwith a uniform

prior from -1 to 10. The modelled and measured TOI and V_mca were compared. The details of the ABC MCMC method can be found in Toni et al. [270].

3.8 3.85 3.9 3.95

Figure 7.2: Example of ABC parameter estimation. Estimated posterior distribution for model parameters [Hbtot] and RCgenerated by comparing measured and simulated TOI and V_mcasignals.

A population size of 500 was used, and an automated tolerance schedule was run, with the final tolerance set to a small value which was not reached. The process was stopped after 120 populations by which time the acceptance rate had fallen to 0.004 indicating that the tolerance was very close to its minimum possible value. The resulting sample of 500 points, which serve as an estimate of the posterior distribution, is shown in Figure 7.2. There is a slight relationship between the two parameters, but both have been identified to within a fairly narrow range. It is difficult to compare this with the optimisation from Chapter 4 since the weightings used in that optimisation affect the results.

This is ongoing work. In particular, the effect of using different distributions to represent uncertainty in the input will be investigated. Also, different numbers and groups of

successfully identified.

Modelling language

Improvements can be made by extending and enhancing the BRAINCIRC modelling environment. Improving the compatibility with SBML and CellML would make exper-imentation with other models simpler, and could also enable inclusion of other models in a modular way. It would also improve the ability to make use of SBML software tools developed by other groups, of which there are many [187]. In particular, the Systems Biology Workbench is an opensource project that provides a framework for different simulation software to communicate in order to encourage code reuse and tool sharing [273]. Making BRAINCIRC compatible with this framework would help to integrate its use with other tools.