A Bayesian framework for the analysis of systems biology models of the brain

Abstract

Systems biology models are used to understand complex biological and physiological systems. Interpretation of these models is an important part of developing this understanding. These models are often fit to experimental data in order to understand how the system has produced various phenomena or behaviour that are seen in the data. In this paper, we have outlined a framework that can be used to perform Bayesian analysis of complex systems biology models. In particular, we have focussed on analysing a systems biology of the brain using both simulated and measured data. By using a combination of sensitivity analysis and approximate Bayesian computation, we have shown that it is possible to obtain distributions of parameters that can better guard against misinterpretation of results, as compared to a maximum likelihood estimate based approach. This is done through analysis of simulated and experimental data. NIRS measurements were simulated using the same simulated systemic input data for the model in a 'healthy' and 'impaired' state. By analysing both of these datasets, we show that different parameter spaces can be distinguished and compared between different physiological states or conditions. Finally, we analyse experimental data using the new Bayesian framework and the previous maximum likelihood estimate approach, showing that the Bayesian approach provides a more complete understanding of the parameter space.

Fig 1. Generalised analysis process.

A simplified representation of the Bayesian analysis process.

Fig 2. Simplified structure of a typical BrainSignals model.

A typical BrainSignals model can be split into four compartments or submodels. The blood flow submodel represents blood flow from arteries to veins via the capillary bed and the oxygen transport submodel estimates diffusion of dissolved O₂ from the capillary blood to the brain tissue. Delivered oxygen is then utilised by the metabolism submodel. Finally, the measurement submodel translates the internal states of the blood flow and metabolism submodels into observable outputs. Model inputs are shown in red and consist of arterial blood pressure (ABP), arterial oxygen saturation (SaO2), partial pressure of CO₂ (PaCO2) and a parameter specifying relative demand, whilst measurable outputs are shown in blue, including NIRS signals as well as middle cerebral artery velocity (Vmca) and cerebral metabolic rate of oxygen (CMRO₂).

Fig 3. Healthy and impaired brain simulations.

Figures a)-e) show simulations of a healthy brain’s response to hypoxia, whilst f)-j) show the impaired brain’s response. The input variable of arterial oxygen saturation is shown in blue and is the same for both simulations, whilst the outputs of TOI, ΔHbO₂, ΔHHb and ΔCCO clearly differ between the two brain states.

Fig 4

Fig 4a shows the effect of different r_t values on the shape of the muscular tension curve for a range of vessel radii. It can be seen that reducing r_t widens the curve, leading to increased muscular tension for the same vessel radius. Figures 4b, 4c and 4d show the effect of both increasing and decreasing model inputs on cerebral blood flow for different values of r_t. Cerebral blood flow (CBF) is given as a proportion of the normal CBF (40 ml 100g⁻¹ min⁻¹). Changing r_t has a significant effect on the brain’s ability to autoregulate within the model. Fig 4b shows that higher blood pressures causes a decrease in cerebral blood flow for lower r_t, as opposed to an increase at the normal value of r_t = 0.018 cm. Fig 4c shows that for lower r_t values, CBF decreases quicker as PaCO₂ is decreased. Fig 4d shows that across all considered oxygen saturations, lower r_t gives a lower CBF.

Fig 5. Experimental hypoxia data.

Data collected from a healthy adult during a hypoxia challenge. Systemic data used as model inputs are shown in figures a), b) and c), with broadband NIRS measurements shown in figures d), e), f) and g).

Fig 6

Fig 6a shows data generated from the same test function y_i = a x sin(x) + b + ϵ, where a, b are both model parameters and ϵ is random Gaussian noise. x was varied from 0 to 2π, producing data y₀, y₁ and y₂ for the parameter sets Θ₀: a = 0, b = 0, Θ₁: a = 1, b = 0 and Θ₂: a = 0, b = 2.5 respectively. Despite both y₁ and y₂ being qualitatively very different they are very similar when summarised using only the Euclidean distance, with y₁ having a Euclidean distance ε_euc,1 = 35.58 and y₂ having a Euclidean distance ε_euc,2 = 35.44. If we instead look at the scaled baseline-to-peak (SBTP) distance we find that y₁ has a SBTP distance SBTP(y₁) = 240.5 and y₂ has a SBTP distance SBTP(y₂) = 0.27, giving ε_SBTP,1 = 240.2 and ε_SBTP,2 = 0.11. Fig 6b illustrates how the scaled baseline-to-peak distance is defined using x sin(x) + ϵ as the example signal. The baseline-to-peak distance is the absolute distance from the baseline to max ({|y_max|, |y_min|}). This is then divided by the range of the ‘default’ data, y₀, to get the distance as a proportion of the total change seen within the data. In this example, baseline-to-peak distance is 4.82 and the range is 0.02, giving the previously mentioned SBTP distance of 240.5.

Fig 7. Sensitivity analysis across all outputs for simulated data set.

Bar charts showing μ_* for the 10 most sensitive parameters across all model outputs, with values plotted on a log scale where appropriate. Distance used for calculation is the sum of ε_SBTP across all model outputs. All outputs except cytochrome-c-oxidase alone have μ_* values that vary on a logarithmic scale. Fig 7a shows results for all outputs combined, Fig 7b for TOI, Fig 7c for HbO₂, Fig 7d for HHb and Fig 7e for CCO.

Fig 8. Sensitivity analysis across all outputs for experimental data set.

Barplots showing μ_* values for the 10 most sensitive parameters across all model outputs, with the x-axis plotted using a log scale where appropriate. Distance used for calculation is the sum of ε_SBTP across all model outputs. Fig 8a shows results for all outputs combined, Fig 8b for TOI, Fig 8c for HbT, Fig 8d for HbD and Fig 8e for CCO.

Fig 9. Comparison of posterior distributions for healthy and impaired simulated data.

Fig 9 shows the posteriors for healthy and impaired data based on an acceptance rate of 0.01%. Posterior are shown over the full prior range as defined in S1 and S2 Tables.

Fig 10. Comparison of predictions for healthy and impaired simulated data.

Figures 10a and 10b show the predicted time series data from the healthy and impaired posteriors respectively. Each posterior was sampled 25 times and the resulting runs aggregated, with the median and 95% credible intervals plotted in dark blue and light blue respectively. Figures 10c and 10d show a zoomed in view of each output in order to show the credible interval of the posterior predictive distribution.

Fig 11. Posterior distributions for the experimental data set.

Fig 11 shows the posterior distribution for the experimental data set, based on an acceptance rate of 0.01%. The posterior median is shown in black and the OpenOpt predicted value is shown in red. Posterior are shown over the full prior range as defined in S3 Table.

Fig 12. Predicted fits for the experimental data set.

Fig 12 shows the predicted time series for all output based on the posterior shown in Fig 11. The posterior was sampled 25 times with the resulting time series aggregated, with the median and 95% credible intervals plotted in dark and light blue respectively. Overall behaviour is reflected in the predicted trace, with 3 distinct periods of hypoxia visible as periodic behaviour within all signals. The fit obtained using OpenOpt is shown in red.

Fig 13. Distribution of ε_NRMSE values for the posteriors of each dataset.

It can be seen here that the three datasets had very different distributions ε_NRMSE values for the samples that made up their respective posteriors. Despite this, the posterior predictive distributions for all datasets were good fits.