A three-compartment model of the hemodynamic response and oxygen delivery to brain
Introduction
The mathematical modelling of the hemodynamic response is an important part of the research in understanding brain function in human neuroscience. There are, in the literature, many mathematical models of the coupling between the hemodynamic responses such as cerebral blood flow, blood volume and blood oxygenation state to the underlying neural activity or cerebral metabolic rate of oxygen (CMRO2). We will classify these models into two groups: steady state models and dynamic models. Steady state models are concerned only with magnitude changes, whereas dynamic models take into account the transient characteristics as well as the steady state behaviour, hence are typically in the form of differential equations with respect to time. Once the transient characteristics are settled, the coupling between the input and output of a dynamic model describes their steady state relationship.
An influential steady state model of oxygen delivery to brain was presented by Buxton and Frank (1997). They used the diffusion equation to describe the oxygen loss along the capillary bed at steady state (that is, the partial temporal derivatives are zero) and thus modelled the oxygen extraction fraction (OEF) from the capillary as a function of the normalised cerebral blood flow (CBF) based on the assumptions that tissue oxygen tension was zero and that blood oxygen concentration decreased exponentially along the capillary. A simple steady state nonlinear relationship between the normalised CBF and the normalised CMRO2 was obtained, and the model required large changes in CBF to support small changes in CMRO2.
Hudetz (1999) and Mintun et al. (2001) modelled the radial and axial gradients of oxygen tension in tissue surrounding a single capillary. The oxygen diffusion equation was then solved numerically for absolute CMRO2 given absolute CBF. Both models used the assumption of linear decrease of blood oxygen concentration down the capillary and predicted a proportional relationship between oxygen delivery and utilisation.
Another steady state model of oxygen diffusion along the capillary was developed by Gjedde et al. (1999) and Vafaee and Gjedde (2000). Again, the model related absolute CMRO2 to absolute CBF as opposed to the normalised quantities used by Buxton and Frank. A parameter related to oxygen diffusion capacity was included, whereas in Buxton and Frank's model, this parameter was assumed constant and thus eliminated during normalisation. The model assumptions included negligible tissue oxygen tension and linear decrease of oxygen along the capillary. The model was applied to positron emission tomography (PET) data and explained the increase in CMRO2 as the combined effect of increased CBF and increased oxygen diffusion capacity in the region of activation. Other similar steady state models were presented by Hyder et al. (1998) and Hayashi et al., 2003.
As the methods for measuring hemodynamic responses became more sophisticated with good time resolution, in particular as the functional magnetic resonance imaging (fMRI) technique was increasingly used as a means of inferring changes in neural activity, dynamic modelling of the hemodynamic responses became increasingly important as it could be used to uncover the neural signal which underlies the blood oxygen level dependent (BOLD) signal as measured by fMRI. One of the first dynamic models linking the hemodynamic responses to neural activity was developed by Friston et al. (2000). They used (i) the Balloon/Windkessel model (Buxton et al., 1998, Mandeville et al., 1999b) which was a dynamic model linking normalised cerebral blood volume (CBV) to normalised CBF in the venous compartment, (ii) a second order linear dynamic system linking normalised CBF to stimulation, (iii) a conservation equation for the normalised deoxy-haemoglobin in the venous compartment and (iv) the steady state relationship between the oxygen extraction fraction (OEF) and normalised CBF derived by Buxton and Frank (1997). These combined to produce a fourth order nonlinear dynamic system linking the normalised deoxy-haemoglobin (Hbr) in the venous compartment to stimulation. Finally, a static nonlinear function was used (Buxton et al., 1998) to link the normalised Hbr and CBV to the BOLD signal.
The above model was extended by Zheng et al. (2002) by considering the oxygen diffusion equation including both the spatial and temporal partial derivatives, thus incorporating dynamics in the OEF and the capillary oxygen concentration. It also assumed a nonzero tissue oxygen tension that was related to an additional state variable. The result was a nonlinear dynamic model with seven state variables most of which were normalised quantities. The model was used to fit optical imaging spectroscopy (OIS) data from a brief stimulation paradigm and was found to produce physiologically plausible model parameters.
Instead of using normalised quantities, Valabregue et al. (2003) developed a dynamic model using absolute quantities for the hemodynamic responses, similar to the steady state model of Vafaee and Gjedde (2000). The model made full use of the oxygen diffusion equation, including nonzero tissue oxygen concentration. Hill's equation was used, and the dynamic equations were solved numerically to yield time series for CMRO2 as well as venous Hbr in response to changes in CBF. It also examined the effects of the assumption of exponential decay of oxygen along the capillary, as used by Buxton and Frank and applied in the dynamic models by Friston et al. and Zheng et al. Other dynamic models developed include those by Aubert and Costalat (2002) and Hathout et al. (1999).
Most, if not all, of the existing hemodynamic models assume either explicitly or tacitly that the measured CBF, CBV and BOLD signals originate from the venous compartment. Although the major contribution of the BOLD signal comes from the deoxy-haemoglobin which is primarily in the veins, the CBF and CBV measurements obtained via laser Doppler flowmetry (LDF) and OIS are likely to come from the arterial, capillary and venous compartments. The adequate interpretation of the BOLD signal will depend on an appropriate hemodynamic model taking into account the hemodynamic changes in all three compartments. It is for this purpose that we developed the three-compartment hemodynamic model presented in this paper. It is an extension to the one-compartment model (Zheng et al., 2002).
The theoretical framework of the three-compartment model will be presented in the next section. The sensitivity of the model output to the model parameters will be studied through simulations, and the model will be fitted to experimental data obtained from LDF and OIS. Both the simulation environment and the experimental procedures will be given in the Materials and methods section. The results will be compared to that obtained using the one-compartment model. We show that there are significant differences between the predictions of the two models.
Section snippets
Theory
This section introduces the mathematical equations used for the three-compartment (3C) model and discusses the appropriate partitioning of the hemodynamic measurements. We will denote absolute quantities by capital letters and normalised quantities by lower case letters, e.g., f = F/F0 and v = V/V0 represent the normalised CBF and CBV respectively, and the subscript “0” denotes the baseline value. Furthermore, the subscripts “a”, “c” and “v” denote the arterial, capillary and venous
Summary
The major differences between the 1C and the 3C models are that in the 3C model the arterial compartment is modelled by a dynamic equation allowing the arterial CBV to change due to changes in CBF and that the arterial saturation can be less than unity. The dynamic equations modelling the capillary compartment are the same in the two models. The partitioning of total blood flow, volume, saturation and deoxy-haemoglobin into the three compartments are shown in Table 1.
In this paper, we do not
Parameter settings
The following parameters were used in both simulation and experimental data analysis.
Simulations
Simulated CBF and CBV data were analysed using the 3C model whilst manipulating the parameters which distinguished the 3C and 1C models. These parameters were: (i) baseline CBV proportions; (ii) fractional CBV changes in the arterial, and hence the venous compartments; (iii) arterial oxygen saturation (Sa). The objective of the simulation study was to investigate how these parameters may affect the dynamics of the deoxy-haemoglobin (Hbr) in the venous compartment in order to determine how this
Discussion
In this paper, we have introduced a three-compartment hemodynamic model which relates the normalised time series of Hbr to neural activity induced changes in CBF. The main novel characteristics of the model are: (i) arterial baseline blood volume is included in the calculation of the changes in the normalised CBV, and significantly, arterial CBV changes with changes in CBF; (ii) arterial saturation can be less than unity; (iii) capillary CBV is also involved in the calculation but remains
Conclusion
We have shown in this paper that both the 1C and 3C model were able to fit the time series of the change in volume very well, however, the important finding was that, when the fitted parameters values were then used to fit the times series of the changes in Hbr, the 3C model far outperformed the 1C model. This really is not surprising because the 1C model makes the unrealistic simplifying physiological assumption that the volume and saturation measurements derive from a single compartment of
Acknowledgments
The authors thank Myles Jones and John Martindale for helpful discussions and access to their data. This work was supported by EPSRC grant No. GR/R46274/01, NIH grant No. RO1-NS44567 and MRC Grant Nos: G0100538 and G9825307.
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