Relationship Between Flow and Metabolism in BOLD Signals: Insights from Biophysical Models

Abstract

In many physiological or pathological situations, the interpretation of BOLD signals remains elusive as the intimate link between neuronal activity and subsequent flow/metabolic changes is not fully understood. During the past decades, a number of biophysical models of the neurovascular coupling have been proposed. It is now well-admitted that these models may bridge between observations (fMRI data) and underlying biophysical and (patho-)physiological mechanisms (related to flow and metabolism) by providing mechanistic explanations. In this study, three well-established models (Buxton’s, Friston’s and Sotero’s) are investigated. An exhaustive parameter sensitivity analysis (PSA) was conducted to study the marginal and joint influences of model parameters on the three main features of the BOLD response (namely the principal peak, the post-stimulus undershoot and the initial dip). In each model, parameters that have the greatest (and least) influence on the BOLD features as well as on the direction of variation of these features were identified. Among the three studied models, parameters were shown to affect the output features in different manners. Indeed, the main parameters revealed by the PSA were found to strongly depend on the way the flow(CBF)-metabolism(CMRO₂) relationship is implemented (serial vs. parallel). This study confirmed that the model structure which accounts for the representation of the CBF–CMRO₂ relationship (oxygen supply by the flow vs. oxygen demand from neurons) plays a key role. More generally, this work provides substantial information about the tuning of parameters in the three considered models and about the subsequent interpretation of BOLD signals based on these models.

Appendix A

In contrast to local approaches (like derivative-based methods) which provide results around relevant “working points”, variance-based methods can deal with extended ranges of parameters (Sobol’ 2001).

These methods are based on the ANOVA-decomposition of the output (features) variance. Assuming that any feature extracted from simulated BOLD signals (typically A _type or D _type , type standing for dip, peak or under) is a nonlinear function Φ of an unknown vector of parameters θ, the ANOVA-decomposition (Rabitz and Alis 1999; Sobol’ 2001) of Φ can be written as

$$ \Upphi \left(\theta \right) = \Upphi_{0} + \sum\limits_{i} {\Upphi_{i} \left({\theta_{i}} \right) + \sum\limits_{i,j} {\Upphi_{ij} \left({\theta_{i},\theta_{j}} \right) + \ldots + \Upphi_{12 \cdots n} \left({\theta_{1},\theta_{2}, \ldots,\theta_{n}} \right)}}. $$

(A.1)

If Φ₀ = ∫Φ(θ)dθ represents the mean value and if $\int\nolimits_{0}^{1} {\Upphi_{{i_{1} \ldots i_{s}}} d_{{\theta_{{i_{p}}}}} = 0}$ for 1 ≤ p ≤ s and 1 ≤ s ≤ n and 1 ≤ i ₁ <···< i _s  ≤ n, then the expression of Eq. A.1 is defined in a unique way. As explained in (Li et al. 2002), the first order component function Φ _i (θ _i ) in Eq. A.1 represents the independent contribution of the ith input parameter θ _i to the feature Φ(θ). The second order component function Φ _ij (θ _i , θ _j ) represents the joint contribution of the parameter pair (θ _i , θ _j ) to the considered feature and so on until the last term which contains the residuals at the nth order (joint contribution of all input parameters). The marginal and joint variances (at 2nd order) are given by

$$ \begin{aligned} \sigma_{i} = \int {\Upphi_{i}^{2}} d\theta_{i}, \\ \sigma_{i,j} = \int {\Upphi_{i,j}^{2}} d\theta_{i} d\theta_{j}. \\ \end{aligned} $$

(A.2)

The total variance of the output Φ(θ) is given by σ _T  = ∫Φ²(θ)dθ − Φ ²₀ . Starting from the exact expansion of Eq. A.1 and the previous variances, one can theoretically calculate the relative variance of each contribution with respect to the total variance σ _T of the output:

$$ \begin{aligned} S_{i} &= \frac{{\sigma_{i}}}{{\sigma_{T}}},1 \le i \le n, \\ S_{ij} &= \frac{{\sigma_{ij}}}{{\sigma_{T}}},1 \le i \le n,1 \le j \le n,i \ne j. \\ \end{aligned} $$

(A.3)

The sensitivity indices S _i and S _ij characterize respectively the marginal influence of parameter θ _i and the joint influence of parameters θ _i and θ _j on the feature of interest. In practice, the total variance is easy to compute, but the computation of the many integrals involved in the marginal variances gets very difficult when increasing the number of input parameters. Improvements can be done by using ad-hoc screening methods (Monte Carlo methods in (Saltelli 2002; Sobol’ 2001)) in order to maximize the number of samples without increasing too much the resulting computational cost. In the purpose of reducing the sampling effort, (Rabitz and Alis 1999) proposed an alternative approach which approximates the output by some projections functions φ _r (that can be seen as a metamodel representing the output) before computing the sensitivity indices. For example using a polynomial representation for the projection functions, it can be shown that Φ _i and Φ_i,j can be expressed as polynomials of the projection functions

$$ \begin{array}{l} \Upphi_{i} \left({\theta_{i}} \right) \approx \sum\limits_{r = 1}^{k} {\alpha_{i}^{r} \varphi_{r} \left({\theta_{i}} \right)}, \\ \Upphi_{ij} \left({\theta_{i},\theta_{j}} \right) \approx \sum\limits_{p = 1}^{l} {\sum\limits_{q = 1}^{m} {\beta_{ij}^{pq} \varphi_{p} \left({\theta_{i}} \right)\varphi_{q} \left({\theta_{j}} \right)}}. \\ \end{array}$$

(A.4)

Later, (Li et al. 2002) improved the method showing that in this case of polynomial approximation, one can have a direct access to the marginal and joint variances $\sigma_{i} \approx \sum\nolimits_{r = 1}^{k} {(\alpha_{r}^{i} )^{2} }$ and $\sigma_{ij} \approx \sum\nolimits_{p = 1}^{l} {} \sum\nolimits_{q = 1}^{m} {\left( {\beta_{pq}^{ij} } \right)^{2} }$.