We present the results of the different linear regression models for the prediction of early and late functional post-stroke outcome in terms of explained variance (\(R^2\) and \(R^2_{adj}\)) and statistical significance in Tables \ref{tab:early_prediction} and \ref{tab:late_prediction}, respectively. Model \ref{eq:third} outperforms model \ref{eq:first}, with \(R^2=0.60\) for both the early and late outcome.
Importantly, the estimated coefficient of this model for \(N_{RC}\) is significant (\(\beta_{early}=2.79, p=0.03\) for the early and \(\beta_{late}=3.25, p< 0.001\) for the late outcome), while this is not the case for \(DWI_V\) and \(AGE\) after including the interaction term. Model metrics improve further after introducing the path length (\(L\)) as a measure of network efficiency and its interaction term with \(N_{RC}\) in model \ref{eq:fourth}, yielding \(R^2=0.69\) for the early outcome prediction and \(R^2=0.72\) for the late outcome (a 5-fold increase over the baseline model), while \(N_{RC}\) remains statistically significant for the early outcome.
Additionally, the linear regression models are compared by means of two commonly used information criteria, i.e. Akaike information criterion (AIC) \cite{Akaike_1974} and Bayes information criterion (BIC) \cite{stone1979}, in Table \ref{tab:model_comparison}. It should be noted that the model with the lowest information criterion value is considered the best-fitting model. As can be observed, model \ref{eq:fourth} is the best-fitting one according to AIC, for both early and late outcomes. BIC, however, which measures the trade-off between model fit and complexity of the model, slightly favours model \ref{eq:second} (4 estimated parameters), which is much less complex than model \ref{eq:fourth} (7 estimated parameters). For both outcomes and information criteria, though, there is a drastic improvement of the model fit compared to model \ref{eq:first}, which disregards connectivity measures.

Discussion

We explored the importance of the integrity of the rich club backbone on functional outcomes after AIS, as well as the topology of functional networks obtained with different anatomical atlases. Functional outcomes vary significantly in the early and late phase of stroke recovery and are difficult to predict at AIS onset. Using the acute stroke lesions visible on the admission DWI, we determined that the greater the number of rich club nodes affected by the stroke lesion is, the worse the functional outcome, both when assessed early (2-5 days) and late (90-days) post-stroke. Importantly, this association was independent of age and infarct volume. Our findings align with a previous report \cite{Munsch_2015}, which identified stroke location as an independent predictor of cognitive outcome as measured at 3 months post-stroke. Similar results were demonstrated by \cite{Wu_2015}, highlighting the importance of joint modelling of DWI volume and topography for stroke outcomes, while our approach also takes network topology into account.
Rich club nodes comprise brain areas responsible for distributing a large fraction of the brain's neural communications. This underpins the importance of these brain regions for recovery after brain damage, as local disruptions to these central hubs of information flow most likely affect the brain more severely at a global level. The predictive model was further improved after introducing a measure of functional network efficiency, as measured by the network's characteristic path length, which directly describes how focal lesions affect the brain network at a global level. Furthermore, our results suggest that a reduced path length could facilitate recovery or make the brain less vulnerable to damage.
These data underscore the importance of efficient brain connectivity in functional recovery and resilience to brain damage after ischemic stroke. Our model accounts for some of the most commonly reported confounding factors that are available in the acute setting, i.e. age \cite{Saposnik_2012,Rost_2016} and lesion volume \cite{Vogt_2012a} as measured on the acute DWI image. Our results indicate that infarct location improves the model independent of infarct size, which has been previously suggested by \cite{Wu_2015} in a study of infarct topography. We found that, although the number of rich club nodes affected by the infarct is correlated with the DWI lesion volume, the volume alone is not enough to explain the early and late outcomes as measured by the NIHSS score.
We further explored how focal lesions affect functional brain connectivity at a global level in the acute stroke phase, and whether network measures can be used to reflect the brain's resilience to damage and potentially facilitate recovery. We observed a statistically significant positive correlation between the characteristic path length and early functional outcome. We, further, reported significant differences between good and poor outcome, as determined by dichotomizing mRS and \(L\), which could indicate that the brain may benefit from a low characteristic path length with regards to facilitating recovery. Nevertheless, the assumption of direct causation between these two measures needs to be carefully explored.
There are certain limitations that need to be taken into consideration when interpreting these results. First, regional delays have been identified in rsfMRI fluctuations in stroke and cerebrovascular disease patients (hemodynamic lag), which takes place due to vascular occlusion \cite{Siegel_2017}. These delays are measured by time shift analysis of regional BOLD time series with respect to a reference signal. Approaches have been proposed to correct for lags in FC analysis, such as shifting the time series in the lagged tissues prior to the estimation of functional connectivity \cite{Siegel_2016}. However, there is no consensus on how such lags should be accounted for. Importantly, the effect of hemodynamic lag and the subsequent drop in the estimated FC are an integral part of the observed differences between patients, and may be utilized to determine variations in outcome. Another limitation that should be taken into account results from potential registration errors, due to relatively low through-plane resolution of the anatomical scans and the fact that the lesions were not masked out when registering the anatomical scans to the MNI template. Among the strengths of this study is the thoroughly ascertained and well-characterized hospital-based dataset of patients with AIS and consecutive assessments of functional post-stroke outcomes.
Overall, this is the first study exploring functional network topology and rich club topology of brain connectivity in AIS patients, as well as their association with early and late post-stroke outcomes. Our findings highlight the impact of stroke location on functional recovery and the importance of structural connectivity hubs and functional connectivity integration for efficient information transmission underlying the mechanisms of stroke recovery. The proposed model yields a remarkable 5-fold improvement in explained variance over the baseline model, based on age and lesion volume.

Acknowledgements

This work was funded by NIH Grant 5R01NS082285. Sofia Ira Ktena was supported by the EPSRC Centre for Doctoral Training in High Performance Embedded and Distributed Systems (HiPEDS, Grant Reference EP/L016796/1) and an EMBO short-term fellowship (Reference 7284). Markus D. Schirmer was supported by the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 753896.