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1.  Mathematical model for describing cerebral oxygen desaturation in patients undergoing deep hypothermic circulatory arrest 
Background
Surgical treatment for aortic arch disease requiring periods of circulatory arrest is associated with a spectrum of neurological sequelae. Cerebral oximetry can non-invasively monitor patients for cerebral ischaemia even during periods of circulatory arrest. We hypothesized that cerebral desaturation during circulatory arrest could be described by a mathematical relationship that is time-dependent.
Methods
Cerebral desaturation curves obtained from 36 patients undergoing aortic surgery with deep hypothermic circulatory arrest (DHCA) were used to create a non-linear mixed model. The model assumes that the rate of oxygen decline is greatest at the beginning before steadily transitioning to a constant. Leave-one-out cross-validation and jackknife methods were used to evaluate the validity of the predictive model.
Results
The average rate of cerebral desaturation during DHCA can be described as: Scto2[t]=81.4−(11.53+0.37×t) (1−0.88×exp (−0.17×t)). Higher starting Scto2 values and taller patient height were also associated with a greater decline rate of Scto2. Additionally, a predictive model was derived after the functional form of a×log (b+c×δ), where δ is the degree of Scto2 decline after 15 min of DHCA. The model enables the estimation of a maximal acceptable arrest time before reaching an ischaemic threshold. Validation tests showed that, for the majority, the prediction error is no more than ±3 min.
Conclusions
We were able to create two mathematical models, which can accurately describe the rate of cerebral desaturation during circulatory arrest at 12–15°C as a function of time and predict the length of arrest time until a threshold value is reached.
doi:10.1093/bja/aep335
PMCID: PMC2791548  PMID: 19933513
brain, ischaemia; brain, oxygen consumption; hypothermia

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