Development of a Model to Aid NIRS Data
Interpretation: Results from a Hypercapnia
Study in Healthy Adults
Tracy Moroz1, Murad Banaji1, Martin Tisdall1, Chris E. Cooper2, Clare E.
Elwell1, Ilias Tachtsidis1
1Biomedical Optics Research Laboratory, Department of Medical Physics and
Bioengineering, University College London, Gower Street, London WC1E 6BT,2Department
of Biological Sciences, University of Essex, Wivenhoe Park, Colchester CO4 3SQ, UK
Abstract The use of a mathematical model of cerebral physiology and metabo-
lism may aid the interpretation of experimentally measured data. In this study,
model outputs of tissue oxygen saturation (TOS) and velocity of blood in the mid-
dle cerebral artery (Vmca) were compared with experimentally measured signals
(TOS using near infrared spectroscopy and Vmca using transcranial Doppler) ac-
quired during hypercapnia in healthy volunteers. Initially, some systematic dis-
crepancies between predicted and measured values of these variables were identi-
fied. The model was optimised to best fit the measured data by adjusting model
parameters. To improve the fit, three additional model mechanisms were consid-
ered. These were: an extracerebral contribution to TOS, a change in venous vol-
ume with CO2levels, and a change in oxygen consumption with CO2levels. Each
mechanism, when used alone, improved the fit of the model to the data, although
significant parameter changes were necessary. It is likely that a combination of
these mechanisms will improve the success of modelling of TOS and Vmca
changes during hypercapnia.
Changes in carbon dioxide levels are known to alter cerebral blood flow . Hy-
percapnia studies have been carried out in healthy volunteers to characterise brain
tissue oxygenation and blood flow changes, measured with near-infrared spectros-
copy (NIRS) and transcranial Doppler (TCD) .
Here we apply the BrainSignals model , a physiological model of brain cir-
culation and metabolism, to data from a hypercapnia study in healthy adults .
The model predicts several physiological variables, including those which can be
measured with NIRS and TCD. It has previously been used successfully to de-
scribe the effects of hypoxia, also in healthy adults. We aimed to reproduce the
experimental results as closely as possible with the model, and in doing so, en-
hance our understanding of the measurements, and the effects of hypercapnia.
2 The BrainSignals model
The structure of the model is illustrated in Figure 1. The circulatory part of the
model comprises three compartments: arteries and aterioles, capillaries and veins.
The venous and capillary volumes are fixed, but the arterial/arteriolar compart-
ment has variable resistance which is sensitive to four input variables: the arterial
pressure of carbon dioxide (PaCO2), the arterial oxygen saturation, the mean arte-
rial blood pressure (MBP), and a parameter representing neuronal activation.
PaCO2affects the resistance via the following equations.
Here, τ is a time constant, and ν represents a low-pass filtered version of PaCO2
with normal value νn. RCis a parameter controlling the magnitude of the response
to PaCO2changes and has a default value of 2.2. The muscular tension in the arte-
rial wall depends on η, which is the sum of the PaCO2term shown and three simi-
lar terms for the other input variables listed above. An average vessel radius is
calculated from the balance of pressures and tensions in the vessel wall. This in
turn determines the resistance of the arterial/arteriolar tree via Poiseuille’s law. An
increase in vessel radius leads to an increase in blood volume and blood flow. The
model output of velocity of blood in the middle cerebral artery (Vmca) is propor-
tional to cerebral blood flow.
All blood compartments have a fixed haemoglobin concentration [Hb] whose
default value in the model is 2.275mM. In each compartment, a fraction of this
haemoglobin is oxygenated. These fractions are determined from the arterial oxy-
gen saturation (a model input), and from the rate of oxygen transport to the mito-
chondria for respiration. The tissue oxygen saturation (TOS) is the overall per-
centage of oxygenated haemoglobin in the arteries and veins. The steady state
changes of Vmca and TOS with default model parameters and varying PaCO2are
shown in Figure 2.
Data were analysed from a hypercapnia study of fourteen healthy adult volunteers
. This involved a 1.5 kPa increase in end tidal CO2(EtCO2) for 10 min, with 5
min at baseline before and after. Throughout the study, the subjects' heart rate,
mean blood pressure (MBP) and arterial oxygen saturation (SaO2) were moni-
tored. The blood velocity in the middle cerebral artery (Vmca) was also moni-
tored, using transcranial Doppler. The tissue oxygen saturation (TOS), a measure
of the percentage of oxygenated haemoglobin, was obtained using the NIRO 300
(Hamamatsu Photonics KK). All signals were smoothed and filtered, and any pe-
riods with obvious instrumentation noise were identified by inspection and re-
placed by a linear interpolation. SaO2, EtCO2and MBP were input to the model,
and its outputs compared with the measured Vmca and TOS.
Parameter optimisation was carried out using a version of Powell's method im-
plemented in SciPy . The rms difference between a measured and simulated
signal was calculated by a numerical integration of the squared difference between
the two signals, over all time points. The aim of optimisation was to minimise this
rms difference for each subject, using TOS, Vmca, or a weighted combination of
the two. For Vmca, the simulated data from each parameter set was rescaled so
that its average value matched that of the measured data, prior to error calculation.
Initially, two model parameters were optimised: blood haemoglobin concentration
[Hb] (or haematocrit), chosen for its influence on the absolute TOS value, and RC,
which represents the sensitivity of the flow response to PaCO2changes.
The success of a simulated dataset was judged by its rms difference, and also
by comparing its response with that of the measured data. To calculate the re-
sponse, a period of hypercapnia and a subsequent baseline period were identified
by inspection of the EtCO2trace for each subject. The response was then calcu-
lated from the means during these periods, after resampling to 1 Hz, as follows
TOS response = TOS(hypercapnia) - TOS (baseline)(1)
ine) Vmca(basel- capnia)Vmca(hyper
After analysing the results, three new mechanisms were added to the model in
turn. Firstly, to simulate an extracerebral contribution to TOS, TOS(corrected)
was calculated as the weighted sum of intra and extracerebral compartments
TOS(corrected) = (1- t) TOS(i) + t . TOS(e)(3)
where t is the fractional contribution of the extracerebral compartment, and
TOS(e) its fixed TOS value. TOS(i), the intracerebral TOS, was calculated as be-
fore. TOS(e) and t were optimised together with RC. Secondly, venous volume
was varied with CO2levels. Previously, it was fixed at 0.75 of the normal total
blood volume (Vblood,n). This was changed to
venous volume = ( 0.75 + v (PaCO2- PaCO2,n) ) Vblood,n
where PaCO2,nis the model’s normal value of PaCO2. The constant v was in-
cluded as an optimisation parameter. Finally, a change linking metabolic rate to
CO2levels was introduced, via a parameter representing the demand. This was
varied in a similar way to venous volume
demand = 1.0 + d (PaCO2- PaCO2,n) (5)
and d was optimised. Changes in demand also have a direct effect on the blood
flow; but this was removed here by setting the relevant parameter Ruto zero.
The measured data are summarised in Table 1. The mean (± SD) TOS and Vmca
responses were 1.1 ± 0.8% and 26 ± 11%, respectively.
Table 1 Summary of the measured data, mean (SD) across the 14 subjects.
The six differently optimised datasets are summarised in Table 2. Parameter
values not given in the table were set at their defaults, except for [Hb]. In optimi-
sations 2 and 4–6 [Hb] was fixed at its value from optimisation 1. The errors in
TOS and Vmca response for each subject in each of these optimisation sets, and
for no optimisation, are shown in Figure 3. With no optimisation, mean simulated
TOS response was 7.1 ± 1.1%. Vmca was better predicted, with a mean response
of 32 ± 6%. The response of each signal could be matched well when optimising
to that signal alone. The parameter [Hb] was not included in the Vmca optimisa-
tion since it only had a small effect on the response. The mean value of RCresult-
ing from optimisation 2 was significantly larger than that from optimisation 1; i.e.
a greater sensitivity of blood flow to CO2was required to explain the changes
measured in Vmca, than that required to explain the changes measured in TOS.
Consequently, optimisation 3, which attempted to match both signals, was less
successful: in every subject, simulated TOS response was too large, whilst simu-
lated Vmca response was too small.
Table 2 Details of the optimisation methods and results. The first row contains the mean (SD)
across the 14 subjects of the optimised parameter values. No value given indicates that a parame-
ter was fixed.
All three new model mechanisms reduced this discrepancy. The additional
compartment for TOS was the most successful, leading to simulated TOS and
Vmca responses of 1.1 ± 0.6% and 22 ± 6%. The mean optimum weight of the ex-
tracerebral compartment was 0.8 (range 0.45-0.95). With a varying venous vol-
ume, simulated response was reduced to 1.6 ± 0.9% for TOS and 20 ± 7%, for
Vmca, and therefore matched better the measured signals. However, this corre-
sponded to a mean venous volume change of 100%, (range 20-230%). When op-
timising the change in oxygen metabolism, the mean resulting cerebral metabolic
rate of oxygen consumption (CMRO2) increase was 18 ± 8.5%. Simulated TOS
and Vmca responses were 1.5 ± 0.9% and 21 ± 8% which matched well with the
measured signals. An example of the CMRO2change in one subject, along with
the measured and modelled TOS and Vmca, is shown in Figure 4.
As expected, Vmca and TOS increased during hypercapnia. The model behaviour
was qualitatively correct, but consistently overestimated the ratio of TOS response
to Vmca response. All three additional mechanisms reduced this discrepancy;
however, the magnitude of the changes required for optimum fitting suggested that
no single mechanism is likely to be successful in its own right.
TOS has been shown to have a high sensitivity and specificity to intracerebral
changes . It is surprising therefore, that an 80:20 extracerebral to intracerebral
weighting was required to optimise the fitting of the TOS data.
The method of changing venous volume was very simplistic. A more realistic
method could be incorporated, as in other models . However, a large change in
venous volume would still be required to fit TOS response accurately. Evidence
from PET studies indicates that the cerebral blood volume changes seen during
hypercapnia are caused primarily by arterial volume changes . Our optimisation
suggests a doubling of the venous volume during the hypercapnia challenge,
which seems unlikely.
Finally, linking of CMRO2 to CO2levels allowed improved simulations of
Vmca and TOS. But as before, the 18% increase was unexpectedly large. Changes
in brain blood flow and oxygenation during CO2challenges are well documented;
however, there are still open questions regarding the changes in metabolism .
CO2is usually assumed to be metabolically neutral. For example, in fMRI studies,
hypercapnia is often used with this assumption to estimate oxygen metabolism
changes from the BOLD signal . However, recently Tachtsidis and colleagues
have analysed the NIRS measured change in concentration of cytochrome c oxi-
dase (CCO), and reported an increase in the CCO redox state in healthy volunteers
during hypercapnia, that cannot be exclusively attributed to the increase in oxygen
delivery . We are currently exploring whether CCO measurements can provide
additional data to help discriminate between the different hypotheses presented
In conclusion, there is an anomalous (small) change of TOS in response to hy-
percapnia that cannot be explained by optimisation of a single parameter in our
model. We are currently attempting to optimise to a combination of factors simul-
taneously. If this does not prove possible it will suggest that TOS as measured op-
tically does not report on the cerebral oxygen saturation as defined in our model,
Acknowledgments The authors would like to thank the Wellcome Trust (088429/Z/09/Z) and
the Centre for Mathematics and Physics in the Life Sciences and Experimental Biology for the
financial support of this work. This work was undertaken at University College London Hospi-
tals and partially funded by the UK Department of Health's National Institute for Health Re-
search Centres funding scheme.
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Fig. 1. A diagram of the BrainSignals model. Inputs are shown in solid ovals and outputs in
dashed ovals. Model processes are shown in rectangles. Figure reproduced from Banaji et al. .
Fig. 2. Prediction o f Vmca and TOS by the model at different PaCO2values.
Fig. 3 Modelled response minus measured response for TOS vs Vmca. The response is the dif-
ference (TOS) or percentage change (Vmca) of the mean at hypercapnia from the mean at base-
line. The legend refers to the six optimisation methods described in Table 2. Series 0 represents
no optimisation. Each point within a series represents a subject. The box at the origin surrounds
all the points from optimisations 4-6, where new mechanisms were introduced.
Fig. 4. Examples of TOS, Vmca and CMRO2from one volunteer. The graphs show the measured
signal (solid black), the modelled signal after optimisation 3 (dashed) and the modelled signal af-
ter optimisation 6 (solid grey).