An Extended Model of Intracranial Latency Facilitates Non-Invasive Detection of Cerebrovascular Changes

2-1- A Review on the Latency Modeling of Arterial Blood Pressure and Cerebral Blood Flow Velocity Pulses

A schematic representation of the components of ABP latency (LT_ABP) and CBFV latency (LT_CBFV) has been illustrated in figure 1, where LT_Shared (a shared latency component that is common to both the ABP and CBFV waveforms) represents the time interval from the QRS components of electrocardiograph (ECG) to the time when blood pressure pulse reaches the juncture point between the extra- and intracranial components. LT_Radial and LT_MCA represent the traveling time of blood pressure pulse from the terminal point of the LT_Shared to the radial artery assessment and to the CBFV measurement point, respectively.

As a result, we have the following equation,

Given the fact that flow velocity pulse usually leads pressure pulse (Nichols and O’rourke, 2005), LT_CBFV includes the intrinsic negative value phase difference between pressure and flow velocity pulse measured at the same site. Note that LT_Radial and LT_Shared are related to the pulse wave velocities of their representative vascular segments (v_Radial and v_Shared) based on the following equations,

where PEP is the cardiac pre-ejection period (the period between the ventricular contraction and the opening of semilunar valves and blood ejection into the aorta), and D_Shared and D_Radial are lengths of the vascular segments from the heart to the origin of the juncture node and from this juncture node to the location of measurement of ABP on the radial artery, respectively. By defining α as the constant ratio of pulse wave velocities at these two vascular segments ( $\frac{v_{\mathit{Radial}}}{v_{\mathit{Shared}}}$), it can be shown that

Following some mathematical derivations, LT_CBFV can be expressed in terms of LT_ABP with the following equation

Now, a corrected beat-by-beat CBFV latency (cLT_CBFV (i), i = 1,…) can be obtained by implementing a subtraction procedure where scaled systemic pulse latency and a constant term were extracted from the measured LT_CBFV (i),

$\overline{{LT}_{\mathit{MCA}}}$ is a component of LT_MCA that does not change from sample to sample. This corrected CBFV latency represents the information more directly related to the sample-to-sample changes in the cerebrovasculature by essentially removing the extracranial confounding influence and consequently improving the positive correlation between CBFV and the CBFV latency time series. For more details on this model and its validation process, the readers are referred to (Hu et al., 2009).

2-2- Extension of the Existing Model in order to Interpret Slope and Intercept as Cardio-Cerebrovascular Metrics

The aforementioned model has been shown to reduce the influence of confounding factors of extracranial origins for characterizing the cerebral vasculature. This impels us to further investigate the physiological interpretation of the parameters of the model. But since the slope and intercept of the linear model in its present form are not the optimally condensed expression of the CBFV latency in terms of ABP latency, we need to modify the model by expressing LT_MCA as

where D_MCA and v_MCA are the lengths and pulse wave velocities of the vascular segments from the juncture node to the location of measurement of CBFV on the MCA, respectively. β is the constant ratio of pulse wave velocity at the aforementioned segment to the pulse wave velocity of the vascular segment from the heart to the origin of the juncture node ( $\frac{v_{\mathit{MCA}}}{v_{\mathit{Shared}}}$). Substituting the equivalent of v_Shared from equation (2-a) in equations (6) led to

Since LT_ABP = LT_Radial + LT_shared, combining equations (3-a) and (7) resulted in

Now, by merging equations (4) and (8), we will have the following equation

which can be rewritten as the following linear equation

where

and LT_ABP (i) and LT_CBFV (i) are ABP and CBFV latencies of the i^th beat (i = 1,2,…, N) in the data segment. The derivation for the Intercept can be further simplified as

Now replacing the equivalent of Slope from equation (11) to equation (12) will result in

The extended latency model of equations (10), (11) and (13) indicates the following interesting remarks which were not apparent in the original model of (Hu et al., 2009).

1. Intercept is itself a function of Slope and PEP. In a segment of data with a short time duration, since the value of PEP does not change substantially (Payne et al., 2006), Intercept would be a linear function of Slope.

2. Since D_MCA, D_Radial and D_shared are constant in an individual patient, Slope could be useful in detecting relative cerebrovascular changes because it is a function of pulse wave velocities of the cerebral vasculature and the upper-limb vasculature. Particularly, it can be concluded that pulse wave velocity, and hence blood vessel stiffness and vascular resistance in the cerebral vasculature, would be negatively correlated with the slope. For example, a selective vasodilation of cerebral arteries would decrease cerebrovascular resistance and PWV in the related cerebral vasculature and result in, based on equation (10), an increase in the value of slope. Note that this analysis provides a measure to capture the static property of the cerebral vasculature while the previous methodology (Hu et al., 2009) to estimate LT_CBFV (i) (equation (5)), provided means to capture the dynamic changes. However, extracting Slope using consecutive segments of data would enable a continuous surveillance of the cerebral vasculature using continuous ABP, CBFV, and ECG measurements.

3. Using the extended model of equations (10) and (13), one can estimate the average value of PEP ( $\overline{\mathit{PEP}}$) over a specific time duration. This can be achieved by implementing the following procedure; the beat by beat estimated ABP and CBFV latency data will be divided to consecutive overlapping data segments where each segment has a short time duration (consisting of N beats). If we use index j, j ∈ {1,2,…} to indicate the j ^th of such a segment, then the Slope(j) and Intercept(j) for that segment can be calculated using equation (10). Now let us define $\mathbf{S}={\left[\begin{array}{ccccc}\mathit{Slope}(1)& \mathit{Slope}(2)& \dots & \mathit{Slope}(j)& \dots \end{array}\right]}^T$, and $\mathbf{I}={\left[\begin{array}{ccccc}\mathit{Intercept}(1)& \mathit{Intercept}(2)& \dots & \mathit{Intercept}(j)& \dots \end{array}\right]}^T$. A linear regression model could then be applied to the vectors of calculated slopes (S) and intercepts (I) as I = γ × S + δ. Now based on equation (13), the average pre-ejection period would be equal to the slope of the obtained regression model with the negative sign. In other words, $\overline{\mathit{PEP}}=-\gamma$.

In the next section, we will describe the dataset which has been used in this paper to test whether data during re-breathing test will support the above points.

2-3- Testing the Model

We tested the hypothesis that the refined latency model would be sensitive to changes in cerebral resistance, where increased pial vessel diameter induced by CO₂ rebreathing, would reduce cerebral resistance, increase LT_MCA and thus result in a steeper slope metric.

3-1- Model Derivation/Description

Figure 2 shows the plots of ABP, estimated ABP latency, CBFV, estimated CBFV latency, heart rate, estimated slope vector (S) and estimated intercept vector (I) (obtained from equation (10)) and their one-to-one relationship for one subject at rest. Each point on any of these plots is obtained by applying a moving average window of 20 beats with 95% overlapping on the beat-to-beat time series of the corresponding physiological parameter.

We observe that all the depicted parameters are stable over time; as is expected at rest, and the estimated values of CBFV latency (with mean of 135.43 ms) are smaller than those of ABP latency (with mean of 193.68 ms). As figures 2-f and 2-h show, estimated slope (with mean of 1.03) is always a positive number, while intercept (with mean of −64.30) has a negative value most of the time. Please note that, disregarding the scaling of the y-axis, the plots of estimated slope and intercept over time are mirror image of each other relative to the x-axis. Figure 2-g demonstrates the estimated intercept values versus their corresponding slope and also a line that has been fitted to these points as Intercept = −196.7×Slope+138.5. The three formerly described parameters for this fitted line are as the following; (R² = 0.99), (p≤10⁻³) and (SD = 1.15). In fact, the mean and standard deviation of these three parameters over all 18 subjects are (R² = 0.99± 0.01), (p≤10⁻³) and (SD = 1.47±0.45). These numbers show that the estimated Slope and Intercept are linearly related.

Figure 3 demonstrates the estimated values of $\overline{\mathit{PEP}}$ calculated following the procedure described in section 2.2. and compares those with the estimated values from Weissler model of equation (14).

We observe that although the estimated pre-ejection values from the proposed latency model and Weissler equation do consistently depict different estimation biases (the mean difference between estimated $\overline{\mathit{PEP}}$ values of both methods is 101.67±12.74 ms), they correlate to a modest degree (correlation coefficient of 0.67 and R-squared value of 0.45) and follow the same changes. As a result, the estimated $\overline{\mathit{PEP}}$ values from the proposed model could be a good qualitative measure of $\overline{\mathit{PEP}}$ values obtained from Weissler equation.

3-2- Comparison of Computed Slope Metric during CO₂ rebreathing and Resting State

Figure 4 shows the plots of ABP, estimated ABP latency, CBFV, estimated CBFV latency, heart rate, estimated slope vector (S), estimated intercept vector (I) and PETCO₂ when the subject of figure 2 undergoes the CO2 rebreathing test.

Each point on these plots is obtained by applying a moving average window of 20 beats with 95% overlapping on the beat-to-beat time series of the corresponding physiological parameter. Please note that the time length of the CO₂ rebreathing experiment (which was set based on the level of subject’s tolerance) and the starting/ending level of PETCO₂ are different from subject to subject, but the following phenomena were observed for all the subjects during the CO₂ rebreathing test; as the subject rebreathed, the level of PETCO₂ and consequently CBFV increases over time, whereas the ABP signal does not either change significantly or increases slightly comparing to that of CBFV (Vantanajal et al., 2007). Heart rate usually decreases first, but then starts to recover and finally stabilizes at a rate that is different from subject to subject. We also observe that CBFV latency and ABP latency are highly correlated at the beginning of the CO₂ rebreathing test (both increase).

Figure 5 shows the mean and standard deviation of the estimated slope and intercept calculated over all 18 subjects during the last T_Rest =190 seconds of rest and the first T_Rebreath=100 seconds of CO₂ rebreathing, respectively.

We observe that during the CO₂ rebreathing test the slope increases while the intercept decreases, reflecting the reduction in cerebrovascular resistance. The mean of slope during rest and CO₂ rebreathing are 0.99 and 1.12 respectively, whereas the mean of intercept during rest and CO₂ rebreathing are −67.50 and −94.10 respectively. Also a t-test on the increase of slope and decrease of intercept when the subject goes from the resting test to CO₂ rebreathing test results in p= 0.006 and p= 0.011, respectively. This indicates that the value of slope significantly increases and the value of intercept significantly decreases, when the subjects at resting begin to rebreath CO_2.

3-3-Comparison of the Computed Slope Metric with Other Conventional Measures of Cerebrovascular Resistance

Figure 6 compares the average slope metric calculated over all 18 subjects with two of existing measures of cerebrovascular resistance (RAP and CCP) during the resting and CO₂ rebreathing. We observe that all the three aforementioned parameters are relatively constant during the resting, but as the subjects start to rebreath CO_2, slope increases while the other two measures of cerebrovascular resistance demonstrate a decreasing pattern. A simple correlation analysis indicates that the slope metric is highly correlated with RAP and CCP with correlation coefficients of −0.89 and −0.89, respectively.

Table 1 displays the mean and standard deviation of all the related parameters calculated over all subjects during resting and CO₂ rebreathing and highlights the ones which are significantly different between resting and CO₂ rebreathing; CBFV and CBFV latency, slope and intercept of the proposed model, RAP and CCP

4-1-Model Validation

The fact that the plots of estimated slope and intercept over time are mirror image of each other demonstrates that the two parameters must be linearly related. Note that, based on equation (13), it was already expected for the estimated slope and intercept to be linearly related. But the former equation also indicates that the line plot of Intercept versus Slope should have a slope equal to −PEP and must intersect the y-axis at the value equal to PEP. However, this is not consistent with the fitted line of figure 2-g as Intercept=−196.7× Slope+138.5. This is related to the fact that flow velocity pulse usually leads pressure pulse (Nichols and O’rourke, 2005) and consequently LT_CBFV includes the intrinsic negative value phase difference between pressure and flow velocity pulse measured at the same site. In other words, considering this phase difference (τ), one can correct equation (13) to relate Intercept and Slope as the following

This correction could further justify the mostly negative values of estimated intercept in figure 2-h even when the estimated slope is smaller than one. Given this reasoning, we can then focus on investigating the quality of the PEP estimate to validate the proposed model. This is a challenging task given that no direct measurement of PEP is available for this dataset. We therefore resorted to using an empirical but widely accepted PEP estimation method (Weissler et al., 1968) that is based on the heart rate. Similar to Weissler’s estimate, the PEP values from our model are negatively correlated with the heart rate and follow the same changes as the Weissler estimate (figure 3-b). We also noted that the absolute values of the two PEP estimates do not match. However, this could be attributed to the ambiguity of the definition of PEP in Weissler model where PEP was defined as the interval from QRS complex to the beginning of the upstroke of the carotid arterial pulse. But the “beginning of the upstroke” has been never accurately described, which is likely to be different from the definition used by us (the interval from QRS complex to the characteristic point of the employed tangent latency estimation (Chiu et al., 1991) on the pressure pulse).

Please note that although PEP may be estimated as a by-product of the proposed model, PEP estimation analysis in this paper has been just employed to validate the model and as a result, its clinical significance and implication are not the focus of the present work.

4-2-Comparison of Computed Slope Metric during CO₂ Rebreathing and Resting State

The CO2 rebreathing test is a cerebral vasomotor reactivity test, where a steeper slope between MCA velocity and PETCO₂ indicates greater vessel reactivity. Figures 4-f and 4-h show that the estimated value of slope increases when the subjects enter CO₂ rebreathing state. CO₂ rebreathing introduces vasodilation causing a decrease in cerebral pulse wave velocity (v_MCA) and hence a decrease in the value of $\beta =\frac{v_{\mathit{MCA}}}{v_{\mathit{Shared}}}$. Because $\mathit{Slope}=\frac{1+\frac{D_{\mathit{MCA}}}{\beta \times D_{\mathit{Shared}}}}{1+\frac{D_{\mathit{Radial}}}{\alpha \times D_{\mathit{Shared}}}}$, if β decreases, then slope would increase. Now one may argue about the possibility of a decrease in α as a result of a possible decrease in v_Radial during the CO₂ rebreathing test. Please note that hypercapnia during re-breathing test is expected to have more pronounced vasodilation influence on the cerebral vasculature than the systemic arterial bed. So even if we assume both α and β decrease as CO₂ level rises, one can expect for a larger decrease of β compared to that of α. In addition, D_MCA (the length of the vascular segment from the juncture node to the location of measurement of CBFV on the middle cerebral artery) is almost less than 30% of the D_Radial (length of the vascular segments from the juncture node to the location of measurement of ABP on the radial artery) (Giller and Aaslid, 1994; Nichols and O’rourke, 2005). So it is expected that the slope increases during CO₂ rebreathing test, as long as the decrease in β is greater than only 30% of the decrease in the parameter α, which is evidently the case.

4-3-Comparison of the Computed Slope Metric with Other Conventional Measures of Cerebrovascular Resistance

Correlation between Slope and two existing metrics of cerebrovascular tone including RAP and CCP, has been shown to be significant (correlation coefficient = −0.89, p≤10₋₃ ). A negative correlation between Slope and these two metrics is also expected. As we discussed in the previous sub-section, Slope increases with cerebral vasodilation. On the other hand, CCP and RAP are expected to decrease with cerebral vasodilation, because vasodilation loosens cerebrovascular tone. Therefore, the correct polarity of correlation and a significant correlation value indicate that Slope metric is a valid measure in characterizing cerebrovascular tone and resistance. It should be noted that Slope and RAP/CCP are based on completely different theories. This also suggests that pulse wave propagation theory is a valid alternative theoretical foundation that can be leveraged for future studies of cerebral vasculature.

4-4- Implication for Continuous Monitoring of the Cerebral Vasculature

Since many of the same principles in terms of latency detection, analysis, and interpretation are fundamentally applicable to both intracranial pressure (ICP) and CBFV pulse waves, the Slope metric could also be calculated by using ICP pulse latency instead of CBFV pulse latency. The continuous acquirement of ICP pulse in the intensive care unit (relative to sporadic measurement of CBFV signal) would then enable the continuous assessment of the cerebral vasculature. Such a capability will be helpful in detection of several physiological events causing the cerebral vasculature changes relative to that of the systematic arterial bed, e.g. cerebral vasospasm which happens after aneurysmal subarachnoid hemorrhage (Gonzalez et al., 2007).