Noninvasive Prediction of Intracranial Pressure Curves Using Transcranial Doppler Ultrasonography and Blood Pressure Curves
Abstract
Background and Purpose Until now the assessment of intracranial pressure (ICP) required invasive methods. The objective of this study was to introduce an approach to a noninvasive assessment of continuous ICP curves.
Methods The intracranial compartment was considered a “black box” system with an input signal, the arterial blood pressure (ABP), and an output signal, the ICP. A socalled weight function described the relationship between ABP and ICP curves. Certain parameters, called transcranial Doppler (TCD) characteristics, were calculated from the cerebral blood flow velocity (FV) and the ABP curves and were used to estimate this weight function. From simultaneously sampled FV, ABP, and (invasively measured) ICP curves of a defined group of patients with severe head injuries, the TCD characteristics and the weight function were computed. Multiple regression analysis revealed a mathematical formula for calculating the weight function from TCD characteristics. This formula was used to generate the ICP simulation. FV, ABP, and ICP recordings from 11 patients (mean age, 46±14 years) with severe head injury were studied. In each patient, ICP was computed by a simulation procedure, generated from the data of the remaining 10 patients. The simulation period was 100 seconds.
Results Corresponding pressure trends with a mean absolute difference of 4.0±1.8 mm Hg between computed and measured ICP were observed. Shapes of pulse and respiratory ICP modulations were clearly predicted.
Conclusions These results demonstrate that this method constitutes a promising step toward a noninvasive ICP prediction that may be clinically applicable under welldefined conditions.
Acute cerebral diseases frequently lead to a rise in ICP. Monitoring of this parameter could be of vital importance in ensuring the efficacy of therapeutic measures in the case of foudroyant ICP changes, for example. Quantitative data, in particular continuous ICP waves, have only been recorded thus far through invasive methods (eg, an epidural ICP monitoring device). On the other hand, various studies have shown that an increased ICP leads to changes of FV waveforms in intracranial arteries, which are accessible by means of TCD. Using Fourier analysis of FV and ABP curves, Aaslid et al^{1} developed an estimation formula for ICP. Klingelhöfer et al^{2} ^{3} ^{4} showed that the MFV decreases with increasing ICP, while the Pourcelot Index (R) increases. A strong correlation (r=.873, P>.001) between the product, mean systemic ABP×R/MFV, and the ICP was observed. This facilitates, under certain conditions that were laid out in the study, a simple quantitative estimation of the ICP. Similar findings concerning the relationships between flow velocity parameters, ABP, and ICP were published by other authors.^{5} ^{6} ^{7} ^{8} Chan et al^{5} showed that a reduction of CPP resulted in a greater fall in diastolic flow velocity than other flow parameters. As CPP decreased below a critical value of 70 mm Hg, they observed a strongly correlated progressive increase in FV pulsatility index. Furthermore, a couple of mathematical models, derived from electrical or mechanical analogues of intracranial dynamics, have been developed in recent years^{9} ^{10} ^{11} ^{12} ^{13} ^{14} ^{15} to explain the pathogenetic basis of ICP waves under certain physiological and pathological conditions. For example, Ursino’s model was used to simulate aspects of the ICP waveforms, such as the response to various clinical tests,^{16} ^{17} the dependence of ICP pulse amplitude on the mean value,^{17} the pattern of cerebral autoregulation,^{18} ^{19} and the genesis of Lundberg A and B waves.^{19} ^{20} ^{21} In contrast to these investigations, our study is not concerned with the inner structure of the intracranial compartment. The intracranial compartment is considered a “black box” system, which is described indirectly by its transformation of the input signal, the ABP curve, into the output signal, the ICP curve. The black box approach and some of the mathematical techniques in our study were formerly used by Kasuga et al,^{22} who described in an animal study with six dogs the dependence of ICP waves on ABP waveforms for each dog individually. The aim of the current study, however, is to introduce a mathematical model that leads, using the ABP curve and certain characteristics derived from the simultaneously recorded FV and ABP curves, to a noninvasive prediction of the ICP curve in a defined group of patients with severe acute brain damage.
Theoretical Considerations
The mathematical model for the noninvasive ICP simulation is based on results from systems analysis.^{23} Systems analysis provides a method to describe systems, in particular physiological systems, with input and output signals. The outgoing signals are considered the system’s responses to its stimulation by incoming signals. Systems analysis does not analyze the inner structure or the realization of a system but characterizes the system by means of its responses to incoming signals. The basic mathematical tools to describe the system’s response to the incoming signal are the weight function and the transfer function. The weight function transforms the input signal into the output signal and, on the other hand, can be computed from the input and the output signals. A detailed description of the concepts of systems analysis and the systems’ properties is given by Marmarelis and Marmarelis.^{23} In this study the system under investigation is the intracranial compartment, with the ABP as the input signal and the ICP as the system’s response. The applicability of this model was shown by Kasuga et al^{22} in an animal study, in which the weight function was computed from a short time interval of the ABP and the (invasively measured) ICP curves. It was shown that even beyond this time interval, the weight function transformed the ABP curve into the ICP curve quite accurately. The basic idea of our procedure is to use certain hemodynamic parameters, the socalled TCD characteristics, to approximate the weight function. This allows the transformation of the ABP curve into the simulated ICP curve without the need of invasive measurements. To differentiate between the weight function and its approximation, the approximation will be called the simulation function.
Subjects and Methods
Patients
We studied 11 intensive care patients (age, 20 to 59 years; mean age, 46±14 years; 8 men, 3 women). At the time of recording all the patients were mechanically ventilated. Their Paco_{2} ranged from 30 to 35 mm Hg. One patient had an infarct in the right territory of the MCA, 2 had a gunshot head injury, and the others suffered from severe head injury (Glasgow Coma Scale ≤7). No patient showed a vasospasm or a stenosis of the intracranial or extracranial brainsupplying arteries. A Doppler CO_{2} test^{24} was performed in each patient to estimate the state of cerebral vasoreactivity. All patients showed a relative CO_{2} reactivity less than 2.5% ΔMFV/mm Hg Paco_{2}; in 3 patients (patients 6, 8, 11) it was less than 2%. In addition, the relationship between CPP and FV in the recorded data was analyzed by a method introduced by Czosnyka et al^{25} for an assessment of cerebral autoregulation. When we adapted his categorization of autoregulation into “intact,” “gray zone,” and “impaired,” 7 patients belonged to the gray zone and 4 patients (patients 2, 6, 8, 11) belonged to the impaired category. Considering these two tests, we concluded that none of the patients had an intact autoregulation. The disturbance ranged from medium to severe impairment.
Transcranial Doppler Ultrasonography
TCD measurements were taken by a 2MHz pulsed Doppler device^{26} (TC 2–64B, EME). Flow patterns of the MCA were continuously recorded on the same side that the epidural device was implanted. A sampling frequency of 25 Hz was used. The probe was fixed mechanically with a specially developed probe holder with elastic bands and fixation strips.
Blood Pressure Recording
Blood pressure was measured with a blood pressure monitoring device (Gould Statham 23 ID), which was implanted into the radial or femoral artery. Together with the FV and ICP curves, it was continuously and simultaneously transmitted to a computer system.
ICP Measurement
The ICP was measured with an implanted epidural ICP monitoring device with an air pouch probe and an hourly automatic recalibration^{27} ^{28} ^{29} (Spiegelberg Plc/Ltd/Co). The following plausibility criterion had to be met when measuring: the measured ICP should fit the overall clinical presentation of the patient, lifting the head should lead to a decrease in ICP, and a shortlived increase in ICP should occur when the airway is suctioned.
Recording and Evaluation of the Curves
Two PC/i 486 computers were used for recording and analyzing the FV, APB, and ICP curves. One of the computers was portable, and each was fitted with a data acquisition system (DAP 2400/Microstar Laboratories). The used sampling frequency was 25 Hz. The mathematical and statistical calculations were supported by a software tool (Real Time Graphics and Measurement Tools/Quinn Curtis).
Noninvasive ICP Simulation
With the use of patients’ data, consisting of simultaneously recorded FV, ABP, and (invasively measured) ICP curves, the procedure was generated in three steps. A more precise description of the calculations performed can be found in the “Appendix.”
Step 1: Computing the Weight Function From the Given ABP and ICP Curves
The weight function between ABP and ICP curves was computed at different times of recording. To transform the ABP into the ICP curve with maximum precision during a defined time interval (in this study an interval of 14 heart cycles was chosen), a system of linear equations had to be calculated. The solution of this system of equations resulted in a vector containing the coefficients of the weight function. In a representation of ABP and ICP curves by a sequence of distinct sample values, the weight function is a vector of numbers. The weight function generally allowed any number of coefficients to be selected. However, for technical reasons (regarding precision and calculation time), 25 coefficients were chosen. For a given weight function (f_{0}, f_{1}, …, f_{24}), the ICP value at point k in the time sequence could be computed by the values of the ABP recorded at times k−24, k−23, …, k−1, k according to the formula ICP_{k}=f_{0}*ABP_{k}+f_{1}*ABP_{k−1}+f_{23}*ABP_{k−23}+f_{24}*ABP_{k−24}.
Step 2: Calculation of TCD Characteristics
In this study the coefficients of a weight function between FV and ABP curves were used as TCD characteristics. The computation was similar to the one described in Step 1 and performed at the same times. For technical reasons six coefficients were used here to define the weight function instead of 25.
Step 3: Statistical Processing to Calculate the Relationship Between Weight Functions and TCD Characteristics
The relationship between the TCD characteristics of step 2 and the 25 coefficients of the weight function in step 1 was described by an approximating linear function (ie, a matrix A and a vector B), which was calculated through a sequence of 25 multiple regression analyses (one for each of the 25 coefficients of the weight function) of the patients’ data. This process was similar to a standard linear regression between one dependent and one independent variable. In contrast to the standard situation, we had 25 dependent variables, and each of them was related to six independent variables (the TCD characteristics). This process established a nonindividual relationship between TCD characteristics and the simulation function by means of a linear function.
After steps 1 to 3 were performed, the noninvasive ICP simulation procedure worked as follows (Fig 1⇓): While the FV and ABP curves were recorded, the TCD characteristics were computed every 10 seconds and transferred to the simulation function. Finally, the simulation function transformed the ABP curve into the simulated ICP curve.
Application of Noninvasive ICP Simulation
To test the predictive capability of the ICP simulation, a group of 11 patients was studied. In each patient, measured and predicted ICP were compared during a time interval of 100 seconds. For each patient tested, the remaining 10 patients were taken as a reference group to generate the ICP simulation (by performing steps 1 to 3). This ICP simulation was then used to predict the ICP of the patient to be tested. In addition to this shortterm study, in 6 patients the ICP simulation was repeated after 1 hour. In 3 of these 6 patients, two additional ICP simulations were obtained on the following day 1 hour apart from each other. As a measure of the simulation’s precision, the mean of the absolute values of the differences between measured and predicted ICP values (MADICP) was taken. The mean was calculated over the sample points during the 100second time period. Since data were sampled at a frequency of 25 Hz, the average was taken from 2500 ICP values. As a measure of the simulation’s capability of predicting the mean ICP averaged over one cardiac cycle (ICP_{CC}), we calculated the mean of the absolute differences between measured and predicted ICP_{CC} (MADICP_{CC}). Both parameters MADICP and MADICP_{CC} are presented in the results together with their corresponding 95th percentiles, 95%ICP and 95%ICP_{CC}, which are the upper limits of 95% of the absolute differences between measured and predicted ICP and between measured and predicted ICP_{CC}, respectively.
Results
A comparison between typical examples of predicted and measured curves in the shortterm study (Figs 2 through 5⇓⇓⇓⇓) shows that waveform modulations caused by pulse and respiration were clearly predicted by the simulated curves. However, the size of the amplitudes could not be preserved. The amplitudes varied between half and double their original size. An example of the simulation’s capability of predicting changes in the mean ICP is seen in Fig 3⇓ (patient 11), in which the sudden ICP decrease after 40 seconds, as well as its gradual increase afterward, was repeated in the ICP simulation. As mentioned above, the amplitudes of the pulse and respiratory waveform modulations were larger in the simulation curve than in the original ICP recording. When we used the MADICP to measure the simulation’s precision, patient 11 had the most incorrect prediction, having an MADICP of 7.5±2.6 mm Hg. In summary of the 11 patients in the shortterm study, the MADICP was less than 3 mm Hg in 4 patients and less than 5 mm Hg in 8 patients, and the maximum MADICP was 7.5 mm Hg. The MADICP_{CC} was slightly lower than the MADICP; it was less than 3 mm Hg in 4 patients and less than 5 mm Hg in 9 patients, and again the maximum was 7.5 mm Hg. On average the patients’ MADICP was 4.0±1.8 mm Hg, and the MADICP_{CC} was 3.8±1.9 mm Hg (Table⇓ ). The 95th percentiles specify the maximum distances to the measured ICP in 95% of the predicted values. The 95%ICP was less than 5 mm Hg in 3 patients and less than 10 mm Hg in 8 patients, and the maximum was 12.8 mm Hg. The 95%ICP_{CC} was less than 5 mm Hg in 4 patients and less than 10 mm Hg in 11 patients, and the maximum was 11.7 mm Hg. Repeated ICP simulations after 1 hour in 6 patients and additionally in 3 patients after 24 hours showed prediction errors (MADICP, MADICP_{CC}) similar to the first ICP simulation (Table⇓). Furthermore, in each of these patients mean ICP was either always underestimated or always overestimated, with similar error margins. Therefore, trends in ICP changes were reflected in the ICP simulation.
Discussion
The ICP simulation was performed in patients with severe head injuries, impaired autoregulation, nondetectable vasospasm or stenosis, and a Paco_{2} between 30 and 35 mm Hg. Our results suggest that for this defined category of patients, a noninvasive ICP simulation based on FV and ABP recordings may have value for clinical applications. In 4 of 11 patients in the shortterm test, the MADICP was less than 3 mm Hg, in another 4 patients the MADICP was between 3 and 5 mm Hg, and only in 3 patients was the MADICP greater than 5 mm Hg, with a maximum of 7.5 mm Hg. The MADICP_{CC}, which measured the simulation of cardiac cycle mean ICP_{CC}, was slightly lower. Considering 5 mm Hg as a critical threshold for the MADICP as well as the MADICP_{CC}, we therefore conclude that in 8 of the 11 patients the simulation is sufficiently precise to measure ICP changes and to facilitate therapeutic decisions. The 95%ICP_{CC} in these patients varied from 1.9 to 7.9 mm Hg, which means that even in the most inaccurate simulation regarding this parameter (patient 7: 95%ICP_{CC}=7.9 mm Hg), 95% of the predicted cardiac cycle means differed less than 7.9 mm Hg from the measured values. The evaluations of the mediumterm (1 hour) and longterm (24 hours) followup tests confirmed these results. In view of the potential clinical application it was important to verify that even less precise simulations of the absolute values still accurately predicted ICP_{CC} changes and trends. In our study this held true for single recording intervals of 100 seconds as well as repeated measurements within 24 hours (eg, patient 11). In the repeatedly measured patients the behavior of the ICP simulation was consistent within one patient, ie, in the same patient the predicted means of ICP (Table⇑) were either always lower or always higher than the measured ones, with similar error margins. Therefore, trends of ICP were reproduced. In particular, this observation may be important because ICP simulations on the basis of TCD are problematic for extensive continuous measurements (eg, ≥24 hours) because of potential adverse effects of ultrasound exposure and the impairment of the patient by longterm probe fixations. Prediction should be done by longterm followup simulations over short or medium time periods. However, since the number of repeated measurements was small, investigations are continuing to confirm this observation.
In this study the accuracy of ICP prediction was lowest in patients 10 and 11. At the same time the measured ICP was lowest in patient 10 and highest in patient 11. To search for possible explanations of this fact, sources of potential errors in the used method should be regarded. According to the American national standard of the Association for Advancement of Medical Instrumentation, errors of 10% in the assessment of mean ICP are tolerated even in accurately working ICP monitoring devices.^{30} In our study patients with obviously erroneous ICP measurements, contradicting the patient’s clinical presentation, had been excluded. Despite these plausibility checks, minor errors in the assessment of ICP by the epidural device might have occurred. Since the accuracy of measurement decreases with increasing ICP,^{28} this might affect the results in patients with high ICP. Although it is impossible to decide in which patients these potential errors have influenced the accuracy of ICP prediction, these effects might explain the differing results in patient 11. Errors in ICP measurements additionally have a more subtle effect. Since each patient is also a member of the reference group, which is used to generate the ICP simulation procedure, errors influence the behavior of the simulation procedure itself. If these errors are not systemic and only happen occasionally, this influence should be small because of the statistical process in the generation of the ICP simulation. Another problem arises from the variability of normal values of MFV in the MCA; standard deviations of ±12 cm/s were stated by several authors.^{31} ^{32} ^{33} This variation in MCA must be considered, particularly in patients with low ICP, such as patient 10, and might lead to individually differing results of ICP prediction.
As mentioned before, our main criteria for the precision of simulation were the MADICP and the MADICP_{CC}. A simple comparison between measured and predicted mean ICP (means over the whole simulation period) could result in an overestimation of the precision of simulation. This happens, for instance, if the predicted curve oscillates around the measured curve with similar deviations above and below the invasive recording, thereby canceling each other out and pretending a false precision. In the present study this occurred in patient 9, in whom similar means of ICP (measured: 30.8 mm Hg; predicted: 32.0 mm Hg) but a high MADICP (5.8 mm Hg) and MADICP_{CC} (4.9 mm Hg) could be observed. Such behavior can be discovered through high standard deviations of MADICP (±5.3 mm Hg) and MADICP_{CC} (±4.3 mm Hg). On the other hand, a simulation curve that only differs from measured ICP by, eg, the sizes of pulse waveform amplitudes, having identical ICP_{CC}, leads to a nonzero MADICP. This is the reason why the MADICP_{CC} and 95%ICP_{CC} values were lower than the corresponding MADICP and 95%ICP values. By definition the chosen TCD characteristics provide a precise description of the dependencies between ABP and intracranial FV curves, detecting the relationship between the waveform modulations as well as the mean values. Since the “driving force” for blood flow is the pressure gradient between ABP and ICP, these TCD characteristics also implicitly carry information about ICP. This might explain the suitability of these characteristics for determining the simulation function. Our own earlier studies, in which we used Fourier coefficients of the FV curve as TCD characteristics, were less successful. The use of weight functions for the estimation of ICP allowed a precise prediction of the shapes of the ICP waveform modulations. Some parameter choices, such as the number of weight function coefficients and number of TCD parameters, are made after various different combinations are tested. The influence of these choices on the results should not be overestimated. We found that small changes of these parameters, eg, 5 instead of 6 TCD characteristics, 20 instead of 25 weight function coefficients, led to similar results.
Clear limitations appertaining to the procedure’s feasibility arise from the type of patients studied. All patients suffered from severe head injuries. From the patients’ performances on the tests, we determined impaired autoregulation.^{24} ^{25} At the time of recording, no cerebral vasospasm or intracranial stenosis was detected. The Paco_{2} ranged from 30 to 35 mm Hg. Any changes in these conditions might affect the interdependencies of the three parameters FV, ABP, and ICP and lead to differing results. In a patient with a pronounced vasospasm and correspondingly increased FV values,^{34} ^{35} the ICP simulation would result in an erroneously low predicted ICP. Therefore, the described ICP simulation may not be assumed to be valid in patients differing in any of the aforementioned points from the studied group. Changes in parameters such cerebral vasoreactivity, vasomotor tone, and vessel diameter would influence hemodynamics and affect the accuracy of ICP prediction. A rising ABP, for example, would cause an increase of ICP in the case of impaired autoregulation, while the opposite would happen in the case of intact autoregulation. On the other hand, the chosen restrictions for the studied patients seem to be weak enough to still be of practical use.
As a possible step toward a broader applicability of the ICP simulation, it is planned to group patients into different categories of diseases and vascular status and to develop specific simulation procedures. One potential problem will be to ensure that these categories are not too specific, so they can be established by simple clinical tests during routine examinations. When the degree of autoregulation is considered, two or three subdivisions seem reasonable. If a clinician had to go to great lengths (perhaps even invasive tests) to determine to which category a patient belongs, then the predictive value of the ICP simulation would be lost. Further studies must show how many subdivisions of disease categories will be needed and whether, for example, it is possible to design a procedure that works in patients with vasospasm as well as in those without, or whether separate simulation procedures are needed. In our study we used a Doppler CO_{2} test^{24} to assess the CO_{2}induced vasomotor reactivity. In view of the fact that in patients with head injuries in particular the relationship between autoregulation and CO_{2} reactivity is problematic, which was the subject of different studies,^{36} ^{37} we also performed the autoregulation test of Czosnyka et al^{25} to obtain further information regarding vasoreactivity. In continuing studies we are additionally performing the cuffs test,^{38} ^{39} using thigh blood pressure cuffs.
Whereas former studies were concerned either with estimating mean values of ICP^{1} ^{2} ^{3} ^{4} ^{5} ^{6} ^{7} ^{8} or with constructing technical analogues that were of theoretical interest but too complex for a general clinical application,^{9} ^{10} ^{11} ^{12} ^{13} ^{14} ^{15} the aim of our study was to introduce a procedure that determines the continuous ICP curves and can be used as a bedside procedure. It produces both mean ICP values and the continuous course of ICP, including pulse and respiratory waveform modulations. All of these properties may be of clinical relevance. Although additional extensive testing is necessary to further validate this method, it seems to be a promising approach in the development of a useful tool in the management of patients with increased ICP.
Selected Abbreviations and Acronyms
ABP  =  arterial blood pressure 
CPP  =  cerebral perfusion pressure 
ICP  =  intracranial pressure 
FV  =  blood flow velocity 
ICP_{CC}  =  mean ICP averaged over one cardiac cycle 
95%ICP  =  95th percentile of absolute differences between measured and predicted ICP 
MADICP  =  mean of absolute values of differences between measured and predicted ICP 
MCA  =  middle cerebral artery 
MFV  =  mean flow velocity 
TCD  =  transcranial Doppler sonography 
Appendix A1
Generating the Procedure of Noninvasive ICP Simulation
Step 1: Computing the Weight Function From the Given ABP and ICP Curves
To calculate the relationship between the TCD characteristics and the weight function (from the ABP to the ICP curves), ABP is considered the input signal and ICP the output signal of the intracranial compartment.
The weight function between the ABP and the ICP curve may be regarded as a vector of numbers (f_{0}, f_{1},. . ., f_{n−1}). Its coefficients f_{0}, f_{1},. . ., f_{n−1} are computed by using the approach where ICP_{k} denotes the estimation of ICP_{k}.
Equation 1 states that the kth value of the ICP curve ICP_{k} is estimated by a number of n former values of the ABP curve ABP_{k}, ABP_{k−1}, ABP_{k−2},. . ., ABP_{k−n+1}. The weight function is characterized by its property to give the best possible estimation of the ICP_{k}. Assuming the computed values (ICP_{k}) to be the best possible estimations of the actual (measured) ICP values (ICP_{k}) during a time period of p points means that the expression is minimal. Substituting Equation 1 for Equation 2 leads to As a result, from standard calculus expression 3 is minimal if the partial derivatives are zero, ie, ∂G/∂f_{i}=0. Straightforward computation results in the equation system or in There is a unique solution for system (4′) if, and only if, the (n×n) matrix is regular. In this case the coefficients f_{j} can be uniquely determined.
From Equation 1 it can be seen that the parameter n describes the time period in which the ABP is considered to influence the current ICP value. In this study we use n=25. The length of time steps (the distance in time between the sample points k−1 and k) depends on the length of the cardiac cycles. The 25 sample points ABP_{k}, ABP_{k−1}, …, ABP_{k−n+1} are equally spread over three cardiac cycles. This means that in our model the current ICP value is determined by ABP over a period of three heart cycles. The idea of using a heart rate–dependent technique for calculating the weight functions is to make them independent of the heart rate. This might at first sound paradoxical, but imagine a sudden increase of heart rate with remaining shapes of ABP and ICP waveforms, just compressed by the decreasing waveform length. Then the time steps used in the weight functions would become smaller, and the calculated weight function would be exactly the same as the weight function calculated at a lower heart rate. Equation 2 shows that p sets the time period during which the computed ICP curve should be the best approximation of the original (measured) curve. p is chosen in such a way that the length of this period is 14 cardiac cycles. From Equation 4′ the weight function coefficients f_{j} are computed. Until now the (n×n) matrix always happened to be regular. This regularity is not based on theory, and therefore nonregular cases might occur. The solution to Equation 4′ leads to the weight function f= (f_{0}, f_{1},. . ., f_{n−1}) and establishes the relationship between ABP and ICP by means of the formula The certain choices of parameters such as 25 coefficients of weight function, 14 cardiac cycles for minimizing the estimation error, and a threecardiac cycle estimation period were made in view of some additional properties the weight function should have: (1) It should estimate both the means and the pulse waveform modulations of the ICP. (2) Once calculated, it should be able to predict ICP precisely for a long time period (ie, 10 to 20 seconds; actually 10 seconds are sufficient because during ICP simulation every 10 seconds the TCD characteristics are computed and used to make a new estimation of the weight function). (3) Artifacts or single irregularities of the ABP should have only little influence to the estimated ICP.
These properties of the weight function seem plausible in view of its use in the ICP simulation. Property 1 could be achieved by a high number of coefficients of the weight function and a small step length for its calculation, while properties 2 and 3 need large intervals for minimization of the estimation errors and also large estimation periods. During the development of this procedure a couple of parameter combinations had been tested, and the shown parameters were chosen.
Step 2: Calculation of TCD Characteristics
In this study the coefficients of a weight function between FV and ABP curves are used as TCD characteristics. The computation is similar to the one described in step 1 and performed at the same times. For technical reasons six coefficients are used here to define the weight function instead of 25. It describes the relationship between FV and ABP with sufficient precision and reduces the complexity of the statistical process in step 3. For reasons similar to those described in step 1, a heart rate–dependent step length (6 points spread equally over one cardiac cycle) for calculating the weight function is used. The FV weight function should not only describe the relationship between MFV and mean ABP but also relate short time changes during a cardiac cycle interval of both signals, ie, relate the pulse waveform shapes of FV and ABP. This should be done depending only on the shapes of ABP and FV pulse waves and should not depend on the heart rate. Otherwise we had to create different patient groups for different heart rates. A further distinction between step 1 and step 2 is that the six coefficients (f_{0}, f_{1}, …, f_{5}) are calculated from the noncausal approach ABP_{k}=f_{0}*FV_{k}+f_{1}*FV_{k+1}+ … +f_{4}*FV_{k+4}+f_{5}*FV_{k+5}. This approach is noncausal in the sense that the ABP is considered to be determined by FV values of later times (FV_{k}, FV_{k+1}, …, FV_{k+5}). It accounts for the physiological fact that blood pressure causes blood flow. Therefore, the current FV value depends on former ABP values, which, formally expressed, means that the current ABP value is determined by FV values of later times.
To be suitable for the approximation of the weight function between ABP and ICP, the following properties of the TCD characteristics are important: (1) They should be independent of pulse rate. (2) They should precisely describe the relationship between MFV and mean ABP and also the relationship between short time changes of the FV and ABP values during a cardiac cycle. This provides the control information for estimating the weight function between ABP and ICP and is therefore essential. (3) Artifacts or single irregularities of the ABP and FV curves should have little influence on the estimated ICP. (4) The number of TCD characteristics should be small. They are used as independent variables during a multiple linear regression, and there should not be too many degrees of freedom to keep the number of needed data pairs small.
Similar to step 1, these properties and a couple of performed tests of parameter combinations lead to the choice of six TCD characteristics, the step length of 6 points spread equally over one cardiac cycle, and the interval of 14 cardiac cycles for minimizing the estimation error of the weight function between FV and ABP.
Step 3: Statistical Processing to Calculate the Relationship Between Weight Functions and TCD Characteristics
A linear relationship between given pairs of weight functions (f_{0}, f_{1}, …, f_{n−1}) and TCD characteristics (tcd_{0}, tcd_{1}, …, tcd_{m−1}) shall be established. This gives rise to the search of a (n×m) matrix A and n dimensional vector B; thus, on average over all data pairs, the calculated data (f̃_{0}, f̃_{1}, …, f̃_{n−1})=A*(TCD_{0}, TCD_{1}, …, TCD_{m−1})+B is the best estimation for (f_{0}, f_{1}, …, f_{n−1}). The above expression is equivalent to the n equations where A_{j*} is the jth column of matrix A, and B_{j} is the j^{th} coefficient of vector B, or or where j=0, 1, …, n−1. If m=1 and j is fixed (f_{j}≈f̃_{j}=A_{j0}*tcd_{0}+B_{j}), then A_{j0} and B_{j} would be computed by a linear regression analysis relating the dependent variable f_{j} to the independent variable tcd_{0}. In equation (1″), f_{j} has to be related to m independent variables tcd_{0}, tcd_{1}, …, tcd_{m−1}. This is done by a generalized linear regression called multiple regression. For each of the coefficients f_{j} a multiple regression analysis is performed calculating the parameters A_{j0}, A_{j1}, …, A_{jm−1}, and B_{j}, resulting in the desired (n×m) matrix A and the n dimensional vector B.
Acknowledgments
This study was supported by the Deutsche Forschungsgemeinschaft (KL 960/1–1).
Footnotes

Reprint requests to Prof Dr med DiplIng J. Klingelhöfer, Department of Neurology, Technical University of Munich, Möhlstraße 28, 81675 München, Germany.
 Received June 20, 1997.
 Accepted August 24, 1997.
 Copyright © 1997 by American Heart Association
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 Noninvasive Prediction of Intracranial Pressure Curves Using Transcranial Doppler Ultrasonography and Blood Pressure CurvesBernhard Schmidt, Jürgen Klingelhöfer, Jens Jürgen Schwarze, Dirk Sander and Ingo WittichStroke. 1997;28:24652472, originally published December 1, 1997https://doi.org/10.1161/01.STR.28.12.2465
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 Noninvasive Prediction of Intracranial Pressure Curves Using Transcranial Doppler Ultrasonography and Blood Pressure CurvesBernhard Schmidt, Jürgen Klingelhöfer, Jens Jürgen Schwarze, Dirk Sander and Ingo WittichStroke. 1997;28:24652472, originally published December 1, 1997https://doi.org/10.1161/01.STR.28.12.2465