**The ****contents of the ****manuscript**** was ****submitted and published in several international medical conferences and journals, and was the basis of the author’s dissertation thesis, which was defended on November 21, 2003 at the Scientific Council of the Yerevan State Medical University.**

**Abstract **

** Objective**: The objective of this study was to present and evaluate an alternative formula for non-invasive intracranial pressure (ICP) and cerebral perfusion pressure (CPP) estimations based on haemodynamic and physiological criteria using transcranial Doppler (TCD) ultrasonography and arterial blood pressure (BP) measurements.

** Methods**: TCD measurements of middle cerebral artery (MCA) blood flow velocities (n=81) were performed in 39 randomly selected neurointensive care (NIC) patients (21 females and 18 males; mean age 57 years, range 22-77) with invasive ICP and BP monitoring. An alternative formula for non-invasive CPP/ICP estimations based on TCD and BP measurements were derived. The average and maximum absolute differences were calculated between the estimated CPP and the invasively measured CPP, using x-y scatter plots and linear regressive analysis.

*Results**:* The following insights were gained: **(a)** the starting point of non-invasive intracranial pressure estimation may be the end-diastolic flow velocity, **(b)** the principle of non-invasive estimation of intracranial pressure is limited to non-invasive estimation of cerebrovascular resistance in end-diastole (Rd), **(c)** in the end-diastolic state the Rd could be calculated as Rd =BPm / FVm (R^{2}=0.9765; average absolute difference = 0.088, max absolute difference = 0.55) and correspondingly CPPe = FVd x BPm / FVm + BPm – BPd (R^{2}=0.9391; average absolute difference = 3 mmHg, max absolute difference = 13 mmHg), **(d) **the method was improved by general as well as by selective statistical and linear regression analysis.

** Conclusions**: Haemodynamic criteria such as the end-diastolic flow (FVd) and cerebrovascular resistance in the end-diastolic state (Rd) were shown to be the key parameters in non-invasive ICP estimation. Such an approach represents a more flexible method of non-invasive ICP estimation with the potential of further development.

** Key words**: non-invasive estimation, intracranial pressure, cerebral perfusion pressure, transcranial Doppler, cerebrovascular resistance, craniovertebral reserve capacity.

**1. ****Introduction **

The classic definition of intracranial pressure (ICP) is the cerebrospinal fluid pressure, i.e. the pressure that must be exerted against a needle introduced into the cerebrospinal fluid space just to prevent escape of fluid (Cohadon et al., 1975)^{1}. Lundberg (1960) introduced intracranial pressure (ICP) recording in neurosurgical practice by using direct puncture of the ventricular system^{2}. Intraventricular pressure is still the standard reference for ICP. Alternatively, intracranial pressure can be measured in the epidural space, the subdural space, the subarachnoid space or in cisterna magna, or in the brain tissue. However, invasive measurements of ICP, using implanted catheters and/or transducers, are undesirable under certain conditions, e.g. coagulopathy and infections. Obviously, a reliable non-invasive method of ICP estimation would be convenient also when continuous ICP monitoring is not necessary, in the emergency room, and for departments without facilities for invasive ICP measurements (e.g. in gynaecology, paediatry, etc), respectively^{3,4,5,6}. Several attempts to use the transcranial Doppler (TCD) ultrasonography for non-invasive ICP assessments have been made^{3,7,8,9,10,11}. The estimations have usually been derived statistically and oftentimes without any physiological or haemodynamic interpretation. The objective of this study was to present and evaluate an alternative formula for non-invasive ICP estimation based on haemodynamic and physiological criteria using transcranial Doppler ultrasonography and arterial blood pressure (BP) measurements.

**2. ****Material and Methods **

All types of neurointensive care (NIC) patients (irrespectively of medical history, diagnosis, age, sex, etc) were eligible, provided that ICP and BP were monitored invasively. In total, 39 patients were examined, 16 with subarachnoid haemorrhage, 8 with traumatic brain injury, 8 with intracerebral hematoma, 2 with cerebellar hematoma, 2 with meningitis, 1 with cerebral infarction, 1 with meningeoma and 1 with rhinorrhea.

The material comprised 21 females and 18 males. The average age was 57 years (range 22-77). Intraventricular catheters were used in 34 patients and intracerebral probes in 5 for ICP monitoring.

A DWL Elektronische System Doppler machine (GmbH Multi-Dop® x 2, Germany) equipped with a 2 MHz TCD probe was used. The transtemporal approach was used for middle cerebral artery (MCA) insonation (mean depth = 50mm). TCD measurements of MCA blood flow velocities were performed 1-3 times in each patient. The average time between the samples taken from each patient was 26 h (range 5 min – 117 h). In total, 81 measurements were conducted (40 on the left side and 41 on the right side). The patients were randomly selected as well as the time point and the side of the TCD measurements.

The data were stored in Microsoft Excel program and were analysed using x/y scatter-plotting. The average and maximum absolute differences between the estimated CPP and the calculated CPP (from the invasive ICP and BP measurements) were calculated. The estimated values of CPP in our study were compared with those obtained by using the formula proposed by Czosnyka et al.^{8}.

**3. ****Results **

a) Basic formula

Hypothetically, if ICP is high, the external pressure to the vessel wall will be high. Consequently, if the autoregulation mechanisms are exhausted, the resistance to blood flow will be high too. In a result of this, most of energy, which is necessary for flow realization, will be spent to overcome that resistance. Eventually the flow will be low, and vice versa. On the other hand reduced flow at elevated ICP can be due to vasocompression of the entire cerebrovascular bed or just vasocompression of the venous bed, which elevates the transmural pressure of the arterial-arteriolar bed and prevents vasocompression. Hence, the correlation between the ICP and routine TCD parameters, such as – end-diastolic flow velocity (FVd), mean flow velocity (FVm) and peak-systolic flow velocity (FVs), as a rule, should be reciprocal. As in the end-diastolic state the vascular wall in its equilibrium, and the influence of the heart, as well as the age-dependent elasticity, and other properties of the vascular wall could be ignored, it could be presumed that from those three routine TCD parameters certainly the end-diastolic flow velocity will be correlated with the ICP to the greatest degree.

As it has been reported previously^{12, 13}, high correlation between ICP and end-diastolic flow velocity (FVd) exists.

Proceeding from these reasons, expediently to accept the end-diastolic flow velocity (FVd) as a starting-point for non-invasive ICP estimation.

According to the classic definition, the pressure responsible for the end diastolic flow (CPPd) can be calculated as a difference between the diastolic arterial blood pressure (BPd) and ICP (CPPd = BPd – ICP). However, according to Hagen-Poiseuille’s law some energy is spent to overcome the resistance (Rd) of cerebral vessel net in end diastole. Hence,

*FVd = (BPd – ICP) / Rd *(1)

It is logical to presume from (1) that the main task of non-invasive ICP evaluation will come to estimation of the cerebrovascular resistance in the end diastole (Rd). Therefore, the successful ICP calculation will eventually be confined by the estimation of Rd. If (1) is rearranged, the following relationship is obtained:

*Rd = (BPd – ICP) / FVd *(2)

Using the MCA FVd obtained from the TCD examinations (n=81) and the simultaneously recorded BP and the invasive ICP values, revealed that Rd, as calculated invasively in (2), is highly correlated to the ratio BPm / FVm (FVm=mean flow velocity).

(R^{2}=0.9765; average absolute difference = 0.088, max absolute difference = 0.55) (Fig1).

Using instead of Rd in (2) the ratio BPm / FVm, the following relationship is derived where only the ICP is unknown:

ICPe_{1}= BPd – FVd x BPm / FVm (3.1)

or

CPPe_{1}= FVd x BPm / FVm + BPm – BPd (3.2.)

The non-invasively estimated values for CPP from (3.2) were compared to those measured invasively and the following correlation was observed (R^{2}=0.9391; average absolute difference = 3 mmHg, max absolute difference = 13 mmHg) (Fig. 2)

The calculated values for CPPc according to the formula proposed by Czosnyka et al.^{8 }are summarised in Fig. 3.

(R^{2}= 0.7467; average absolute difference = 11 mmHg, max absolute difference = 30 mmHg).

*b) **General statistical development *

The better the Rd is estimated, the more precise will the calculated ICP be. Using statistical and linear regression analysis (Fig. 1), it was found that the correlation between *Rd* and *BPm / FVm *could be expressed as:

*Rd = 1.0429 * BPm / FVm – 0.0549 *(4)

(R^{2}=0.9765; average absolute difference = 0.086, max absolute difference =0.48) (Fig 4).

The more cases, the more precise is this correlation.

Using this algorithm (4) for estimation of Rd in (2), we will obtain the following relationship for estimation of ICP non-invasively:

*ICPe*_{2}* = BPd + FVd x 0.0549 – FVd x BPm x 1.0429 / FVm *(5.1)

or

*CPPe*_{2}* = FVd x BPm x 1.0429 / FVm + BPm -BPd – FVd x 0.0549 *(5.2)

Using (5.2) the CPPe_{2} was calculated (Fig. 5) and compared with invasively estimated CPP (R^{2}=0,9406; average absolute difference = 3 mmHg, max absolute difference =11.6 mmHg).

*c) **Selective statistical development*

Statistical and linear regression analysis between the arterial compliance

<<100% x (Rd – Rm) / Rd>>

and the error of the ratio BPm / FVm in relation to Rd

<<100% x (Rd – BPm / FVm) / Rd>>

revealed the following three main conditions:

· Under the condition of arterial compliance less than 7%

(i.e. 100% x (Rd – Rm) / Rd < 7%),

the cerebrovascular resistance in the end diastolic state could be estimated as:

Rd = 0.93 x BPm / FVm.

Respectively, the estimated CPP would be

CPPe = 0.93 x BPm x FVd / FVm + BPm – BPd.

· If arterial compliance is more than 15%

(100% x (Rd – Rm) / Rd > 15%),

the Rd could be calculated as:

Rd = 1.07 X BPm / FVm, and respectively

CPPe = 1.07 x BPm x FVd / FVm + BPm – BPd.

· When the arterial compliance is in the range from 7% to 15%

(7% ≤ 100% x (Rd – Rm) / Rd ≥ 15%),

the Rd is equal to the ratio

BPm / FVm,

and the CPP could be estimated as:

CPPe = BPm x FVd / FVm + BPm – BPd.

Using these three main principles the CPPe was estimated and was compared with the invasively measured CPP (R^{2}=0,9629; average absolute difference = 2.5 mmHg, max absolute difference = 8.5 mmHg) (Fig 6).

Theoretically, the estimated coefficient between the Rd and the ratio BPm / FVm during several pathological processes will make the estimation of the CPP much more precise.

**4. ****Discussion**

The results suggest that the FVd may be used as the starting point for non-invasive estimation of ICP. Hence, the cerebrovascular resistance in the end-diastole (Rd) is the limiting factor for successful non-invasive ICP estimation. The more precise Rd can be estimated, the more precise can the ICP be calculated. It was found that in the end diastolic state the

Rd = BPm / FVm

(R^{2}=0.9765; average absolute difference = 0.088, max absolute difference = 0.55) (Fig1).

Correspondingly,

CPPe_{1}= FVd x BPm / FVm + BPm – BPd (3.2)

(R^{2}=0.9391; average absolute difference = 3 mmHg, max absolute difference = 13 mmHg) (Fig2).

After performing general statistical and linear regression analysis, the following relationship was observed:

Rd =1.0429 x BPm / FVm – 0.0549 (4)

(R^{2}=0.9765; average absolute difference = 0.086, max absolute difference =0.48) (Fig4),

respectively:

CPPe_{2 }= FVd x BPm x 1.0429 / FVm + BPm – BPd – FVd x 0.0549 (5.2)

(R^{2}=0,9406; average absolute difference = 3 mmHg, max absolute difference =11.6 mmHg). (Fig5).

After selective statistical and linear regression analysis between the ratio

<<100% x (Rd-BPm / FVm) / Rd>>

and the arterial compliance represented as

100% x (Rd – Rm) / Rd

the following three main conditions were derived (Fig. 6):

· Under the condition of arterial compliance less than 7%

(100% x (Rd – Rm) / Rd < 7%),

the cerebrovascular resistance in the end diastolic state could be estimated as

Rd = 0.93 x BPm / FVm.

Respectively, the estimated CPP would be

CPPe = 0.93 x BPm x FVd / FVm+BPm – BPd.

· If arterial compliance is more than 15%

(100% x (Rd – Rm) / Rd > 15%),

the Rd could be calculated as

Rd = 1.07 x BPm / FVm,

and respectively

CPPe=1.07 x BPm x FVd / FVm + BPm – BPd.

· When the arterial compliance is in the range from 7% to 15%

(7% ≤ 100% x (Rd – Rm) / Rd ≥ 15%),

the Rd is equal to the ratio

BPm / FVm,

and the CPP could be estimated as:

CPPe = BPm x FVd/FVm + BPm – BPd.

Besides, by a customary mathematical manipulation, our formula can be rearranged for non-invasive estimation of the mean (Rm) and the peak-systolic (Rs) cerebrovascular resistances and respectively for non-invasive determination of the cerebral perfusion pressure in the peak-systolic state (CPPs).

Thus:

Rm = (FVd x BPm) / (VFm x FVm) + (BPm – BPd) / FVm

(R^{2}=0.9891; average absolute difference = 0.05, max absolute difference =0.3)

and

Rs = (FVd x BPm) / (VFm x FVs) + (BPs – BPd) / FVs

(R^{2}=0.9964; average absolute difference = 0.026, max absolute difference =0.13)

Correspondingly,

CPPs = FVd x BPm / FVm + BPs – BPd

(R^{2}=0.9809; average absolute difference = 3 mmHg, max absolute difference = 13 mmHg).

Hence, the percentages of the arterial compliance, which are described in the mentioned three main conditions, could be calculated non-invasively using the non-invasive estimates of the Rd and the Rm.

The formula (3.2) is very similar to that previously statistically derived by Czosnyka et al^{8}

CPPc = FVd x BPm / FVm + 14

(R^{2}= 0.7467; average absolute difference = 11 mmHg, max absolute difference = 30 mmHg). (Fig. 3).

Thus, our approach to the given problem of non-invasive ICP/CPP estimation has revealed, that the statistically proposed algorithm «FVd x BPm / FVm» by Czosnyka et al.^{8}, is non other than a cerebral perfusion pressure in the end-diastolic state (CPPd).

In the other words:

FVd x BPm / FVm = BPd – ICP = CPPd (3.1)

(R^{2}=0.8922; average absolute difference = 3 mmHg, max absolute difference = 13 mmHg).

It may be concluded thus that, if 14 is substituted by *(BPm – BPd*), a more flexible and a truly patient orientated approach is derived.

Therefore, the formula proposed by Czosnyka et al.^{8 }represents the dynamics rather than exact values of CPP, especially in situations when BPm – BPd ≠ 14.

Another formula has been developed for estimation of CPP by Belfort et al^{3 }

«CPP = FVm x (BPm – BPd) / (FVm – FVd) ».

According to Czosnyka and Steiner^{15} this formula could be interpreted as a simplification of Aaslid’s^{7 }approach by substitution of the amplitude of the fundamental frequency components with the differences between mean and diastolic values.

The original Aaslid formula is

CPP = FVm x F1_{BPm }/ F1_{FVm }^{.}

(F1=amplitude of the fundamental frequency components).

If formula

« CPP = FVm x (BPm – BPd) / (FVm – FVd) »

proposed by Belfort et al^{3 }is rearranged for estimation of Rd we will obtain the following equation:

Rd = (BPm – BPd) / (FVm – FVd),

which is a questionable value according to our dataset. Moreover, if formula proposed by Belfort et al^{3 }is rearranged for estimation of Rm we will obtain the same equation:

Rm = (BPm – BPd) / (FVm – FVd)

So, according to formula proposed by Belfort et al^{3 }Rd = Rm, which is not logical for normal cases.

It could be possible only in those situations, when the intracranial arterial compliance is exhausted. Usually, but not necessary, these cases are associated with high level of ICP, and/or with low CPP.

Though the equation « (BPm – BPd) / (FVm – FVd) » is not characterizing the exact values of Rd, it will somehow be correlated with the Rd, due to the constituent part BPm / FVm, which as we found is representing the real Rd.

Therefore, the formula proposed by Belfort et al^{3 }represents the dynamics rather than exact values of CPP, especially in situations when Rd ≠ Rm.

For example the relationship between the arterial compliance (100% x (Rd – Rm) / Rd) and the error of the Belfort’s formula (CPP invasive – CPP Belfort) could be expressed as it is shown in the following x/y-scatterplot :

The more the difference between the Rd and the Rm, the more the error of the formula proposed by Belfort and colleagues.

According to the theory proposed by Bergsneider et al.^{14}, the conversion of pulsatile arterial to non-pulsatile venous flow occurs as a result of arterial compliance, which in turn is subserved by the pulsation capability of the intracranial arterial system proximal segment (i.e. large and medium arteries). Exactly the proximal segment of the intracranial arterial system is the reachable part for the TCD examination. On the other hand, as adult cranial compartment is practically indistensible, the dynamic CSF movement across the foramen magnumis the primarily factor by which intracranial arterial proximal segment expansion occurs. Interference of the displacement of CSF during systole results in decrease in arterial compliance.

Hence, it could be suspected from this theory that the less the intracranial volume buffering system, the less the difference between the Rd and the Rs.

In those situations when the overall intracranial volume buffering system is exhausted, the Rs will equal to Rd, or even will become higher than Rd.

Based on this presumption, the formula

100% x (Rd – Rs) / Rd, similarly 100% – Rs x 100% / Rd

will represent the craniospinal reserve capacity in percentage.

Other essential singularity of our method is that having data of TCD analysis (e.g. FVd and FVm), it is possible to compute that value of BP, which will maintain the CPP at an optimal level. Decrease in BP (spontaneous, oligemia, cardiogenic, pharmacological, heightening of viscosity, hypoxia, hypercapnia, the heightening of oxygen consumption by the brain) results in decrease in CPP, which cause vasodilatation, that results in increase in cerebral blood volume, which augments ICP, starting up the process of the ischemia. In turn, it results in further increase in ICP, starting up a vicious circle of the brain ischemia. It is necessary to abort this process by increasing the BP, which, accordingly, will result in increase in CPP, causing a vasopressor cascade.

Increase in BP (spontaneous, hypervolemia, ischemia, pharmacological, lessening of oxygen consumption by the brain, decrease in viscosity, hyperoxia, hypocapnia) results in increase in CPP which cause vasoconstriction, which results in decrease in the cerebral blood volume and ICP.

However a very high level of BP can also lead to the cerebral ischemia due to the vasoconstriction and following decrease in cerebral blood volume.

Accordingly, it is very important to estimate that value of BP, which will maintain the CPP at an optimal level.

The following formula could be used for this purpose:

FVd/FVm = CPPd / BPm

For example, the value of BPm, which will maintain the CPPd above a critical limit let’s say 60mmHg, could be calculated as ABPm ≥ FVm x 60 / FVd. In the other words the ABPm shouldn’t be less than the ratio FVm x 60 / FVd in order to maintain the CPPd above the critical limit 60mmHg.

The method would be of special interest also for non-invasive assessment of the cerebral blood circulation autoregulation level. For this reason, using ”body tilt test”, or applying inflation and subsequent deflation of blood pressure cuffs, placed on the upper thighs, it could be possible to change non-invasively the ICP and ABP values respectively, and simultaneously to supervise the variations of

Rd = BPm / FVm,

Rs = (FVd x BPm) / (VFm x FVs) + (BPs – BPd) / FVs,

Rm = (FVd x BPm) / (VFm x FVm) + (BPm – BPd) / FVm

and

CPPd = FVd x BPm / FVm,

CPPm = FVd x BPm / FVm + BPm – BPd,

CPPs = FVd x BPm / FVm + BPs – BPd,

as well as the ratio << 100% – Rs x 100% / Rd>> and <<100% – Rm x 100% / Rd>>

While the values of Rd, Rs and Rm as well as the ratio << 100% – Rm x 100% / Rd >> will be varying in response to the ICP and/or BP variations, the CBF should remain relatively stable owing to autoregulation mechanisms.

Thus, contrary to the other statistically proposed earlier methods, our formula

«ICP = BPd – FVd x BPm / FVm»

has several potential possibilities for noninvasive estimation of cerebrovascular haemodynamics, and explains an operational principle of statistically maneuvered analogs.

**5. ****Conclusion**

Haemodynamic criteria such as the end-diastolic flow (FVd) and cerebrovascular resistance in the end-diastolic state (Rd) were shown to be the key parameters in non-invasive ICP estimation. Such an approach represents a more flexible method of non-invasive ICP estimation with the potential of further development.

**Acknowledgements**

We wish to express our appreciation and sincere gratitude to all those who helped us with this work, in particular: the Swedish Institute is acknowledged gratefully for the financial support of one of the authors (A. Z.); the Department of Neurosurgery, University Hospital, Uppsala, Sweden and Professor Lennart Persson personally are acknowledged for the facilities and opportunities provided; Professor Michael L. Daley from the Deptartment of Electrical Engineering, University of Memphis, U.S.A. is acknowledged gratefully for useful comments and suggestions.

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**Legends**:

Figure 1 – x-y scatterplot of the cerebrovascular resistance in end diastole (Rd) compared with the ratio BPm/FVm, using linear regressive analysis.

n (number of samples) = 81

R^{2}(correlation coefficient) = 0,9765

Average absolute difference = 0,088

Max absolute difference = 0.55

Figure 2 – x-y scatterplot of the invasively measured cerebral perfusion pressure (CPP) compared with cerebral perfusion pressure estimated non-invasively (CPPe_{1}), using linear regressive analysis.

n (number of samples) = 81

R^{2}(correlation coefficient) = 0,9391

Average absolute difference = 3 mmHg

Max absolute difference = 13 mmHg

Figure 3 – x-y scatterplot of the invasively measured cerebral perfusion pressure (CPP) compared with cerebral perfusion pressure estimated non-invasively by the statistically derived formula from Czosnyka et al (1998) (CPPc), using linear regressive analysis.

n (number of samples) = 81

R^{2 }(correlation coefficient) = 0,7467

Average absolute difference = 11 mmHg

Max absolute difference = 30 mmHg

Figure 4 – x-y scatterplot of the cerebrovascular resistance in end diastole (Rd) compared with the ratio 1.0429 x BPm / FVm-0.0549, using linear regressive analysis.

n (number of samples) = 81

R^{2}(correlation coefficient) = 0,9765

Average absolute difference = 0.086

Max absolute difference = 0.48

Figure 5 – x-y scatterplot of the invasive measured cerebral perfusion pressure (CPP) compared with cerebral perfusion pressure estimated non-invasively by statistically developed formula (CPPe_{2}), using liner regressive analysis.

n (number of samples) = 81

R^{2}(correlation coefficient) = 0,9406

Average absolute difference = 3 mmHg

Max absolute difference = 11.6 mmHg

Figure 6 – x-y scatterplot of the invasive measured cerebral perfusion pressure (CPP) compared with cerebral perfusion pressure estimated non-invasively by selective statistical development of the formula (CPPe), using liner regressive analysis.

n (number of samples) = 81

R^{2}(correlation coefficient) = 0,9629

Average absolute difference = 2.5 mmHg

Max absolute difference = 8.5 mmHg