- Research
- Open Access

# Effect of non-linearity in predicting doppler waveforms through a novel model

- Aman Gayasen
^{1}Email author, - Sunil Kumar Dua
^{1}, - Amit Sengupta
^{2}and - D Nagchoudhuri
^{3}

**2**:16

https://doi.org/10.1186/1475-925X-2-16

© Gayasen et al; licensee BioMed Central Ltd. 2003

**Received:**10 July 2003**Accepted:**18 September 2003**Published:**18 September 2003

## Abstract

### Background

In pregnancy, the uteroplacental vascular system develops de novo locally in utero and a systemic haemodynamic & bio-rheological alteration accompany it. Any abnormality in the non-linear vascular system is believed to trigger the onset of serious morbid conditions like pre-eclampsia and/or intrauterine growth restriction (IUGR). Exact Aetiopathogenesis is unknown. Advancement in the field of non-invasive doppler image analysis and simulation incorporating non-linearities may unfold the complexities associated with the inaccessible uteroplacental vessels. Earlier modeling approaches approximate it as a linear system.

### Method

We proposed a novel electrical model for the uteroplacental system that uses MOSFETs as non-linear elements in place of traditional linear transmission line (TL) model. The model to simulate doppler FVW's was designed by including the inputs from our non-linear mathematical model. While using the MOSFETs as voltage-controlled switches, a fair degree of controlled-non-linearity has been introduced in the model. Comparative analysis was done between the simulated data and the actual doppler FVW's waveforms.

### Results & Discussion

Normal pregnancy has been successfully modeled and the doppler output waveforms are simulated for different gestation time using the model. It is observed that the dicrotic notch disappears and the S/D ratio decreases as the pregnancy matures. Both these results are established clinical facts. Effects of blood density, viscosity and the arterial wall elasticity on the blood flow velocity profile were also studied. Spectral analysis on the output of the model (blood flow velocity) indicated that the Total Harmonic Distortion (THD) falls during the mid-gestation.

### Conclusion

Total harmonic distortion (THD) is found to be informative in determining the Feto-maternal health. Effects of the blood density, the viscosity and the elasticity changes on the blood FVW are simulated. Future works are expected to concentrate mainly on improving the load with respect to varying non-linear parameters in the model. Heart rate variability, which accounts for the vascular tone, should also be included. We also expect the model to initiate extensive clinical or experimental studies in the near future.

## Keywords

- Heart Rate Variability
- Transmission Line
- Uterine Artery
- Normal Pregnancy
- Total Harmonic Distortion

## Background

## Method & Model Formulation

### The anatomy and pathophysiology of the utero-placental system

For the maintenance of the normal pregnancy, the placenta, a fetomaternal organ, plays an important role. It is developed by the interaction of the maternal spiral end vessels and the trophoblastic villi. The spiral end vessels are the branches of the radial artery, which comes out from the arcuate arteries at right angles from the main uterine artery. Brosen (1967) proposed that for normal fetoplacental flow, the growing trophoblasts invade the spiral vessels during formative stages beyond the deciduo-myometrial junction converting the entire system into a low-resistance shunt. The spiral vessels lose the normal architecture and the elasticity and become a low resistance, dilated, straight, vein like vessel. The diameter increases from 0.5 mm to 1.0 mm. The main proximal uterine artery, supplying most of the blood to the system, also gets dilated with a relative fall in the resistance, and the diameter increases from 2.0 mm to approximately 5.0 mm. In total, 120 such vessels are affected, discharging the blood into intervillous space and displacing the villi with the maternal arterial pressure head. The capacitance increases to allow the excess demand of blood flow during pregnancy. In abnormal condition such as Intra-Uterine Growth Restriction (IUGR) and pre-eclampsia (PE), the increase in diameter has not been observed. The detailed architectural geometric patterns are given in Fig. 1.

### The electrical model

- 1.
Current Source: A current source in the model represents the blood flow input to the biological system. It consists of 4 frequencies, which we have taken from the TL model. The actual implementation was done using 4 sinusoidal current sources connected in parallel.

- 2.
Matching Impedance (R

_{m}, C_{m}): The matching impedance is needed to absorb the waves that come after reflection from the load. - 3.
Uterine Artery: As in case of the TL model, the uterine artery is represented by an RLC ladder-type structure. The difference is the implementation of resistance (MOSFETs are used to implement resistance, to give more flexibility), and also the inductance, which is made constant. Implementation details are discussed in Appendix.

- 4.
Load: The uteroplacental system is lumped with the intervillous sinus and approximated by a capacitive bank in parallel with a MOSFET resistance (Appendix).

The uteroplacental system is considered in isolation from the rest of the circulatory system. External influences on the system have not been considered.

### Modification over transmission line model

The architecture of the proposed model is based on the TL model. It has been modified in the following ways.

The proposed model simulates the uteroplacental system at any given pregnancy time. Pregnancy time is an input to the model, which internally affects the value of a voltage source, (which in turn change the circuit parameters). This must not be confused with the real time, and the system remains time invariant. The system can be simulated for an arbitrary pregnancy time by changing the "pregnancy time" input accordingly.

In the TL model, the uterine artery is modeled as a transmission line with the uteroplacental portion forming its load, which is also linear, consisting of a resistance and a capacitance. In the proposed model, it is assumed that as pregnancy matures, load of the TL model changes non-linearly.

The elements of transmission line are made to vary with respect to a voltage, which is controlled by the input "pregnancy time". These elements have been shown in boxes in Fig. 2. MOSFETs have been used to implement such voltage-controlled parameters.

In the TL model, inductance of the vessel also varies with pregnancy time. It is difficult to implement a voltage-controlled inductance using MOSFETs [10]. Therefore, we removed this variation in inductance through some mathematical manipulations (Appendix), thereby, making both, inductance and capacitance of the vessel constant with pregnancy time. The variation in them is effectively mapped onto the load of the transmission line. Thus, now the load of the transmission line does not represent only the uteroplacental arteries and the intervillous sinus. Rather, some variations in the uterine artery also affect the load. Modeling of the load has been improved. In our model, load capacitance decreases as pregnancy matures. This voltage varying capacitor has been implemented using a capacitor bank. MOSFET switches control the actual number of capacitors that add in parallel in the circuit. This is the major non-linear element in the model and all the non-linearities like elasticity, viscosity, density are studied using the equations governing the capacitors (Appendix).

### Actual doppler flow velocity waveform analysis

## Results and Discussion

### Density of the blood and the flow velocity profile

### Vessel elasticity and the flow velocity profile

### Blood viscosity and the flow velocity profile

Variation in blood FVW with viscosity appears to be negligible. We varied the viscosity from 0.1 to 10 times the normal value. On decreasing viscosity from normal, waveforms are indistinguishable from each other. On increasing the viscosity, though, there is a small decrease in the systolic peaks. Viscosity per se may not effect the flow velocity waveforms in arteries as inertial forces predominate. Only in microvessels does it predominate over inertial forces. Viscosity has been shown not to influence the Doppler flow profile in pregnancy [9].

## Conclusions

Advancement in the field of non-invasive Doppler image analysis and simulation incorporating non-linearities may unfold the complexities associated with the inaccessible uteroplacental vessels. An abnormality in the non-linear vascular system is known to trigger a serious pathological condition of pre-eclampsia and/or intrauterine growth restriction (IUGR); one of the leading causes of Feto-maternal deaths the world over. A non-linear mathematical model of the uteroplacental system has already been developed and validated using the available Doppler FVW's studies only to a certain extent [4]. As a further advancement in modeling, a non-linear MOSFET electrical model using the load as derived by the mathematical model for the utero-placental system has been proposed in the present study, which gives expected results and matching FVW's outputs for the normal as well as the abnormal pregnancy. Effects of the blood density, the viscosity and the elasticity changes on the Doppler FVW's profile are simulated. Total harmonic distortion (THD) is found to be informative and valuable in determining the Feto-maternal health. The Future works is expected to concentrate mainly on improving the load with respect to varying non-linear parameters in the model. These shapes may be incorporated in the model. Also, the heart rate has been assumed to be constant in our model. But in actual system, the heart rate varies with the time and is known to play an important role in determining the underlying cardiovascular sympathetic activity. The heart rate variability should also be included in the future model.

## Appendix

Replacing a time varying inductor with constant inductor: In the transmission line model, transmission line parameters vary as derived below [6, 18] --------- (1).

Where *r*
_{0} is the vessel radius, *E* is the Young's modulus of the vessel wall, *h* is the vessels wall thickness, ρ is the mass density of blood, and Η is the blood viscosity. We notice that the transmission line model has an inductive element that varies with pregnancy time. It is difficult to make an inductance using active elements [10]. So, we removed this variation in inductance by mapping the change in inductance to other circuit elements. This was achieved through some mathematical manipulations, as explained below.

*The transfer function for RLC-circuit implementation of transmission line model is:*

where *I*
_{
out
}is the output current; *I*
_{
in
}is the input current; *R, C* and *L* are transmission line resistance, capacitance and inductance respectively; and *R*
_{
l
}and *C*
_{
l
}are load resistance and capacitance respectively. In order to make the inductance of transmission line independent of pregnancy time, we make the following transformations in the circuit.

*L* → 1; *C* → *LC*; *C*
_{
l
}→ *LC*
_{
l
}; *R* → *R/L*; *R*
_{
l
}→ *R*
_{
l
}/*L*

Note that this does not alter the transfer function of the TL model. Thus, equations for the transformed circuit elements become:

*As, h = 0.1 r*
_{
0
}
*for normal pregnancy, the capacitance becomes independent of r*
_{
0
}
*. This enables us to use a passive capacitor to represent the transmission line capacitance. It may be observed that with this transformation, the transfer function is effectively represented in terms of time constants of the circuit.*

This transformation resulted in an equivalent model for the micro-vascular system. But the disadvantage of this exercise is that the model is no more directly analogous to the biological system. For example, in the modified model, the load capacitance is a function of radius of the vessel as well as the mass density of blood. This has no direct analogy in the biological system. Also, there is no longer a direct analogy between pressure in biological system and voltage in the model. But the resulting simplification in implementation justifies this transformation. Thus, in the modified model, inductance remains constant throughout the pregnancy period. The value of inductance is kept at 10 uH.

### Implementation of circuit elements

#### Transmission line resistance

The transmission line resistance is implemented using channel resistance of an NMOS. Channel resistance has the following expression [17], when the NMOS is in triode region.

Where *k* is gain factor of NMOS, *W* is width of channel, *L* is length of channel, *V*
_{
GS
}is gate voltage, *v*
_{
ds
}is drain voltage, *V*
_{
T
}is threshold voltage of MOSFET.

Assuming low *v*
_{
ds
},

Thus, if the gate voltage were made to vary proportional to pregnancy time,

the resistance of uterine artery would decrease as pregnancy matures. This is as expected, in normal pregnancy. Assuming a linear variation of radius in equation (3) with pregnancy time, we get

where, *t* refers to pregnancy time (expressed as periods of 8 weeks) and the vessel radius is assumed to vary as (*a* + *bt*).

In normal pregnancy, vessel radius varies from 0.14 cm to 0.20 cm in 25 weeks. Thus, *b* is small. Therefore,

(*a* + *bt*)^{2} ≈ *a*^{2} + *2abt* = *A* + *Bt*

*Substituting the above expression in equation. (6), we get*

Comparing equation (5) and (7), we get an expression for V_{GS} in terms of pregnancy time. Thus, the transmission line resistance is directly controlled with a voltage that is in turn proportional to pregnancy time.

#### Transmission line capacitance

A passive capacitance is used to model the capacitance in transmission line. Its value is proportional to density of blood and radius of vessel, and inversely proportional to Elasticity of arterial wall and the thickness of arterial wall.

#### Transmission line inductance

A passive inductance is used to model inductance in transmission line.

#### Load resistance

The load resistance is also implemented using an NMOS. Load resistance decreases as pregnancy matures.

#### Load capacitance

#### Matching resistance

Variation in matching resistance is neglected.

#### Matching capacitance

Matching capacitance is also implemented using a bank of capacitors. This capacitance varies linearly with pregnancy time.

#### Matching Impedance (C_{m}, R_{m})

The matching impedance is needed at source to absorb the waves that come after reflection from the load.

## Declarations

### Acknowledgements

We would like to thank all the anonymous reviewers for their helpful comments.

## Authors’ Affiliations

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