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Uterine Contraction Modeling and Simulation

Miao Liu, Lee A. Belfore, Yuzhong Shen, and Mark W. Scerbo

Virginia Modeling, Analysis, and Simulation CenterOld Dominion University, Norfolk, VA, USA, 23529

(mliux003, Ibelfore, yshen, mscerbo}@odu.edu

Abstract. Building a training system for medical personnel to properly interpret fetal heart rate tracing requiresdeveloping accurate models that can relate various signal patterns to certain pathologies. In addition to modelingthe fetal heart rate signal itself, the change of uterine pressure that bears strong relation to fetal heart rate andprovides indications of maternal and fetal status should also be considered. In this work, we have developed agroup of parametric models to simulate uterine contractions during labor and delivery. Through analysis of realpatient records, we propose to model uterine contraction signals by three major components: regular contractions,impulsive noise caused by fetal movements, and low amplitude noise invoked by maternal breathing and measuringapparatus. The regular contractions are modeled by an asymmetric generalized Gaussian function and leastsquares estimation is used to compute the parameter values of the asymmetric generalized Gaussian functionbased on uterine contractions of real patients. Regular contractions are detected based on thresholding andderivative analysis of uterine contractions. Impulsive noise caused by fetal movements and low amplitude noise bymaternal breathing and measuring apparatus are modeled by rational polynomial functions and Perlin noise,respectively. Experiment results show the synthesized uterine contractions can mimic the real uterine contractionsrealistically, demonstrating the effectiveness of the proposed algorithm.

1. Introduction

Uterine contractions are stimulated by uterine musclecells. Uterine contraction (UC) variations reflect thephysiological changes of the uterus during bothpregnancy and labor [1]. As a critical component infetal heart rate (FHR) monitoring during labor anddelivery, uterine contractions provide importantinformation regarding maternal and fetal wellbeing.There are three main methods to record uterinecontractions [2]: tocography, electrohysterography,and using intrauterine pressure catheter.Tocography measures the strain exerted by uterus onthe maternal abdomen via external a tocotransducer.Electrohysterography records the electrical uterineactivities from the maternal abdomen. Theintrauterine pressure catheter (IUPC) measuresintrauterine pressure invasively, and is mostly usedduring labor. Regardless of their differences, allthree methods aim at providing records of thecontraction patterns and their relationship to FHR.Proper interpretations of fetal heart rate and uterinecontractions require special training, while monitoringof both are only available when pregnant women arehospitalized in parturiency. Building a trainingsystem that can simulate fetal heart rate and uterinecontractions can help medical personnel learn criticalpatterns of both signals without putting patient indanger. To gain better understanding of uterineactivities, it is necessary to develop mathematicalmodels to quantitatively describe various uterinecontraction patterns and this is the problem to beaddressed in this paper.

Even though FHR monitoring is now the standard

practice during labor and delivery, surprisingly, therewere only a few studies on uterine contractionmodeling and simulation. Young used polynomialsto model five characteristics of uterine contractions: 1)gradual onset, 2) a linear rising segment, 3) a plateauregion, 4) a symmetrical fall, and 5) gradual offset,and fitted the simulated contractions with recordedIUPC data [3]. Their results matched their physicalanalysis. However, their simulations were notperfect especially in the tail region of the contractioncurve. Vauge et al. [4] developed a system ofdifferential equations that describe the dynamics ofuterine pressure during human parturition. Thismethod was based on three simplified assumptions:identical contractile properties of all myometrial cells,intrauterine pressure proportional to the number ofcontracted myometrial cells, and that all cells havethree states, namely, contraction, recovery, andresting. Their model was simple and effectiveespecially for normal contractions which begin in thefundus, reach the apex, and then proceedsymmetrically downward toward the fundus. But thismethod did not consider the fact that asymmetry canoccur when the uterine cells function independentlycausing ineffective uterine contractions and minimaldilatation [2]. Recently, Kemal et al. [5] employedtwo methods to simulate uterine contractions. Thefirst one is based on the same mathematical modelproposed by Vauge [4]. The second approach isbased on recorded patient data. They first appliedHilbert-Huang Transform (HHT) [6] to identify thecontraction locations from real patient data, and thendeveloped spatial-temporal simulations of uterinecontractions. All these methods discussed aboveare capable of illustrating the dynamics of uterineactivities. However, these methodologies are

https://ntrs.nasa.gov/search.jsp?R=20100012865 2020-06-19T06:25:51+00:00Z

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deterministic and no comprehensive parameterestimation for these models was developed.Furthermore, the noise caused by fetal movementsand maternal breathing were not considered andmodeled. To address the problems in existinguterine modeling and simulation, the paper proposesa novel algorithm that integrates three majorcomponents: asymmetric generalized Gaussianfunction (AGGF) for modeling contractions, Perlinnoise for modeling maternal breathing and instrumentnoise, and impulsive noise for modeling fetalmovements. The parameters of the asymmetricgeneralized Gaussian function are estimated usingthe least square method based on detected uterinecontractions from real patient records.

The remainder of this paper is structured as follows.Section 2 first introduces the proposed asymmetricgeneralized Gaussian functions for modeling uterinecontractions and then estimate the parameters ofgeneralized Gaussian functions based detectedcontractions. Section 3 describes Perlin noisegeneration for maternal breathing and low amplitudenoise modeling. Section 4 presents impulsive noisegeneration for simulating disturbances caused byfetal movements. Section 5 summarizes thesimulation procedure and compares the simulationresults. Section 6 concludes this paper anddiscusses future research directions.

2. Uterine Contraction Modeling

2.1 Asymmetric Generalized Gaussian Function

A typical uterine contraction curve of a real patient isdepicted as a continuous waveform in Figure 1. Thiscurve is characterized by a basal tone varying from 0to 20 units, and a deflection of the contraction curveabove the baseline, whose amplitude and durationare within a certain range of values [2]. Moreover,one should note this curve is asymmetric.

100

00 Q2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8time (rrirue)

Figure 1: Typical uterine contractions. Also shownare parameters used for contraction detection.

Since the asymmetry of contraction curve matches

real cases of ineffective contractions and comprisesthe symmetrical case, we propose to use anasymmetric generalized Gaussian function to modeluterine contractions as follows.

f(t) = AI exp{ - (t _;~tl }[u (t -tl ) -u (t -to)]

+A, exp { - (t-~:t· }[u(t -fo)-u(t-t,)] (1)

+b(f)[u(t-tl)-u(t-f,)],

where the parameters of (1) are shown below.

AI, A, Amplitudes for the left and right sides0./, a, Exponents for the left and right sides[JI, [J, Variances for the left and right sidesu(t) A unit step functionIt, t, Left and right cut off timeto The position wheref(t) reaches its maximumb(t) Baseline representing some basal strain

exerted by the uterine muscle whencontractions do not occur

To simulate uterine contractions, we need todetermine the range within which the aboveparameters lie and how they vary with time. So inthe next step we will detect uterine contractions fromreal patient data and estimate the parameters for theasymmetric generalized Gaussian function from thedetection results.

2.2 Uterine Contraction Detection

Several methods were proposed to detect uterinecontractions for different purposes. Radhakrishnanet al. (7) developed a higher-order zero crossingbased method and studied the frequency ofoccurrence of contractions in different pregnancystages. Novak et al. (8) described two UC detectionapproaches: amplitude- and derivative-basedalgorithm. By comparing the results from these twomethods, they suggested combining both methodstogether to achieve better detection results. Aimingat quantitatively analyzing uterine contractions in timedomain, Jezewski et al. (9) introduced a statisticalmethod to determine the threshold and alsoconsidered duration condition for UC detection.These methods were used to calculate the regularparameters of uterine contractions such as amplitude,duration, frequency of occurrence. Our method isbased on a combination of the last two methods, inwhich thresholding is first performed to detect thepresence of uterine contractions, andderivative-based method is applied subsequently toinclude the samples whose amplitudes are below thethreshold, but still belonging to the contractions.

The algorithm proposed by Jezewski et al. (9) startswith low pass filtering with cutoff frequency of 0.04 Hz

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(3)

to suppress the artifacts caused by fetal movementsand maternal breathing. Then the record isanalyzed by using a moving window with a length of 4minutes and 1-minute step. Within each window, thehistogram of uterine pressure samples is constructedfirst and the mode of the histogram is then selectedas the baseline value. Finally, the threshold level isset as 10 units above the baseline and the validity ofdata segment is examined. A valid contractionshould remains above the threshold level for aduration longer than 30 seconds and the amplitude ofcontraction exceeds 20 units.

:~uJo 2 .. e a 10 12 14

:F:JKmy\uT\~1J]m!kmJo 2 .. e a \0 12 ~

(q WtiIf UC d1ttea.il:tl ,. ....

:~T'--6-------v\-----I'A---'--I'--"Af----"'8---'''''lo 2 .. 0 I 10 12 l-\

:~Jo ., .. e 8 10 ·12 14

:K is. 7\A~-K, K, Jo 2 .. e I lO 12 ~

RHi:fual

Figure 2: Intermediate results in uterine contractiondetection. (a) Raw UC data. (b) Filtered UC. (c)UC detection by thresholding. (d) The derivatives offiltered UC. (e) Improved UC detection results. (f)The residual between raw UC and filtered UC which isdetected.

After the uterine contractions are detected, there arestill two remaining problems. The first is that theamplitude of detected samples are mostly above thethreshold, thus the estimated parameters may not beaccurate enough to model the tail areas. We need toinclude more samples from tail areas. On the otherhand, the tail areas are prone to other varioussources of noise. Before including them for parameterestimation, we must differentiate contaminatedsamples and uncontaminated or less-contaminatedsamples. Since the derivatives in noisy area changedrastically, we propose to employ derivative-baseddetection method to differentiate them. In otherwords, for less contaminated contractions, thederivative of the left side of the contraction should bepositive, while the derivative of right side should be

negative. Starting from the peak of the contraction,we proceed to its left and compute the derivatives ofthe smoothed uterine pressure signal and then searchthe position where the derivative changed to negativefor left side curve, and denote it as the tail point. Thesample points between the peak and the tail point willbe utilized for curve parameters estimation. Thesame principle applies to the right side of contractionwith positive derives being searched.

The results of uterine contraction detection are shownFigure 2, where (a) is the original uterine pressuresignal; (b) is its filtered version, the red solid linerepresents the base line; (c) illustrates the samplewhose values are above the threshold; (d) is thederivatives of filtered UC data; (e) is the improveduterine contraction detection result; and (f) shows theresidual between the detected contractions andoriginal signal (a).

2.3 Uterine Contraction Parameter Estimation

After the uterine contractions are detected, we needto compute the parameters of the asymmetricgeneralized Gaussian function (1) in order to simulatethe detected contractions. Here we only considerthe left half of the asymmetric generalized Gaussianfunction (1), that is,

f(l) " A, exp{ J(t -;; JI"' }+b, [U(I-I, )-U(I-I.)J

(2)for the parameter estimation, while the right half canbe handled similarly and thus is omitted in this paper.

First, the baseline b, is estimated as the contractionvalue at the onset point /, which is the lowest pointbetween two filtered contractions. Thus

A, =f(to)-b,.

After simple manipulation, the remaining knownvariables and unknown parameters are separated bytaking the logarithm of both sides of (2),

In J(t)-b, =-'(I-to tA, /3,

Since J(t) - b, < A, ' adding minus sign to both sidesof (3) and taking logarithm again gives

Now we denote the detected uterine contraction data

set as {f(t,),t,}, 15,i5,N. where N is the

number of samples. Substituting the data set intoequation (4) and writing each term in matrix form, we

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By adding the above generated series together, wehave the Perlin noise

where tE [1, N], and p is called persistence factor

which controls the amplitude.

have

y=

lo ( -lo f(t~/-bl)

In(-in f(t~/-blJloltl -tol, -1

In It2 - to I, -1,X=

s

fp(t) =I/n,(t) ,1=0

(9)

In(-In f(t~-bl)

P =[lnap](5)

Then equation (4) can be transformed into a conciselinear equation represented by a matrix product,

Y=Xp, (6)

where Xand Yare matrices containing detectedsample information, p is the parameter vector to beestimated. The parameter estimation problem canbe expressed as the following minimization problem,

Po =arg min Ily - Xpll~, (7)pelf

where Po is the solution. This is a least square

estimation problem whose solution is

Po =(XTXf' XTy . (8)

Thus we have obtained the parameter estimator forthe proposed asymmetric generalized Gaussianfunction for modeling uterine contractions.

3. Perlin Noise

The low amplitude noise caused by maternalbreathing and measuring apparatus are random yetexhibiting both low and high frequency characteristicsas shown in Figures 2(a) and (t). Common randomnumber generators cannot be used directly togenerate such noise, since they are too random toexhibit the natural outlook of continuity andself-similarity of the noise. To address this problem,we propose to use Perlin noise generator (10) tosimulate the low amplitude noise.

A Perlin noise generator is composed of twocomponents: a noise function and an interpolationfunction. The basic procedure of Perlin noisegeneration is(1) Generate a series of random numbers nl of

length from uniform distribution U[-0.1 0.1];(2) Decimate the series to size 12;(3) Upsample the decimated series to size by

B-spline interpolation and increase the amplitudeof the new series by a factor of p;

(4) Repeat (2) and (3) until reaching the specified

level S, and we obtain a set of series {nl. n2....,ns}.

(a)

(b)

(c)

(d)

l~. .

-~

(e)

(f)

!j~(g)

Figure 3: (a) - (t) are waveforms corresponding to 6levels noise series {n1. n2.... ,n6}. (g) is the Perlin noise.

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4. Impulsive Noise Modeling

The last component to simulate is the spikes or largemagnitude impulsive noise in uterine contractions,which suggest possible fetal movements [2]. Afterexamining the shape of those spikes, we propose touse the following rational polynomial function tomodel them

....

5. Uterine Contraction Synthesis

Uterine contraction simulation is finalized bysuperimposing the components generated by theasymmetric generalized Gaussian function, Perlinnoise, and impulsive noise. First, a typical segmentof 20 minutes was extracted from real patient recordand the parameters for the asymmetric generalizedGaussian function were estimated. Figure 5compares the parameters estimated from twodetected data sets. The first data set contains onlysamples whose amplitude is above the threshold.The second data set is the expanded version of thefirst set by incorporating samples whose amplitudeare below the threshold but are belong to thecontraction. Figures 5 (a) and (b) plot parametersestimated from the first set. Figures 5(c) and (d) areestimated from the second set. It can be seen thatthe parameters of the left side and right side aredifferent, validating the asymmetry of uterinecontractions.

To compare the impact of different parameters onuterine contraction simulation, we select one set ofparameters from Figures 5(a) and (b), and anotherset from Figures 5(c) and (d) to simulate twocontractions, namely, contraction 1 (green dotted line)and contraction 2 (red dashed line), as shown inFigure 5(e). It can be seen that the red dash lineachieves a better fit to the real contraction at both thepeak and tail area of the contraction and has smallernormalized root mean square error (NRMSE), whichis defined as

2::1 (J; (i) - fo (i)YNRMSE = N (10)

max{fo(j)} - min {fo(k)}lS.jS.N 1s.!<s.N

This result indicates that the derivative-basedanalysis could help to recruit more effective data forparameter estimation, and thus increase theestimation accuracy. Finally, by adding together theasymmetric Gaussian function, the Perlin noise, andthe impulsive noise, we obtain the final simulationresults in Figure 6(b). Comparing the original uterinecontraction record in Figure 6(a) and the simulatedresults in Figure 6(b), it can be easily seen that theproposed algorithm is very effective and producessuperb results.

(8)a

5

0

5

I ~ I

J(t) = ( )21+ t - to

b

where a is the amplitude which follows uniformdistribution with value between 0 and 50 units, b is thescale parameter following uniform distribution withvalue between 0 and 10, and to is the spike position.A typical simulation result of the impulsive noise isshown in Figure 4.

50

10

One fact worth of explaining is the mechanism forhow the continuous noise is produced. Since n, isgenerated from LLd. (independent andidentically-distributed) uniform distribution, it isactually white noise. Through repeateddownsampling and upsampling, the new noise seriesbecome the low-pass filtered version of the previousnoise series. In other words, decimation by a factorof 2 reduces one half the Nyquist frequency ofprevious noise, then B-spline interpolation restoresthe sample number of noise without incurring newfrequency contents. Thus the noise ns generated inthe last step, occupies the lowest frequency band.So the waveforms of the noise series from n\ to ns ,

become increasingly smoother. Figure 3 illustratesthe components of the Perlin noise generated in thiswork, in which (a) is the noise generated from uniformdistribution U[-O.I, 0.1], (b) - (f) are low-pass filteredversion of noise generated from one level before withp = 2. Note that the sub-figures in Figure 3 havedifferent vertical scales and low-frequencycomponents have much larger amplitudes thanhigh-frequency components. The final synthesizedPerlin noise is shown in Figure 3(g).

Figure 4: Impulsive noise simulation••~-,;,--~;o---.:.....----.:..,.-;;;\;,..........-..!"...

•..

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(a) fllC ofllad dala

F FerreUterineHuman

References

Parameter estimation for the asymmetric generalizedGaussian function is derived based on real uterinecontractions. The proposed algorithm is effectiveand produces realistic simulation results. Futurework includes parameter estimation for the Perlinnoise and impulsive noise in order to further improvethe quality of uterine contraction simulations.

Acknowledgments

This work is sponsored by Eastern Virginia MedicalSchool (EVMS). The authors appreciate the dataand support provided by EVMS.

1. M Holger, M William, S George, G Robert (2003):The physiology of uterine contractions. Clinics inPerinatology 30:665.

2. M Michelle, H Gayle, R Patricia (2007):Essentials of Fetal Monitoring, Springer, NewYork.

3. RC Young (1997): A computer model of uterinecontractions based on action potentialpropagation and intercellular calcium waves.Obstetrics and Gynecology 89:604.

4. C Vauge, B Carhonne, E Papiernik,(2000): A Mathematical Model ofDynamics and Its Application toPartruition. Acta Biotheoretica 48:95.

5. A Kemal, H Tansel, B Coskun, L Kamran (2006):Computer Simulation of Uterine ContractionDynamics. In: Proceedings of the 5th WSEASInternational Conference on Circuits, Systems,Electronics, Control & Signal Processing,WSEAS, Dallas, Texas, 1790.

6. NE Huang, Z Shen, SR Long, MC Wu, HH Shin,Q Zheng, N-C Yen, CC Tung, HH Liu (1998): TheEmpirical Mode Decomposition and HilbertSpectrum for Nonlinear and Non-stationary TimeSeries Analysis. In: The Royal Society, 903.

7. N Radhakrishnan, JD Wilson, C Lowery, HEswaran, P Murphy (2000): A NoninvasiveMethod Utilizing Higher-Order Zero Crossing forSignal Analysis. IEEE Eng. Med. BioI 19:89.

8. D Novak, A Macek-Lebar, D Rudel, T Jarm(2007): Detection of contractions during labourusing the uterine electromyogram. In: IFMBE,148.

9. J Jezewski, K Horoba, A Matonia, J Wrobel(2005): Quantitative analysis of contractionpatterns in electrical acitivity signal of pregnantuterus as alternative to mechanical approach.Phyiological Measurement 26:753.

10. Perlin Noise,http://freespace.virgin.netlhugo.elias/models/m_perlin.htm

13.51312.5

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Figure 5: Parameters estimation and uterinecontraction simulation. (a) is the scatter plot of a,and (b) is the scatter plot of fJ for left and right curves.(c) and (d) are a and fJ plots of the second data set.(e) shows the simulations of one uterine contractionfrom two data sets.

Figure 6: Simulation results. (a) The original uterinecontraction recording. (b) Simulated uterinecontraction based on (a). It can be seen that theproposed algorithm is very effective, producingrealistic simulations.

6. Conclusion

This paper proposed a set of parametric models tosimulate uterine contractions. The proposedalgorithm contains three major components: AGGFmodel for contractions, Perlin noise for maternalbreathing and instrument noise, and rationalpolynomial functions for fetal movements.

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