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Year : 2012  |  Volume : 27  |  Issue : 2  |  Page : 101-104  

Updated anatomical data and mathematical models for embryo/fetus dosimetry

Department of Health Research, National Institute of Medical Statistics (ICMR), Ansari Nagar, New Delhi, India

Date of Web Publication18-Apr-2013

Correspondence Address:
Suresh Mehta
National Institute of Medical Statistics (ICMR), Ansari Nagar, New Delhi - 110 029
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Source of Support: None, Conflict of Interest: None

DOI: 10.4103/0972-3919.110693

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Purpose of the Study: It is proposed to fill in the gaps in the existing data matrix of mass/volume of uterus, its contents as well as mass of fetal organs by mathematical techniques down to 6 week gestation and relate this dynamic target mass during in-utero growth to recently revised Medical Internal Radiation Dose (MIRD) 21 schema. Materials and Methods: The existing data is subjected to numerical interpolations using a standard 4 degree polynomial for certain set of variables. Interpolations of mass, volume, etc., of various components of the uterus (placenta, embryo/fetus, brain, uterine wall, etc.) at weekly/biweekly intervals have been carried out. Subsequently, the step wise regression starting with three predictors - placental mass (W p ), total fetal mass (W f ) and greatest length (H) for the augmented data set led to identification of "H" and "W f" as the most significant predictors for 10 fetal organ masses W i using standard software "MS Excel." Results and Discussion: Further analysis utilizing allometric equations reveal that there is strong evidence in favor of W f compared to H for predicting (P < 0.001) the individual organ mass "W i". The prediction of W i -liver, heart, thymus, pancreas, and thyroid fall under the linear case of prediction (predictor is ln [W f ]); whereas the brain, lung, kidney, spleen, crown-heel length, etc., fall under linear-quadratic case (where ln (W f ) plus [ln (W f )] 2 ) are the predictors) respectively. The estimates indicate a rapid decline of "brain mass/total mass" ratio from 80% to 39% during 7-9 weeks. Information on specific absorbed fraction Φ (=φ/m T ) is required to arrive at the dose estimates (φ being the absorbed fraction). The very small target mass m T -few milligrams (for 90% of organs) to a maximum 11 g for brain during early pregnancy; the fetal thyroid, with its mass variation of about 300% during 10-13 weeks can impact Φ. Reported standardized doses are presented and variation of Φ with source-target distance for individual specific scaling of Φ is discussed. Conclusion: Time dependent mass m (t) of the target and consequently Φ(t) [=φ(t)/m T (t)] of the revised MIRD dose expression can be of relevance in fetal dosimetry when source-target distances are in reasonable limits.

Keywords: Fetus organs, interpolation, mathematical models, regression

How to cite this article:
Mehta S. Updated anatomical data and mathematical models for embryo/fetus dosimetry. Indian J Nucl Med 2012;27:101-4

How to cite this URL:
Mehta S. Updated anatomical data and mathematical models for embryo/fetus dosimetry. Indian J Nucl Med [serial online] 2012 [cited 2022 Jan 21];27:101-4. Available from:

   Introduction Top

In-utero growth is characterized by multiple parameters including anatomical as well as biological which serves as input information for embryo/fetus dosimetry. In the absence of one single data source or standard curve for reference, "pooling of data" from various sources is necessary to cover full range of pregnancy. This also allows to approximate the growth curve to some mathematical function like a polynomial which is rather simple and crude whereas more sophisticated ones are the "Gompertz" or "Logistic" function. [1] Simple statistical models for prediction utilizing few reliable predictors is another approach for interpolating or extrapolating data. [2]

The basic anatomical data for use in radiological protection were recently revised and extended by ICRP 89. New compilations of up-to-date and reliable information of human embryo/fetus have become available. [3],[4] Utilizing these data, the mathematical models of various components of the uterus, e.g., embryo, fetus, placenta, uterine wall, etc., at different prenatal ages have also been reported for applications in dosimetry. [5]

The body mass index (BMI), surface area (SA)-are two useful related parameters of general use in biomedical work, have also been computed in present work. SA estimates of the placenta up to about 34 weeks of gestation are also useful in internal dosimetry. [6] The BMI-1 (wt/ht 2.88 ) and BMI-2 (wt/ht 3 ) are recommended for smaller and growing bodies. [7],[8]


Embro/fetus dosimetry poses quite a formidable challenge to deal with. The complexity arises due to peculiar scenario of in-utero growth vis-a-vis dosimetric constraints. For example, there can be several and changing sources (e.g., placenta/uterus); changing targets (e.g., fetus and its organs); formation, maturation/functionality of different organs and tissues at different stages of gestation; evolution of placenta over time and its transport quality etc., The distance between maternal source and fetal targets are constantly varying so are the sizes and shapes of fetal organs. The placenta, which plays a dominant role in transferring essential elements to enable growth is itself varying over time both in terms of size and its transport mechanism. Although, the mechanism of transport across para placenta for growth is fairly known, the bio-kinetic details for computing accumulated activity due to radionuclides of interest in pregnant female are severely limited. The detailed anatomical and biological description of the placenta and its role in fetal dosimetry can be found elsewhere. [6] A brief overview is presented for some simple mathematical models adopted in the past and some more realistic but complex ones based on refined reference standards. MIRD pamphlet 21 has recently revised its generalized schema of average absorbed dose expression to include time variable in SAF in recognition of the dynamic target mass in many dosimetry situations like in the case of fetal growth and/or shrinking tumors during radionuclide therapy. [9],[10]

Anatomical models of human pregnancy and uterus

Simple models

For the purpose of dosimetry, in utero growth could be classified into the following four broad stages with notable growth feature in brackets namely-(Ovum: 1 st week after conception [solid trophoblast]); (Embryo: 7-56d [~few mg to g]-a period of major organogenesis for more than 90% organs and tissues, max. chorion diameter = 65 mm); (Fetus/later fetus: 56d - term [few g-3500 g]-period of growth and maturation, brain being highly radiosensitive, 56d-105d is the window of Cortical sensitivity). The developing embryo and fetus are radiosensitive during whole of the prenatal period though the effect decreases with age. In particular, the brain shows pronounced sensitivity to long-term damage due to prolonged cell formation and maturation process. [11]

As mentioned earlier, during pregnancy, the sizes and geometric shape of the target fetus in relation to the maternal sources is complex and differ greatly from the small to the term fetus. Simple geometric shapes are preferred to more complex ones in model development because it is easy to use in Monte Carlo simulations. Smith and Warner have modeled a 10-41 d conceptus as a sphere of radius 0.13 cm and mass 9.2 mg. The sphere representing embryo has been assumed located at the origin of semi-axes of a 66 cm 3 ellipsoid uterus in 70 kg heterogeneous adult phantom. [12] Cloutier, et al., have modeled a 3 month pregnant of 58 kg in which the uterus, the embryo, and the other contents of the uterus are non-distinguishable as a homogeneous mixture of soft-tissue. [13] The models for maternal whole body (40-70 kg), fetus (1-4 kg) and its thyroid (1-3 g) were assumed as ellipsoid with axes ratio 1:1.8:9.27; 1:0.667:1.337; and 1:2:4 respectively by Sastry, et al., a 30 week uterus was modeled as a sphere of 20 cm diameter; placenta as a truncated spherical shell (1.7 cm thick, 9 cm radius, corresponding to mass 430 g) with both ends subtending at an angle 60° at the center. [14] Elsasser, et al., have modeled the embryo/fetus of mass ranging from few mg to 100 g sphere at the center of the uterus or at different points along the semi-major axis of the uterus in a 15-year-old anthropometric child phantom of Henrichs and Kaul. [15] Simple approximations have been replaced by more realistic but complex ones using updated data on reference masses of organs and tissues published in ICRP 89 report.

Refined models

In both, non-pregnant as well as pregnant woman, the uterus is a pear shaped structure about 7.5 cm in length initially. At the end of pregnancy, the size of uterus and its contents increases, the position changes but the shape remains the same. The model of uterus has been approximated by frustum of right circular cone capped at both ends by hemispheres, similar to the model of uterus at the end of third trimester developed by Stabin, et al. [16] While this effort was directed to develop three different geometrical models of uterus at age 13.26 and 38 weeks; Chen unified these three into one which assumes a single shape but has different size and location at these stages of pregnancy. [5] Stabin's model of placenta has now been extended to trimester 1 also. For fetus and its organs, more than one simple geometric shape in combination were utilized to develop symmetric models for brain and whole body. The symmetry is chosen to account for constant movement of the fetus within the uterus. For example both an embryo stage brain and whole body of term fetus are approximated by right circular cylinder capped by hemispheres. Another minor but important change in earlier 3 month model of uterine contents was to distinguish fetus as separate entity from other contents.

Placenta increases in size but remains in same shape represented by hemispherical shell quite similar to the one used by Sastry, et al. The 3-dimensional mathematical equations representing the geometrical models of embryo, total fetus and the associated relevant organs-placenta, uterus, uterine walls at 8W (early fetus), 13W (trimester 1), 26W (trimester 2), 38W (term) representing four stages of pregnancy respectively have been reported. [5]

   Materials and Methods Top

Interpolations on ICRP data on mass, volume and densities of components of the uterus (e.g., placenta, embryo/fetus, brain, uterine wall, etc.,) at each week of gestational age have been carried out by a numerical recipe-a FORTRAN subroutine "POLINT". [17] A polynomial of degree 3 or 4 was applied. Incomplete data matrix was treated with either "pooling" of data or "smoothing" in order to generate complete sets of data for prediction of each fetal organ. As the pooled data came from diverse sources as well as collected over many years, it suffers from both - the unknown measurement errors and the model error estimation. Regression analysis was done in two steps with the help of Microsoft package, "LINEST". First, was to identify three variables W f , H, and W p generally believed to be important predictors, with a step down approach, fetal body mass was identified as statistically most significant predictor. The crown-rump length is often been used as substitute for time variable in human prenatal development as it is difficult to identify the day of fertilization. Where possible, the prediction data is extended to 6 weeks gestation. In the second step, mass "W f" in its log transformation was used as predictor. Where possible, prediction was done up to 6 weeks gestational age. The two models fitted, popularly known as allometric equations are explained below.

The biology of scaling is a well-known method of modeling relationship of individual organ/tissue mass or size as a function of total body mass. The mathematical model is given by equations

Where, W i is the measured weight of the i th organ; W f is the measured weight of the embro/fetus and a i , b i , c i parameters that are determined empirically by regression over measurements. Both equations are treated with "LINEST" as above.

   Results and Discussion Top

In [Table 1], it is seen that the "mass to volume" ratio for fetus at week 8, which is < 1 is reversed to > 1 at week 38, indicating an accelerated growth velocity of mass compared to volume. This can be largely due to rapid fat accumulation in the fetus from week 16 onwards up to 36 weeks. Similar variable growth rates are seen in [Table 2] for fetal organs like brain, lungs, kidney and spleen due to additional coefficient of {ln (W f )} 2 . The estimates of thyroid mass range from 0.021 to 0.85 g during 10-14 weeks showing a significant 40 times increase. Similar increase or more can be seen for most (~90%) of the organs with the exception of brain mass (~80% of 4.7 g fetus at week 8; 39% at week 9). So at 3 months gestation, the mass of the fetus excluding brain comes to about 75 g which is distributed among 90% of organs and other tissues; fetal thyroid only about 0.048 g. The time of radioiodine concentration in thyroid is 10-13 weeks when it can cross placenta. With changing mass, the source-target distance can also change especially, in the case of a large organ like liver. This can impact on the value of SAF and the resultant doses. A mean value would be appropriate for distance.

[Table 3] shows standardized doses to the fetal thyroid per unit activity administered to the mother. Values are shown for four radionuclides from 3 m to 9 m gestation. [18] In another study, Millard et al., used MIRDOSE3 software for computing doses to fetal thyroid as well as whole body. The doses to the majority of fetal organs is 10 times higher, for fetal thyroid up to 100 times when compared to the corresponding organs of the mother. [19]
Table 1: Numerically generated interpolated and computed values of the relevant quantities for dosimetry

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Table 2: Coefficients of prediction equation 1 and equation 2

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Table 3: Doses to the fetal thyroid per unit activity administered to the mother mGy/MBq*

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To arrive at the individual-specific doses, simple linear mass scaling is employed on standard dose factor in case of alpha and beta radiations because absorbed fraction is 1.0. For photons, a simple relation Φ2 = Φ1 (m 1 /m 2 ) 0.667 , in case of self-absorption but its use is limited to few regions and radionuclides. For internal electron emitters, new set of Φ values for electrons have been generated as function of mean distance by fitting a power curve for a range of energies. However, the Φ value may increase for larger (such as liver) and for distributed (such as muscle) source organs. However, for organs that are very close to the fetus (such as uterus and uterine contents) and organs whose distances to the fetus are "wide spread" (e.g., muscles, skin and skeleton), the power fit will underestimate the Φ value to a large extent. [20]

   Conclusion Top

The radiation risks are known to depend upon time of exposure and the absorbed radiation dose during different stages of in-utero growth. The time dependent mass "m T (t)" and consequently "Φ(t)" of revised MIRD 21 can be useful in fetal dosimetry.

   References Top

1.Naito K, Udagawa J, Otani H. Multidimensional standard curve for the development process of human fetuses. Stat Med 2010;29:2235-45.  Back to cited text no. 1
2.Luecke RH, Wosilait WD, Young JF. Mathematical representation of organ growth in the human embryo/fetus. Int J Biomed Comput 1995;39:337-47.  Back to cited text no. 2
3.Basic anatomical and physiological data for use in radiological protection: Reference values. A report of age- and gender-related differences in the anatomical and physiological characteristics of reference individuals. ICRP Publication 89. Ann ICRP 2002;32:5-265.  Back to cited text no. 3
4.O'Rahilly R, Muller F. Human Embryology and Teratology. New York: Wiley-Liss; 2001.  Back to cited text no. 4
5.Chen J. Mathematical models of the embryo and fetus for use in radiological protection. Health Phys 2004;86:285-95.  Back to cited text no. 5
6.Reddy AR, Jain SC. Approaches to assessing doses to the embryo and fetus. Radiat Prot Dosimetry 1998;79:289.  Back to cited text no. 6
7.Bailey BJ, Briars GL. Estimating the surface area of the human body. Stat Med 1996;15:1325-32.  Back to cited text no. 7
8.Mordenti J. Man versus beast: Pharmacokinetic scaling in mammals. J Pharm Sci 1986;75:1028-40.  Back to cited text no. 8
9.Goddu SM, Howell RW, Rao DV. Generalized approach to absorbed dose calculations for dynamic tumor and organ masses. J Nucl Med 1995;36:1923-7.  Back to cited text no. 9
10.Bolch WE, Eckerman KF, Sgouros G, Thomas SR. MIRD pamphlet No. 21: A generalized schema for radiopharmaceutical dosimetry - Standardization of nomenclature. J Nucl Med 2009;50:477-84.  Back to cited text no. 10
11.International Commission on Radiological Protection: Biological Effects after Prenatal Irradiation (Embryo and Fetus). Oxford: ICRP Publication 90, Pergamon Press; 2003.  Back to cited text no. 11
12.Smith EM, Warner GG. Estimates of radiation dose to the embryo from nuclear medicine procedures. J Nucl Med 1976;17:836-9.  Back to cited text no. 12
13.Cloutier RT, Watson EE, Snyder WS. Dose to the fetus during the first three months from gamma sources in maternal organs. In: Cloutier RT, Coffey TL, Snyder WS, Watson EE, editors. Radiopharmaceutical Dosimetry Symposium. Rockville: BRH; 1976. p. 370-5.  Back to cited text no. 13
14.Sastry KG, Reddy AR, Nagaratnam A. Dosimetry in radioisotope placentography. Indian J Med Res 1976;64:1527-36.  Back to cited text no. 14
15.Elsasser U, Henrichs K, Kaul A, Reddy AR, Roedler HD. Specific absorbed fraction and S-factors for calculating absorbed dose to embryo/fetus. In: Stelson AT, Watson EE, editors. Fourth International Radiopharmaceutical Dosimetry Symposium. Springfield NTIS conf-85113; 1986. p. 155-6.  Back to cited text no. 15
16.Stabin MG, Watson EE, Cristy M, Ryman JL, Eckerman KF, Davis JL, et al. Mathematical models and specific absorbed fraction of photon energy in the non-pregnant adult female and at the end of each trimester of Pregnancy. ORNL/TM-12907. Tennesse: Oak Ridge National Lab, Oak Ridge; 1995.  Back to cited text no. 16
17.Press WH, Flannery BP, Teukolsky SA, Vetterling WK. Numerical Recipes Art of Scientific Computing-Fortran Version. Cambridge, England: Cambridge University Press; 1990.  Back to cited text no. 17
18.Watson E. Radiation absorbed doses to the human fetal thyroid. In: 5 th Int Radiopharmaceutical Dosimetry Symposium. Tennesse: Oak Ridge; 1992.  Back to cited text no. 18
19.Millard RK, Saunders M, Palmer AM, Preece AW. Approximate distribution of dose among foetal organs for radioiodine uptake via placenta transfer. Phys Med Biol 2001;46:2773-83.  Back to cited text no. 19
20.Guo B, Xu XG, Shi C. Specific absorbed fractions for internal electron emitters derived for a set of anatomically realistic reference pregnant female models. Radiat Prot Dosimetry 2010;138:20-8.  Back to cited text no. 20


  [Table 1], [Table 2], [Table 3]


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