Chapter 4 Interpreting magnetic resonance images of placental blood flow
Imaging the spatial distribution of placental perfusion is not straightforward. As such, diagnosis of abnormal placental resistance is normally inferred from 2D Doppler ultra-sonography . The gold-standards for imaging perfusion distributions use ionising radiation imaging and contrast agents (e.g. X-ray computed tomography and positron emission tomography), both of which are generally contraindicated in pregnancy . However, these issues are avoided in ultrasound and MRI. Ultrasound is able to produce measures of perfusion including vascularisation indices and flow indices , but their relationship to actual flow and tissue properties is uncertain [191, 192]. Ultrasound is limited by the penetration depth of the ultrasound beam, which makes it challenging to reliably image the placenta if it is in a posterior location, or if the transducer is obstructed by tissues (like in obesity) . The field of view is also limited, so visualisation of the entire placenta is diﬃcult as the size of the placenta increases over gestation . On the other hand, MRI of the placenta allows a greater tissue contrast than ultrasound and a wider field of view for functional imaging [193, 194].
A promising approach for analysing placental blood flow without the use of contrast agents is to employ the pulsed gradient spin echo (PGSE) sequence and the intravoxel incoherent motion (IVIM) technique. The IVIM method has been used successfully to quantify changes in vascularisation between population groups in the brain, renal tissue, and other vascularised tissues . IVIM has been applied to understand the nature of placental perfusion [196, 197], and has shown diﬀerences in the perfusion heterogeneity between normal and growth restricted pregnancies .
The IVIM method was developed based on a mathematical model of the magnetisation of tissue following the PGSE signal. The model assumes that perfusion of tissue (whatever the type of tissue) is a quasi-diﬀusion process, as illustrated in Figure 4.1a. Essentially, the assumption relies on a vascularised tissue approximating a network of randomly oriented tubes. This means that blood flow through the tubes is close to random, and so the governing equations for magnetisation in this network can be decomposed into two diﬀusion processes, that of magnetisation in stationary tissue, and that of magnetisation in the moving blood [195, 198, 199].
Unlike other tissues where the IVIM method has been applied to a micro-circulatory network embedded within the tissue (Figure 4.1a), imaging of the placenta must take into account its unique structure at the interface of two independent blood circulations (maternal and fetal, Figure 4.1b). Even if we assume that the fetal circulation in the placenta behaves in a quasi-diﬀusive manner as it resides in the complex villous tree structure, it is unclear whether the maternal circulation is also quasi-diﬀusive, and if it has the same properties as the fetal circulation. Conventionally, analysis of IVIM images is broken down to a biexponential fit to signal data under diﬀerent imaging protocols by assuming a vascularised network is embedded in stationary tissue . This fit approximates two apparent diﬀusion coeﬃcients: D, representing the diﬀusion of water molecules in stationary tissue, and D∗, representing the apparent diﬀusion of moving blood. In the placenta, it is unknown whether a single D∗ representing all moving blood (fetal and maternal) is suﬃcient, and indeed D∗ has been found to be subjected to high fitting errors in placental IVIM . Also, there are potentially some regions within the placenta where flow is non-uniform within a voxel but cannot be approximated as a quasi-diﬀusion process, for example, close to the openings of the maternal arteries. It remains unknown how PGSE signal is aﬀected by such blood pattern and if classical assumptions made in the IVIM method are applicable to such flow.
To investigate these issues, we introduce a component of fetal blood flow to the maternal blood flow model described in Chapter 3, and present a mathematical model that generates the PGSE signals that would be expected in a normal, healthy placenta. Using the predicted signals, we aim to provide an interpretative tool for understanding how IVIM parameters relate to the diﬀerent blood flow patterns in the placenta.
Before modelling the influence of placental structure on IVIM signal, a knowledge of nuclear magnetic resonance (NMR) theory is required. Here, we introduce important con-cepts, including how signals are generated and how the IVIM method was developed.
Nuclear spin and magnetic fields
Using the principles of NMR, MRI allows visualisation of internal organs by using magnetic fields to influence the spin of atoms in the tissues of the body. Spin is an intrinsic property of atoms with an odd number of protons, neutrons or both; atoms with an even number of protons, neutrons or both do not possess an intrinsic spin. In clinical imaging, signals generated from the hydrogen atoms are measured as they are the most abundant isotope in the human body, since they are found in fats and water.
Nuclear spin is quantified by the spin quantum number, I, which is a half-integer if the total number of protons and neutrons is odd, and an integer if the total number of protons and neutrons is even. I is used to calculate the magnitude of the total intrinsic angular momentum of the nucleus, I, defined by.
where h is Planck’s constant (6.63×10−34 m2kg/s). The magnetic dipole moment of the nucleus, µn, is given by 
where γ is the gyromagnetic ratio of the nucleus, which is unique for each nucleus and is calculated using where qn is the charge, mn is the mass and gf is the dimensionless magnetic moment (g-factor) of the nucleus (for hydrogen, γ = 2.68×108 rads−1T−1(42.58 MHzT−1) ).
Nuclear spins are randomly oriented in an unmagnetised state, and align like tiny compass needles when subjected to an external magnetic field, B. Locally, the magnetic dipole moment of the nucleus experiences a torque, τn, given by which aligns the magnetic moment with the magnetic field . Note that a nu-cleus needs to have spin in order to be sensitive to a magnetic field (as is applied in MRI), otherwise µn = τn = 0.
As a result of torque, the magnetic moment precesses around the axis of the applied magnetic field (or wobbles like a spinning top). This is known as the gyroscope effect and the resonant angular frequency of the nuclear spin is known as the Larmor frequency, ωL , with To obtain an NMR signal, an electromagnetic radiofrequency (RF) pulse is applied, and energy from the RF pulse is absorbed by the spins which causes them to be in phase with each other and precess at their Larmor frequency. Upon removal of the RF pulse, an NMR signal is generated as the absorbed energy is released when the excited spins which were aligned in the direction of the applied magnetic field begin to dephase (T2 relaxation) and return to their original orientation (T1 recovery).
This equation describes overall magnetisation rather than the motions of individual nuclear moments. Equation 4.6 was modified by Bloch  to factor in the eﬀects of spin excitation and relaxation on the macroscopic behaviour of M due to an RF pulse. Assuming a net magnetisation of M0 along the z-axis at equilibrium, application of the RF pulse causes the spins to precess in the transverse x–y plane. Magnetisation is hence represented by the Bloch equation as where T2 relaxation occurs in the x–y plane at a rate of 1/T2 and T1 recovery occurs along the z-axis at a rate of 1/T1.
Pulsed Gradient Spin Echo (PGSE)
PGSE is a technique used in MRI for generating images with diﬀusion weighted contrast. The PGSE sequence involves implementing a pair of π/2 rad and π rad RF pulses to create a spin echo, as shown in Figure 4.2. Gradient pulses of duration δ and magnitude G are applied along the z-axis before and after the π rad pulse. The first gradient
pulse is applied at time t1 while the second gradient pulse is implemented at time t1 + . The π rad RF pulse is applied at half of echo time (T E/2) and the PGSE signal is acquired at the end of echo time T E.
Figure 4.3 illustrates how an ensemble of spins behave in response to the diﬀerent pulses implemented during a PGSE sequence for fluid undergoing diﬀerent types of motion. At the start of the sequence, nuclear spins are aligned along the magnetic field B0 and possess a net magnetisation (M0) along the z-axis before the application of a π/2 rad RF pulse. With the introduction of a π/2 rad RF pulse, the macroscopic magnetisation is flipped from the z-axis to the x–y plane. As the π/2 rad RF pulse is switched oﬀ, the spins attempt to revert to their equilibrium state through loss of the energy gained from the RF pulse and they dephase to realign themselves with the B0 field. When a rad RF pulse is applied at time T E/2, the spins are reversed by twice the amount they have moved since t = 0. By thinking of the spins as hands of a clock, this spin reversal essentially cancels the response of the spins induced by the π/2 rad RF pulse if the spins are not influenced by any form of fluid movement. The spins revert to their original orientation by signal acquisition time T E, and there is essentially no signal attenuation (i.e. S(T E) = S(0)) (Figure 4.3a). However, if dephasing of the spins is confounded by atomic diﬀusion before the application of the π rad RF pulse, the spins are unable to revert back to their original starting positions by signal acquisition time, and the degree of dephasing varies within the material being imaged  (Figure 4.3b), resulting in a detectable spin-echo amplitude attenuation. In the presence of coherent flow without any atomic diﬀusion, outflowing spins are constantly being replaced by incoming spins. As shown in Figure 4.3c, the spins return to being in phase by signal acquisition time . However, they experience a net phase shift which is proportional to the velocity of flow (v), resulting in a signal represented by Mathematically, magnetisation throughout the PGSE sequence is described by a modified Bloch equation which accounts for how magnetisation of the spins are altered by both fluid movement and diﬀusion 
where D is the diﬀusion coeﬃcient and v is the velocity of fluid movement. The magnetic field, B, consists of the static magnetic field (B0) applied along the z-axis superimposed with a spatially varying magnetic gradient field (Gz), which is also applied along the z-axis:
where G is the magnitude of the gradient pulses applied during the PGSE sequence. As the change in magnetisation in the transverse x–y plane provides the NMR signal, we consider transverse magnetisation, m = Mx + iMy, to reduce Equation 4.9 to where ω0 = γB0. Additional detail on the derivation of Equation 4.11 from Equation 4.9 is provided in Appendix B.1.
Since m can be decomposed into a component that is unaﬀected by T2 relaxation (ψ) and a component that is aﬀected by T2 relaxation (e−iω0t−t/T2 ) :
Equation 4.13 solved over the course of the PGSE sequence generates a solution for ψ in the form of ψ = Ae−iϕ, where A is the magnitude of ψ, and ϕ is the phase of ψ (ϕ = γGzt). The total signal strength from an imaging voxel is determined by integrating the magnetisation of the population of spin phases in the voxel space:
ZS = Ae−iϕe−iω0t−t/T2 dV, (4.14)
where V is the volume of a voxel. Therefore, signal attenuation due to fluid movement and diﬀusion is obtained by normalising the signal intensity at acquisition time by the signal at the start of the PGSE sequence.
When an imaging voxel contains fluid movement by diﬀusion only (v=0), Equation 4.13 can be solved analytically. The signal at the end of the PGSE sequence is con-stant for any given diﬀusion sensitivity parameter b and the relationship between the signal amplitude S and b is [198, 199]
where D is the diﬀusion coeﬃcient of fluid molecules, and
By altering G (and so b), a range of signal magnitudes is measured and the diﬀusion coef-ficient D is estimated via an exponential fit to signal data on a voxel-by-voxel basis.
In vascularised tissue where blood flows in randomly oriented capillaries, flow is typically conceptualised as a quasi-diﬀusion process with a diﬀusion coeﬃcient D∗ and Le Bihan et al.  demonstrated that Equation 4.15 is modified to
S(b) = S(0) h(1 − f)e−bD + fe−bD∗ i , (4.17)
where f is the fraction of moving blood in an image pixel. For data acquired across a range of b values, a biexponential fit is used to determine D and D∗, with D∗ and D influencing S predominantly at low and high values of b, respectively .
Here, we solve the classical magnetisation theory (the Bloch equation, Equation 4.13) numerically under the PGSE protocol using the model of placental blood flow as described below. This allows us to investigate whether flow patterns derived from placental structures behave in a quasi-diﬀusive manner and to predict signal attenuation due to flow profiles that might be seen in the placenta.
Simulations of PGSE signal were conducted in a geometry reflective of an image ‘slice’ that sits within the placental boundary. As a basis for modelling flow profiles that are representative of placental blood flow, we constructed a 3D hexahedral domain with thickness τ and width to represent a placental subunit comprising a fetal blood circulation, and a maternal circulation which is supplied through a central SA and drained by two DVs. To gain a better understanding of how blood flow in the placenta contributes to signal attenuation, we used five diﬀerent models as shown in Figure 4.4 to examine the individual behaviour of the fetal and maternal circulations as well as their combined eﬀects on PGSE signal generation.
Model 1–Fetal blood flow in randomly oriented network of vessels
The theory developed to analyse IVIM signals assumes a quasi-random flow profile . Using a model composed of vessels that are connected to form a random path, we aim to verify our model framework. For the first vessel segment, five random numbers were generated using a random number generator (the RAND function embedded in Matlab), to define the midpoint of the vessel (its x-, y-, and z-coordinates), the angle between the longitudinal axis of the vessel and the z-axis (0 < θ < π), and the angle of rotation about the x–y plane (0 < σ < 2π). The first vessel was assigned a radius r and length l. The end coordinate of the vessel was used as a starting point for the next vessel segment, and new values for θ and σ were generated using the random number generator. The process was continued until the required number of vessel segments was attained. All vessels were assigned the same length and radius as the first vessel segment.
Model 2–Fetal blood flow in villous vessel tree structure
No attempt has been made in the literature to capture the nature of PGSE signal generated from flow in a tree-like structure. In order to verify whether flow in a villous vessel tree behaves in a quasi-diﬀusive manner, we generated a villous tree structure in 3D using the method described in Section 3.2.1. Here we assumed that the fetal vascular tree follows the same course of branching as the generated villous tree structure, and constructed a fetal arterial tree by adjusting the branch diameters of the generated villous tree structure to reflect the diameter of fetal vessels. The diameter of villus artery (dA) in the trunk of the villous tree was derived from measurements presented in the literature , while diameter of subsequent blood vessels were controlled using the Strahler diameter ratio (DSR), which was fitted to ensure that the diameter of the smallest vessel is at least the size of a capillary. DSR was also fitted to generate a total fetal vascular volume that matches the volume estimated using stereological methods [42, 205]. To obtain the total fetal vascular volume, we assumed that the fetal venous tree follows the arterial tree, with each venous vessel (dv) having twice the diameter of its associated artery . In this model, we only examined how spin magnetisation and signal generation are aﬀected by the structure of the fetal arterial tree since we do not anticipate additional insights with the inclusion of the venous tree.
List of Figures
List of Tables
List of Abbreviations
List of Symbols
1.1 Research aims
1.2 Thesis overview
1.3 Research contributions
1.4 Research outputs
2.1 Basic anatomy of the human placenta
2.2 The placental villous tree
2.3 The maternal uterine vasculature
2.4 Maternal blood flow and its interaction with villous tree structure
2.5 Oxygen and substrate exchange
2.6 Placental pathology and its relationship to placental structure
2.7 Existing computational models and knowledge gap
3 Predicting maternal blood flow in a placental subunit
3.5 Summary and conclusions
4 Interpreting magnetic resonance images of placental blood flow
4.2 Theoretical background
4.6 Summary and conclusions
5 Predicting oxygenation in a placental subunit
5.5 Summary and conclusions
6 Towards predicting placental oxygenation on an organ level
6.5 Summary and conclusions
7 Conclusions and future research directions
7.1 Thesis summary
7.2 Future work
7.3 Final word
GET THE COMPLETE PROJECT
Multiscale modelling of blood flow and transport in the human placenta