Mathematical Modeling of Rhesus Cytomegalovirus (RhCMV) Placental Transmission in Seronegative Rhesus Macaques

Approximately 1 in 200 infants is born with congenital cytomegalovirus (CMV), making it the most common congenital infection. About 1 in 5 congenitally-infected babies will suffer long-term sequelae, including sensorineural deafness, intellectual disability, and epilepsy. CMV infection is highly species-dependent, and the Rhesus CMV (RhCMV) infection of rhesus monkey fetuses is the only animal model that replicates essential features of congenital CMV infection in humans, including placental transmission, fetal disease, and fetal loss. To better understand the determinants and dynamics of congenital CMV transmission, we developed a mathematical model for placental transmission, comprising of maternal, placental, and fetal compartments using parameters from literature and experimental data from RhCMV seronegative rhesus macaques inoculated with RhCMV at 7.7-9.0 weeks of pregnancy. The model was then used to study the effect of the timing of inoculation, maternal immune suppression, and hyper-immune globulin infusion on the risk of placental transmission in the context of primary and reactivated chronic maternal CMV infection. Author summary Congenital cytomegalovirus (CMV) is the most common congenital infection in humans. Congenial CMV affects 1 in 200 infants, and can result in sensorineural deafness, intellectual disability, epilepsy, and death. The Rhesus CMV (RhCMV) model is the only animal model that replicates essential features of congenital CMV infection and fetal sequelae in humans and provides a critical experimental system to develop mechanistic insight. We propose a novel mathematical model for CMV transmission that integrates viral dynamics in the maternal, placental, and fetal compartments. We calibrate the model using data from RhCMV transmission experiments and show that the model can recapitulate experimental observations of primary versus reactivated chronic CMV infection in pregnancy, primary infection at different stages in pregnancy, and infection in the presence of varying degrees of immune suppression and hyper-immune globulin infusion. Our in-silico model provides a means to rapidly explore mechanistic hypotheses for the physical, viral, and immune determinants of CMV transmission to complement and support expensive and difficult experiments on non-human primates.

Cytomegalovirus, or CMV, is the causative agent for of the most common human 2 congenital infection and can result in multiple long-term deficits in infants, especially 3 neurological defects and hearing loss. Congenital CMV occurs in 0.7% of all 4 pregnancies [1] and is the main cause of non-genetic hearing loss, permanent sensory 5 nerve and neurocognitive impairment in infants worldwide. Primary infection with CMV 6 during pregnancy can result in transmission rates above 30%, but the transmission rate 7 of congenital CMV is less than 2% among mothers with reactivated chronic infection. 8 Species-specific CMV infections results in species-specific pathology, and only 9 non-human primate (NHP) CMV models have recapitulated the neurological deficits 10 seen in congenital human CMV infections [2]. This model has been used to demonstrate 11 the importance of maternal CD4+ T cell and pre-existing antibodies in prevention of 12 fetal transmission and disease [2,3]. The need to use experimental NHP models such as 13 the Rhesus CMV macaque (Macaca mulatta) to perform relevant proof-of-concept 14 studies of the determinants of congenital CMV transmission and disease poses a major 15 challenge to understanding and developing effective therapies to prevent this disease. 16 There have only been a limited number of previously published work on the 17 mathematical modeling of CMV infection. Kepler et al developed a system of ordinary 18 differential equations (ODE) to model CMV infection or reactivation in solid organ 19 transplant recipients who are given immunosuppressive therapy [4]. The paper focused 20 on the effect of the level of immune suppression on the viral load. Bryan et al developed 21 several deterministic model of the cytolytic immune pressure during primary CMV 22 infection to predict the dynamics of oral shedding [5], as well as a stochastic model to 23 simulate early stages CMV infections in infants. 24 However, none of these previous studies directly model the process of CMV 25 transmission from the maternal blood across the placenta to the fetus, which we believe 26 is essential for simulating therapeutic interventions to prevent congenital infection. In 27 this paper, we develop a mathematical model for CMV transmission that encompasses 28 host-virus dynamics in the mother, placenta, and fetus. This model allows us to explore 29 the interaction of immune control of virus in the mother, virus clearance in the placenta, 30 placental growth, and immune interventions (CD4+ T cell depletion or hyperimmune 31 globulin therapy) to predict the number of transmitted viruses, timing of infection, and 32 overall risk of congenital infection in the fetus. Such a model can help the future design 33 of human CMV (HCMV) studies based on small NHP intervention experiments, while 34 minimizing the use of resource intensive NHPs.

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Experimental procedure 37 The model was calibrated using data from primary RhCMV infection of seronegative 38 Rhesus macaque dams [2,3]. In these experiments, twelve pregnant RhCMV 39 seronegative rhesus macaques were assigned to an immunocompetent group (n=3) or an 40 immune-suppressed group (n=9). The immune-suppressed group was depleted of their 41 CD4+ T cell via anti-CD4 monoclonal antibody infusion prior to inoculation. Both RhCMV viral loads in maternal plasma and fetal amniotic fluid were measured 44 longitudinally using quantitative PCR. We consider transmission to occur when fetal 45 amniotic fluid was PCR-positive. Two of three of the immunocompetent group animals 46 had detectable RhCMV DNA in amniotic fluid [2]. All 9 CD4+ T cell depleted animals 47 had detectable RhCMV DNA in amniotic fluid, indicating 100% placental 48 transmission [2,3]. The experimental measurements are presented in Fig. 1 To model immune and viral dynamics in the mother, we use a model originally developed for CMV infection in kidney transplant recipients on immune suppression [4]. This model builds on the standard virus dynamics model [6], with an additional term regulating the degree of immune suppression. The model structure is shown in Figure 2, and can be written as a system of ordinary differential equations (ODE): In the ODE system (1), V denotes viral load (free virus) per µl-blood, E denotes 55 virus-specific immune effector cells per µl-blood, R I , R S , and R L denotes actively 56 infected, susceptible cells, and latently infected cells per µl-blood respectively. A 57 description of all model parameters is given in Table. 1.  Placental compartment 59 We model the transmission of CMV through the placenta as a diffusion process with 60 decay. While physical diffusion is an unrealistic physical model for CMV in the placenta, 61 we treat this as a phenomenological representation intended to capture the coarse grain 62 dynamics of placental transmission. Similar PDE models have been shown to accurately 63 capture the dynamics of HIV passage across the urogenital mucus epithelium [7]. For 64 April 5, 2022 4/18 simplicity, we assume homogeneous dynamics in the radial directions and model 65 diffusion through the placental depth-wise as a 1D diffusion process, from maternal 66 blood lake to the fetal circulation. This gives: with boundary conditions: and initial condition: Here, Q denotes the concentration of CMV. Constant D denotes the diffusion into the fetus, it is carried away rapidly by the fetal blood flow.

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With simplified versions of these boundary conditions (e.g., maternal viral 76 concentration is constant or can be approximated by a simple polynomial, (2) -(5) can 77 be solved analytically using Fourier series to obtain a closed form solution (See S1 File 78 in Supplementary Materials). This analytic solution is used to verify the correctness of 79 numerical simulations needed for more complex maternal viral load dynamics.

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The number of viruses that enter the infant, denoted by C,is estimated by: where G(t) refers to the growth curve of placenta throughout pregnancy. Here we 82 assume that G(t) has the form of a logistic curve in the range where we have data 83 points, and has a linear expression that connects to the origin: By treating the viral flux as the rate for an inhomogeneous Poisson point process, we can obtain the time that 86 each virus arrives in the fetal compartment.

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Fetal compartment 88 We model the arrival and subsequent fate of each CMV virion that arrives in the fetus 89 using a stochastic process to accurately reflect the dynamics of small populations. Since 90 we only model the early infection dynamics in the fetus, we assume that the number of 91 susceptible cells is constant. Once the virus enters a susceptible cell, we consider two 92 stochastic models for the subsequent dynamics, with and without latent viral infection 93 stages, respectively: Growth function G(t) for placenta surface area over time. Using data from [8], the area of the placenta of fetal rhesus macaques, from the fortieth day of gestation to term were plotted and fitted using a logistic curve. Note that the dashed line is a linear interpolation between the predicted value at the first available data point and the origin since a logistic curve cannot pass the origin.  The maternal dynamics comes from the population fits of immunocompetent and CD4+ T cell depleted groups of rhesus macaques. We have used different transplacental scales for the y-axis to show the daily viral counts clearly -note that the difference of CMV viral load between immunocompetent and CD4+ T cell depleted dam groups is more than one order of magnitude.
Model 1: Susceptible cell with latent stage: Model 1 corresponds to a target-cell limitation model given in [5], which has a 95 reproductive number R 0 given by: Model 2: Susceptible cell without latent stage: Model 2 corresponds to a target infectiion virus model given in [9], which has a 97 reproductive number R 0 given by: For each virion, we choose between Model 1 and 2 as follows: Assuming that the ratio 99 between the susceptible cells that go through latent stage and the susceptible cells that 100 become infected cells is r : (1 − r), we draw a uniform random variable s between 0 and 101 1, and if s < r, we run model 1 given by Equation (9), else if s > r, we run model 2 102 given by (11) (In our sample runs, we use r = 0.5). Since we assume that the number of 103 susceptible cells remains constant over the period of interest, we do not allow the 104 production and death of susceptible cells, i.e., S = S 0 and then βS = βS 0 is a constant 105 0.0012 (virions · day) −1 .

Model calibration 107
Maternal compartment 108 We used the parameters for human CMV infection given in Kepler et al [4] as initial (SAEM) to perform parameter estimation [11], and can accommodate the use of viral 117 loads below the limit of detection (left censoring). Population parameter estimates for 118 both the control and immune suppressed grouped described by ODE system Equation 119 (1) are summarized in Table. 1.

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Placental compartment 121 We first estimate the growth curve for the placenta given by Equation (7). Using the 122 data from [8] for pregnant rhesus macaques, we fitted a logistic function to obtain Fig. 123 3 with a = 132434, b = 0.028, c = 90 g = 666.93.

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For boundary conditions on the maternal-placental interface, we need to estimate a 125 continuous function from discrete experimental measures of viral load over time. Where 126 there are sufficient data points, we fit the experimental measures directly using 127 Piecewise Cubic Hermite Interpolating Polynomial (PCHIP) interpolation [12]. Unlike 128 the more commonly used cubic spline interpolation, PCHIP avoids overshoots and can 129 accurately connect the flat regions without creating artificial oscillations. For rhesus 130 macaques with insufficient data points, we use the ODE model proposed by [4] to 131 obtain a smooth trajectory for each animal, as well as population fits for 132 immunocompetent and CD4+ T cell depleted dam groups using a nonlinear mixed 133 effect model. These fits are shown in Fig. 4.

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For the PDE Equation (2), we used the diffusion coefficient of HIV through stroma as our D [7]. We observe from the experimental data that all fetal infections occurred within 2-4 weeks after the mothers are infected with RhCMV. Using this, we can approximate the distance l that the virus needs to travel. The diffusion coefficient determines the time it takes a solute to diffuse a given distance in a medium: Then we have: We can also calibrate µ in (2) using the experimental data so that the probability of 135 transplacental RhCMV transmission for immunocompetent rhesus macaques using 136 stochastic simulation is between 30% to 40%, and we obtain µ = 1 (day) −1 . Equations (9) and (11), all parameters are from [5] and [13]. These parameters are 140 shown in Table 2.

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With the arrival time of each virus, we run the Gillespie algorithm [14] to simulate 142 viral dynamics in the fetus, and define a fetus as infected once the viral load exceeds a 143 prespecified threshold. Repeated runs of simulations then yield a distribution for the  [2]. At this time, fetal weight is measured to be around 155 0.14 kg [8]. Using the blood volume of rhesus macaques at 60 mL/kg [15], we can 156 calculate the fetus has ≈ 8.4mL blood in total. Since the limit of detection of CMV is 157 100 copies/ml [2], we calculated the threshold M , total number of viruses in fetus, to be 158 around 800 copies.

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While the threshold is arbitrary, our simulations show that once this threshold is 160 crossed, spontaneous clearance of virus is extremely unlikely. We first estimate the 161 probability p that a single replicating virus that passes through the placenta will lead to 162 sustained infection. In fact, theoretically, we can obtain the extinction probability when 163 we start with one virus using a continuous time branching process. This means the 164 probability of any number of viruses surviving is d(R0−1) ω with target infection virus 165 model derived in [16] and with R 0 given by Equation (12). However, we consider there 166 is sustained infection only if the number of viruses is above a threshold. Therefore, we 167 run stochastic simulations to directly estimate the value of p. Using stochastic 168 simulation is more realistic since we can estimate the probability of viruses count 169 reaching a specified threshold value M , as well as incorporate the switching between 170 models with (Equations (9)) and without (Equations (11)  Then, if we consider that every virus can cause persistent infection independently, we 176 can estimate on the probability of infection if N viruses travel through the placenta as: 177 As a result, we can obtain a cumulative probability over time since we can calculate 178 the number of viruses using (8): higher probability of trans-placental transmission as shown in Fig. 9 and Fig. 10. While 213 we cannot rule out that changes in the immune milieu also play a role [19], the model 214 provides a parsimonious explanation and a major contribution by placental growth that 215 is obvious in hindsight. The red solid line refers to CD4+ T cell depleted mother that is infected during 1st trimester of pregnancy.