Mathematical modeling

Mathematical modeling of twin–twin transfusion syndrome

Twin–twin transfusion syndrome (TTTS) is a unique complication of monochorionic twin pregnancies, diagnosed by discordant amniotic fluid volume (oligo/anhydramnios–polyhydramnios sequence). Often, but not always, serious cardiovascular sequelae develop, resulting in the assessment of TTTS as having a widely variable and unpredictable clinical presentation. 

TTTS is a consequence of placental anastomoses, which can be arteriovenous from donor to recipient (AVDR), arteriovenous from recipient to donor (AVRD), arterioarterial (AA), and venovenous (VV). These anastomoses allow a net fetofetal transfusion to develop from one twin (the donor) to the other (the recipient). While about 96% of all monochorionic placentas have anastomoses, only 5–10% of them develop TTTS (see Chapter 3). 

TTTS severity has been classified.1 Stage I includes the oligo–polyhydramnios sequence without further complications. Stage II also includes lack of donor bladder filling. Stage III includes critically abnormal arterial or umbilical venous flow and/or ductus venosus patterns in either twin. Stage IV includes hydrops, and stage V intrauterine fetal demise in either or both twins. Although ultrasonography can study the fetal and placental anatomy, and Doppler sonography the blood flow of major fetal, umbilical, and placental vessels, other parameters contributing to TTTS presentation and sequelae cannot be studied directly (i.e. the fluid flows responsible for the amniotic fluid discordance, cardiovascular parameters including fetal blood pressures, and the net fetofetal transfusion). 

Because an animal model of TTTS is not available, understanding the complex pathophysiology of TTTS to its full extent is problematic. As an alternative, mathematical models of monochorionic twin pregnancies have been developed with the hope that they can aid in identifying and understanding the sequence of events that leads to the various TTTS manifestations and the potential efficacy of therapies. 

The first TTTS model was developed by the British medical physicist Dr David G Talbert.2,3 It consists of two identical pulsating fetoplacental units, comparable to 28 weeks of gestation, which are abruptly connected by AVDR, AVRD, AA anastomoses. The subsequent progression of the two fetoplacental circulations and their amniotic fluid volumes towards a new steady state is then computed. This model identified for the first time a sequence of events that related AVDR transfusion with onset of the TTTS stages I and II. 

Our group has markedly expanded upon this initial model, developing three consecutive generations of TTTS mathematical models, using nonpulsating circulations, but including fetoplacental and anastomotic growth and varying placental sharing and amnionicity.4–6 Table 6.1 summarizes the parameters used in the three models in the form of first-order differential equations. The purpose of this chapter is to present our modeling of TTTS pathophysiology in a tutorial way. We have not included simulations of TTTS therapies and refer to our publications for details.