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Am J Physiol Regul Integr Comp Physiol 292: R1519-R1531, 2007. First published December 7, 2006; doi:10.1152/ajpregu.00534.2006
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NEUROHUMORAL CONTROL OF CARDIOVASCULAR FUNCTION

A mathematical model of twin-twin transfusion syndrome with pulsatile arterial circulations

¹Laser Center and Department of Obstetrics and Gynecology, and ²BMEYE Cardiovascular Monitoring Company, Academic Medical Center-University of Amsterdam, The Netherlands; and ³Department of Obstetrics and Gynecology, Harbor-University of California, Los Angeles Medical Center, Torrance, California

Submitted 25 July 2006 ; accepted in final form 9 November 2006

	ABSTRACT

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The twin-twin transfusion syndrome (TTTS) is a severe complication of monochorionic twin pregnancies caused by a net transfusion of blood from one twin (the donor) to the other (the recipient) through placental anastomoses. To examine the pathophysiology of TTTS evolving through clinical stages I to IV, we extended our mathematical model to include pulsating circulations propagating along the arterial tree as well as placental and cerebral vascular resistances, and arterial wall thickness and stiffness. The model demonstrates that abnormal umbilical arterial flow (TTTS stage III) in the donor twin results from increased placental resistance as well as reduced resistance in the cerebral arteries. In contrast, recipient twin abnormal umbilical arterial flow requires a significantly greater increase in placental resistance, resulting from the compressive effects of high amniotic fluid pressure. Thus simulated abnormalities of donor umbilical arterial pulsations occur in the donor more commonly and earlier than in the recipient. The "normal" staging sequence (I, II, III, IV) correlates with the presence of compensating placental anastomoses, constituting the majority of monochorionic twin placentas. However, TTTS stage III may occur before manifestations of stage II (lack of donor bladder filling), in our model correlating with severe TTTS from a single arteriovenous anastomosis, an infrequent occurring placental angioarchitecture. In conclusion, this mathematical model describes the onset and development of the four stages of TTTS, reproduces a variety of clinical manifestations, and may contribute to identifying the underlying pathophysiology of the staging sequence in TTTS.

stage III; quintero staging sequence

TTTS is believed to progress through four worsening stages of severity with both twins alive (36). Stage I TTTS is diagnosed by the oligo-polyhydramnios sequence, where the donor has almost no amniotic fluid and the recipient simultaneously has excess amniotic fluid. More severe stages of TTTS often develop subsequently: stage II includes lack of donor bladder filling; stage III, abnormal umbilical flows in either twin (i.e., reduced or reversed end-diastolic flow) (21); and stage IV, congestive cardiac failure and hydrops in the recipient.

The syndrome has a largely inaccessible pathophysiology, because invasive fetal study is unethical and an animal model is lacking. Accordingly, mathematical models of TTTS have been developed to aid in identifying the pathophysiology of the various TTTS presentations (42, 43, 46, 50) and the response to therapy (44, 47). However, none of these models described TTTS stage III, probably the most complex of the four stages, because the abnormal umbilical flows can occur in either or both of the twins. Therefore, stage III pathophysiology likely comprises a variation of different mechanisms that may develop independently in each of the twins (4, 23, 26, 33, 37). We hypothesized that multiple cardiovascular abnormalities [e.g., in blood viscosity, arterial pressure, arterial wall stiffness (10, 18), redistribution of arterial blood flow to the brain, vasoconstrictive peptides, and placental resistance] contribute to the onset of stage III TTTS.

We recently modeled (48) pulsating arterial flow of the fetoplacental circulation, using viscoelastic tube segments and three-element windkessels to simulate the arteries and fetal organs, respectively. In the present study, we sought to extend the TTTS model, including our model of pulsating arterial circulations and multiple cardiovascular parameters, to simulate the onset and development of abnormal umbilical arterial pulsations (stage III) and to examine the progression of TTTS from stage 1 to stage IV in relation to the placental angioarchitecture.

	METHODS

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The present model is based on combining two previous mathematical models. First, our pulse arterial propagation model (48), and second, our TTTS model with nonpulsating circulations (46). The resulting new model incorporates 13 differential equations for each twin, coupled by the net fetofetal transfusion of blood volume and blood constituents, i.e., colloids, osmoles, and vasoconstrictive peptides. These 26 equations describe the development of and interactions occurring between the volumes of fetal arterial and venous blood, interstitial fluid, intracellular fluid and amniotic fluid, the colloid osmotic pressures of fetal blood and interstitial fluid, the osmolalities of fetal blood and amniotic fluid, the concentration of vasoconstrictive peptides in the fetal blood, generically referred to as renin-angiotensin-system (RAS) mediators, the blood hematocrit, and the elastin content and thickness of the arterial walls. Output parameters of the TTTS model, i.e., cardiac output, arterial wall thicknesses and Young's moduli (stiffness related to the elastin content), placental and cerebral vascular resistances, and blood viscosities related to blood hematocrits, at a certain gestational age, are used as input for the pulse propagation model to compute the donor and recipient pulsating flow waves in their respective arterial trees at that gestational age. An overview of both models is schematically given in Fig. 1. Possible differences in diameter and length of arterial segments as a consequence of discordant developing donor and recipient twins were neglected because differences in blood volume as large as a factor of two, for example, would cause a much smaller difference of a factor of 2^1/3 1.26 in diameters and lengths.

View larger version (13K): [in this window] [in a new window]   Fig. 1. Schematic representation of the 2 models and their input and output parameters. For the twin-twin transfusion syndrome (TTTS) model, types and radii of anastomoses, degree of placental sharing and amnionicity, and initial values of all parameters are entered so that all parameters can be computed as a function of gestation . For the pulsatile model, the hematocrit, Young's modulus, vessel wall thickness, and placenta and brain resistances are used at one specific gestational age to obtain the flow pulsations in the arterial tree at that gestational age. D, donor; R, recipient.

Analysis showed placental and cerebral resistances to have the highest influence on the pulse waveforms along the fetal arterial tree. In addition, blood viscosity and vessel wall stiffness had a much smaller influence, whereas organ, i.e., windkessel, compliance had negligible influence. The model of pulsatile flow and pressure is written in Mathematica 4.0 (Wolfram Research, Champaign, IL) and uses 25 harmonics, multiples of the normal fetal heart rate, to compute the frequency spectrum of the input flow waveform. One computer run took 2.5 min on a Pentium 4.8 GHz personal computer. Details of the model can be found elsewhere (48).

Blood hematocrit dynamics. The normal fetal blood hematocrit, Hct_N(t), as a percentage at gestational age t (weeks), is given by (53)

(1)

and ranges from 30% at 10 wk to 42% at 40 wk. Because it is the ratio of the volume of red blood cells, V_RBC, to total blood volume, V_b, the normal volume of red blood cells can be written as

(2)

When fetal hematocrit is lower than normal, e.g., in anemia, the production of red blood cells is upregulated by erythropoietin from the fetal kidneys (35). The maximal increase of red blood cell production in acutely hemorrhaged fetal lambs was two- to threefold (19, 41). In addition, a similar upregulation follows from the results of reticulocyte count (19). In a recent study in human fetuses with the twin anemia polycythemia sequence (25), the donor had a 2.2–2.7 times larger absolute reticulocyte count than the recipient. Because the exact amount of upregulation due to chronic blood loss is unknown, the maximal upregulation was chosen as 1.5. We further assumed that this upregulation has a time constant of ₁ = 3 wk after onset of a lower hematocrit than normal so that the upregulation is 1.25 after 2 wk vs. 1.4 after 7 wk. Using t (weeks) as the time period between the gestational age considered and that where the hematocrit became reduced, the dynamics of upregulation for donor and recipient have been described as

(3)

The time derivative of V_RBC,N follows directly from Eq. 2; and D and R represent donor and recipient, respectively. For clarity, the time dependence in the parameters was omitted.

In case of an excess hematocrit at gestational age t₀, defined as

(4)

the excess red blood cells are removed from the circulation by an exponential function with a time constant of ₂ = 7.2 wk, corresponding to the actual measured maximal life span of 10 wk (8) and hence a half-life time of 5 wk. As shown previously (46), this can be expressed as

(5)

The differential equation of donor and recipient hematocrit can now be written as

(6)

with X = D, R. The minus sign refers to the donor, the plus sign to the recipient. For clarity, the time dependence in all parameters was omitted. The first right-hand side term includes the red blood cell volumetric growth, i.e., Eq. 3; the second contains the fetofetal transfusion through the primary AV and the opposite transfusion through the VA, AA, and VV; and the last term describes the decay of the excess red blood cells. In an anemic donor twin, an excess hematocrit does not occur, hence ExHct_D = 0. As part of a sensitivity analysis, the 1.5-fold upregulation in Eq. 3 was varied from zero to five, described in the DISCUSSION below.

Resistance to mean blood flow of the normal fetus and placenta. The normal total resistance to mean blood flow as experienced by the fetal heart is defined as (46)

(7)

where P_art,N and P_ven,N are the normal mean arterial and venous pressures (mmHg), and CO_N (ml/wk) is the normal combined cardiac output (31). Under normal conditions, we assumed that 70% of the cardiac output perfuses the fetal body, of which 25% is directed to the fetal brain, 45% to the rest of the body (for convenience, called periphery), and 30% to the placenta (48); see Fig. 2. We considered the normal total resistance to consist of the parallel circuit of the normal resistances of the fetal brain, the periphery, and the placenta. Thus

(8a)

(8b)

The normal placental resistance is the sum of the precapillary and postcapillary resistances, R_{PlacentaPre,N} and R_{PlacentaPost,N}. Thus, using fetoplacental capillary blood pressure, P_cap,N, it follows that

(9a)

(9b)

(9c)

View larger version (15K): [in this window] [in a new window]   Fig. 2. Normal distribution of blood flow in the fetus and placenta in our nonpulsatile model. The brain, periphery, and placenta resistances are in parallel (see Eq. 9). Percentages indicate the fraction of cardiac output through these segments. The placenta consists of the pre- and postcapillary parts, with Pcap defined as the fetoplacental capillary pressure between these parts. Pven, mean venous pressure; Part, mean arterial pressure.

(10)

The placental resistance has been expressed as shown below, including the influence of RAS, expressed by factor f_RAS, the viscosity of the fetal blood, expressed by factor f_Viscosity, and the amniotic fluid pressure, expressed by factor f_AFP.

(11)

We modeled an increased placental resistance in response to increased blood viscosity. In contrast however, we assumed that a reduced hematocrit, as in the TTTS donor, has no influence on placental resistance, hence, f_Viscosity,D = 1. This was chosen because in donor twins, the placental resistance is not expected to decrease as strongly as that based on the decreased viscosity only. This is a consequence of donor twins having reduced placental vascularization (52) and delayed villous maturation (34, 52), both of which have an opposite, i.e., increasing, effect on the placental resistance, counteracting the decreasing effect of a low viscosity.

Placental resistance for donor and recipient depending on their RAS concentrations. In analogy to the work of Faber and Anderson (Ref. 15; their Eq. 4), we chose arteriolar precotyledonary resistance to increase asymptotically to nine times the normal resistance, which agrees with the range of values obtained by the measurements of Adamson et al. (1).

(12)

Placental resistance for donor and recipient depending on the amniotic fluid hydrostatic pressure. Polyhydramnios increases the intra-amniotic fluid pressure that compresses the placenta (placental squeezing) (9, 16, 39). The equal amniotic fluid pressure in donor and recipient sacs was defined as (43)

(13)

where normal amniotic fluid pressure is 10 mmHg, and AF_N is the normal amniotic fluid volume of a singleton fetus.

To develop the relation between increased amniotic fluid volume/pressure and placental resistance, we used clinical data obtained from two amnioreductions of a TTTS polyhydramniotic recipient, where the umbilical venous flow was found to increase immediately following amnioreduction (unpublished observations). On average, placental flow increased 1.5-fold, implying a 1.5-fold reduction in placental resistance. Assuming that polyhydramnios increases the sum of the amniotic fluid volumes at least three times, based on normal AF_N values of 0.4 liter at 20 wk, and the amnioreductions remove 1 liter, Eq. 13 predicts the amniotic fluid pressure to increase to at least 15 mmHg. This can subsequently be used to correlate with the decrease in umbilical flow of 1.5 times. Because polyhydramnios shows a large variation in amniotic fluid pressures (16, 51), the relation of f_AFP with amniotic fluid pressure includes the adjustable parameter k as follows

(14)

and with f_AFP = 1 otherwise. In the simulations below, the adjustable parameter k was chosen as one and five, where k = 1 implies that f_AFP = P_AF/P_AF,N; hence, the placental resistance is proportional to the actual amniotic fluid pressure. With k = 5, a significantly stronger influence of amniotic fluid pressure on placental resistance is assumed, i.e., a twofold increased amniotic fluid pressure increases the placental resistance sixfold. In the model, the placental resistance is then roughly increased 2.5-fold at 22 wk due to the predicted level of polyhydramnios in a severe case of TTTS (single AV, stuck donor at 20 wk).

Cerebral resistance for donor and recipient depending on their placental flow. Consistent with "brain sparing" blood flow responses, Jensen et al. (22) showed that cerebral perfusion in fetal lambs increased 0.6% when placental perfusion decreased by 1%. Following Myers and Capper (32), we decreased cerebral resistance accordingly when the placental resistance was increased, also resulting in decreased placental flow, F_Placenta. With X = D, R, we used

(15a)

(15b)

(15c)

Vascular Young's moduli in donors. Hypovolemia and growth retardation in the donor twin leads to a decreased elastin deposition in the large arteries (29, 30) and hence increased Young's moduli (stiffer arterial walls). Previously, the normal Young's modulus (in mmHg) for the thoracic aorta was derived as (48)

(16a)

where we combined measured pulse wave velocities (17) with estimated anatomical values for vessel wall radius and thickness from literature data (48). From the latter data and using an arterial wall tissue density of 1.05 g/ml (12), we assessed the weight of the thoracic aorta wall (in g) as (48)

(16b)

From measurements by Berry et al. (7), we fitted the normal weight content of collagen, W_Coll,N, and elastin, W_Elas,N, in the thoracic aorta wall as

(17)

Subsequently, using the normal vascular Young's modulus and the weight contents of elastin and collagen of the thoracic aorta, we could fit the following relation for the normal Young's modulus:

(18)

with an R² value of 0.994 (Microcal Origin, version 6.0; Northampton, MA).

Fetal elastin synthesis is abnormal in cases of a blood volume smaller than normal (in donor twins). We modeled this as

(19a)

(19b)

(19c)

where dW_Elas,N/dt follows from Eq. 17. Subsequently, from Eq. 18, the donor Young's modulus of the thoracic aorta can be calculated by replacing the normal elastin weight with W_Elas,D, solved from Eq. 19. In recipient twins, the Young's modulus remained as given by Eq. 18.

This approach produces an increased Young's modulus in hypovolemia or growth retardation in TTTS donors, approximating the observations described in the literature (18, 29). The factor of increase in thoracic aorta Young's modulus was subsequently used to increase the Young's moduli of the other arteries correspondingly (48).

Arterial wall thickness in recipients. Clinical data suggest that severe hypertension may develop in the recipient twin (27). According to experiments in the hypertensive rat (56) and fetal sheep (6), hypertension increases the vascular wall thickness with almost no change in relative wall constituents. As a result, the Young's modulus itself remains unaltered, but pulse wave velocity increases proportionally to the square root of the increase factor of the wall thickness. Using pathology data by Naeye (33), which showed vessel wall muscular mass to increase in recipient twins in 11 severe cases of TTTS, it can be estimated that the maximal wall thickness increase in the recipient twin was 1.5-fold. Accordingly, we implemented the arterial wall thickness to increase as a consequence of increased arterial pressure as follows, using h (in cm) to represent wall thickness, where h_N(t) is known from literature data (48).

(20a)

(20b)

This produces a thickness increase of almost 1.5-fold in a simulated severe case of TTTS (single AV, stuck donor at 20 wk). We assumed that the collagen and elastin contents of the vascular wall remain normal; hence, the Young's moduli remain normal. The corresponding increase in pulse wave velocity, which increases with the square root of h, is of the same magnitude as observed in clinical cases (18).

Maternofetal fluid flow for donor and recipient depending on their placental resistance. Adamson et al. (1) postulated an important regulatory mechanism for angiotensin, suggesting that by increasing the precapillary resistance, 1) the placental capillary hydrostatic pressure that is encountered by the maternal blood, P_cap, is decreased, and 2) the maternofetal transplacental fluid flow is enhanced. Statements 1 and 2 can be proven as follows.

Capillary hydrostatic pressure P_cap (Fig. 2) follows from the difference between the arterial pressure and the pressure drop over the precapillary resistance as

(21)

We dropped the X = D, R in all terms and used F_Placenta as defined in Eq. 15c. An increase in the precapillary resistance indeed decreases the capillary pressure, i.e., statement 1.

As described previously (43), the maternofetal flow (transplacental flow) depends on the P_cap as

(22)

with X = D, R, where L_pl is the placental filtration coefficient in ml·wk^–1·mmHg^–1, P_mat is the maternal hydrostatic villous blood pressure, and COP_mat and COP_fet,X are the maternal and fetal colloid osmotic blood pressures (in mmHg). A decreased P_cap indeed increases the transplacental flow, i.e., statement 2.

By allowing a low capillary hydrostatic pressure to develop in the donor twin as a consequence of RAS-mediated increased precapillary resistance, the maternofetal flow to the donor is increased compared with the previous model (43, 46). This stimulates development of polyhydramnios, which is confirmed by the observation that low-dose infusions of angiotensin produce polyhydramnios (15).

Description of TTTS staging in the model. The clinical stages of TTTS are defined as follows. The first stage is a stuck donor twin and simultaneously occurring polyhydramnios in the recipient. In our model, this is defined as a donor amniotic fluid volume of 10 ml and a recipient amniotic fluid volume of at least twice the normal volume. TTTS stage II commences when donor urine production has become zero. TTTS stage III, i.e., abnormal umbilical flow, is characterized by absent or reversed end-diastolic flow in the distal umbilical artery (48). TTTS stage IV is defined by hydrops in the recipient, defined at least an 18% increase of the interstitial fluid volume.

Model simulations. Model simulations included three selected cases (Table 2), all with an equally shared placenta. Case 1 has one single uncompensated AV anastomosis; this angioarchitecture represents severe TTTS (45). We used k = 1 in Eq. 14. The radius of the AV anastomosis was varied until the model outcome predicted a stuck donor twin at 22 wk of gestation. Case 2 has an AV anastomosis, inadequately compensated by one oppositely directed smaller VA anastomosis and one small AA anastomosis. This placental angioarchitecture, i.e., an AV inadequately compensated by opposite anastomoses, is found in a large number of TTTS placentas (11). In this case, we increased the AV radius 1.3-fold compared with the previous case; thus the AV resistance was reduced 3-fold (from 1/1.3⁴), and compensating VA and AA anastomoses were included so that a stuck donor twin also developed at 22 wk. An example of a non-TTTS placenta with this type of anastomoses is depicted in Fig. 3. Compared with case 1, this angioarchitecture allows for return of colloids, osmoles, and hematocrit from the recipient to the donor, causing milder TTTS (34). We used k = 1. Case 3 has the anastomotic pattern of case 2, but the influence of amniotic fluid pressure on the placental resistance was increased. We used k = 5 in Eq. 14. As a result, a stuck donor twin developed at 22.5 wk.

View larger version (85K): [in this window] [in a new window]   Fig. 3. A: schematic overview of a biamniotic-monochorionic twin pregnancy with a polyhydramniotic recipient and a stuck donor twin. [Adapted with permission from Fig. 1 of Saade et al. (40a)]. B: a dye-injected monochorionic non-TTTS placenta showing arterioarterial (AA), arteriovenous (AV), and opposite arteriovenous (VA) anastomoses.

	RESULTS

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View larger version (19K): [in this window] [in a new window]   Fig. 4. Model flow waves for case 1 (single AV) in the distal umbilical artery (UA) and middle cerebral artery (MCA) for donor and recipient twins as a function of gestational age. Normal flows are indicated with dashed lines. The predicted staging sequence is I, III(D), II, and IV.

View larger version (15K): [in this window] [in a new window]   Fig. 5. Model flow waves for case 2 (AV, VA, and AA) in the distal UA and MCA for donor and recipient twins as a function of gestational age. Normal flows are indicated with dashed lines. The predicted staging sequence is I, II, III(D), and IV.

View larger version (15K): [in this window] [in a new window]   Fig. 6. Model flow waves for case 3 (AV, VA, and AA; k = 5 in Eq. 15) in the distal UA and MCA for donor and recipient twins as a function of gestational age. Normal flows are indicated with dashed lines. The predicted staging sequence is I, III(D), II, III(R), and IV.

	DISCUSSION

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In our model, abnormal umbilical artery flows (stage III) develop in the donor twin more commonly and earlier than in the recipient. This results from an increased donor placental vascular resistance in conjunction with decreased cerebral vascular resistance. The increased placental resistance occurs in both donor and recipient twins from two primary mechanisms: 1) increased levels of RAS developing in the hypotensive donor that are transfused by the AV anastomosis to the recipient, which increases the precapillary resistance of the cotyledons; and 2) placental squeezing due to polyhydramnios-induced increased amniotic fluid pressure, which is distributed equally to both placentas. A third mechanism, which increases placental resistance only in one twin, is the increased blood viscosity of the polycythemic recipient. In addition, although not addressed in the present report, unequal placental sharing increases donor or recipient placental resistance, depending on which twin has the smaller placental part. In our model, abnormal umbilical flows, developing only in the recipient and not in the donor, would result from a small recipient placental part. We assume the placenta can be described by a set of parallel capillary resistances; hence, a smaller placenta contains less capillaries, and its resistance is inversely proportionally increased. In addition, our model indicates a decreased mean umbilical blood flow for donor twins compared with recipient twins, which is comparable to clinical observations (5).

In contrast to the similar effects of RAS and polyhydramnios on both twins, effects of viscosity differ greatly. As the decreased donor blood viscosity (anemia) increases the arterial pulse wave amplitude (48), the donor twin more readily develops reduced or absent end-diastole flow. In addition, the "normal" staging sequence where stage II precedes stage III additionally requires that donor anemia is limited as a result of sufficient mixing of donor and recipient blood by opposite anastomoses. This ameliorates the severity of the viscosity discordance, hence causing a smaller increase of donor arterial pulse wave amplitudes, and thus postpones the onset of donor TTTS stage III, in our model beyond onset of donor anuria (stage II). This trend was simulated in case 2, which represents an inadequately compensated AV anastomosis by opposite anastomoses, the angioarchitecture of the majority of TTTS cases (11). As opposed to donor twins, the recipient has an increased blood viscosity (polycythemia) that decreases the arterial pulse wave amplitudes and hence tends to prevent the onset of abnormal arterial pulsations (48). As a consequence, RAS mediators alone were unable to simulate abnormal umbilical artery flows in the polycythemic recipient. Only with an additional strong increase in placental resistance, from polyhydramnios, or a small placental fraction (unequal placental sharing, not shown) could abnormal umbilical flows be simulated in the recipient. The clinical implications of these simulations are described below.

First, donor arterial stage III TTTS occurs more commonly (and earlier) than in the recipient. This model prediction agrees with observations by Martínez et al. (28), which show that arterial stage III develops more frequently in the donor than in the recipient, i.e., 15 of 110 vs. 1 of 110 cases of TTTS, respectively. In addition, stage III in recipients virtually always refers to venous instead of arterial flow abnormalities such as reversed flow in the ductus venosus or pulsatile umbilical venous flow. Of 110 cases of TTTS, at least 30 recipients developed ductus venosus reversed flow during atrial contraction, and at least 33 developed pulsatile umbilical venous flows, compared with 1 recipient with arterial stage III (28). Pulsating umbilical venous or ductus venosus flows may develop as a consequence of an impaired relaxation of the right ventricle (40), resulting from a decreased compliance of the myocardium. Our model obviously cannot simulate the significant presentation of abnormal venous flows in recipients at the present time, because venous circulations were not included.

Second, the current model demonstrates that stage III TTTS (in the donor) may develop before stage II (lack of donor bladder filling). Although TTTS stages are based on clinical observations, it is clear that not all pregnancies follow the consecutive pattern of stage I through IV. Simulation of donor stage III preceding stage II required conditions of severe TTTS (i.e., a single AV anastomosis without compensating anastomoses, cases with severe polyhydramnios, or a small placental fraction). Thus stage III preceding stage II is prognostic for severe cases of TTTS. Clinically, these cases occur infrequently.

Interestingly, our simulations suggest that under conditions of high blood viscosity and severe blood flow redistribution, recipient middle cerebral arterial flow nevertheless may be normal. This is the result of two simultaneous pathological mechanisms that affect the pulsations in opposite directions, namely, reduced cerebral vascular resistance and increased blood viscosity. Reduced cerebral vascular resistance reduces the reflection of the pulsatile blood flow against the brain vascular bed; hence, both the pulse amplitude and the mean flow in the middle and anterior cerebral arteries increase (48). In contrast, an increased blood viscosity increases the cerebral vascular resistance but reduces the mean flow through the brain as well as the amplitudes of arterial pulsations. Thus, in our model, the degree of severity of the recipient's condition is not reflected by abnormal cerebral arterial pulsations.

Our model has several limitations. First, we included only the fetal arterial circulation to simulate stage III TTTS in donor twins, obviously with a limited number of arteries and organs and an estimated flow distribution (20, 32). Because we have used our previous pulsatile model (48), the same limitations as previously apply. One important simplification is the fixed distribution of the combined cardiac output, which, in the in vivo situation, changes during gestation (38). A model analysis with flow distribution depending on gestational age was, however, included in the sensitivity analysis (see below). Second, our model simulates a highly complex and still incompletely understood syndrome that affects many parameters of pathophysiology in both fetuses, often in opposite ways, and at different points in time. Therefore, as discussed previously (43, 46, 50), we were sometimes forced to include a simplified and at times even empirical or pragmatic description of fetal pathophysiological mechanisms. Examples are described below.

First, the influence of the blood viscosity on the placental resistance was not considered, where low blood viscosity in the donor twin has not been simulated to reduce the placental resistance. This results from two opposite and simultaneous mechanisms, both of which increase the placental resistance, i.e., a reduced number of cotyledonic capillaries as well as delayed villous maturation (52). Second, other examples include pragmatic choices of parameter development in mechanisms that are essentially correct but not known in sufficient detail: 1) the placental resistance is increased by excess concentrations of RAS and increased amniotic fluid pressures; 2) hypovolemia and growth retardation reduces the production of elastin in arterial walls, which increases the Young's moduli, and 3) hypertension causes an increased thickness of recipient arterial walls. Third, possible differences in diameter and length of arterial segments as a consequence of abnormal donor and recipient development were not accounted for, because differences in linear dimensions are much smaller than in, for example, blood volume, i.e., length/diameter develop proportional to the one-third power of volume. Fourth, circulatory adaptations such as cardiac hypertrophy as a response to the increased preload in the recipient twin due to volume loading have not been accounted for, resulting in a somewhat lower combined cardiac output for recipients compared with clinical observations (5).

We performed a sensitivity analysis. First, varying the maximal hematocrit upregulation between zero and fivefold caused slight differences in donor blood viscosity. Increased hematocrit upregulation of more than threefold in case 1 prevented early development of abnormal umbilical artery flow in the donor, i.e., before anuria. In the recipient, upregulation caused a strongly increased viscosity that reduced the placental flow. The staging sequence remained unchanged for hematocrit upregulation of threefold or less. In cases 2 and 3, increased hematocrit increased both donor and recipient blood viscosities, but without altering the staging sequence. Second, the influence of amniotic fluid pressure in cases 2 and 3, with k = 1 and k = 5 in Eq. 14, respectively, increased the placental resistance 1.2- and 1.9-fold at 12 mmHg amniotic fluid pressure (at 22 wk) and 1.6- and 3.8-fold at 16–17 mmHg (32 wk). The overall result of increased placental resistance due to k variation as illustrated by cases 2 and 3 is earlier onset of abnormal donor umbilical artery flows and occurrence of abnormal umbilical flows in the recipient when k was 4 or higher. Third, excluding changes in vessel wall thickness and stiffness, and, fourth, excluding changes in brain resistance, produced the normal staging sequence in cases 1 and 2 and unchanged staging in case 3. Fifth, we included that a percentage of the combined cardiac output, depending on gestation, enters the pulmonary circulation (38). This part of the combined cardiac output was removed from the ascending aorta flow. In addition, placental flow and descending aorta flow were modified according to clinical measurements (13, 14) to respectively 25 and 50% of the combined cardiac output, retaining 3% that enters each kidney as measured by Guettouche et al. (20). To account for the change in pulmonary flow, we varied carotid artery flow between 20 and 31% during gestation. As a result of the modified flow distribution, abnormal umbilical artery flows presented earlier, leaving the staging sequence in cases 1 and 3 unchanged. In case 2, however, onset of absent end-diastolic flow in the distal umbilical artery presented earlier than donor twin anuria. Overall, however, severe cases present with abnormal flows in the umbilical artery earlier. Interestingly, therefore, subtle differences in fetal parameters, within natural physiological variation, cause large variability in TTTS staging, confirming that clinical TTTS presentation often is perceived as widely variable and at times unpredictable.

In conclusion, our mathematical model of TTTS is the first that describes the sequence of TTTS stages I, II, III, and IV. The model explains the more common finding of donor vs. recipient stage III TTTS and predicts that the majority of TTTS pregnancies progresses in the normal consecutive staging sequence I through IV, but some have an abnormal sequence, i.e., donor stage III preceding stage II. These findings may be applied to the prospective assessment and management of TTTS pregnancies, providing clinicians with insight into both pathophysiology and prognosis, and aiding in guiding therapeutic decisions.

Perspectives

Monozygotic twins result from embryonic splitting of a fertilized ovum producing identical twins in two amniotic sacs, although often, i.e., in 75% of cases, competing for one monochorionic placenta. Monochorionic twin placentas virtually always include vascular anastomoses, which may result in TTTS, from a chronic net fetofetal transfusion from donor to recipient twin, with significant cardiovascular consequences to either or both twins. For clinical management and prediction of outcome, TTTS has been categorized into four stages of worsening severity: stage I exhibits oligo- and polyhydramnios; stage II includes lack of donor bladder filling; stage III includes abnormal umbilical flows in either twin; and stage IV includes congestive cardiac failure and hydrops in the recipient. Stage III likely is the most complex of these, because it refers to abnormal umbilical arterial or venous pulsations in either or both twins. So far, the underlying pathophysiology of these abnormal pulses in the donor or recipient, as well as the order of appearance of the various stages, has not been identified. Therefore, we present our mathematical model on abnormal donor and recipient umbilical arterial pulsations, which enables an improved understanding of the TTTS pathophysiology and may identify the sequence of events that determines the progression of TTTS stages as well as the efficacy and outcome of TTTS therapies.

	APPENDIX: OVERVIEW OF NONPULSATILE TTTS MODEL

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The first part of the model describes the normal physiology of fetal and amniotic fluid development. The second part incorporates the consequences of the net fetofetal transfusion of blood and blood constituents along the anastomoses. The growth equations of donor and recipient blood volumes are the essential equations. Blood is part of the fetal total body fluid volume, TBF, together with the interstitial and intracellular volumes, V_Inter and V_Intra, as follows:

(A1)

Normal growth of the fetal total body and amniotic fluids is caused by the net transplacental fluid flow from the maternal to the fetoplacental circulation. We described this flow by Starling forces as the difference between the fetal and maternal hydrostatic (P_fet and P_mat) and colloid osmotic pressure (COP_fet and COP_mat) gradients in the intervillous space. Thus

(A2)

(A3)

Including the net fetofetal transfusion (I_net = I_AV – I_VA – I_AA – I_VV), growth of fetal blood volumes is described from Eqs. A1 and A2 as

(A4)

The plus sign is the recipient equation, the minus sign is the donor equation. The rate of change of the amniotic, interstitial, and intracellular fluid volumes is given by

(A5)

(A6)

where U is urine production, L is lung secretion, S is swallowing, and IM is the intramembranous flow. Interstitial and intracellular volumetric growth was modeled by Starling forces between the vascular and the interstitial compartment, comparable to Eq. A3, and including the lymph flow from the interstitial to the vascular compartment. To separately calculate the interstitial volume, we assumed the rate of change of intracellular fluid is its normal rate of change times normalized fetal blood volume.

To derive equations for the arterial and venous blood volumes separately, we calculated venous return and arterial cardiac output separately. Excess venous blood from cardiac failure is removed from the arterial blood. Thus

(A7a)

(A7b)

Note that Eq. 12 of our group's previous report (46) contained the venous excess blood volume rather than its rate of change as in Eq. A7, which is the correct description included in the numerical code.

Colloids in blood and interstitium, and osmoles and RAS in blood, are incorporated in control functions of U and L, and amniotic fluid osmoles is one of the factors that determine the IM flow, also by Starling forces. These parameters follow from

(A8)

(A9)

(A10)

(A11)

(A12)

In Eq. A12, the plus sign refers to the recipient, and the minus sign refers to the donor. Parameters in the equations not specified are defined in our previous report (46). In the present report, we have added equations for blood hematocrit (Eq. 6), elastin concentration in arterial wall (Eq. 19), and arterial wall thickness (Eq. 20).

In our model, tubes connecting with the umbilical arterial or venous compartments represent the anastomoses. We used Poiseuille anastomotic resistances

(A13)

Length and diameter of anastomotic tubes grow proportional to gestational age, because we assumed that their volumes develop proportionally to the placental volume, proportionally to gestation to the third power. As a consequence, the anastomotic resistances depend on gestational age as a strongly decreasing function of gestation, with a fixed relation to their values at 40 wk

(A14)

Blood vessels become functional at t = 4 wk. The ratio term at the right-hand side assures that it becomes 1 at t = 40 wk. Anastomotic resistances at 40 wk, based on their length and diameters, are one of the input parameters in the model.

We used simple algebraic descriptions of the normal values of all model parameters, summarized in tables in our previous reports (Table 4 of Ref. 50, Table 1 of Ref. 43, Table 3 of Ref. 46). Also, we used (other) simple algebraic relations to express the abnormal development of various parameters, divided by their normal values, vs. another parameter, also divided by its normal value, taking these relations independent of gestation. Nevertheless, gestation is included, because the normal values of the parameters depend on gestation. An example is abnormal arterial and venous blood pressures vs. abnormal blood volumes (i.e., Fig.. 2 of Ref. 46). Input parameters in the model are vascular anastomotic radii at 40 wk, anastomotic lengths of 15 cm at 40 wk, the degree of placental sharing between donor and recipient twin, and the amnionicity.

In our first model (50), an initial condition was that all donor and recipient parameters are zero at zero gestational age, the moment of embryonic splitting. The later models, including the present one, use the output of the first model at 11 wk as well as added estimated or measured values of all other parameters. Thus all parameters at 11 wk are known, including the I_net. A short time step later, a small blood volume, equal to the product of I_net and the time step, is subtracted from the donor blood volume and added to that of the recipient. These new blood volumes also generate new pressures, osmoles, colloids, I_net, RAS, and other parameters, at that slightly later gestational age. This process continues, beginning at 11 wk and ending at 40 wk.

The 26 coupled differential equations were solved by a standard forward finite-difference method, using a time step of 10^–6 wk, which is comparable to 0.6 s. The computer program is written in Delphi 5.0 (Borland Enterprise, Cupertino, CA). One computer run, from 0 to 40 wk, took about 9 min on a Pentium 4.8 GHz personal computer, and an additional 2.5 min for each run of the pulse model (thus simulating the pulse propagation waveforms for 5 gestational ages, as in Fig. 4, took 21.5 min).

	GRANTS

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This work was supported in part by the European Community Euro-Twin-2-Twin Project (to J. P. H. M. van den Wijngaard) and National Heart, Lung, and Blood Institute Grant HL-40899 (to M. G. Ross).

	ACKNOWLEDGMENTS

 
We acknowledge fruitful communications with Dr. Liesbeth Lewi (Catholic University, Leuven, Belgium), Prof. Rubén A. Quintero (St. Joseph's Women's Hospital, Tampa, FL), Prof. Yves Ville (University Paris-Ouest Versailles-St. Quentin, Poissy, France), and Prof. Nico Westerhof (Free University, Amsterdam, The Netherlands).

	FOOTNOTES

 

Address for reprint requests and other correspondence: J. P. H. M. van den Wijngaard, Laser Center, Academic Medical Center, Univ. of Amsterdam, Meibergdreef 9, 1105 AZ, Amsterdam, The Netherlands (e-mail: j.p.vandenwijngaard{at}amc.uva.nl)

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

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