The mass transport that occurs during dialysis can be roughly explained using body fluid exchange and solute kinetics as in the model of Ursino et al.5 Body fluid compartments can be approximately classified into intracellular and interstitial fluid and plasma compartments. Intracellular and interstitial fluid exchange usually occurs as a consequence of osmotic differences that arise at the cell membrane due to differences between solute concentrations inside and outside the cell. Fluid exchange between interstitial fluid and plasma arises from differences between oncotic and hydrostatic pressure. Proteins generate an osmotic pressure difference between plasma and interstitial fluid, and this difference participates in the fluid exchange between interstitial fluid and plasma. This pressure, called "oncotic pressure", was calculated using the Landis-Pappenheimer equation. Furthermore, assuming that the elastic coefficient of tissues is constant, hydrostatic pressure of the interstitial fluid was calculated as a function of interstitial fluid volume and elastic coefficient of tissues.5 The movement of solutes into and out of cells provides the main mechanism of diffusion and involves both passive channels and active pumps.

Fig. 2 is a schematic diagram of the mass transport model used in this study. The solid lines show the exchange of fluids and the dotted lines show the transport of solutes. M_s,ic and M_s,ex represent the mass of solute s inside and outside the cell, respectively; the mass transport coefficients for this solute from the inside and outside of the cell are η_s and η_s·β_s, respectively. β_s is an equilibrium ratio that sets the ratio c_s,ic/c_s,is at the state of the resting potential through the cell membrane. The model considers the solute urea, creatinine, Na⁺, K⁺, and HCO₃^-. V_ic, V_is, and V_pl are the volumes of intracellular fluid, interstitial fluid, and plasma, respectively, and the extracellular volume V_ex is the sum of V_is and V_pl. Q_inf and Q_f are the substitute fluid injection and ultrafiltration rate, respectively, and C_s,inf is the solute concentration dissolved in the substitute fluid. J_s is the transport rate of the solutes through the dialyzer, and F_a(t) and R_v(t) are the plasma filtration of arterial capillaries and reabsorption rate of venous capillaries, respectively. The plasma filtration and reabsorption rates are calculated using equations (6)-(9). While Ursino et al.4 assumed constant arterial and venous capillary pressures, we assumed that arterial and venous capillary pressure were not constant and used time-varying values from Heldt et al.7: P_ac1 as P_up, P_ac2 as P_sp, P_ac3 as P_ll, P_vc1 as the difference between P_up and P_svc, P_vc2 as the difference between P_sp and P_avc, and P_vc3 as the difference between P_ll and P_avc. By coupling dependent variables in the cardiovascular system model with time-varying parameters in the mass transport model, the integrated model achieves more realistic results. Water permeability through the cell membrane is represented by η_f.

The governing equations for the mass transport model used in the actual numerical calculations are listed in Appendix 1.

A cardiovascular system model using a lumped parameter method was used to evaluate the cardiovascular system. Using components to express the crucial factors that participate in blood circulation, this model calculates pressure and volume at each component along with blood flow among components. Owing to conceptual similarities, this model can easily be represented by using electric circuit diagrams. Fig. 3 shows a representative component in which blood flow corresponds to electric current and blood pressure corresponds to voltage. Electric resistance expresses flow resistance inside blood vessels due to viscosity, and the storage battery is equivalent to the compliance of blood vessel walls. Therefore, changes in the state of a discretionary component are expressed by using the blood flow resistance (R) at the entrance and exit to the component, capacitance (C), blood flow (q), blood pressure (P), and pressure acting on the exterior wall of the blood vessels (P_is), which is calculated in the above model as a function of interstitial fluid volume and elastic coefficient of tissue. To facilitate interpretation, all resistance elements and storage batteries were assumed to be linear elements.

Based on earlier research and anatomic information, we expressed all human blood flow relationships using a simplified closed circuit consisting of 12 components (Fig. 4): the left ventricle (lv), aorta (a), right ventricle (rv), pulmonary artery (pa), pulmonary vein (pv), and components of the tissue circulation system, including 3 parts of the vena cava-the superior vena cava inside the thoracic cavity (svc), inferior vena cava inside the thoracic cavity (ivc), and vena cava outside the thoracic cavity or abdominal vena cava (avc) and 4 arterial branches: the upper body (up), kidney (kid), splanchnic organs (sp), and lower limb (ll). The contraction and relaxation of the heart to serve its pumping function used a predetermined value that changed with time. More detailed information is available in the study by Heldt et al.7 The effect of the atrium is only as an accessory, therefore, it was included in the calculations of adjacent nodes. Diodes at the entrances and exits of ventricles represent the heart valves that prevent blood reflux. Pressure that changes with time, P_th, is the pressure in the thorax. In addition, exterior pressure affecting a vessel is indicated by the subscript is, which means interstitial. Finally, ultrafiltration rate is Q_f, injecting flow rate of the substituting fluid is Q_inf, and the difference between the 2 values equals the actual volume change of the plasma through the dialyzer. To express the actual phenomenon more accurately, this dialysis component was located at the wrist (the cephalic vein), where the AV shunt is. The governing equations of the 12-parameter model of the cardiovascular system are given in Appendix 2.

When the cardiovascular system is disturbed and components such as arterial pressure are altered, the autonomic nervous system (ANS) performs several control functions. First, average arterial blood pressure (P_A) is sensed through baroreceptors in the carotid sinus artery. This information is transmitted to the ANS, where an error signal for the difference between the transmitted pressure and standard value is generated. This signal is transmitted to various cardiovascular components via sympathetic and parasympathetic nerves, changing the heart rate at the SA node and increasing or decreasing heart contractility by changing the calcium concentration inside cardiac myocytes. Furthermore, alteration of the contractility of vascular smooth muscle regulates vascular resistance and venous tone. This series of processes forms a feedback control system to restore the blood pressure changed (error signal) sensed in the carotid artery to normal.

We used a baroreceptor reflex model based on a beat-to-beat method (Fig. 5) to mimic the response of the human autonomic control function to the short-term disturbance of the cardiovascular system caused by dialysis. P_A represents the average arterial pressure that is a substitute for carotid artery pressure, and is a variable input to the system. The effective pressure deviation (P_A^eff) sensed by the autonomic nervous system can be expressed by the equation (32), as given by DeBoer et al.6

This nonlinear equation expresses the tested observation that the error signals are not amplified by the neural system in response to an excessive percentile range in pressure. Any significant pressure percentile range is transmitted through sympathetic and parasympathetic nerves. However, because the transmission speed differs between the 2, the stimulus-reaction curves also differ, such that the response time of the parasympathetic nerve is fast (0.5 - 1 s) and that of the sympathetic nerve is slow (3 - 10 s). This model, developed by DeBoer et al.6 for the command transmission of autonomic control function, was successfully introduced in studies by Heldt et al.7 and Shim et al.8 The ganglion transmission model and autonomic control function used here have thoroughly been discussed in a review by Shim et al.9

It is known that disturbances of ANS function are common in chronic renal failure (CRF) patients and its mechanism is multifactorial. Reduced end-organ response to norepinephrine appears to be a major factor underlying these abnormalities in CRF patients. Derangements in the parasympathetic nervous system and/or disturbances in cardiac function may also be involved.10 However, there are few quantitative data on the variations of cardiovascular variables such as heart rate and arterial resistance according to ANS disturbance. Because of this reason, we did not consider the ANS disturbance during hemodialysis, which may be one of the main limitations of the present study.

A dialysate with a constant Na⁺ concentration is usually used for dialysis treatment. However, in patients requiring dialysis of a large volume of blood, a dialysate with a Na⁺ concentration that changes over time is used to slow Na⁺ exchange in the body.6 In these cases, Na⁺ concentration in the extracellular fluid decreases rapidly at later dialysis times. Therefore, the change in Na⁺ concentration with time is represented by an upwardly convex curve. In clinical tests, Ursino et al.11 verified that the rates of intracellular and extracellular volume changes could be reduced in patients using this method.

In the present study, we wanted to predict how the exchange rate of a patient's intracellular and extracellular volume and cardiovascular response change during the course of dialysis as a function of the Na⁺ concentration profile. Thus, we evaluated 3 profiles: (1) a convex downward profile, (2) convex upward profile, and (3) linear profile intermediate between the two. By comparing cardiovascular response in these 3 cases, we expected to predict physiological effects of plasma Na⁺ concentration profile on the cardiovascular system and determine the optimal Na⁺ profile based on a patient's characteristics.

The values of the basic parameters and initial conditions of the numerical model that were used to investigate cardiovascular system response during hemodialysis were estimated from general conditions in dialysis patients. For an average adult with a body weight of 70 kg, hemodialysis is performed for 4 hours and involves about 2 L of fluid at an ultrafiltration rate of 0.5 L/hr. The overall mass transfer coefficient, KoA, of the simulated dialyzer is 700. The other parameters and initial conditions used in the models were obtained from Ursino et al.11 and Heldt et al.7

To validate the mass transport model in terms of body fluid exchange and solute transport, the calculated results were compared to the results of Barth et al.,12 which assumed a constant body fluid and used a 2-compartment model to calculate urea concentration profile. To facilitate the comparison, we used a urea clearance rate identical to that used by Barth et al.,12 As shown in Fig. 6, the 2 models give essentially identical urea concentration profiles and both show the concentration rebound phenomenon for extracellular fluid. In the study by Barth et al.,12 urea concentration in extracellular fluid decreased from an initial value of 100 mg/dL (the BUN in a chronic renal failure patient) to 40 mg/dL after 5 hrs of dialysis and then increased by 5 mg/dL after an additional hr. In our study, the concentration decreased to about 37 mg/dL and then rebounded by about 5 mg/dL after 1 hour.

To validate the 12-parameter model of the cardiovascular system, we simulated the shape of the blood pressure wave. The simulated blood pressure waves in the left ventricle, right ventricle, aorta, pulmonary artery, and pulmonary vein (Fig. 7) matched the hemodynamic values for a normal person.12

The results of the 3 different dialysate Na⁺ concentration profiles calculated using the inverse algorithm are shown in Fig. 8B. Inversely, we used the dialysate Na⁺ concentration profile in the model to determine the above plasma Na⁺ concentration profile (Fig. 8A). In profile A, it decreases rapidly, but the rate of decrease slows down after 2 hours and then increases to a concentration of 142 mEq/L after 3 hours. In profile B, the concentration decreases at a constant rate. In profile C, the plasma Na⁺ concentration increases for a short period early in dialysis, decreases gradually after about 1 hour, and then decreases rapidly after 2 hours to 142 mEq/L.

Fig. 9 shows the temporal changes in overall blood volume according to Na⁺ concentration profile. The calculation time was set at 6 hours, consisting of 4 hours of dialysis and 2 hours of observation after dialysis. Although the differences among blood volumes are not large, they increased according to the profile in the order A < B < C. For profile A, the blood volume is minimized while for the opposite case, i.e., profile C, the blood volume is maximal. Profile B has a blood volume intermediate between the two.

The physiological response of the cardiovascular system and autonomic nervous system to hemodialysis were evaluated. Again, the calculations were for a 6-hour period, reflecting 4 hours of hemodialysis and 2 hours of observation.

Fig. 10 shows the temporal variation in systemic arterial pressure (SAP). Under autonomic nervous system control, arterial pressure decreased linearly to 96 mmHg. Among the 3 different Na⁺ concentration profiles, SAP at the end of the dialysis session, decreased the most with profile A and least with profile C.

Fig. 11 illustrates the temporal change in heart rate. Heart rate increased at an almost constant rate of 1.5 bpm per hour over 4 hours of dialysis. After dialysis, heart rate decreased gradually but remained markedly elevated 2 hours after dialysis. Consequently, comparing heart rate and sodium concentration, heart rate was greatest with profile A, lower with B, and lowest with C.

Fig. 12 shows temporal change in systolic compliance of the left and right ventricles during dialysis. Although the right ventricle is more compliant than the left, both ventricles had the lowest compliance with profile A and the highest with C.

Fig. 13 shows the percentage change in total peripheral resistance, which consists of resistance in the lower limbs, splanchnic organs, and upper body compartment. During hemodialysis, peripheral resistance increased by about 15%. On comparing profiles, the magnitude of resistance was in the order A > B > C.