Model Based Sensitivity Analysis of Arterial Pressure Response to Hemodialysis Induced Hypovolemia

Abstract

The role of hemodynamic and regulatory factors in the arterial pressure response to hemodialysis induced hypovolemia was investigated by means of a computer model of the cardiovascular system, including the main short-term pressure regulatory mechanisms. The model mimics the arterial and venous systemic circulation, Starling’s law and inotropic heart regulation, arterial and cardiopulmonary baroreflex controls of resistance, and capacitance vessels. All of the model parameters have a clear physiologic meaning: 10 represent the systemic circulation, 4 describe cardiac pump performance, and 3 characterize baroreflex regulation. Sensitivity analysis is performed to determine the effect of each parameter on the pressure response to mild hypovolemia (a 10% blood volume reduction after 4 hours). The results demonstrate that circulatory parameters, such as resistances and compliances, have no relevant effect upon the pressure response. Conversely, regulation of venous capacity seems to play a pivotal role in sustaining arterial pressure during hemodialysis induced hypovolemia. Regulation of systemic peripheral resistance exerts a compensatory action only as long as the blood volume reduction is < 5%, but it is inadequate to compensate for a larger blood volume reduction when venous capacity regulation is absent. A paradoxical arterial pressure increase during hypovolemia can be referred to a prevalence of cardiopulmonary afferences in the regulatory process.

Despite the recent technologic advances in the artificial kidney, symptomatic arterial hypotension and related cardiovascular instability are still the most frequent intratreatment complication of hemodialysis. ^1–4 Ultrafiltration occurring in the course of treatment induces a reduction in circulatory blood volume (5–15%), and this transient state of hypovolemia is regarded as the main determinant of hypotension. In fact, it represents a stress for the cardiovascular system, and the physiologic mechanisms devoted to short-term regulation of cardiovascular function may fail in their compensatory mission.

To prevent excessive circulatory stresses, online blood volume monitoring in the course of a hemodialysis session is rapidly gaining acceptance in clinical practice. The modern artificial kidney is equipped with devices for the continuous measurement of hemoglobin concentration, ⁵ hematocrit, ⁶ or blood density, ⁷ each of which allows the changes in circulating blood volume to be estimated. Simultaneous monitoring of changes in circulating blood volume and arterial pressure allow the pressure response to hypovolemia to be characterized.

Recently, a new approach based on computer simulation for analyzing the pressure response to hemodialysis induced hypovolemia has been proposed. ⁸ This approach endeavors to combine blood volume, arterial pressure, and heart rate data collected during a conventional hemodialysis session using a computer model of the cardiovascular system. In previous work, ⁸ the model was used to compare the pressure responses of hemodynamically stable and unstable patients. Blood volume and heart rate measured in the course of treatment were used as model inputs, while simulated arterial pressure response was fitted to the measured data by tuning model parameters, accounting for the efficiency of regulatory mechanisms. Such model identification obtained an accurate data fit, and it was possible to characterize the pressure response recorded in the course of hemodialysis with four parameters, each one related to a specific pressure control mechanism. In this second, related paper, the same model was used to underline the factors that can predispose to hypotension. In particular, the computer simulation is used to focus on the role of short-term pressure controls and to test the implications of different hypotheses on the effectiveness of such regulatory mechanisms. For this purpose, the model sensitivity to parameter assignment was analyzed. Comparative simulations were performed by assigning different values to model parameters to assay the effects of parameter changes on the pressure response. The results of this study also constitute the rationale that supports the choice of parameters to be estimated when the model is used to fit the pressure response recorded during a treatment.

Methods

Model of Arterial Pressure Response to Hemodialysis Induced Hypovolemia

Accurate description of the model equations, as well as the physiologic rationale, are presented in the related work. ⁸ The main aspects of the model are summarized in this section. In particular, the physiologic meaning of parameters involved in the sensitivity analysis is given. The main physiologic aspects included in the model, and their mutual interactions, are depicted in Figure 1. In this scheme, the plus sign indicates a direct influence (e.g., increase in heart rate causes an increase in cardiac output), whereas the minus sign denotes an inverse influence (e.g., increase in unstressed venous volume causes a decrease in central venous pressure). In accordance with this scheme, plasma water depletion occurring during hemodialysis reduces circulatory blood volume, after which venous and arterial pressures tend to drop. In response to the pressure fall, the regulatory mechanisms operate a negative compensatory feedback by increasing vasoconstriction. As a consequence, peripheral resistance increases and capacitance of the venous circulation decreases. To mimic these physiologic behaviors, the model includes (1) arterial and venous systemic circulation, represented by capacitive and resistive elements; (2) Starling’s law and inotropic heart regulation; and (3) arterial and cardiopulmonary baroreflex controls of peripheral resistance and capacitance vessels.

Figure 1: Interaction between the main hemodynamic variables involved in the model. Shaded boxes refer to model inputs (heart rate and blood volume) and model output (arterial pressure). The + sign indicates a direct influence, (i.e., heart rate increase causing an increase in cardiac output), whereas the - sign denotes an inverse influence (i.e., increase in unstressed venous volume causing a central venous pressure decrease).

An equivalent representation of model equations is given by the electric analogue shown in Figure 2 and by the block scheme of Figure 3. All model parameters appear in these figures. The meaning of parameters and principle hemodynamic variables is briefly given in the Glossary, whereas the model equations are reported in the Appendix. Since pressure response to hemodialysis induced hypovolemia involves slower time changes than the cardiac cycle, all the hemodynamic quantities relate to mean values over the cardiac cycle, thus neglecting the pulsatile nature of the cardiac pump.

Figure 2: Electric analogue of the cardiovascular model. The heart is represented by a single compartment (right atrium) and by the relationship between cardiac output (CO) and right atrial pressure (P _ra). Systemic circulation is depicted by three compartments in series: arterial, peripheral, and venous. The capacities C _a, C _m, C _v, and C _ra represent the circulatory compliance of the arterial, peripheral, venous, and atrial compartments, while the resistances R _m, R _v, and R _ra represent the hydraulic resistance of peripheral, venous, and atrial compartments;P _a, P _m, and P _v are arterial, peripheral, and venous pressure. Voltage generators V _au, V _mu, V _vu, and V _rau account for the unstressed volumes of each compartment. Dashed elements, R _m and V _vu, indicate quantities under regulation.

Figure 3: Block diagram of the baroreflex control of unstressed venous volume and peripheral resistance. Variations in arterial (P _a) and atrial (P _ra) pressure compared with the values at the beginning of treatment (P _ae and P _rae, respectively) actuate the vasomotor tone (T) through balanced afferent arterial and atrial baroreceptor activity. A first order filter accounts for the typical dynamic of this reflex. Vasomotor efferent tone (T _b) affects the vasoconstrictive state through peripheral resistance (R _m) and unstressed venous volume (V _vu) (dashed elements in Figure 2).

The peripheral resistance R_m and the unstressed venous volume V_vu (see shaded elements in Figure 2) depend upon the arterial and atrial pressure according to the block scheme shown in Figure 3. This scheme approximates the baroreflex regulation of vasomotor activity. Two distinct afferent pathways are considered: one for the arterial and the other for the cardiopulmonary baroreceptors. The parameter K_aff ranges from −1 to 1 and balances the afferent values with a different relative weight. When K_aff is equal to −1, the cardiopulmonary afference is completely inhibited and control is exercised by the arterial side only. When K_aff is equal to 0, the afferences are exactly balanced. Finally, when K_aff is equal to 1, control is completely shifted toward the cardiopulmonary side. The control effects the unstressed venous volume, V_vu, and the peripheral resistance, R_m, which are, respectively, considered representative of the venous capacity and the vascular resistance. The gains K_v and K_r express the efficiency of the efferent static regulation of unstressed venous volume and peripheral resistance, respectively. Both parameters may range between 0 and 1: When the parameter is equal to 1 the corresponding regulatory pathway is maximally efficient. Conversely, no regulation takes place when the parameter is equal to 0. The balance K_aff of the afferent pathways and the gains K_v and K_r of the efferences express the efficiency in overall regulation of the vasoconstrictive state.

Sensitivity Analysis

Although the model has a very simple structure, it involves several parameters: 4 for the heart curve; 10 for arterial, peripheral, and venous systemic circulation and the right atrium; and 3 for controls (see Appendix and Glossary). Most of these parameters can neither be directly measured nor their values well established in humans. Because of such uncertainties in parameter assignment, we performed a sensitivity analysis to evaluate the effects of parameter perturbation on the model computed arterial pressure response. As one expects, sensitivity analysis establishes the more critical parameters, i.e., those that have the greatest influence on the model response. Of course, this is useful information for planning model identification, in the sense that these parameters are those that are recursively estimated to best fit measured data. At the same time, since the model parameters all have a precise physiologic meaning, sensitivity analysis also provides a framework for understanding the physiologic role of each parameter in determining the pressure response.

To perform a sensitivity analysis, we considered a basal set of values for the parameters related to the circulation and cardiac pump (see Glossary). A dialysis treatment with a constant rate of blood volume loss causing a 10% initial blood volume reduction after 4 hours was simulated. Circulating blood volume and arterial pressure at the beginning of simulated treatment was close to the mean value estimated in stable patients in a previous study. ⁸ To emphasize the isolated role of vasoconstriction, without including heart inotropic regulation, heart rate, i.e., the other input to the model, was kept to a constant value. When heart rate does not change with respect to the initial conditions, the parameter K_co that takes into account the sensitivity of cardiac output saturation level to heart rate changes (see equation A2 in the Appendix) had no influence on the pressure response. Therefore, this parameter was not included in the present analysis.

To establish how the assignment of heart and circulatory parameters affects model pressure response to hypovolemia, we increased these parameters one by one and, for each parameter change, we evaluated the change in the final arterial pressure after a 10% decrease in blood volume. This analysis was repeated for different configurations of controls and afference balance. We imposed a 10% change in each parameter, since it is small enough to consider model response linearly dependent on the parameter variation, and it is in the range of physiologic variability.

Subsequently, we widened the study to consider the effects of control parameters on the overall model response, not only on the final pressure. For this purpose, circulatory and heart parameters were fixed at their baseline values, and arterial pressure response was computed at many different control settings. In particular, we considered (1) inhibited controls (K_v =K_r = 0); (2) medium active controls (K_v =K_r = 0.5); (3) maximally active controls (K_v =K_r = 1); (4) three distinct degrees of peripheral resistance regulation (K_r = 0, 0.5, 1) with unstressed venous volume control at medium activity (K_v = 0.5); and (5) three distinct degrees of unstressed venous volume regulation (K_v = 0, 0.5, 1) with peripheral resistance control at medium activity (K_r = 0.5). In addition, we examined how the final arterial pressure changes in response to independent variations of efficiency of peripheral resistance and unstressed venous volume controls.

Results

Effects of Parameter Change with Inhibited Controls

In total absence of control (K_v =K_r = 0), arterial pressure after a 10% blood volume reduction (hereafter termed final arterial pressure) falls to a very low level: 57 mm Hg (see Figure 4). The simulated 10% increase in CO_m, V_au, V_mu, or V_vue, as well as the change in P_ras from 0 to -0.4, has no effect on the final arterial pressure (Figure 4). P_ran, C_a, C_m, C_v, C_ra, R_v, or R_ra increase causes a modest increase in final arterial pressure (<5.5%). The capacitance of the peripheral compartment (C_m) determines the largest increase (from 57 to 60 mm Hg). Final arterial pressure drops further (from 57 to 55 mm Hg) only when increasing R_me (Figure 4). It is noteworthy that the final pressure rise, after increasing the parameters, is in general directly sustained by a cardiac output at the end of the simulated dialysis higher (from 0.5 to 4.2%) than the one calculated with the unperturbed parameter set. Only in the case of an R_v increase is the final cardiac output unchanged. Reduction in final arterial pressure, due to the increase in R_me, is consistent with a lower cardiac output (10.7%).

Figure 4: Pressures at the end of simulated treatments (10% blood volume reduction after 4 hours) with inhibited controls (K _v =K _r = 0). The central line indicates the final arterial pressure corresponding to basal parameter values (see Glossary), while the black trapezium depicts an absolute variation in the final arterial pressure after a 10% increase in each parameter (except for P _ras that decreases from 0 to −0.4). Parameter changes do not affect the initial pressure (110 mm Hg).

Effects of Parameter Change with Active Controls and Balanced Afferences

When peripheral resistance or unstressed venous volume controls are operative, the simulated 10% blood volume reduction induces a limited change in the final arterial pressure, strictly dependent upon the values of the control parameters (see Figure 5). Final arterial pressure is 84 mm Hg when only the peripheral resistance control is active (K_v = 0, K_r = 1), 105 mm Hg when only the unstressed venous volume control is active (K_v = 1, K_r = 0), and 112 mm Hg when both the controls are operative (K_v =K_r = 1). Interestingly, in the latter condition the, final arterial pressure is slightly higher than the initial pressure.

Figure 5: Pressures at the end of simulated treatments (10% blood volume reduction after 4 hours) with balanced afferences (K _aff = 0). Three different control configurations are compared: only peripheral resistance control active (K _v = 0, K _r = 1), only unstressed venous volume control active (K _v = 1, K _r = 0) and both controls active (K _v =K _r = 1). Straight lines indicate the final arterial pressures corresponding to basal parameter values (see Glossary) in the different control configurations, while the black trapezia indicate absolute variation in final arterial pressure after a 10% increase in each parameter (except for P _ras that falls from 0 to -0.4).

As long as the unstressed venous volume control is active (K_v = 1), the effect of parameter change on the final arterial pressure is negligible and the only appreciable variation is due to P_ras (see Figure 5). When the control on unstressed venous volume is absent (K_v = 0), the final arterial pressure shows a weak sensitivity to the changes in P_ras, CO_m, C_m, C_v, R_me, and R_ra. Only the increase in R_me produces a final pressure lower than in the case of the unperturbed parameter set, similar to the case with both controls inhibited (K_v =K_r = 0). The lower final arterial pressure corresponds to a lowered cardiac output (11.4%).

Effects of Parameter Change with Active Controls and Arterial Afference Only

Final arterial pressure slightly decreases when only the arterial side presides over the control of peripheral resistance and unstressed venous volume (compare Figure 6 with Figure 5). As in the case of balanced afferences, final pressure is insensitive to parameter changes when the control of unstressed venous volume is active (K_v = 1), just minimal dependence on the V_vue is evident (1% increase in final pressure for a 10% increase in parameter). Final arterial pressure becomes sensitive to P_ran, C_a, C_m, C_v, C_ra, R_me, or R_ra when only the control of peripheral resistance is active (K_v = 0, K_r = 1). However, the percentage change in the final arterial pressure, due to the parameter increase, is smaller than in the case of total absence of control (compare Figure 4 and Figure 6).

Figure 6: Pressures at the end of simulated treatments (10% blood volume reduction after 4 hours) with arterial afference only (K _aff = -1). Three different control configurations are compared: only peripheral resistance control active (K _v = 0, K _r = 1), only unstressed venous volume control active (K _v = 1, K _r = 0) and both controls active (K _v =K _r = 1). Straight lines indicate the final arterial pressures corresponding to basal parameter values in the different control configurations, while the black trapezia depict absolute variation of final arterial pressure after a 10% increase in each parameter (except for P _ras that is lowered from 0 to -0.4).

Effects of Parameter Change with Active Controls and Cardiopulmonary Afferences Only

When the afferences are unbalanced toward the cardiopulmonary side (Figure 7) and both the controls (peripheral resistance and unstressed venous volume) are active (K_r =K_v = 1), final arterial pressure is even higher than that run at baseline (119 vs 110 mm Hg). In this condition, final pressure has no evident sensitivity to changes in model parameters except for P_ras, V_vue, and R_me. It is worth noting that the effect of R_me is opposite to the previous cases, and the final pressure rises only slightly (0.8%) when K_v =K_r = 1. Parameter change has a more evident influence when the control of unstressed venous volume is off (K_v = 0). Under these conditions, final arterial pressure is especially sensitive to the increase of C_m or C_v, as well as to R_me or R_ra (Figure 7).

Figure 7: Pressures at the end of simulated treatments (10% blood volume reduction after 4 hours) with cardiopulmonary afference only (K _aff = 1). Three different control configurations are compared: only peripheral resistance control active (K _v= 0, K _r = 1), only unstressed venous volume control active (K _v = 1, K _r = 0), and both controls active (K _v =K _r = 1). Straight lines indicate the final arterial pressures corresponding to basal parameter values in the different control configurations, while the black trapezia depict absolute variations in the final arterial pressure after a 10% increase in each parameter (except for P _ras which drops from 0 to -0.4).

Effects of Simultaneous Change in Resistance and Capacity Control Gains

Simultaneous change in the control gains K_v and K_r from 1 to 0 causes a drastic modification of pressure response to hypovolemia (see Figure 8). Until the control gains are > 0.5, arterial pressure during hypovolemia exhibits modest changes with respect to the initial value. In contrast, when complete lack of regulation (K_v =K_r = 0) is simulated, arterial pressure falls sharply during the circulating blood volume depletion, and final pressure dramatically drops to 57 mm Hg after a 10% blood volume loss.

Figure 8: Simulated mean arterial pressure in response to a constant rate of blood volume loss, for different sets of control parameters (K _v =K _r = 0;K _v =K _r = 0.5;K _v =K _r = 1). The left panel refers to only arterial afference activity (K _aff = −1), the middle panel to balanced afferences (K _aff = 0), while the right panel to only cardiopulmonary afference activity (K _aff = 1).

When both controls are off (K_v =K_r = 0), a compensatory pressure reflex does not take place and arterial pressure response does not change to vary the afference balance parameter K_aff (see Figure 8). Therefore, arterial pressure response to hemodialysis induced hypovolemia is not sensitive to variations in K_aff, and the identification of this parameter is insignificant.

Arterial pressure remains below the initial pressure when only the arterial side is operative (K_aff = −1). Conversely, when cardiopulmonary afference is active, arterial pressure may overcome the initial value (compare left and right panels in Figure 8).

The arterial pressure response to blood volume loss exhibits a typical non linear dependence upon control gains K_v and K_r: Pressure response is more sensitive to changes in K_v and K_r from 0 to 0.5 than from 0.5 to 1 (see Figure 8). Moreover, the sensitivity of arterial pressure response to blood volume loss increases, unbalancing the afferences toward the cardiopulmonary side (K_aff = 1). For instance, in the case of moderate activation of controls (K_v =K_r = 0.5), the difference between final and initial arterial pressure is -9 mm Hg, with afferences unbalanced toward the arterial side (see left panel in Figure 8), and + 3 mm Hg, with afferences unbalanced toward the cardiopulmonary side (see right panel in Figure 8).

Effects of Change in Peripheral Resistance Control Gain

The effects of only changing K_r, with K_v fixed to 0.5, depend strictly on the balancing of arterial and cardiopulmonary afferences (Figure 9). When the compensatory reflex is exerted only by the arterial afference (K_aff = −1), the strength of peripheral resistance control does not influence arterial pressure response (see left panel in Figure 9). The more cardiopulmonary afference takes place in the control action, the greater the sensitivity of the arterial pressure response to K_r increases, as evident by comparing the left and right panels of Figure 9. In particular, with balanced afferences (K_aff = 0), there is a moderate variation in the pressure response by varying the gain of peripheral resistance control, K_r, from 0 to 1; final arterial pressure moves from 101 mm Hg to 112 mm Hg (see central panel in Figure 9). When arterial afference is completely inhibited and peripheral resistance regulation is maximally active (K_aff = 1, K_r = 1), the final pressure rises to 126 mm Hg (see right panel in Figure 9). This extreme condition is characterized by a paradoxical hyperefficient response to hypovolemia because of the lack of a limiting factor due to arterial afference, which generally opposes the arterial pressure increases. It is worth noting that this hyperefficient regulation is limited by strengthening the control of unstressed venous volume (compare K_r = 1 curves in the right panels of Figure 9 and Figure 8).

Figure 9: Simulated mean arterial pressure in response to a constant rate of blood volume loss, for different values of K _r (K _r = 0;K _r = 0.5;K _r = 1) and K _v fixed to 0.5. The figure is arranged as for Figure 8.

Effects of Change in Unstressed Venous Volume Control Gain

Pressure response exhibits great sensitivity to changes in the K_v gain with K_r fixed to 0.5 (see Figure 10). Indeed, when this regulatory mechanism does not take part in the control process (K_v = 0), arterial pressure dramatically falls to hypotensive levels (final arterial pressure equals 70 mm Hg), although peripheral resistance regulation is moderately active (K_r = 0.5). On the contrary, maximum activation of venous capacity regulation (K_v = 1) allows for efficient stabilization of arterial pressure response. With balanced afferences (central panel in Figure 10) the final arterial pressure moves from 71 mm Hg (K_v = 0) to 109 mm Hg (K_v = 1), which is very close to initial pressure (110 mm Hg).

Figure 10: Simulated mean arterial pressure in response to a constant rate of blood volume loss, for different values of K _v (K _v = 0;K _v = 0.5;K _v = 1) and fixed K _r (K _v = 0.5). Figure is arranged as for Figure 8.

Pressure response exhibits clear, non linear behavior with respect to K_v: when K_v changes from 0 to 0.5, it causes substantial pressure responses, whereas changes from 0.5 to 1 produce limited variations (see Figure 10). Pressure response manifests greater sensitivity to K_v than to K_r. This is evident by comparing pressure response with control of peripheral resistance only moderately active (K_v = 0, K_r = 0.5 in Figure 10) and pressure response with control of unstressed venous volume only moderately active (K_r = 0, K_v = 0.5 in Figure 9). In particular, with respect to the case of lack of regulation (K_v =K_r = 0, Figure 8), moderate activation of control on peripheral resistance (K_r = 0.5) moves the final arterial pressure from 57 to 71 mm Hg (K_v = 0 curve in Figure 10), while similar activation of control on unstressed venous volume (K_v = 0.5) moves the final pressure from 57 to 100 mm Hg (K_r = 0 curve in Figure 9).

Interestingly, pressure response behaves strongly in the case of completely inhibited arterial afference (K_aff = 1) and inactive unstressed venous volume regulation (Figure 10, right panel, K_v = 0 curve). Under these conditions, pressure initially increases (while blood volume loss is smaller than 2%), afterward it quickly decreases. In this second phase, pressure goes down more steeply than in the case of a compensatory reflex unbalanced toward the arterial side (compare left and right panels in Figure 10).

Effects of Control Gain Changes on the Final Pressure

When unstressed venous volume or peripheral resistance controls are off (K_v = 0 or K_r = 0), final arterial pressure is not affected much by variation in K_aff (left top and bottom panels in Figure 11). Sensitivity to K_aff becomes important when either of the control gains is different from zero (see upper and lower middle and right panels in Figure 11).

Figure 11: Final pressure sensitivity to independent changes in K _v and K _r. In the lower panels, sensitivities of final arterial pressure to changes in K _v are compared for three different values of K _r (K _r = 0, K _r = 0.5 and K _r = 1). In the upper panels, sensitivities of final arterial pressure to changes in K _r are compared for three different values of K _v (K _v = 0, K _v = 0.5, and K _v = 1). Circles in the right lower panel indicate two different values of K _v corresponding to the same final arterial pressure.

Final arterial pressure shows a linear dependence on K_r gain, as evidenced by the straight lines in the upper panels of Figure 11. Unbalancing K_aff from the arterial (K_aff = −1) to cardiopulmonary side (K_aff = 1), sensitivity of final arterial pressure to K_r increases. The maximum sensitivity to K_r is for K_aff = 1 and K_v = 0.3 (see right lower panel). In this condition, final pressure is 20 mm Hg greater than the initial pressure. On the other hand, when the afferences are unbalanced toward the arterial side (K_aff = −1) and K_v is > 0.5, the sensitivity of final pressure to K_r is negligible (see middle and right upper panels). Moreover, the sensitivity of the final pressure to K_aff has a non linear dependence on K_v: it is negligible when K_v = 0, maximum with K_v = 0.3, and it decreases again when K_v = 1 (see lower panels in Figure 11).

The sensitivity to changes in K_v strictly depends on the value of K_v, and final pressure shows high non linearity with an evident saturation for high values of K_v (lower panels in Figure 11). For values of K_v lower than 0.2, final arterial pressure is highly sensitive to changes in K_v, while no significant changes in the final arterial pressure can be observed for a K_v > 0.6.

When the control of peripheral resistance is completely active (K_r = 1) and afferences are unbalanced toward the cardiopulmonary side (K_aff = 1), the final pressure curve exhibits an evident hump for K_v = 0.3 (see lower right panel in Figure 11). In this state, the same final pressure after a 10% blood volume loss can be reached for two distinct values of K_v. As an example, the two circles in the lower right panel identify two distinct values of K_v for which the final pressure is equal to 120 mm Hg. Actually, pressure responses corresponding to these two values of K_v are very different (Figure 12): Thus, considering the overall response and not only the final pressure values, no problem exists in the model identification.

Figure 12: Simulated mean arterial pressure in response to a constant rate of blood volume loss, for two different values of Kv producing the same arterial pressure after 240 min (see circles in lower right hand panel of Figure 11).

Discussion

A novel approach to analyzing pressure response to hemodialysis induced hypovolemia was recently proposed. ⁸ It is based on computer simulation and endeavors to integrate a time-series of hemodynamic data recorded during conventional hemodialysis using a simple model of the cardiovascular system for better understanding patient hemodynamic behavior. The present study shows how pressure response to hemodialysis induced hypovolemia computed by this model depends upon circulatory and heart parameters, as well as on parameters representing the effectiveness of short-term pressure regulatory mechanisms. In particular, a computer simulation approach was used to gain more information on the role of pressure compensation reflexes, as well as to test possible consequences or implications of different hypotheses on the exertion of such mechanisms.

Ursino and Innocenti ⁹ also performed a model based sensitivity analysis to underline the principal factors that may predispose a patient to acute hypotension during hemodialysis. Since that model was intended to investigate the role of osmotic factors on vascular refilling, the sensitivity analysis principally accounts for (1) the elastance of the interstitial space, (2) the initial blood composition and the sodium in the dialysate, and (3) the ultrafiltration rate. The effect of changes in circulatory parameters were not considered, ⁹ except for the systemic compliance that was found to have no relevant influence on the pressure response, according with the present results. Concerning cardiovascular regulatory mechanisms, the model of Ursino and Innocenti included more details by distinguishing for each mechanism the gain of arterial and cardiopulmonary sides, as well as the afferent and efferent gains. In fact, there were 19 parameters to completely characterize the control mechanisms. The significant differences in the models do not allow for a direct comparison of the sensitivity analysis results. However, the previous study underlined the pivotal role of the effectiveness of the control mechanisms. To perform a more detailed sensitivity analysis, in the present study the effects on the pressure response of each heart and circulatory parameter were analyzed first for different control configurations, after which our attention was focused on the control parameters.

Sensitivity analysis reveals that circulatory and heart parameters have a limited influence on pressure response. In particular, after varying P_ran, C_a, C_m, C_v, C_ra, R_v, or R_ra, final pressure tends to increase slightly, while the same percentage increase in R_me produces a paradoxical reduction in final pressure (see Figure 4). Since the initial pressure is the same before and after the parameter change, the increase in P_ran, C_a, C_m, C_v, C_ra, R_v, or R_ra causes arterial pressure to be less sensitive to blood volume reduction. Conversely, the increase in R_me causes a greater pressure response sensitivity to blood volume changes. For P_ran, C_a, C_ra, and R_v, the reduction of sensitivity is insignificant (<2%), whereas the peripheral capacitance C_m causes a larger reduction in sensitivity (approximately 5.3%). To better understand how each parameter affects the sensitivity of the pressure response to blood volume changes, one can resort to an analytic sensitivity expression, obtained in the total absence of control (K_v =K_r = 0) and for small variations of blood volume V

A 10% increase in P_ran causes a 9% reduction in the cardiac output slope (dCO/dP_ra), while dV_ra/dV increases by about 8.8%. As a whole, according to equation 1, sensitivity shows a very limited decrease: from 13.9 to 13.7 mm Hg/100 ml. The increase in compartment capacitances does not change the cardiac output slope, and the sensitivity reduction (1.9%, 5.7%, 3.8%, and 1.9%, respectively, for C_a, C_m, C_v, and C_ra) is due to the reduction in dV_ra/dV only. A 10% increase in R_v or R_ra produces a slight increase (2.6% or 1%) in the sum of resistances (R_me +R_v +R_ra) that moderately contributes to increase the sensitivity, S. At the same time, the increase in R_v or R_ra shifts P_rae and CO_e (i.e.,P_ra and CO at the beginning of dialysis) toward a region of the cardiac curve with a higher slope. As a consequence, dCO/dP_ra increases (1.6% or 0.8%), thus indirectly contributing to increase the sensitivity, S, whereas dV_ra/dV falls (3.8% or 4.9%). As a whole, after increasing R_v or R_ra resistance, the sensitivity of the arterial pressure response to blood volume reduction drops, respectively, from 13.9 to 13.7 or 13.4 mm Hg/100 ml. For R_me, a 10% increase produces a 6.3% increase in the sum of resistances, which directly affects the sensitivity, S (see Equation 1). The cardiac output slope (dCO/dP_ra) also increases (3.7%), while dV_ra/dV falls (3.9%). As a whole, the sensitivity, S, increases from 13.9 to 14.5 mm Hg/100 ml (about 4.3%).

The influence of resistances on the sensitivity, S, does not depend upon the manner of assigning the initial value of variables involved in the model. For instance, maintaining the same CO_e after changing R_me, neither dV_ra/dV nor dCO/dP_ra cause appreciable changes, and the increase in sensitivity (about 5.4%) is due to the increase in peripheral resistance alone. Naturally, assigning the same CO_e, the initial pressure increases (117 vs. 110 mm Hg) and consequently final pressure increases as well, but because of the high sensitivity, final pressure in any case reaches very low values (about 61 mm Hg). Therefore, whether initial pressure or initial cardiac output is kept constant, after increasing R_me, a similar increase in S takes place, thus denoting that resistance alone and not the manner of assigning the initial conditions has an effect upon pressure sensitivity to blood volume reduction.

Nakamura et al.¹⁰ studied the arterial pressure response to hemodialysis in hemodynamically stable and unstable patients. In the unstable group, systemic vascular resistance before dialysis was higher than in the stable group, indicating, in accord with the sensitivity analysis, that a higher resistance could be associated with a higher sensitivity to blood volume reduction. Actually, we found that a 10% increase in R_me resistance, when the controls are inhibited (K_v =K_r = 0), causes the final pressure to decrease from 57 to 55 mm Hg. Therefore, such increase in the resistance does not change the unstable nature of the pressure response. This finding is especially true when the control on venous capacitance is active (K_v = 1): In this condition, the change in peripheral resistance at the beginning of dialysis (R_me) does not affect the final arterial pressure at all. Accordingly, Nakamura et al.¹⁰ found a difference of about 13% in systemic vascular resistance when comparing stable and unstable groups; this was, however, not statistically significant, since the standard deviation of systemic resistance was 30% in both groups.

The present analysis shows that final arterial pressure has a low sensitivity to circulatory and heart parameter variations, whereas it is largely influenced by the parameters K_v, K_r, and K_aff. Moreover, sensitivity to circulatory and cardiac parameters strictly depends on these parameters. In particular, when the controls of peripheral resistance and unstressed venous volume are absent (K_r =K_v = 0), the final pressure was sensitive to changes in P_ran, C_a, C_m, C_v, C_ra, R_me, R_v, and R_ra (see Figure 4). The activation of peripheral resistance control (K_r = 1, K_v = 0) makes the final pressure less sensitive to parameter assignment than the case of complete absence of controls (K_v =K_r = 0), except for R_me. Even when venous capacity regulation is active (K_v = 1), final arterial pressure is not affected by the parameter change (see Figure 5–7). In any case, the greatest variations in final pressure are due to capacitance and resistance of the peripheral compartment (C_m and R_me) and are limited at 5.3%. On the other hand, by changing the configuration of the control gains, the final pressure displays very large variations (from 57 mm Hg in absence of control to 120 mm Hg when both controls are maximally activated and afferences are unbalanced toward the cardiopulmonary side). It is worth noting that by changing the afference balance, the final pressure changes but the sensitivity to parameter K_aff is unaffected. More specifically, with control activation being equal, the final pressure tends to increase, unbalancing the afference toward the cardiopulmonary side. After this sensitivity analysis, one can conclude that K_v, K_r, and K_aff represent the parameters that maximally influence the pressure response, making the assignment of the remaining model parameters does not seem to be critical. As a rational consequence, model identification can only be based on the estimation of the control parameters K_v, K_r, and K_aff, using a like set for the other parameters. ⁸

This analysis significantly demonstrates that when both venous capacitance and peripheral resistance regulations are absent (K_v =K_r = 0) arterial pressure is largely sensitive to blood volume reduction: a very small blood volume decrease (<6%) is sufficient to induce acute hypotension (see Figure 8). Sensitivity analysis also shows that the pressure response has a greater sensitivity to changes in parameter K_v than to changes in parameter K_r (compare Figure 10 with Figure 9). When active capacitance vessel constriction is not involved in the regulatory process (K_v = 0), although peripheral resistance regulation is active (K_r = 0.5), the pressure response displays a continuous trend toward hypotension (Figure 10). When, instead, peripheral resistance regulation is off (K_r = 0), the pressure decrease is limited as long as the compensation in venous volume is adequate (Figure 9). This result is in agreement with the hypothesis that the regulation of capacitance vessels is the major component of hypovolemic compensation. ^11,12

Some authors have emphasized the role of peripheral arteriole regulation in sustaining arterial pressure during hypovolemia. ^13,14 In particular, it has been suggested that the progressive arterial pressure decrease observed late during the hemodialysis session, in some instances, may be caused by an impairment of this mechanism. Actually, we found that, when the control of peripheral resistance alone is active (K_r = 1, K_v = 0), the final pressure falls to 85 mm Hg, thus decreasing by at least 25 mm Hg relative to baseline. Hence, if we hypothesize as being unique compensatory mechanisms those based on increase in peripheral resistance, it is impossible to account for the data collected during hemodialysis where arterial pressure remains stable or increases throughout the treatment.

During simulated hypovolemia, we also considered the changes in cardiac output. A significant reduction in cardiac output (about 48%) was computed in the case of inhibited venous capacity regulation (K_v = 0), whereas cardiac output remained substantially unchanged (differences < 10%) in the case of K_v different from zero. This result suggests that venous capacity regulation plays a pivotal role in sustaining cardiac output during hemodialysis; indeed, the more efficient the control the more limited the cardiac output reduction, thus avoiding an arterial pressure decrease toward lower values. This model prediction closely agrees with the results of Nakamura, ¹⁰ who observed a stable cardiac index in seven patients whose pressure response to hemodialysis was stable, while the cardiac index fell in a group of 10 patients with a hypotensive pressure response.

When the case of control completely mediated by arterial baroreceptors (K_aff = −1 in Figure 8) is simulated, response to hypovolemia appears to be less efficient than the case when both arterial and cardiopulmonary afferences are active (K_aff = 0), demonstrating the importance of the cardiopulmonary baroreflex in keeping arterial pressure high. In fact, the arterial baroreflex alone cannot avoid a slight decrease in arterial pressure and, by the end of the simulated treatment, the pressure is 4.5% less than the initial value. As a matter of fact, a paradoxical increase in arterial pressure can be observed during hemodialysis. On the grounds of the current sensitivity analysis, to have such an increase in arterial pressure, there has to be a prevalence of cardiopulmonary over arterial baroreflex control (K_aff = 1, see Figures 8–10). The reflex originating in cardiopulmonary baroreceptors can cope with hypovolemia through strong sympathetic vasoconstriction, which causes a significant decrease in unstressed venous volume and a consequent paradoxical increase in arterial pressure. The hyperefficient regulatory action due to cardiopulmonary baroreceptors is instead limited by the vasodilation reflex due to the loading of arterial baroreceptors (compare middle and right panels in Figure 8). Under these conditions (K_aff = 0), the two afferences operate competitively, so the arterial baroreflex is enabled and avoids an arterial pressure increase.

It is also interesting to observe that, when the controls are maximally active (K_v =K_r = 1), and afferences unbalanced toward the cardiopulmonary side (K_aff = 1, see Figure 8), arterial pressure increases, while cardiac output decreases (about 4.3%). Hence, it is not necessary to hypothesize a cardiac output increase to justify the pressure increase observed during hemodialysis induced hypovolemia.

The study presented here has been based upon the simulation of dialysis treatments with a constant blood volume loss rate (about 95 ml/h), producing a 10% reduction in the initial blood volume after 4 hours. Thus, we have not considered the typically swift change in vascular refilling occurring in the course of the treatment. To assess the influence of the rate of blood volume reduction, we also simulated a 10% change in initial blood volume in just 1 hour and no significant change in pressure response versus blood volume loss curve was observed.

The simulation of a 10% blood volume reduction was sufficient to highlight differences in pressure response in respect to different sets of control parameters. However, by the end of a hemodialysis treatment, blood volume changes > 10% (as much as 20%) were frequently observed. ^4,8,15,16 To determine whether a larger hypovolemia stress can cause hypotension, a dialysis treatment producing a 20% blood volume reduction after 4 hours was also simulated for two different degrees of effectiveness in the regulation of peripheral resistance and unstressed venous volume. With both controls fully active (K_v =K_r = 1), the arterial pressure showed no substantive changes, and with both controls moderately active (K_v =K_r = 0.5), only a slight decrease in the arterial pressure compared with the initial value (−10.3%) was computed. Thus, according to the results presented in Figure 10, when the regulatory mechanisms are operative, the pressure response remains stable even given a larger blood volume loss (20%). However, examples of pressure responses to large blood volume losses have been reported in a previous study. ⁸

In the present analysis, inotropic cardiac regulation was not included to emphasize the role of peripheral and venous vasoconstriction alone. This condition represents the worst case: indeed, when the inotropic cardiac regulation is active as well, the increase in pumping capability sustains the arterial pressure through an increased cardiac output. For instance, a 15% increase in heart rate during the simulated dialysis causes a higher final pressure (from + 3.5% in the case of inhibited controls to + 18.7% when the controls are maximally active), with respect to the case of unchanged heart rate.

In the sensitivity analysis, no critical cardiac alteration was considered, and the simulated changes in cardiac parameters cause little perturbations in cardiac pump performance. Actually, it is well known that cardiac performance dysfunction in the course of treatment may significantly alter pressure stability. ^17,18

It is worth pointing out that the analysis is limited to the effects of parameter variations on pressure response but does not test the effect of the choice of the initial condition. Moreover, an important limit to this study is that the computer model considers the cardiovascular functions as regulated solely through the arterial and cardiopulmonary baroreflex. Actually, more inter-related mechanisms interfere with the baroreceptor mediated change in vascular tone. As an example, large modifications in ionic concentrations or pH occurring in the course of treatment may directly affect the baroreflex response, as well as vascular tone and cardiac pump performance. ¹⁹

Conclusions

On the basis of our analysis, we can conclude that circulatory parameters, such as resistances and compliances, have a limited effect on the pressure response to hemodialysis induced hypovolemia. Conversely, venous capacity regulation seems to play a pivotal role in sustaining arterial pressure. The regulation of systemic peripheral resistance exerts a compensatory action only as long as the blood volume reduction is < 5%, but it is inadequate to sustain arterial pressure after larger blood volume reduction when venous capacity regulation is absent. Paradoxical arterial pressure increases during hemodialysis induced hypovolemia can be referred to a prevalence of cardiopulmonary afferences in the regulatory process.

Glossary

C_a: Compliance of arterial systemic circulation (1 ml/mm Hg)

C_m: Compliance of peripheral systemic circulation (9.2 ml/mm Hg)

CO: Cardiac output (ml/s)

Co_m: Maximum cardiac output at the beginning of simulation (154.4 ml/s)

C_ra: Compliance of right atrium (33.12 ml/mm Hg)

C_v: Compliance of venous systemic circulation (20.6 ml/mm Hg)

f: Heart rate (bpm)

f_e: Heart rate at the beginning of simulation (72 bpm)

K_aff: Parameter that balances arterial and cardiopulmonary afferent pathway (—1 - 1)

K_co: Efficiency of inotropic regulation of heart (0 - 1)

K_r: Efficiency of static regulation of peripheral resistance (0 - 1)

K_v: Efficiency of static regulation of unstressed venous volume (0 - 1)

P_a: Mean arterial pressure (mm Hg)

P_ae: Mean arterial pressure at the beginning of simulation (110 mm Hg)

P_m: Microcirculation pressure (mm Hg)

P_ra: Right atrial pressure (mm Hg)

P_ran: Parameter that determines the sensitivity of cardiac output to right atrial pressure (2.5 mm Hg)

P_ras: Right atrial pressure for null cardiac output (0 mm Hg)

P_v: Venous pressure (mm Hg)

R_m: Inflow resistance in peripheral systemic compartment, see Figure 3 (mm Hg*s/ml)

R_me: Inflow resistance in peripheral systemic compartment at the beginning of simulation (1.2 mm Hg*s/ml)

R_ra: Inflow resistance in right atrium (0.2 mm Hg*s/ml)

R_v: Inflow resistance in venous systemic compartment (0.5 mm Hg*s/ml)

T: Static vasomotor efferent tone

T_b: Dynamic vasomotor efferent tone

V_au: Unstressed volume of arterial systemic circulation (493 ml)

V_mu: Unstressed volume of peripheral systemic circulation (102 ml)

V_rau: Unstressed volume of right atrium (ml)

V_tote: Total systemic circulatory blood volume at the beginning of simulation (3,807 ml)

V_vu: Unstressed volume of venous systemic circulation, see Figure 3 (ml)

V_vue: Unstressed volume of venous systemic circulation at the beginning of simulation (1,700 ml)