Phosphate Kinetics During Weekly Cycle of Hemodialysis Sessions: Application of Mathematical Modeling

Phosphate kinetic modeling as an estimate of daily ingested phosphate in hemodialysis patients with or without residual kidney function

BACKGROUND AND AIM

Daugirdas suggested a 2-pool phosphate kinetic model based on his previously established urea kinetic model. The current study aims to assess the level of agreement between the modeled daily ingested phosphorus (DIP) values and the routine method of dietary recall calculations in hemodialysis patients.

METHOD

The study was conducted on 100 hemodialysis patients; 50 were anuric, and the others had residual kidney function (RKF). The level of correlation and agreement between the dietary calculated and modeled DIP were assessed in both study groups.

RESULTS

A statistically significant positive correlation existed between the calculated and modeled DIP (r = 0.79 for the anuric group, r = 0.84 for the RKF group, p < 0.001). There was a significant level of agreement between calculated and modeled DIP in RKF patients only.

CONCLUSION

These findings suggest that phosphate modeling can estimate phosphate intake in RKF patients and be cost-effective in their management.

Evaluation of a phosphate kinetics model in hemodialysis therapy-Assessment of the temporal robustness of model predictions

In-depth understanding of intra- and postdialytic phosphate kinetics is important to adjust treatment regimens in hemodialysis. We aimed to modify and validate a three-compartment phosphate kinetic model to individual patient data and assess the temporal robustness. Intradialytic phosphate samples were collected from the plasma and dialysate of 12 patients during two treatments (HD1 and HD2). 2-h postdialytic plasma samples were collected in four of the patients. First, the model was fitted to HD1 samples from each patient to estimate the mass transfer coefficients. Second, the best fitted model in each patient case was validated on HD2 samples. The best model fits were determined from the coefficient of determination (R2 ) values. When fitted to intradialytic samples only, the median (interquartile range) R2 values were 0.985 (0.959-0.997) and 0.992 (0.984-0.994) for HD1 and HD2, respectively. When fitted to both intra- and postdialytic samples, the results were 0.882 (0.838-0.929) and 0.963 (0.951-0.976) for HD1 and HD2, respectively. Eight patients demonstrated a higher R2 value for HD2 than for HD1. The model seems promising to predict individual plasma phosphate in hemodialysis patients. The results also show good temporal robustness of the model. Further modifications and validation on a larger sample are needed.

Analytical solution of phosphate kinetics for hemodialysis

Chronic kidney diseases imply an ongoing need to remove toxins, with hemodialysis as the preferred treatment modality. We derive analytical expressions for phosphate clearance during dialysis, the single pass (SP) model corresponding to a standard clinical hemodialysis and the multi pass (MP) model, where dialysate is recycled and therefore makes a smaller clinical setting possible such as a transportable dialysis suitcase. For both cases we show that the convective contribution to the dialysate is negligible for the phosphate kinetics and derive simpler expressions. The SP and MP models are calibrated to clinical data of ten patients showing consistency between the models and provide estimates of the kinetic parameters. Immediately after dialysis a rebound effect is observed. We derive a simple formula describing this effect which is valid both posterior to SP or MP dialysis. The analytical formulas provide explanations to observations of previous clinical studies.

Bayesian parameter estimation for phosphate dynamics during hemodialysis

Hyperphosphatemia in patients with renal failure is associated with increased vascular calcification and mortality. Hemodialysis is a conventional treatment for patients with hyperphosphatemia. Phosphate kinetics during hemodialysis may be described by a diffusion process and modeled by ordinary differential equations. We propose a Bayesian model approach for estimating patient-specific parameters for phosphate kinetics during hemodialysis. The Bayesian approach allows us to both analyze the full parameter space using uncertainty quantification and to compare two types of hemodialysis treatments, the conventional single-pass and the novel multiple-pass treatment. We validate and test our models on synthetic and real data. The results show limited identifiability of the model parameters when only single-pass data are available, and that the Bayesian model greatly reduces the relative standard deviation compared to existing estimates. Moreover, the analysis of the Bayesian models reveal improved estimates with reduced uncertainty when considering consecutive sessions and multiple-pass treatment compared to single-pass treatment.

Recent Advances and Future Perspectives in the Use of Machine Learning and Mathematical Models in Nephrology

We reviewed some of the latest advancements in the use of mathematical models in nephrology. We looked over 2 distinct categories of mathematical models that are widely used in biological research and pointed out some of their strengths and weaknesses when applied to health care, especially in the context of nephrology. A mechanistic dynamical system allows the representation of causal relations among the system variables but with a more complex and longer development/implementation phase. Artificial intelligence/machine learning provides predictive tools that allow identifying correlative patterns in large data sets, but they are usually harder-to-interpret black boxes. Chronic kidney disease (CKD), a major worldwide health problem, generates copious quantities of data that can be leveraged by choice of the appropriate model; also, there is a large number of dialysis parameters that need to be determined at every treatment session that can benefit from predictive mechanistic models. Following important steps in the use of mathematical methods in medical science might be in the intersection of seemingly antagonistic frameworks, by leveraging the strength of each to provide better care.

Acid-base kinetics during hemodialysis using bicarbonate and lactate as dialysate buffer bases based on the H+ mobilization model

BACKGROUND

The H⁺ mobilization model has been recently reported to accurately describe intradialytic kinetics of plasma bicarbonate concentration; however, the ability of this model to predict changing bicarbonate kinetics after altering the hemodialysis treatment prescription is unclear.

METHODS

We considered the H⁺ mobilization model as a pseudo-one-compartment model and showed theoretically that it can be used to determine the acid generation (or production) rate for hemodialysis patients at steady state. It was then demonstrated how changes in predialytic, intradialytic, and immediate postdialytic plasma bicarbonate (or total carbon dioxide) concentrations can be calculated after altering the hemodialysis treatment prescription.

RESULTS

Example calculations showed that the H⁺ mobilization model when considered as a pseudo-one-compartment model predicted increases or decreases in plasma total carbon dioxide concentrations throughout the entire treatment when the dialysate bicarbonate concentration is increased or decreased, respectively, during conventional thrice weekly hemodialysis treatments. It was further shown that this model allowed prediction of the change in plasma total carbon dioxide concentration after transfer of patients from conventional thrice weekly to daily hemodialysis using both bicarbonate and lactate as dialysate buffer bases.

CONCLUSION

The H⁺ mobilization model can predict changes in plasma bicarbonate or total carbon dioxide concentration during hemodialysis after altering the hemodialysis treatment prescription.

Hemodialysis-induced changes in hematocrit, hemoglobin and total protein: Implications for relative blood volume monitoring

Background

Relative blood volume (RBV) changes during hemodialysis (HD) are typically estimated based on online measurements of hematocrit, hemoglobin or total blood protein. The aim of this study was to assess changes in the above parameters during HD in order to compare the potential differences in the RBV changes estimated by individual methods.

Methods

25 anuric maintenance HD patients were monitored during a 1-week conventional HD treatment. Blood samples were collected from the arterial dialysis blood line at the beginning and at the end of each HD session. The analysis of blood samples was performed using the hematology analyzer Advia 2120 and clinical chemistry analyzer Advia 1800 (Siemens Healthcare).

Results

During the analyzed 30 HD sessions with ultrafiltration in the range 0.7–4.0 L (2.5 ± 0.8 L) hematocrit (HCT) increased by 9.1 ± 7.0% (mean ± SD), hemoglobin (HGB) increased by 10.6 ± 6.3%, total plasma protein (TPP) increased by 15.6 ± 9.5%, total blood protein (TBP) increased by 10.4 ± 5.8%, red blood cell count (RBC) increased by 10.8 ± 7.1%, while mean corpuscular red cell volume (MCV) decreased by 1.5 ± 1.1% (all changes statistically significant, p < 0.001). HGB increased on average by 1.5% more than HCT (p < 0.001). The difference between HGB and TBP increase was insignificant (p = 0.16).

Conclusions

Tracking HGB or TBP can be treated as equivalent for the purpose of estimating RBV changes during HD. Due to the reduction of MCV, the HCT-based estimate of RBV changes may underestimate the actual blood volume changes.

A two-pool kinetic model predicts phosphate concentrations during and shortly following a conventional (three times weekly) hemodialysis session

Background

Previous studies have suggested that a conventional two-pool model cannot be used to predict intradialysis and early postdialysis phosphorus concentrations.

Methods

A conventional two-pool urea model was modified by increasing the distal compartment volume from two-thirds to three times the total body water and by the use of a dynamically variable intercompartmental phosphorus clearance during dialysis. The phosphate solver model parameters were derived from an examination of the results in the literature, and fine-tuned using a training set (F4) of 415 Hemodialysis (HEMO) Study patients studied during a dialysis session where phosphorus was measured at 4 months of follow-up. Validation was done in a group of 380 different HEMO Study patients plus 9 from the original F4 group, who were evaluated at 36 months of follow-up.

Results

The model predicted measured median early (1 h) intradialysis, end-dialysis and 30-min postdialysis serum phosphorus levels in the test and validation datasets with little apparent bias, including the highest and lowest deciles of predialysis serum phosphorus. The model tended to underestimate slightly intradialysis serum phosphorus when predialysis serum phosphorus was <3.0 mg/dL (0.97 mmol/L). There was a large scatter and standard deviation among patients, and whether aberrant values represent a patient-specific phenomenon is unclear.

Conclusions

A modified two-pool model using a slightly expanded distal compartment and a dynamically varying intercompartmental clearance, depending on the intradialysis phosphorus concentration, can be used to predict serum phosphorus level during and shortly after dialysis, in patients following a conventional three times per week dialysis prescription.

Volume of urea cleared as a therapy dosing guide for more frequent hemodialysis

INTRODUCTION

With dialysis delivery systems that operate at low dialysate flow rates, prescriptions for more frequent hemodialysis (HD) employ dialysate volume as the primary parameter for small solute removal rather than blood-side urea dialyzer clearance (K). Such delivery systems, however, yield dialysate concentrations that almost completely saturate with blood (water), suggesting that the volume of urea cleared (the product of K and treatment time or Kt) can be readily estimated from the prescribed dialysate volume to target small solute removal. Methods For more frequent HD, we examined the volume of urea cleared per treatment required to achieve a minimal dose of small solute removal, comparing results based on body surface area (BSA) with those based on KDOQI clinical practice guidelines, that is, a weekly stdKt/V of 2.1. Estimates of the target volume of urea cleared were calculated for 4, 5, and 6 treatments per week, and compared for patients with different anthropometric estimates of total body water volume (V_ant ). BSA was assumed proportional to V_ant ^0.8 , and residual kidney function was neglected. Findings Whether based on BSA or weekly stdKt/V of 2.1, the target volume of urea cleared per treatment required to achieve a minimal dose of small solute removal was lower at higher treatment frequency. As with conventional thrice-weekly HD, target volumes of urea cleared for more frequent HD based on BSA were larger for patients with small V_ant and smaller for patients with large V_ant than those based on a weekly stdKt/V of 2.1. Discussion Prescription of more frequent HD using the volume of urea cleared per treatment, calculated from the prescribed dialysate volume, is simple in principle and can be readily implemented in clinical practice when using dialysis delivery systems that operate at low dialysate flow rates. Other aspects of dialysis adequacy require additional consideration.

Phosphate Kinetic Models in Hemodialysis: A Systematic Review

BACKGROUND

Understanding phosphate kinetics in dialysis patients is important for the prevention of hyperphosphatemia and related complications. One approach to gain new insights into phosphate behavior is physiologic modeling. Various models that describe and quantify intra- and/or interdialytic phosphate kinetics have been proposed, but there is a dearth of comprehensive comparisons of the available models. The objective of this analysis was to provide a systematic review of existing published models of phosphate metabolism in the setting of maintenance hemodialysis therapy.

STUDY DESIGN

Systematic review.

SETTING & POPULATION

Hemodialysis patients.

SELECTION CRITERIA FOR STUDIES

Studies published in peer-reviewed journals in English about phosphate kinetic modeling in the setting of hemodialysis therapy.

PREDICTOR

Modeling equations from specific reviewed studies.

OUTCOMES

Changes in plasma phosphate or serum phosphate concentrations.

RESULTS

Of 1,964 nonduplicate studies evaluated, 11 were included, comprising 9 different phosphate models with 1-, 2-, 3-, or 4-compartment assumptions. Between 2 and 11 model parameters were included in the models studied. Quality scores of the studies using the Newcastle-Ottawa Scale ranged from 2 to 11 (scale, 0-14). 2 studies were considered low quality, 6 were considered medium quality, and 3 were considered high quality.

LIMITATIONS

Only English-language studies were included.

CONCLUSIONS

Many parameters known to influence phosphate balance are not included in existing phosphate models that do not fully reflect the physiology of phosphate metabolism in the setting of hemodialysis. Moreover, models have not been sufficiently validated for their use as a tool to simulate phosphate kinetics in hemodialysis therapy.

A Pseudo-One Compartment Model of Phosphorus Kinetics During Hemodialysis: Further Supporting Evidence

A pseudo-one compartment model has been proposed to describe phosphorus kinetics during hemodialysis and the immediate post-dialysis period. This model assumes that phosphorus mobilization from tissues is proportional to the difference between the pre-dialysis serum concentration (a constant) and the instantaneous serum concentration. The current study is exploratory and evaluated the ability of a pseudo-one compartment model to describe the kinetics of phosphorus during two short hemodialysis treatments separated by a 60-min inter-treatment period without dialysis; the latter is the post-dialysis rebound period for the first short hemodialysis treatment. Serum was collected frequently during both hemodialysis treatments and the inter-treatment period to assess phosphorus kinetics in 21 chronic hemodialysis patients. Phosphorus mobilization clearance and pre-dialysis central distribution volume were previously estimated for each patient during the first hemodialysis treatment and the inter-treatment period. Assuming those kinetic parameters remained constant for each patient, serum phosphorus concentrations during the second treatment were used to estimate the driving force concentration (C_df ) for phosphorus mobilization from tissues during the second treatment. Treatment time (117 ± 14 [mean ± standard deviation] vs. 117 ± 14 min), dialyzer phosphorus clearance (151 ± 25 vs. 140 ± 32 mL/min), and net fluid removal (1.44 ± 0.74 vs. 1.47 ± 0.76 L) were similar during both short hemodialysis treatments. Measured phosphorus concentration at the start of the second hemodialysis treatment (3.3 ± 0.9 mg/dL) was lower (P < 0.001) than at the start of the first treatment or C_pre (5.4 ± 1.9 mg/dL). Calculated C_df was 4.9 ± 2.0 mg/dL, not significantly different from C_pre (P = 0.12). C_df and C_pre were correlated (R = 0.72, P < 0.001). The results from this study demonstrate that the driving force concentration for phosphorus mobilization during hemodialysis is constant and not different from that pre-dialysis, providing further evidence supporting a fundamental assumption of the pseudo-one compartment model.

Artificial Organs 2015: A Year in Review

In this Editor's Review, articles published in 2015 are organized by category and briefly summarized. We aim to provide a brief reflection of the currently available worldwide knowledge that is intended to advance and better human life while providing insight for continued application of technologies and methods of organ Replacement, Recovery, and Regeneration. As the official journal of The International Federation for Artificial Organs, The International Faculty for Artificial Organs, the International Society for Rotary Blood Pumps, the International Society for Pediatric Mechanical Cardiopulmonary Support, and the Vienna International Workshop on Functional Electrical Stimulation, Artificial Organs continues in the original mission of its founders "to foster communications in the field of artificial organs on an international level." Artificial Organs continues to publish developments and clinical applications of artificial organ technologies in this broad and expanding field of organ Replacement, Recovery, and Regeneration from all over the world. We take this time also to express our gratitude to our authors for providing their work to this journal. We offer our very special thanks to our reviewers who give so generously of their time and expertise to review, critique, and especially provide meaningful suggestions to the author's work whether eventually accepted or rejected. Without these excellent and dedicated reviewers, the quality expected from such a journal could not be possible. We also express our special thanks to our Publisher, John Wiley & Sons for their expert attention and support in the production and marketing of Artificial Organs. We look forward to reporting further advances in the coming years.

Quantification of Dialytic Removal and Extracellular Calcium Mass Balance during a Weekly Cycle of Hemodialysis

Objectives

The removal of calcium during hemodialysis with low calcium concentration in dialysis fluid is generally slow, and the net absorption of calcium from dialysis fluid is often reported. The details of the calcium transport process during dialysis and calcium mass balance in the extracellular fluid, however, have not been fully studied.

Methods

Weekly cycle of three dialysis sessions with interdialytic breaks of 2-2-3 days was monitored in 25 stable patients on maintenance hemodialysis with calcium concentration in dialysis fluid of 1.35 mmol/L. Total and ionic calcium were frequently measured in blood and dialysate. The volume of fluid compartments was measured by bioimpedance.

Results

Weekly dialytic removal of 12.79 ± 8.71 mmol calcium was found in 17 patients, whereas 9.48 ± 8.07 mmol calcium was absorbed per week from dialysis fluid in 8 patients. Ionic calcium was generally absorbed from dialysis fluid, whereas complexed calcium (the difference of total and ionic calcium in dialysis fluid) was removed from the body. The concentration of total calcium in plasma increased slightly during dialysis. The mass of total and ionic calcium in extracellular fluid decreased during dialysis in patients with the dialytic removal of calcium from the body and did not change in patients with the absorption of calcium from dialysis fluid.

Conclusions

We conclude that about one third of patients on dialysis with calcium 1.35 mmol/L in dialysis fluid may absorb calcium from dialysis fluid and therefore individual prescriptions of calcium concentration in dialysis fluid should be considered for such patients.