Modelling Transcapillary Transport of Fluid and Proteins in Hemodialysis Patients

Optical monitoring of hemodialysis using noninvasive measurement of uric acid in the dialysate

The aim of this study was to present a methodology for predicting changes in uric acid concentrations in the blood of chronically hemodialyzed patients based on an optical measurement of the intensity of selected wavelengths in the dialysate. Blood samples were taken from the arterial line every 30 min throughout the hemodialysis period, to measure uric acid levels. Simultaneously, optical measurements were made on dialysate flowing from the dialyzer. Uric acid concentration can be measured either directly from the blood or from dialyzer outflow with acceptable error. In addition, both methods reveal any increased dynamics in uric acid concentration in the initial phase of hemodialysis. The wavelength of the light was adjusted for optimal uric acid particle detection. Comparing the uric acid concentration measured in the blood of patients with the intensity of wave absorption in the dialysate, the functional relationship between the uric acid concentration levels was determined. Using the optical method for measuring uric acid concentration in the dialysate, the concentration of uric acid in the blood during hemodialysis can be non-invasively and accurately estimated. This method can be used to assess the adequacy of hemodialysis by computer acquisition of uric acid concentrations determined in on-line dialysate.

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.

Impact of the nature of the capillary wall on plasma refilling during hemodialysis

OBJECTIVES

Our aim was to clarify the impact of the nature of the capillary wall, defined by the contribution of large (LP), small (SP), and ultrasmall (UP) pores, on plasma refilling in a hemodialysis session.

METHODS

This study included data from 78 patients. The relative blood volume change (ΔBV%) was monitored using a Crit-Line monitor. A bioimpedance device was used to measure extracellular and intracellular fluid volumes, and the excess fluid mass (M_ExF) was calculated. We simulated blood volume change (sΔBV%) based on a three-pore model. Hydraulic permeability of the capillary wall (LpS) and fractional contribution of LP to LpS (α_LP) were determined by fitting sΔBV to ΔBV. The total refilling volume (TV_ref) was calculated from the total ultrafiltration volume and total blood volume change. Values were standardized to a body surface area of 1.73 m² and are denoted by the subscript BSA.

RESULTS

LpS and α_LP were 3.09 (2.32, 4.68) mL/mmHg/min and 0.069 (0.023, 0.109), respectively. The standardized regression coefficient (β) of the ultrafiltration rate (UFR_BSA) and initial excess fluid mass (M_ExF,_BSA,0) by multiple linear regression analysis of TV_ref,BSA without (Model 1) and with (Model 2) α_LP were as follows: UFR_BSA, 0.714/<0.001 (β/p); M_ExF,_BSA,0, 0.247/<0.001 (Model 1); UFR_BSA, 0.799/<0.001; M_ExF,_BSA,0, 0.066/0.237; and α_LP, -0.327/<0.001 (Model 2).

CONCLUSIONS

The impact of volume overload (M_ExF,_BSA,0) on plasma refilling became insignificant with the addition of α_LP in the model, suggesting that the nature of the capillary wall described by inter-endothelial gaps (LP) may have a greater impact on plasma refilling than volume overload.

Using a Human Circulation Mathematical Model to Simulate the Effects of Hemodialysis and Therapeutic Hypothermia

Background: We developed a hemodynamic mathematical model of human circulation coupled to a virtual hemodialyzer. The model was used to explore mechanisms underlying our clinical observations involving hemodialysis. Methods: The model consists of whole body human circulation, baroreflex feedback control, and a hemodialyzer. Four model populations encompassing baseline, dialysed, therapeutic hypothermia treated, and simultaneous dialysed with hypothermia were generated. In all populations atrial fibrillation and renal failure as co-morbidities, and exercise as a treatment were simulated. Clinically relevant measurables were used to quantify the effects of each in silico experiment. Sensitivity analysis was used to uncover the most relevant parameters. Results: Relative to baseline, the modelled dialysis increased the population mean diastolic blood pressure by 5%, large vessel wall shear stress by 6%, and heart rate by 20%. Therapeutic hypothermia increased systolic blood pressure by 3%, reduced large vessel shear stress by 15%, and did not affect heart rate. Therapeutic hypothermia reduced wall shear stress by 15% in the aorta and 6% in the kidneys, suggesting a potential anti-inflammatory benefit. Therapeutic hypothermia reduced cardiac output under atrial fibrillation by 12% and under renal failure by 20%. Therapeutic hypothermia and exercise did not affect dialyser function, but increased water removal by approximately 40%. Conclusions: This study illuminates some mechanisms of the action of therapeutic hypothermia. It also suggests clinical measurables that may be used as surrogates to diagnose underlying diseases such as atrial fibrillation.

Transcapillary transport of water, small solutes and proteins during hemodialysis

The semipermeable capillary walls not only enable the removal of excess body water and solutes during hemodialysis (HD) but also provide an essential mechanism for maintaining cardiovascular homeostasis. Here, we investigated transcapillary transport processes on the whole-body level using the three-pore model of the capillary endothelium with large, small and ultrasmall pores. The transcapillary transport and cardiovascular response to a 4-h hemodialysis (HD) with 2 L ultrafiltration were analyzed by simulations in a virtual patient using the three-pore model of the capillary wall integrated in the whole-body compartmental model of the cardiovascular system with baroreflex mechanisms. The three-pore model revealed substantial changes during HD in the magnitude and direction of transcapillary water flows through small and ultrasmall pores and associated changes in the transcapillary convective transport of proteins and small solutes. The fraction of total capillary hydraulic conductivity attributed to ultrasmall pores was found to play an important role in the transcapillary water transport during HD thus influencing the cardiovascular response to HD. The presented model provides a novel computational framework for a detailed analysis of microvascular exchange during HD and as such may contribute to a better understanding of dialysis-induced changes in blood volume and blood pressure.

A Whole-Body Mathematical Model of Sepsis Progression and Treatment Designed in the BioGears Physiology Engine

Sepsis is a debilitating condition associated with a high mortality rate that greatly strains hospital resources. Though advances have been made in improving sepsis diagnosis and treatment, our understanding of the disease is far from complete. Mathematical modeling of sepsis has the potential to explore underlying biological mechanisms and patient phenotypes that contribute to variability in septic patient outcomes. We developed a comprehensive, whole-body mathematical model of sepsis pathophysiology using the BioGears Engine, a robust open-source virtual human modeling project. We describe the development of a sepsis model and the physiologic response within the BioGears framework. We then define and simulate scenarios that compare sepsis treatment regimens. As such, we demonstrate the utility of this model as a tool to augment sepsis research and as a training platform to educate medical staff.

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 quantitative in silico platform for simulating cytotoxic and nanoparticle drug delivery to solid tumours

The role of tumour-host mechano-biology and the mechanisms involved in the delivery of anti-cancer drugs have been extensively studied using in vitro and in vivo models. A complementary approach is offered by in silico models, which can also potentially identify the main factors affecting the transport of tumour-targeting molecules. Here, we present a generalized three-dimensional in silico modelling framework of dynamic solid tumour growth, angiogenesis and drug delivery. Crucially, the model allows for drug properties-such as size and binding affinity-to be explicitly defined, hence facilitating investigation into the interaction between the changing tumour-host microenvironment and cytotoxic and nanoparticle drugs. We use the model to qualitatively recapitulate experimental evidence of delivery efficacy of cytotoxic and nanoparticle drugs on matrix density (and hence porosity). Furthermore, we predict a highly heterogeneous distribution of nanoparticles after delivery; that nanoparticles require a high porosity extracellular matrix to cause tumour regression; and that post-injection transvascular fluid velocity depends on matrix porosity, and implicitly on the size of the drug used to treat the tumour. These results highlight the utility of predictive in silico modelling in better understanding the factors governing efficient cytotoxic and nanoparticle drug delivery.

Model of fluid and solute shifts during hemodialysis with active transport of sodium and potassium

Background

Mathematical models are useful tools to predict fluid shifts between body compartments in patients undergoing hemodialysis (HD). The ability of a model to accurately describe the transport of water between cells and interstitium (J_v,ISIC), and the consequent changes in intracellular volume (ICV), is important for a complete assessment of fluid distribution and plasma refilling. In this study, we propose a model describing transport of fluid in the three main body compartments (intracellular, interstitial and vascular), complemented by transport mechanisms for proteins and small solutes.

Methods

The model was applied to data from 23 patients who underwent standard HD. The substances described in the baseline model were: water, proteins, Na, K, and urea. Small solutes were described with two-compartment kinetics between intracellular and extracellular compartments. Solute transport across the cell membrane took place via passive diffusion and, for Na and K, through the ATPase pump, characterized by the maximum transport rate, Jp_MAX. From the data we estimated Jp_MAX and two other parameters linked to transcapillary transport of fluid and protein: the capillary filtration coefficient Lp and its large pores fraction α_LP. In an Expanded model one more generic solute was included to evaluate the impact of the number of substances appearing in the equation describing J_v,ISIC.

Results

In the baseline model, median values (interquartile range) of estimated parameters were: Lp: 11.63 (7.9, 14.2) mL/min/mmHg, α_LP: 0.056 (0.050, 0.058), and Jp_MAX: 5.52 (3.75, 7.54) mmol/min. These values were significantly different from those obtained by the Expanded model: Lp: 8.14 (6.29, 10.01) mL/min/mmHg, α_LP: 0.046 (0.038, 0.052), and Jp_MAX: 16.7 (11.9, 25.2) mmol/min. The relative RMSE (root mean squared error)averaged between all simulated quantities compared to data was 3.9 (3.1, 5.6) %.

Conclusions

The model was able to accurately reproduce most of the changes observed in HD by tuning only three parameters. While the drop in ICV was overestimated by the model, the difference between simulations and data was less than the measurement error. The biggest change in the estimated parameters in the Expanded model was a marked increase of Jp_MAX indicating that this parameter is highly sensitive to the number of species modeled, and that the value of Jp_MAX should be interpreted only in relation to this factor.

Numerical simulations of the microvascular fluid balance with a non-linear model of the lymphatic system

Fluid homeostasis is required for life. Processes involved in fluid balance are strongly related to exchanges at the microvascular level. Computational models have been presented in the literature to analyze the microvascular-interstitial interactions. As far as we know, none of those models consider a physiological description for the lymphatic drainage-interstitial pressure relation. We develop a computational model that consists of a network of straight cylindrical vessels and an isotropic porous media with a uniformly distributed sink term acting as the lymphatic system. In order to describe the lymphatic flow rate, a non-linear function of the interstitial pressure is defined, based on literature data on the lymphatic system. The proposed model of lymphatic drainage is compared to a linear one, as is typically used in computational models. To evaluate the response of the model, the two are compared with reference to both physiological and pathological conditions. Differences in the local fluid dynamic description have been observed using the non-linear model. In particular, the distribution of interstitial pressure is heterogeneous in all the cases analyzed. The resulting averaged values of the interstitial pressure are also different, and they agree with literature data when using the non-linear model. This work highlights the key role of lymphatic drainage and its modeling when studying the fluid balance in microcirculation for both to physiological and pathological conditions, e.g. uremia.

Impact of hemodialysis on cardiovascular system assessed by pulse wave analysis

Valuable information about cardiovascular system can be derived from the shape of aortic pulse wave being the result of reciprocal interaction between heart and vasculature. Pressure profiles in ascending aorta were obtained from peripheral waveforms recorded non-invasively (SphygmoCor, AtCor Medical, Australia) before, during and after hemodialysis sessions performed after 3-day and 2-day interdialytic intervals in 35 anuric, prevalent hemodialysis patients. Fluid status was assessed by Body Composition Monitor (Fresenius Medical Care, Bad Homburg, Germany) and online hematocrit monitoring device (CritLine, HemaMetrics, Utah). Systolic pressure and ejection duration decreased during dialysis. Augmentation index remained stable at 30 ± 13% throughout hemodialysis session despite the decrease of augmented pressure and pulse height. Subendocardial viability ratio (SEVR) determined after 3-day and 2-day interdialytic intervals increased during the sessions by 43.8 ± 26.6% and 26.1 ± 25.4%, respectively. Hemodialysis performed after 3-day and 2-day interdialytic periods reduced significantly overhydration by 2.4 ± 1.0 L and 1.8 ± 1.2 L and blood volume by 16.3 ± 9.7% and 13.7 ± 8.9%, respectively. Intradialytic increase of SEVR correlated with ultrafiltration rate (R = 0.39, p-value < 0.01), reduction in overhydration (R = -0.57, p-value < 0.001) and blood volume drop (R = -0.38, p-value < 0.01). The strong correlation between the decrease of overhydration during hemodialysis and increase in SEVR confirmed that careful fluid management is crucial for proper cardiac function. Hemodialysis affected cardiovascular system with the parameters derived from pulse-wave-analysis (systolic and augmented pressures, pulse height, ejection duration, SEVR) being significantly different at the end of dialysis from those before the session. Combination of pulse-wave-analysis with the monitoring of overhydration provides a new insight into the impact of hemodialysis on cardiovascular system.

In-silico dynamic analysis of cytotoxic drug administration to solid tumours: Effect of binding affinity and vessel permeability

The delivery of blood-borne therapeutic agents to solid tumours depends on a broad range of biophysical factors. We present a novel multiscale, multiphysics, in-silico modelling framework that encompasses dynamic tumour growth, angiogenesis and drug delivery, and use this model to simulate the intravenous delivery of cytotoxic drugs. The model accounts for chemo-, hapto- and mechanotactic vessel sprouting, extracellular matrix remodelling, mechano-sensitive vascular remodelling and collapse, intra- and extravascular drug transport, and tumour regression as an effect of a cytotoxic cancer drug. The modelling framework is flexible, allowing the drug properties to be specified, which provides realistic predictions of in-vivo vascular development and structure at different tumour stages. The model also enables the effects of neoadjuvant vascular normalisation to be implicitly tested by decreasing vessel wall pore size. We use the model to test the interplay between time of treatment, drug affinity rate and the size of the vessels’ endothelium pores on the delivery and subsequent tumour regression and vessel remodelling. Model predictions confirm that small-molecule drug delivery is dominated by diffusive transport and further predict that the time of treatment is important for low affinity but not high affinity cytotoxic drugs, the size of the vessel wall pores plays an important role in the effect of low affinity but not high affinity drugs, that high affinity cytotoxic drugs remodel the tumour vasculature providing a large window for the normalisation of the vascular architecture, and that the combination of large pores and high affinity enhances cytotoxic drug delivery efficiency. These results have implications for treatment planning and methods to enhance drug delivery, and highlight the importance of in-silico modelling in investigating the optimisation of cancer therapy on a personalised setting.

One of the main challenges in optimising cancer therapy is understanding the in-vivo cancer environment and how it changes over time. The efficacy of chemotherapeutic drugs is known to be strongly dependent on blood vessel wall properties and the architecture of the developing tumour vasculature, which in turn are dependent on biochemical and mechanical interactions between cancer cells and their microenvironment. Here we present a novel in-silico modelling framework of dynamic tumour growth, angiogenesis and drug delivery, and we use it to explore biophysical factors governing the efficient delivery of cytotoxic drugs to solid tumours. We find that the time of treatment and vessel permeability are important factors for the efficacy of chemical agents with low binding affinity, that high affinity drugs can impact the tumour vasculature remodelling and bring vascular structure back to a more normalised state, and that the combination of large-sized vessel wall pores and high affinity enhances cytotoxic drug delivery and efficacy. These results have implications for treatment planning and optimisation, and show how in-silico models can be used to help understand and optimise cancer therapy.

Does the plasma refilling coefficient change during hemodialysis sessions?

The filtration coefficient in the Starling equation is an important determinant of plasma refilling during hemodialysis. A method for calculating from clinical data an estimate of the filtration coefficient, called the refilling coefficient, was proposed in the past. The assumption behind this method was that the only drive for refilling is the increase in plasma oncotic pressure, and the remaining Starling forces have negligible effect. The refilling coefficient was observed to decrease during hemodialysis, and this was interpreted as a change in the filtration coefficient. The purpose of our study was providing an alternative explanation for the behavior of the refilling coefficient and, using clinical data and mathematical modeling, to predict the values of the immeasurable Starling forces and provide the theoretical basis for the interpretation of the refilling coefficient as the filtration coefficient. Blood volume and bioimpedance data from 23 patients undergoing hemodialysis were used to calculate the refilling coefficient according to the original formulation and to fit a two-compartment model of protein and fluid transport. The changes in the other Starling forces were non-negligible, ranging from 19% to 60% of plasma oncotic pressure. The results showed that the decrease observed in the refilling coefficient is likely caused by neglecting important changes in the Starling forces while deriving the equation for the refilling coefficient. When these Starling forces were taken into account, constant filtration coefficient and dynamic refilling coefficient provided an equivalent description of the data in most cases. However, this was not true for a subgroup of sessions, which suggests that additional factors may also be responsible for the observed decrease in the refilling coefficient.