Title: Modeling and Estimation for the Renal System
Description: Understanding how a therapy will impact the injured kidney before being administered would be an asset to the clinical world. The work in this thesis advances the field of mathematical modeling of the kidneys to aid in this cause. The objectives of this work are threefold: 1) to develop and personalize a model to specific patients in different diseased states, via parameter estimation, in order to test therapeutic trajectories, 2) to use parameter estimation to understand the cause of different kidney diseases, differentiate between potential kidney diseases, and facilitate targeted therapies, and 3) to push forward the understanding of kidney physiology via physiology-based mathematical modeling techniques. To accomplish these objectives, we have developed two models of the kidneys: 1) a broad, steady-state, closed-loop model of the entire kidney with human physiologic parameters, and 2) a detailed, dynamic model of the proximal tubule, an important part of kidney, with rat physiologic parameters.

To readily aid physicians, a human model would easily fit into the clinical workflow. Since there is a lack of invasive human renal data for validation and parameter estimation, we employ a minimal modeling approach. However, to aid in deeper understanding of renal function for future applications, targeted therapy testing, and potentially replace invasive measures, we develop a more detailed model. The development of such a model requires invasive data for validation and parameter estimation, and hence we model for rodents, where such invasive data are more readily available. The kidneys are composed of approximately one million functional units known as nephrons. The glomerular filtration rate (GFR) is the rate at which the kidney filters blood at the start of the nephron. This filtration rate is highly regulated via several control mechanisms and needs to be maintained within a small range in order to maintain a proper water and electrolyte balance. Hence, fluctuations of GFR are indicative of overall kidney health.

In developing the human kidney model, we also sought to understand the relationship between blood pressure and GFR since many therapies affect blood pressure and subsequently GFR. This model describes steady-state conditions of the entire kidney, including renal autoregulation. Model validation is performed with experimental data from healthy subjects and severely hypertensive patients. The baseline model’s GFR simulation for normotensive and the manually tuned model’s GFR simulation for hypertensive intensive care unit patients had low root mean squared errors (RMSE) of 13.5 mL/min and 5 mL/min, respectively. These values are both lower than the error of 18 mL/min in GFR estimates, reported in previous studies. It has been shown that vascular resistance and renal autoregulation parameters are altered in severely hypertensive stages, and hence, a sensitivity analysis is conducted to investigate how changes in these parameters affect GFR. The results of the sensitivity analysis reinforce the fact that vascular resistance is inversely related to GFR and show that changes to either vascular resistance or renal autoregulation cause a significant change in sodium concentration in the descending limb of Henle.

This is an important conclusion as it quantifies the mapping between hypertension parameters and two important kidney states, GFR and sodium urine levels. Glomerulonephritis is one of the two major intra-renal kidney diseases, characterized as a breakdown at the site of the glomerulus that affects GFR and subsequently other portions of the nephron. This disease accounts for 15% of all kidney injuries and one-fourth of end-stage renal disease patients. The human kidney model is used to estimate renal parameters of patients with glomerulonephritis. The model is an implicit system and in developing an optimization algorithm to use for parameter estimation, we modify in a novel way, the Levenberg-Marquardt optimization using the implicit function theorem in order to calculate the Jacobian and Hessian matrices needed. We further adapt the optimization algorithm to work for constrained optimization since our parameter values must be physiologically feasible within a certain range. The parameter estimation method we use is a three-step process: 1) manually adjusting parameters for the hypertension comorbidity, 2) iteratively estimating parameters that vary from person to person using no-kidney- injury (NKI) data, and 3) iteratively estimating parameters that are affected by glomerulonephritis using labeled diseased data.

Such a process generates a model that is personalized to each given patient. This patient-specific model can then be used to simulate and evaluate outcomes of potential therapies (e.g., vasodilators) on the model in lieu of the patient, and observe how alterations in blood pressure or sodium level affect renal function. Parameter estimation in the presence of glomerulonephritis is a challenging task due to the complexity of the kidney physiology and the number of parameters to estimate. This is further complicated by comorbidities such as hypertension, cardiac arrythmia, and valvular disease, because they alter kidney physiology and hence, increase the number of parameters to estimate. We chose to focus on hypertension since it is very prevalent in hospitals and intensive care units. It was found that over all patients, average model estimates of GFR and urine output rate (UO) were within 9.2 mL/min and 0.71 mL/min for NKI data. These results are expectedly better than those achieved from the non-personalized model since the parameters are now specific to each patient.

The results also demonstrate our ability to non-invasively estimate GFR with less error than the 18 mL/min currently possible. The estimations were validated by ensuring that the estimated parameter values were physiologically sound and matched the literature in terms of expected values for different demographic groups. It is vital for a properly functioning kidney to maintain solute transport throughout the nephron. Kidney diseases in the nephron can manifest themselves via the solute transport mechanisms. To understand how these diseases affect the kidney and to simulate transporter- targeting therapies, we have developed a detailed model, starting from the human model previously developed, of one portion of the kidneys, the proximal tubule. The proximal tubule is the site of the most active transport within the nephron and the target for several therapies. Our goal is to study and understand the dynamic behavior of the proximal tubule when solute transporters breakdown and to investigate treatment therapies targeting certain solute transporters.

The proposed model is dynamic and includes several solutes’ transport mechanisms, with parameters for rats. We chose to investigate diabetic nephropathy and the associated sodium-transporter alteration (knockout) therapy. Diabetic nephropathy is characterized as kidney damage due to diabetes and affects 30% of diabetics. In terms of reducing hyperfiltration, a potential cause of diabetic nephropathy where an overabundance of solutes and fluid are filtered at the glomerulus, the model demonstrates that knockout of this transporter results in a reduction in sodium and chloride reabsorptions in the proximal tubule, thereby preventing hyperfiltration. Further, we conclude that vital flows for maintaining kidney homeostasis, fluid and ammonium reabsorptions, are corrected to healthy values by a 50% knockout (impairment) of the sodium-hydrogen transporter. Next, we use the dynamic model to detect different diseased states of the proximal tubule transporters. We have accomplished this task by using Bayesian estimation to estimate transporter density parameters (a metric for kidney health) using measured signals from the proximal tubule.

This approach is validated with experimental rat data, while further investigations are conducted into the performance of the estimation in the presence of varied input signals, signal resolutions, and noise levels. Estimation accuracy within 20% of true transporter density and within 4% of true fluid and solute reabsorption was achieved for all combinations of diseased transporters. We concluded that including chloride and bicarbonate concentrations improved estimation accuracy, whereas including formic acid did not. This is an important conclusion as it can help physicians determine which blood tests to order for diagnosing kidney disease; to our knowledge, this is a first. It was also found that sodium and glucose proximal tubule concentrations are most affected by changes in the sodium-hydrogen and sodium-bicarbonate transporters. This conclusion provides insight into the interplay between solute transporter density and sodium and glucose concentrations in the proximal tubule. Such knowledge paves the way for new transporter targeted therapies.