Mathematical model of hemodynamic mechanisms and consequences of glomerular hypertension in diabetic mice

Many preclinically promising therapies for diabetic kidney disease fail to provide efficacy in humans, reflecting limited quantitative translational understanding between rodent models and human disease. To quantitatively bridge interspecies differences, we adapted a mathematical model of renal function from human to mice, and incorporated adaptive and pathological mechanisms of diabetes and nephrectomy to describe experimentally observed changes in glomerular filtration rate (GFR) and proteinuria in db/db and db/db UNX (uninephrectomy) mouse models. Changing a small number of parameters, the model reproduced interspecies differences in renal function. Accounting for glucose and Na+ reabsorption through sodium glucose cotransporter 2 (SGLT2), increasing blood glucose and Na+ intake from normal to db/db levels mathematically reproduced glomerular hyperfiltration observed experimentally in db/db mice. This resulted from increased proximal tubule sodium reabsorption, which elevated glomerular capillary hydrostatic pressure (Pgc) in order to restore sodium balance through increased GFR. Incorporating adaptive and injurious effects of elevated Pgc, we showed that preglomerular arteriole hypertrophy allowed more direct transmission of pressure to the glomerulus with a smaller mean arterial pressure rise; Glomerular hypertrophy allowed a higher GFR for a given Pgc; and Pgc-driven glomerulosclerosis and nephron loss reduced GFR over time, while further increasing Pgc and causing moderate proteinuria, in agreement with experimental data. UNX imposed on diabetes increased Pgc further, causing faster GFR decline and extensive proteinuria, also in agreement with experimental data. The model provides a mechanistic explanation for hyperfiltration and proteinuria progression that will facilitate translation of efficacy for novel therapies from mouse models to human.

. Sensitivity of simulation results to τnephronLoss -the time constant governing the effect glomerular pressure on nephron loss. As this time constant is decreased, nephrons are lost more rapidly, causing GFR to decrease and glomerular pressure to increase more quickly as wel. Figure S3. Sensitivity of simulation results to τalbumin -the time constant governing the effect glomerular pressure on damage to the protein sieving membrane. This parameter does not affect GFR or glomerular pressure, but as this time constant gets smaller, the UAER response to increased glomerular pressure increases.

Renal Vasculature
The glomeruli are modeled in parallel, and in series with the preafferent (interlobar, interlobular, and arcuate arterioles) and peritubular vasculature. Glomerular capillary resistance is assumed negligible. Thus, renal vascular resistance RVR is given by: Rpreaff and Rperitubular are lumped resistances describing the total resistance of preafferent and peritubular vasculatures, respectively, while Raa and Rea are the resistances of a single afferent or efferent arteriole, as determined from Pouiselle's law, based on the arteriole's diameter d, length L, and blood viscosity µ: Eq. A2 Nnephrons is the number of nephrons. All nephrons are assumed identical, and the model does not account for spatial heterogeneity.
Renal blood flow (RBF) is a function of the pressure drop across the kidney and RVR, according to Ohm's law: Eq. A3 Renal venous pressure (Prenal-vein) is treated as constant. The second term in this equation accounts for lower flow through the efferent arterioles due to GFR. As an approximation, all filtrate is assumed reabsorbed back into the peritubular capillaries, so that peritubular flow is the same as afferent flow. , = min ( , , ) Eq. 5 Any non-reabsorbed glucose is then excreted, so that the rate of urinary glucose excretion (RUGE) is:

Eq. 6
Na + filtration and reabsorption in the PT Similarly to glucose, Na + is freely filtered across the glomerulus, so that the single nephron-filtered Na + load is given by: where CNa is the plasma Na + concentration.
Assuming that glucose reabsorption through SGLT1 is small compared to reabsorption through SGLT2, the rate of Na + reabsorption through SGLT2 is approximately equal to the rate of glucose reabsorption: , − 2 = , Eq. 8 Total PT Na + reabsorption is then given by: Eq. 9 where ηpt,non-SGLT2 is the fractional rate of PT reabsorption through mechanisms other than SGLT2. Na + flow rate out of the PT is then: Eq. 10 Na + reabsorption along the rest of the tubule is modeled as in [12].

Water Reabsorption along the tubule
Water reabsorption in the PT is isosmotic. Therefore, water leaving the PT and entering the loop

Eq. A18
In the loop of Henle (LoH), water is reabsorbed in the water permeable descending LoH (DLH) due to the osmotic gradient created by actively pumping sodium out of the water-impermeable ascending limb (ALH). We have described modeling of this countercurrent mechanism previously(23), but only accounted for the osmotic effects of sodium. Here we adapt that description to include the osmotic activity of both sodium and glucose. The osmolality along the length of the DLH OsmDLH, which is assumed in equilibrium with the osmolality in the surrounding interstitium OsmIS, is given by: , − Eq. A19 Where RALH is the rate of sodium reabsorption per unit length in the ascending loop of Henle (ALH), as described previously (our reference here). Water flow through the DLH is then given by: ( )

Eq. A20
The ALH and the distal convoluted tubule (DCT) are modeled as impermeable to water, so that the flow through these segments equals the flow out of the DLH: In the collecting duct (CD), water reabsorption is driven by the osmotic gradient between the CD tubular fluid and the interstitium, and is modulated by vasopressin: Φ water,reabs− CD = Φ water,CD (0) * (1 − ( ) ( ) ) Eq. A22 Where the osmolality in the CD OsmCD(L) accounts for sodium reabsorbed in the collecting duct: ( ) = Φ osm,cd (0)−2 * (Φ Na,cd (0)− Φ Na,cd ( )) Φ water,CD (0) Eq. A23 The effect of vasopressin on water reabsorption is modeled as a sigmoidal relationship, such that ηvasopressin is one and water flows freely across the osmotic gradient when vasopressin levels are very high, and ηvasopressin is zero and no water is reabsorbed when vasopressin in absent.

Peripheral Sodium Storage
We incorporated that three compartment model of volume homeostasis into the renal physiology model, to allow evaluation of the potential role of peripheral sodium storage in the renal response to dapagliflozin.
Sodium and water are assumed to move freely between the blood and interstitial fluid. Water and sodium intake rates were assumed constant. Then blood volume (BV) and blood sodium (Nablood) are the balance between intake and excretion of water and sodium respectively, and the intercompartmental transfer.
Eq. A28 Sodium concentrations in the blood and interstitial compartments are assumed to equilibrate quickly. Change in interstitial fluid volume (IFV) is a function of intercompartmental water transfer.

Cardiovascular Function
The mean cardiac filling pressure (Pmf) is a function of blood volume and venous compliance cvenous.

Modeling Tubular Hydrostatic Pressure
Hydrostatic pressure in the Bowman's space is a key factor affecting GFR, and this pressure is influenced by both morphology and flow rates through the tubule. Changes in Na and water reabsorption along the nephron, which can occur either due to disease or treatments, can alter GFR by altering tubular pressures.
Thus dynamically modeling tubular pressures can be critical to understanding GFR changes.
Adapting from Jensen et al(16), tubular flow rates described in the main text can be used to determine tubular pressure. The change in intratubular pressure dP * over a length of tubule dx can be defined according to Poiseuille's law as: Eq. A39 Eq. 22 describes the relationship between transtubular pressure P and tubular diameter D, where Dc is the diameter at control pressure Pc, and β is the exponent of tubular distensibility (16).

Eq. A40
Substituting and assuming uniform interstitial pressure throughout the kidney, we obtain: Integrating over a tubule segment length, we obtain inlet pressure as a function of the outlet pressure and the flow rate:

Eq A42
The pressure calculated at the inlet to the PT is used as PBow in Eq. 4 above.
Because the diameter of the CNT/CD changes as nephrons coalescence, calculating pressure along this segment is challenging. Under normal conditions, pressure drops 5-7mmHg across the CNT/CD (16). Thus, an effective control diameter was calculated to give this degree of pressure drop under baseline conditions.

Modeling Glomerular Capillary Oncotic Pressure
The glomerular capillary oncotic pressure is calculated using the Landis Pappenheimer equation, where Cprot is the concentration of protein at the point of interest. π = 1.629 * C prot + 0.2935 * C prot 2 Eq. A43 Plasma protein (Cprot-plasma) is assumed constant. Protein concentration at the distal end of the glomerulus (Cprot-glom-out) is determined as: Protein concentration is assumed to be varying linearly along the capillary length, and thus the oncotic pressure − is calculated using the average of the plasma protein concentration and protein concentration at the distal end of the glomerulus.
The model does not account for filtration equilibrium, which occurs in some species.

Modeling Regulatory Mechanisms
Multiple control mechanisms act on the system to allow simultaneous control of Cna, CO, MAP, glomerular pressure, and RBF. For each control mechanism, the feedback signal µ is modeled by one of two functional forms. The choice of functional form is determined by whether a steady state error is allowed in the controlled variable X. When a steady state error is not allowed (i.e. X always eventually returns to the setpoint X0), the effect is defined by a proportional-integral (PI) controller. The initial feedback signal is proportional to the magnitude of the error (X-X0), with gain G. But the feedback continues to grow over time as long as any error exists, until the error returns to zero. The integral gain Ki determines the speed of return to steady-state.
= 1 + * (( − 0 ) + * ∫( − 0 ) ) Eq. 45 All other mechanisms, for which the controlled variable can deviate from the setpoint at steady-state, are described by a logistic equation that produces a saturating response characteristic of biological signals: Here, m defines the slope of the response around the operating point, and S is the maximal response as X goes to ±∞.

Control of plasma Na concentration by vasopressin
Changes in plasma osmolality are sensed via osmoreceptors, stimulating vasopressin secretion, which exerts control of water reabsorption in the CNT/CD. To insure that blood sodium concentration CNa is maintained at its setpoint CNa,0 at steady state, this process is modeled by a PI controller: μ vasopressin = 1 + G Na−vp * (C Na + K i−vp * ∫(C Na − C Na,0 )dt) Eq. 47 The parameters GNa-vp and Ki-vp are gains of proportional and integral control, respectively.

Tubular Pressure Natriuresis
For homeostasis, Na excretion over the long-term must exactly match Na intake (the principle of Na balance). Any steady-state Na imbalance would lead to continuous volume retention or loss-an untenable situation. Pressure-natriuresis(2), wherein changes in renal perfusion pressure (RPP) induce changes in Na excretion, insures that Na balance is maintained. It may be partially achieved through neurohumoral mechanisms including the RAAS, but there is also an intrinsic pressure-mediated effect on tubular Na reabsorption, where renal interstitial hydrostatic pressure (RIHP) is believed to be the driving signal. RIHP is a function of peritubular capillary pressure, and is calculated according to Ohm's law: As a simplification, we assume an increase in peritubular pressure will generate a proportional increase in RIHP. Since the kidney is encapsulated, we assume interstitial pressure equilibrates and changes in one region are transduced across the kidney. The relationship between RIHP and fractional Na reabsorption rate of each tubular segment is then modeled as: Eq. 49 where i = PT, LoH, DCT, or CNT/CD. − ,0 is the nominal fractional rate of reabsorption for that tubule segment. RIHP0 defines the setpoint pressure and is determined from RIHP at baseline for normal Na intake. SP-N,i defines the maximal signal as RIHP goes to ∞.

Control of Cardiac Output
CO, which describes total blood flow to body tissues, returns to normal over days to weeks following a perturbation (38). CO regulation is a complex phenomenon that occurs over multiple time scales, but we focus only on long-term control (days to weeks), which is thought to be achieved through whole-body autoregulation -the intrinsic ability of organs to adjust their resistance to maintain constant flow(38). The total effect of local autoregulation of all organs is that TPR is adjusted to maintain CO at a constant resting level. The feedback between CO and TPR is modeled with a PI controller, such that CO is controlled to its steady-state setpoint CO0. TPR = TPR 0 * (1 + G CO−tpr * (CO + K i−tpr * ∫(CO − CO 0 )dt)) Eq. 50

Control of Macula Densa Sodium Concentration by Tubuloglomerular Feedback
Tubuloglomerular feedback (TGF) helps stabilize tubular flow by sensing Na concentration in the the macula densa, which sits between the LoH and DCT, and providing a feedback signal to inversely change afferent arteriole diameter. The TGF effect is defined as: The basal afferent arteriole resistance Raa is then multiplied by μTGF to obtain the ambient afferent arteriolar resistance. The setpoint CNa,MD,0 is the Na concentration out of the LoH and into the DCT in the baseline state at normal Na intake.

Myogenic Autoregulation of Glomerular Pressure
Glomerular hydrostatic pressure is normally tightly autoregulated, and changes very little in response to large changes in blood pressure. This autoregulation is in part through myogenic autoregulation of the preglomerular arterioles. While the pressure drop and thus myogenic response varies along the arteriole length, we make the simplifying assumption that the preafferent vasculature responds to control pressure at the distal end.
Pressure at the distal end of the preafferent vasculature is given by: The basal preafferent arteriole resistance Rpreaff is then multiplied by μautoreg to obtain the ambient preafferent arteriolar resistance.

Renin-Angiotensin-Aldosterone System Submodel
Renin is secreted at a nominal rate SECren,0 modulated by macula densa sodium flow, as well as by a strong negative feedback from Angiotensin bound to the AT1 receptor.
Eq. A54 The macula densa releases renin in response to reduced sodium flow: Eq. A55 We have found that the inhibitory effect of AT1-bound AngII on renin secretion can be well described by the following relationship: Eq. A56 Plasma renin concentration (PRC) is then given by: Where Krenin Kd,renin is the renin degradation rate. PRA can be related to PRC by the conversion factor 0.06 (ng/ml/hr)/(pg/ml).
Angiotensin I is formed by PRA, assuming that its precursor angiotensinogen is available in excess and the plasma renin activity (PRA) is the rate-limiting step. AngI is also converted to AngII by the enzymes ACE and chymase, and is degraded at a rate of Kd,AngI.
Angiotensin II is formed from the action of ACE and chymase on AngI, can be eliminated by binding to either the AT1 or AT2 receptors at the rate CAT1 and CAT2 respective, and is degraded at a rate of Kd,AngII.
The complex of Angiotensin II bound to the AT1 receptor is the physiologically active entity within the pathway, and is given by: AT1-bound AngII has multiple physiologic effects, including constriction of the efferent, as well and preglomerular, afferent, and systemic vasculature, sodium retention in the PT, and aldosterone secretion. Each effect is modeled as: where i represents the effect on efferent, afferent, preafferent, or systemic resistance, PT sodium reabsorption, or aldosterone secretion.
Aldosterone is the second physiologically active entity in the RAAS pathway, acting by binding to mineralocorticoid receptors (MR) in the CNT/CD and DCT to stimulate sodium reabsorption. MR-bound aldosterone is modeled as the nominal concentration Aldo,0 modulated by the effect of AT1-bound AngII, and the normalized availability of MR receptors (1 in the absence of an MR antagonist).
The effects of MR-bound aldosterone on CNT/CD and DCT sodium reabsorption are modeled as: Where i is the CNT/CD or DCT.

Calculation of Dependent Model Parameter Values Required for Steady State
Both peritubular vascular resistance (Rperitubular,0) and CNT/CD fractional reabsorption (ηCNT/CD,0) are difficult to measure precisely, and a wide range of values have been reported in the literature, primarily from non-human sources. For these parameters, also, we can instead calculate what their values must be to give specified steady-state values for more easily measureable parameters. Rperitubular,0 can be calculated based on the RVR0 calculated as given above, and reported values for afferent, efferent, and preglomerular resistances, which are known with greater certainty: Eq. A66 Similarly, ηCNT/CD,0 can be calculated for a given equilibrium GFR, plasma Na concentration, and Na intake, as well as upstream fractional Na reabsorption rates, based on the fact that at equilibrium, Na excretion must equal Na intake (Φsodin). Na flow out of each segment can be calculated as: Na flow out of the CNT/CD must equal sodium intake: Then, solving for ηcnt/cd,0 gives: There is substantial variability in experimental measurements of fractional Na reabsorption rates for each tubular segments. PT fractional reabsorption has been reported across the range of 40-78% (40-44). Under normal flow conditions, LoH fractional Na reabsorption has been reported from 85-93%(23, 45). For this analysis, fractional Na reabsorption rates in the PT, LoH, and DCT were set to 0.70, 0.88, and 0.5, respectively, and CNT/CD fractional reabsorption rate was calculated to be 0.827.
# Assuming that reabsorption rates are known in all but one segment of the tubule, the exact rate # of reabsorption of the remaining segment can be calculated. We chose to calculate the CD rate of reabsorpion based on estimates for # PT, LoH, and DT reabsorption.