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Home , Distal convoluted tubule, Glomerulus (kidney), Loop of Henle, Nephron, Peritubular capillaries, Proximal tubule

Modeling Tori Turkington, Jordan Mattheisen, Cinzia Ballantyne, & Caroline Martel 1. The The is an essential organ in vertebrates, involved in removing waste products from the body. In accomplishing this goal, the kidney maintains in levels, acid/base balance, and salt/ balance. The kidney itself has many parts that serve different purposes, but the large part of the kidney can be understood through an explanation of the nephron. The nephron is the structure that is responsible for filtering water, wastes, and substances such as or from the body and moving them on to redistribute them where appropriate, whether that be where those substances are needed or preparing them for . One kidney itself is made up of about one million .

Figure 1.1: A model of the nephron with labeled parts.

Blood enters the afferent and moves to the , where filter out water and small molecules. From there, fluid moves to the proximal convoluted , where sodium is actively reabsorbed, along with water due to the osmotic gradient that is present. Following that is the , where the sodium concentrations in the fluid start to change. The Loop of Henle is made up of both an ascending and descending limb. In the permeable descending limb, sodium flows in and water flows out, concentrating the amount of sodium in the fluid. Concentration is highest at the end of this process, where the descending limb meets the ascending limb. Sodium is actively pumped out in the impermeable ascending limb, without water following. Because this process moves against a concentration gradient, this is an active process that requires metabolic energy. The end result of this is removal of concentrated or diluted , as well as other fluid that gets circulated throughout the body. Many of the functions in the nephron can be explored via mathematical models, and these concepts will be presented in the following sections.

2. Dynamics of Sodium and Water: Transport Along Renal Let the volume rate of flow along the tubule past � be given by � � . In this model we will assume that the flow is steady and independent of time. The volume flow rate through the walls of the tubule per unit length near � is given by �(�). A positive � will be in the direction of increasing � and positive � will represent the outward flux through the walls of the tubules.

We will begin by considering a segment of tubule that lies between the arbitrary points � = � and � = � as shown in the diagram below to derive the equations for sodium and water transport along the walls in the renal tubule.

Figure 2.1: Diagram of the sodium ion and water transport system (Hoppensteadt)

We define the following variables as: � = distance along the tubule � � = concentration of the sodium ion inside the tubule at position x � � = flow of water (volume per unit time) along tubule at x in the direction of increasing x

� � = outward transport of water (volume per unit time per length of tubule) across the walls of the tubule

� � = outward transport of sodium ions (number of ions per unit time per length of tubule) across the walls of the tubule

Since we are asumming the flow through the tubule is steady, the volume of the segment cannot change. Therefore the flow into the segment must equal the sum of the flow out of the segment and the flux of water out through the walls of the tubule between � = � and � = �. Using the variables we defined earlier, we can rewrite this as: (2.1) � � = � � + � � ��

Keeping � fixed and treating � like a variable, we can differentiate with respect to � getting �� 0 = (�) + � � , (2.2) �� which can be written as �� 0 = + � � , (2.3) �� since � is an arbitrary number.

Using this equation, we see that if the walls of the tubules are impermeable to water,

� = 0 and � is a constant. Considering the same section of tubule, we can let the sodium concentration in the tubule be �(�). Therefore, the amount of sodium ion per unit time transported along the tubule by the flow past the point � is � � �(�). Let the amount of sodium ion per unit time per unit length transported outward through the walls of the tubule be �(�). Using the equation derived above, we argue that � 0 = �� + � � . (2.4) ��

In a tubule whose walls are impermeable to sodium ion, � = 0 and �� is a constant. These previous two equations apply to all tubules in the kidneys although the properties of their walls will be different as reflected by their fluxes � and �.

3. The Loop of Henle We start with the Loop of Henle to construct a model of the nephron. Using the diagram below as reference, the descending limb of the loop is tube 1 and the ascending limb of the loop is tube 2. The concentration of sodium ion is �(�) and the water flow in the tubules are �(�) where � = 1,2. The external concentration of sodium ion is � � . Let the flow be positive in the descending limb and negative in the ascending limb.

Figure 3.1: Schematic diagram of the Loop of Henle. The descending limb is on the left side (called tube 1) and the ascending limb is on the right side (called tube 2). (Hoppensteadt)

1) The first simplifying assumption we will make is that the walls of the descending limb are permeable to water but not to sodium ions. Although this is not entirely true in the body, we will ignore the slight permeability to sodium ions since it is not essential in the function of the Loop of Henle. Furthermore, we will assume that the permeability to water is so large that the flux makes the internal and external concentrations of sodium ion equal. This gives us the following equations: �� 0 = + � � , (3.1) �� � 0 = � � , (3.2) ��

� � = � � . (3.3) 2) Next we assume that sodium ions are pumped from the ascending limb at a ∗ steady rate � per unit time and that the ascending is impermeable to water. This gives: 0 = ��/�� (3.4)

∗ 0 = (�/��)(��) + � (3.5)

3) At the turn in the Loop of Henle given when � = �, we assume that all sodium ions and water leaving the descending limb enter the ascending limb. This gives us

� � = �(�) (3.6)

� � = −� � . (3.7) 4) Additionally, we account for the where sodium ions are actively pumped from the ascending limb and water passively flows from the descending limb of the loop. Making the assumption that the capillaries pick up sodium ions locally and since the model is in steady state, we name the rate at which the peritubular capillaries pick up water as �(�) and the rate at which ∗ they pick up sodium ions as �. We assume that the driving force for the reverse is due to , a form of osmotic pressure exerted by proteins in the surrounding plasma. The proteins are abundant in the peritubular capillaries because they were held back during the process of filtration by the glomerulus. The reverse filtration process that occurs at the peritubular capillaries allows sodium ion to passively be carried by water. The flux of sodium ions and the flow of water is related by the equation:

∗ � = � � � � . (3.8) Using these assumptions and rewriting equations 3.1, 3.2, 3.3, and 3.8, we can derive a differential equation for the interstitial concentration � � .

∗ 0 = (��/��) + �/� � , (3.9)

0 = (�/��) + (��), (3.10) where �� is a constant. To get � � in terms of � � , we have the equation:

� � = � 0 �(0)/� � . (3.11)

Here, � 0 is the sodium ion concentration of the fluid entering the Loop of Henle via the which is the same as the sodium ion concentration in the plasma so it can be considered a given value. � 0 is the volume rate of flow entering the Loop of Henle which is less than the filtration rate by the fluid reabsorbed in the proximal tubule per unit time. From above, we also get the equation �� �� ��/�� � = −� = −� 0 �(0) , (3.12) �� �� �(�) which can be written as: �� �∗ � = . (3.13) �� (� 0 � 0 ) This implies that

�∗ � � � = � 0 ��� , (3.14) � 0 � 0 So

�∗ � � � = � 0 ��� , (3.15) � 0 � 0

∗ where �� is the total rate that the sodium ions are actively pumped out of the walls in the ascending limb in the Loop of Henle and � 0 � 0 is the rate that the sodium ions enter the loop from the proximal tubules.

�∗ � � = < 1 (3.16) � 0 � 0

The ratio of these fluxes can be renamed as α which determines the maximum sodium ion concentrating ability of the nephron given through the equation � � = � 0 exp � . (3.17) Considering the ascending limb of the Loop of Henle which we assumed was impermeable to water, we get

� � = � � = −� � = −� 0 � 0 /�(�) = −� 0 exp (−�). (3.18)

Since � can be thought of as a constant, equation 3.5 becomes: �� �∗ �∗ exp (�) = = . (3.19) �� −� �(0)

Therefore,

∗ � � = � � + � − � � exp � /� 0 . (3.20)

Because � � is equal to equation 3.17..

∗ � � = � 0 exp � + � − � � exp � /�(0), (3.21) Specifically when � = 0,

∗ � 0 = � 0 exp � + 0 − � � exp � /� 0 , (3.22) which simplifies to

� 0 = � 0 exp � 1 − � . (3.23)

We see that exp � 1 − � < 1 when � ≠ 0. � 0 < � 0 so the fluid leaving the top of the ascending limb of the Loop of Henle is more dilute than the . The nephron, however, actually adjusts the �(0) to achieve a certain �(0) at the top of the ascending limb. This requires that �(0) be expressed in terms of �(0) which is treated as a parameter. 4. The Juxtaglomerular Apparatus The juxtaglomerular apparatus (depicted in Figure 4.1 (Marieb)) describes the region of the renal system adjacent to the top of ascending loop of Henle and the afferent arteriole. Three types of cells play important regulatory functions within the apparatus; the first to note are the cells. This group detects changes in sodium concentration of the filtrate as it travels through the ascending limb of the loop of Henle. The second group of cells, granular cells, are enlarged smooth muscles cells that act as mechanoreceptors. These cells sense in the afferent arteriole and secrete the in response to a decrease in blood pressure. The third and final type of sends regulatory signals between the two aforementioned types and are called extraglomerular mesangial cells.

Figure 4.1: An anatomical diagram of the juxtaglomerular complex and the glomerulus (Marieb). Once renin enters the blood it is converted into ; a that constricts blood vessels. This constriction of blood vessels is important to elevate blood pressure throughout body (recall that this hormone was produced in response to the detection of low blood pressure by the granular cells). Overall, this system is relevant to maintain blood pressure throughout the body as well as regulate the concentration of sodium in urine. While the renin-angiotensin system details will not be modeled they are assumed to function normally as they would in a healthy individual. These assumptions are expressed in equations 4.1 and 4.2 which state that the concentration of sodium in the filtrate as it leaves the afferent arteriole is the concentration that the juxtaglomerular apparatus intended to accomplish.

∗ � 0 = � (4.1) � 0 > �∗ (4.2) Utilizing equations 3.23 and 4.1 we derive that � = �(1 − �) (4.3) �∗ � = < 1 (4.4) � 0 0 < � < 1 (4.5) For small values of a… � ≈ 1 (4.6) � � 1 − � ≈ = (4.7) � � With this information we can now simplify the previous section’s equations in terms of c* rather that alpha since in the equations just shown we assume alpha is small. Providing us with the series of equations 4.8 through 4.11 � � 0 = �∗ (4.8) � 0

� −� = �∗ (4.9) �� 0 � � = �� 0 (4.10) ∗ � 0 = � (4.11) The above four equations are used to model the functions of the juxtaglomerular apparatus. Equation 4.8 shows that the amount of sodium that can enter the loop during a specified amount of time is the same amount that can be pumped out over the same amount of time. Equation 4.9 indicates that the flow out of the ascending limb of the loop is smaller than the flow entering the loop of Henle by a factor of e. This is due to the water being reabsorbed throughout the descending loop of Henle (as was previously described) to conserve water. This is also due to the sodium that is being conserved by the body at this point by being pumped out on the journey of the filtrate through the ascending limb of the loop of Henle. Equation 4.10 states that the concentration of sodium inside the descending limb of the loop of Henle and outside the tubules actually increases by a factor of e from the beginning of the filtrate’s journey into the descending limb until it exits. The final equation [4.11] indicates that the final concentration of sodium, once the filtrate is prepared to be excreted is equivalent to the concentration of sodium that the juxtaglomerular apparatus intended to obtain. The vast majority of the filtrate that enters the kidney must be reabsorbed prior to excretion. This discrepancy is so large, in fact, there is a 99:1 percent : error in reabsorption ratio. The one percent of error in reabsorption actually accounts in its entirety for urine formation. This kind of ratio provides one with a sense of the importance of the juxtaglomerular complex. If there are any errors made in reabsorption, the amount of urine that could be excreted could be 100 times what was intended. Mistakes like this could result in life threatening , so the function of the juxtaglomerular apparatus is crucial to survival and to the contribution of each nephron to function in coordinated, nearly homogenous fashion. 5. The Distal Tubule and Collecting Duct: There is, of course, a method for regulating the concentration of urine. Without such a methodology, whatever fluid exited the ascending limb becomes the urine. However, when a regulating hormone (antidiuretic hormone, or ADH) is present, the amount that the urine is concentrated or diluted is regulated. ADH makes the distal tubules and the collecting duct permeable to water, unlike what was previously modeled, which causes equilibrium throughout. Therefore, the Na+ flux at the end of the collecting loop must be the same as that leaving Henle’s loop. This flux can be modeled by: ∗ ∗ ∗ −�� = ��� /(�� 0 ) (5.1)

The flow leaving the collecting loop can then be modelled by: � �∗ � � = − = �∗ ��∗/(�� 0 ) (5.2) � �

In this case, urine is more concentrated than the blood plasma, with a smaller volume flow rate. When ADH is not present, urine is more dilute than the blood plasma, with a higher volume flow rate. Because of the different flow rates, the total amount of Na+ does not change. Instead, ADH only affects the concentration of Na+ in the blood plasma.

6. Nephrons Do Better Than a Factor of E: The model for nephrons in the previous sections does not allow for the kidneys to create a urine in which the solute concentration is greater than e x the solute concentration in blood plasma. The number e is approximately equal to 2.7. We must adjust the current model because this limitation is not observed in real kidneys; however, Harold Layton discovered that an adjustment is not necessary. Rather, nephrons can work in an organized cascade manner to produce increased solute concentrations in the urine. Nephrons do this as a group as opposed to doing it by themselves. Two factors must stay true for this model to work properly: nephrons must be of different lengths and must share interstitial space. Consider: Each loop begins and ends at x=0 Each loop penetrates to a depth of x=L L = lengths of loops. We assume:

0 < L< Lmax

Lmax could be the radius of the kidney We define the entire population of Loops of Henle in which:

� � = population density function

ℒ = number of loops with L in the interval (L1, L2) � � �� ℒ = total number of loops in entire population � � �� Consider that the walls of descending limbs are permeable to water but impermeable to sodium and the walls of ascending limbs are impermeable to water and pump sodium out at a fixed rate f*Na per unit length of tubule. We can assume each loop of Henle comes with a juxtaglomerular apparatus that regulates the inflow to the loop until the concentration of Na+ at the top of the ascending limb is equal to some prescribed value c*. Note: c* < c0, where c0 is the concentration of Na+ in blood plasma. Because of all these factors we call c(x) the external concentration of Na+ ions and the concentration of Na+ in each descending limb. Nephrons of different lengths in the descending limb might have different volume flow rates regardless of depth x. This allows the following to be defined (assuming that for each : � ≤ �)

� �, � = flow rate at x in the descending limb of a loop length L

(�) �, � = flux of water at x out of a descending limb of length L With all this information we have five equations that determine the behavior of the whole � population of loops of Henle (note that �� denotes the partial derivative with respect to x):

� 0 = � (7.1) Equation 7.1 shows that fluid entering all loops have the same Na+ concentration as blood plasma �� �, � + (� ) �, � = 0 (7.2) �� ℋ 7.2 shows the preservation of volume for flows in descending limbs in each loop � (� � � �, � = 0 (7.3) �� 7.3 shows transport of Na+ in each descending limb is constant

∗ ∗ � �, � � � = �� + � �, � � (7.4) 7.4 shows Na+ equilibrium for ascending limbs

∗ (7.5) � � ( �) �, � � � �� = � � � ��

7.5 shows the steady state condition that rate of uptake of Na+ must equal the rate at which Na+ is pumped out of ascending limbs. Right side of the equation shows the ∗ number of loops of length at least x. This number is multiplied by � to get the total flux of Na+ in ascending limbs at the level x. The left side of the equation expresses the total flux of water that leaves the descending limbs at the level x. With above equations we solve for a single equation for the interstitial concentration profile � � by solving equation 7.4 for � �, � : �∗ � Q1 L, L = (7.6) � � − �∗

Then we use equation 7.3 to derive the following:

�∗ � � � � �, � = � � � �, � = (7.7) 1 − �∗/� � 1 �∗ � � �, � = (7.8) �(�) 1 − �∗/� �

Thirdly, from equation 7.2,

�� 1 �� �∗ � (� ) �, � = − �, � = (�) (7.9) ℋ �� (�(�)) �� 1 − �∗/� �

We substitute this result into equation 7.5 and get:

1 �� � � � �� (7.10) (�) ∗ = � � �� � � �� 1 − � /�(�)

Which is rewritten as follows:

� � � � �� (7.11) (log �(�)) = � � �� ( ∗ ) �� 1 − � /�(�)

If we integrate this equation from x=0 to x=x, then apply the exponential function to both sides we achieve

(7.12) � � �� � � = � exp( ��) �� � �� 1 − �∗ �(�)

There is a special case in which this equation reduces to a formula for � � when �∗ = 0 which causes � � to disappear. This gives us:

� � �� (7.13) � � = � exp( ��) �� � ��

�∗ This equation is exact only when �∗ = 0, it is almost correct when �∗ is small ( ≪ �0 1) For an example of a nephron population, consider when p(L) is the same on the whole interval from 0 to Lmax, Let p(L) = p0. This is showing a condition in which the number of loops of length at least L decreases linearly with L until that number reaches 0 at L =Lmax. We would have: (7.14) � � �� = � � − � 1 (7.15) � � � �� = � � − � 2 If you factor the difference of squares and combine the results then simplify the result knowing exp(2 log(z)) = exp(log(z2)) = z2 we will have,

� (7.16) � � = �( ) �

Notice that c(Lmax) = 4c0 which exceeds the concentrating ability of a single nephron by the factor 4/e which is equal to roughly 1.5. This is a 50% improvement. Now we must take into account the fact that water that is reabsorbed from the collecting duct affects the concentration of the interstitium. To do so we assume that the interacting nephrons with different loop lengths, that were considered above all share a single collecting duct. This single duct extends from x = 0 to x = Lmax. Now we must recall two assumptions made previously. One is that in the concentrating mode, enough water is withdrawn from the to make the concentration of Na+ the same as c0 at the distal end, where it joins the collecting duct. The second previous assumption is that the distal convoluted tubule is impermeable to Na+. These assumptions give us,

∗ (7.17) � � �, � � � �� = ��(0)

The left hand side of the above equation is the sum of the rate at which Na+ is leaving all the ascending limbs of the population of loops of Henle (with various lengths L). The right side of the equation is the rate at which Na+ enters the collecting duct. The equations for flow of Na+ and water along the collecting duct: � � � + (� ) � = 0 (7.18) �� � � � � � = 0 (7.19) ��

We modify equation 7.5 to take into account the flow of water out of the collecting duct

∗ (7.20) � � (�) �, � � � �� + �(�)(�) � = � � � ��

+ The new aspect seen in this equation [�(�)(�) � ] expresses the amount of Na picked up by the water that has left the collecting duct as that water flows through the interstitium on its way to being reabsorbed by the peritubular capillaries. This new improved model (equations 7.1-7.4) and 7.17-7.20)) can be reduced to an integral equation for the unknown interstitial Na+ concentration profile c(x). To do so we use equations 7.17-7.19) to show (fH20)3(x) in terms of c(x). Then we substitute into this the expression for Q1(L, L) found in 7.6. Now we have, 1 �� � � � �� (�∗ �(�)) � �(�) �� (7.21) � ( ∗ + ∗ = � � �� � � �� 1 − � � � 1 − � �(�)

We manipulate this equation to arrive at an improved version of 4.7.12:

(7.22) � � �� � � = � exp( ��) � � � �� (�∗ �(�)) � �(�) �� + 1 − �∗ �(�) 1 − �∗ �(�) From this equation we learn as c* → 0, the presence of the collecting duct has less and less influence on the concentrating profile of the interstitial solute. 7. Applications of Modeling the Renal Physiology Modeling the physiology of the renal system is not just interesting; it has potential to assist in diagnosis and suggest treatment for individuals with various renal . For example, in regards to the model of the juxtaglomerular apparatus, one of the descriptors of the system is that the concentration of sodium in the fluid leaving through the afferent arteriole will be equal to the concentration that is sought by the juxtaglomerular apparatus. Any functional problems occurring in this system could be lethal considering 99% of the filtrate that runs through each nephron is reabsorbed into the body. If there was any discrepancy in these aforementioned concentrations, then it is likely that dehydration would occur. In order to combat this severity of dehydration intravenous fluids would have to be administered immediately. That being said, the magnitude of this discrepancy in concentrations can help to indicate how quickly one could excrete too much sodium and water to continue living. Assuming that we do in fact know the concentration that is sought out by the juxtaglomerular apparatus, and that the concentration leaving is measurable, it is possible to estimate the length of time before an individual is lethally dehydrated. These calculations could therefore help to predict how much fluid a person requires and how quickly it must be administered to remove them from a life threatening situation. Works Cited Hoppensteadt, F., & Peskin, C. (2002). Modeling and simulation in medicine and the life sciences (2nd ed.). New York: Springer. Labeled Diagram of Nephron. (2014, April 4). Retrieved December 11, 2015, from http://www.buzzle.com/articles/labeled-diagram-of-nephron.html Marieb, E. (2006). Essentials of human & physiology (8th ed.). San Francisco: Pearson/Benjamin Cummings.