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Understanding the Human Machine - A Primer for Bioengineering - Max E. Valentinuzzi

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128

Understanding the Human Machine

resistance conductor would flow at the speed of light, which provided an important stepping stone towards the electromagnetic theory of light formulated in 1880 by James Clerk Maxwell.

2.4.4.1. Filtration

Let us assume a substance that is only filtrated (as, for example, inulin, frequently used in the determination of glomerular filtration). In such case, secretion and reabsorption are absent and the equation reduces to,

F [Px ] E[U x ] = 0

(2.89)

which corresponds to a straight line when the excreted load E[Ux] is represented as a function of the plasmatic concentration of substance Px (Figure 2.52, left upper panel, line a). The slope F is precisely a measure of the glomerular filtration rate (also called GFR). By definition of ultrafiltrated fluid, the concentration of x in plasma is equal to the concentration in the capsule fluid. Experimentally, it has been found that the inulin filtration rate is about 120 to 130 mL/min, or in the order of 180 L/day (which is abou 30 times the blood volume).

If equation (2.89) is divided through by the plasmatic concentration [Px], we obtain,

C x = F = GFR =

E[U x ]

(2.90)

[Px ]

 

which is the glomerular filtration rate and, by definition, it is the excreted load over the plasma concentration, called also the clearance of substance x. In other words, clearance of x is the excreted load of that substance per unit concentration of the same substance in plasma. The numerator is measured in mg/min and the denominator in mg/mL meaning that clearance is expressed in mL/min, thus, another common definition in medical practice and in renal physiology states that clearance is the volume of plasma that in the unit time (say, one minute) is completely cleared of the substance x. For inulin or for any substance that only is being filtered by the kidneys, Cx is constant with respect to Px (Figure 2.52, right upper panel, horizontal line a).

Study subject: Find the definition and meaning of the terms extraction and filtration fraction. They are rather common in the specialist’s jargon. They are important concepts mainly applicable in renal diseases.

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129

2.4.4.2. Secretion

Essentially, all substances are filtered by the kidneys, and some are also secreted into the tubular fluid but not reabsorbed. The general equation (2.88) becomes,

F [P ] + S[K '

] E[U

x

] = 0

(2.91)

 

x

x

 

 

 

because R = 0. If it is accepted that [Kx] = [Px], we can define

 

T '

= S[P ]

 

 

 

 

(2.92)

x

x

 

 

 

 

 

as the secreted load Figure 2.52, lower panel, line a). That straight line, as the plasma concentration increases, reaches a plateau or saturation value Tmx. Experimental curves show that the breaking point is not

Figure 2.52. THE THREE BASIC RENAL PROCESSES. Left upper panel: Excreted load as a function of the plasmatic concentration. For pure filtration (a), for secretion first (c) and filtration predominance thereafter (b), and reabsorption followed by filtration predominance (d). Rigth upper panel: Clearance as a function of the plasmatic concentration. Filtration (a), secretion (b) and reabsorption (c). At high plasmatic concentrations, all three processes tend to the same constant glomerular filtration rate, which equals the inulin filtration value. Lower panel: Secreted and reabsorbed loads as functions of the plasmatic concentration. Curve (a), secretion, and curve (b), reabsorption. Notice that, in the case of reabsorption, the maximum plasmatic concentration [Pmx] determines the breaking points of (d), excreted load, (c), clearance, and (b), reabsorbed load.

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Understanding the Human Machine

abrupt but rather there is a smooth bending to reach the maximum level (dashed line in Figure 2.52). This phenomenon is due to the spread out or splays of the individual nephron behaviors.

Clearance is also defined from equation (2.91) dividing it through by the plasma concentration and solving for the quotient excreted load to the same concentration, i.e.,

 

E [U

x

]

 

 

T

'

 

 

C x =

 

 

 

= F

+

 

x

]

(2.93)

[P x ]

 

 

 

 

 

 

 

 

[P x

 

which is represented in Figure 2.52 (right upper panel, curve b). When the plasmatic concentration is low, the secretory process dominates and the clearance curve starts at a high value Co (not infinite, as incorrectly predicted by the equation, meaning that the model is far from being perfect). The curve falls rather sharply with increasing plasmatic concentrations tending to the inulin clearance value at high concentrations, when there is a net filtration predominance. This is clearly seen in equation (2.93) when its mathematical limit is taken for [Px] → ∞ .

In turn, the excreted load displays a steep slope first, (F + S), as shown in Figure 2.52 (upper left panel, line c) to become equal to the GFR (same figure, line b) as the plasmatic concentration goes up. Thus, line b runs parallel to line a, and with an upward shift. The passage from the first slope to the second one in actual experimental curves is smooth and the breaking point represents only a theoretical limiting value.

A typical substance frequently used in renal secretion studies is paraaminohippurate (PAH), with a maximum secreted load of 80 mg/min and a clearance of 650 mL/min at low plasmatic concentrations.

2.4.4.3. Reabsorption

The third process does not have secretion and the equation, with S = 0, becomes now,

F [P ] R[K "

] = E[U

x

]

(2.94)

x

x

 

 

 

because the excreted load was moved to the right hand side. This is represented by the straight line d in Figure 2.52, shifted to the right with respect to line a, and crossing the horizontal axis at [Pmx]. If [Kx'' ]=[Px ], it is possible to define the reabsorbed load as,

Chapter 2. Source: Physiological Systems and Levels

131

T " = R[P ]

(2.95)

x

x

 

which is represented by line b in Figure 2.52 (lower panel). Breaking points, as mentioned before, are really defined by the respective projections of the straight lines and experimental curves display a smooth transition from one to the other, as shown in the figure. The reabsorbed load reaches a maximum value (saturation) when the tubules are no longer able to take more substance and that happens beyond a given plasmatic concentration specific for each substance. At lower concentrations, instead, there is full reabsorption. As we did before, clearance is obtained by dividing equation (2.94) by [Px] leading to,

 

E[U

]

T"

 

 

 

x

x

 

 

Cx =

 

 

 

=F[P

]

(2.96)

[P ]

 

 

 

x

 

 

x

 

 

The latter describes a hyperbola (Figure 2.52, upper rigth) that tends to the constant value given by inulin as the plasmatic concentration increases. For low concentrations, the reabsorptive process dominates and the substance in full is returned to the circulation. This is the case for substances of high physiological value, as glucose is. The maximum reabsorbed load is in the order of 380 mg/min at a plasmatic threshold level of about 300 mg/100 mL that, in practice because of the splay phenomenon, lies between 180 and 200 mg/100 mL (Figure 2.52 Ad, Bc and Cb, where A, B and C stand, respectively, for the upper left, upper right and lower panels). A diabetic person will have a high level of glucose in blood (hyperglucemia), usually beyond the renal plasma threshold. The reabsorptive capacity is saturated and glucose appears in urine (glucosuria). In the old days, the physician used to taste the urine for its sweetness in order to determine whether a patient was diabetic. Do not laugh, it is true! Fortunately, techniques are more advanced and sophisticated nowadays.

All substances are filtered by the kidneys, including exogenous substances as drugs are. Almost common knowledge is the fact of a person taking vitamin B who, a few hours later, produces reddish urine, typical of that substance. Many times the attending physician will warn the patient not to be scared in such case. Some substances are reabsorbed, others are secreted and there is a group that is both absorbed and secreted (as urea and creatinine, with a predominance of secretion, the latter much

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Understanding the Human Machine

more than the former). Sodium, instead, is also secreted and reabsorbed but approximately in equal amounts. In a quantitative sense, the filtration and reabsortion of ions and water are by far the most significant operations of the mammalian kidney. The body cannot afford, for example, to lose potassium and, thus, about 93% is reabsorbed and retained. The interplay of these processes, in the end, is responsible for the proper electrolyte balance.

2.4.4.4. Osmosis

When a membrane, which is permeable to solvent but not to solute, separates a solution and pure solvent, the solvent passes into the solution by osmosis (from Greek, meaning “push”). The osmotic pressure is that particular hydrostatic pressure which must be applied to the solution to prevent the entry of solvent. In other more general words, water passes from a solution with lower concentration to a solution with higher concentration so that both concentrations tend to equilibrate.

Osmotic pressure is one of the properties of solutions and depends on the number of particles per unit volume of solvent, not on their chemical characteristics. Similar to the law of gases which relates pressure, volume and temperature, van’t Hoff’s equation states that,

Π = Cm ×R ×T

(2.97)

where Π represents the osmotic pressure or “water attraction” generated by the solution, Cm stands for the molar concentration of the solution, R is the gas constant (= 0.08 atm×L/mole×°K) and T is the absolute temperature. Since the molar concentration Cm = n/V, n being the number of moles and V the volume in liters, thus expressing it in moles/L, equation (2.97) can be also written as,

Π×V = n ×R ×T

(2.98)

which well reminds the law of gases because pressure P, in the latter, is replaced by osmotic pressure Π, in the former. Equation (2.98) defines 1 osmol, a unit, as the osmotic strength generated by a concentration of 1 mole per liter at the measured temperature T in Kelvin degrees, or

1osmol=1

mole

×0.08

atm×liter

 

×310oK =25.4atm=19,304mmHg

(2.99)

liter

mole×o K

 

 

 

 

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133

Above, we take the temperature as equal to the body temperature, 37oC, and adding 273 to obtain the Kelvin degrees. If the temperature instead were 0oC, that is, T = 273oK, 1 osmol becomes equivalent to a pull of 22.4 atm. The osmotic strength of a solution can be found by summing the molar concentrations of all the ions and non-dissociating molecules.

Some numerical examples will help in the understanding of the subject. On purpose, we are using several times the whole word and the abbreviation for a given unit in order to be clearer in its meaning. For any substance, one gram-molecular weight (or one mole = 1 M) contains N = 6.02×1023 molecules (Avogadros number). If the substance is glucose, sucrose or any non-dissociating compound, the osmotic strength is 1 osmol. Now, let us consider a substance that dissociates when in solution. One gram-molecular weight of NaCl consisting of 6.02×1023 molecules dissociate into twice this number of ions in solution. Thus, 1 M of NaCl exerts an osmotic effect of nearly two osmols (we say “nearly” because the degree of dissociation depends on the concentration, the lower the concentration, the higher the dissociation). If it were a substance dissociating in three ions (such as Na2SO4), the effect would approach 3 osmols. The osmotic concentration of plasma, interstitial fluid and intracellular fluid are all kept in man within the band (283 ± 11) milliosmoles/L (mOsm/L), basically by the ingestion and excretion of water. Gain of water induces prompt water diuresis while loss of water induces thirst and antidiuresis. Such value, roughly equal to 300 mOsm/L, produces at body temperature an osmotic effect or pull equivalent to 7.62 atmospheres or 5,791 mmHg, which is quite an impressive value (check the calculation with the relationship given above). In other words, this is the pressure needed to counteract the tendency of plasma (or to any solution similar to plasma) to draw water.

The osmotically active solutes are to a large extent (around 90% or more) the electrolytes such as sodium, chloride, and bicarbonate. Glucose, amino acids and urea contribute not more than 10%. Hence, the osmolar concentration is a way of measuring the “total concentration” of a fluid. Besides, and as practical information, the weight concentration of a substance, in g/L, is given by the molecular weight MW (a number) multiplied by the molar concentration, in moles/L, or Cw = MW×Cm. A 5.4% solution of glucose means a weight concentration of 54 g in 1 liter which,

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Understanding the Human Machine

with a molecular weight of 180 yields the molar concentration Cm = 300 mM/L at 37°C. By the same token, a 0.9% NaCl solution means a weight concentration Cw = 9 g/L which, with MW = 58, produces a molar concentration Cm = 155 mM/L and the same osmolar strength of plasma. Verify these calculations using the relationships given above.

Another piece of useful information refers to the freezing technique already and briefly described above: 1 mole/L of ideal solute will depress the freezing point by 1.86°C, from which the freezing point depression (in °C) experimentally obtained from a given sample divided by 1.86×10–3°C yields directly the number of mOsm/L of the sample solution. “Osmolarity” refers to the number of osmols per liter of solution (the solution contains the solute) while “osmolality” is the number of osmols per kilogram of solvent (the solvent does not contain the solute). These terms are common in the texts as the terms “molarity” and “molality” are. The student can easily deduce the definitions for the two latter.

Amadeo Avogadro (1776–1856), Italian chemist, stated in 1811 that “equal volumes of all gases under the same conditions of temperature and pressure contain the same number of molecules” (Avogadro's Hypothesis), but he never tested it.

The name “Avogadro's Number” is just an honorary name attached to the calculated value of the number of atoms or molecules in a gram-mole of any chemical substance. If we used some other mass unit for the mole, such as “pound-mole”, the “number” would be different than 6.022 x 1023. The first person to have actually calculated the number of molecules in any mass of substance was Josef Loschmidt (1821–1895), an Austrian high school teacher, who in 1865, using the new Kinetic Molecular Theory, obtained the number of molecules in one cubic centimeter of gaseous substance under ordinary conditions of temperature and pressure to be around 2.6 x 1019 molecules. This is usually known as “Loschmidt's Constant.” Check INTERNET to get more details regarding this subject.

Maintenance of osmolarity is essential for life. It is basically kept by water ingestion and excretion. An individual can survive many days without food but not too long without water. If a large amount of water is drunk, a diuresis of diluted, uncolored, and almost odorless urine reestablishes normal osmolarity. Conversely, if there is dehydration (as in diarrhea, or heavy sweating during a sunny day), oliguria restricts urine outflow and the reduced amount excreted is characterized by high concentration (high osmolarity), amber color (yellowish to brownish), and penetrating offensive odor. Besides, the mechanisms of thirst are activated so that the sub-

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135

ject is driven to drink water. The renal system is much more efficient in defending the organism against dilution (water excess) than against dehydration. Children and old persons are particularly sensitive to the latter and sportspeople must be always warned of the risks faced when heavy exercise is practiced under high temperature conditions, especially during summer time. In the end, the thirst mechanism leads to compensating for the water deficit. The kidneys cannot do this by themselves for they only are able to either remove or conserve water. There are special sensors, the osmoreceptors, located in the central nervous system, that constantly check osmolarity to effect thirst and the proper renal actions.

The kidneys maintain a strong osmolar concentration gradient from their deep medullar region, where osmolarity is 1,200 or even 1,400 mOsm/L, to gradually decrease down to 300 mOsm/L at the cortical region. There is a group of nephrons with their glomeruli placed at the level of the renal cortex and the loops of Henle and collecting ducts penetrating deep into the renal medulla. Besides, their peritubular capillaries run almost parallel to Henle’s limbs, which show a U-like shape (Figure 2.50). When a subject is dehydrated, central osmoreceptors located in the hypothalamus order the secretion of antidiuretic hormone (ADH), also called vasopressin, from the posterior hypofisis. ADH acts on the distal convoluted tubules and the collecting ducts increasing their permeability to water which, due to the concentration gradient across the tubular wall, as fluid moves along the tubules and duct, water goes easily from the intratubular fluid to the interstitium by simple passive osmotic gradient and from there to the blood in the peritubular capillaries. Water is, thus, retained. An opposite situation is when the subject drinks too much water. The osmoreceptors suppress the secretion of ADH and the permeability to water of the distal tubules and collecting ducts decreases. Hence, in spite of the concentration gradient, water cannot traverse the walls and there is diuresis.

In these two extreme situations, the total urine excretion in 24 hours can be from 0.5 L in dehydration at a concentration of 1,200 or even 1,400 mOsm/L meaning an excreted load of 600 to 700 mOsm, and up to about 20 or 24 L in dilution because of excess water at a concentration of only 30 mOsm/L, meaning an excreted load also of 600 or 720 mOsm. In

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Understanding the Human Machine

other words, the excreted load is kept essentially constant in both situations.

Jacobus Henricus van 't Hoff was born in The Netherlands, in 1852, and died in 1911, in Germany. In 1885, he came up with a study on the chemical equilibrium in gaseous systems and strongly diluted solutions. Here, he demonstrated that the “osmotic pressure” in solutions, which are sufficiently dilute, is proportionate to the concentration and the absolute temperature so that this pressure can be represented by a formula that only deviates from the formula for gas pressure by a coefficient. Thus, van 't Hoff was able to prove that thermodynamic laws are not only valid for gases, but also for dilute solutions. His pressure laws, given general validity by the electrolytic dissociation theory, are considered comprehensive and seminal in the realm of natural sciences. He can be regarded as one of the founders of physical chemistry. In 1901, van’t Hoff received the first Nobel Prize in Chemistry.

Friedrich Gustav Jacob Henle (1809–1885), German anatomist and pathologist. His name is best known today for the loop-shaped portion of the nephron. It consists of a thin descending limb and a thicker ascending limb. His observation of it in 1862, supported by isolation preparations, was correct in itself but the interpretation was wrong. Nevertheless, his study resulted in a new series of investigations on the kidneys through which, between 1863 and 1865, their structure was definitely determined.

2.4.5. Countercurrent Mechanisms

We have seen in the previous paragraphs that the kidneys require a me- dullo-cortical osmolar gradient to regulate the osmolar-excreted load and, with it, to keep the extracellular fluid osmolarity. Let us explain now how this gradient is built up and maintained. Each operation makes use of a countercurrent system, a principle well known by chemical engineers in many types of industrial exchangers to improve the exchanging efficiency.

2.4.5.1. Countercurrent multiplication

There is a basic mechanism between the ascending and the descending limbs of the loop of Henle, at any level, that by active transport of sodium ions from the ascending branch into the descending one generates a constant transversal difference of 200 mOsm/L. This is a physiological renal property. The difference includes the interstitial fluid, too (Figure 2.53). The ascending limb walls, we underline, are impermeable to water while the descending branch as the collecting duct are not, implying that water cannot get into that portion of the tubules and dilute its content.

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137

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HENLE´S LOOP

COLLECTING

descending

ascending

DUCT

 

 

 

 

limb

 

limb

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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H2O

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H2O

H2O

H2O

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Na

 

 

 

 

H2O

H2O

H2O

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Figure 2.53. MECHANISM OF COUNTERCURRENT MULTIPLICATION. Upper panel: Stage 1 depicts the loop of Henle full of fluid at 300 mOsm/L. At stage 2, due to a basic active process, a transverse osmolar difference of 200 mOsm/L is established between the descending and ascending limbs.

Lower panel: It shows the countercurrent exchange relationship between Henle’s loop and collecting duct. See text for further details. Redrawn after Pitts (1966).

1200 1200 1200

Because of the U-like shape of the loop, the fluid in both limbs flow in a countercurrent way (opposite directions) so that there is a multiplicatory effect that, in the end, becomes a longitudinal gradient.

(Figure 2.53, upper panel, drawn horizontally) explains the build-up process step by step: At stage 1, say that fluid gets into the loop filling it fully at the same concentration, entering via the descending limb and exiting along the ascending portion. Thereafter, at stage 2, due to the basic active transport of sodium, a 200 mOsm/L is transversely generated all along the loop (say that the descending side rises its concentration to 400 mOsm/L while the other side lowers it to 200 mOsm/L). However, flow continues and a moment later, stage 3 depicts the situation when fresh