J. theor. Biol. (1992) 159, 271-286

The Effect of Auxiliary Conditions on Intestinal Unstirred Layer Diffusion Modellc~l by Numerical Simulation M. L. LUCAS, L. SOOD, M. MCGREGOR, N. SATTAR, A. WATT't AND J. C. TAYLORt

Institute of Physiology and Institute of Natural Philosophy, tThe University, Glasgow, G12 8QQ, Scotland, U.K. (Received on 9 July 1988, Accepted in revised form on 26 August 1992) Estimation of intestinal unstirred layer thickness usually involves inducing transmural potential difference changes by altering the content of the solution used to perfuse the small intestine. Osmotically active solutes, such as mannitol, when added to the luminal solution diffuse across the unstirred water layer (UWL) and induce osmotically dependent changes in potential difference. As an alternative procedure, the sodium ion in the luminal fluid can be replaced by another ion. As the sodium ion diffuses out of the UWL, the change in concentration next to the intestinal membrane alters the transmural potential difference. In both cases, UWL thickness is calculated from the time course of the potential difference changes, using a solution to the diffusion equation. The diffusion equation solution which allows the calculation of intestinal unstirred layer thickness was examined by simulation, using the method of numerical solutions. This process readily allows examination of the time course of diffusion under various imposed circumstances. The existing model for diffusion across the unstirred layer is based on auxiliary conditions which are unlikely to be fulfilled in the same mtestine. The present simulation additionally incorporated the effects of membrane permeability, fluid absorption and less than instantaneous bulk phase concentration change. Simulation indicated that changes within the physiologically relevant range in the chosen auxiliary conditions (with the real unstirred layer length kept constant) can alter estimates of the apparent half-time. Consequently, changes in parameters unassociated with the unstirred layer would be misconstrued as alterations in unstirred layer thickness.

1. Introduction A widely used and convenient electrical method for assessing the thickness o f the intestinal unstirred layer from potential difference transients makes use o f a simplified formula derived from a particular solution o f the diffusion equation. When the auxiliary conditions for the particular solution are considered, the limitations in the use of the simple formula become apparent. The auxiliary conditions may be valid for gall-bladder and frog skin, the tissues for which the method was devised (Dainty & House, 1966; Diamond, 1966) but are unlikely to apply in the small intestine, where the method has found greatest application (Dietschy et aL, 1971; Thomson & Dietschy, 1977, 1984). The small intestine absorbs fluid and has permeability to a wide range o f solutes o f varying molecular weight. In addition, it can be difficult to approximate an instantaneous bulk solution change in an in vivo experiment. 271

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Consequently, it is important to consider how these auxiliary conditions affect estimates of small intestinal unstirred water layer (UWL) thickness. As there have been no studies to date which consider this question, the technique of numerical solutions was applied to an unstirred layer model to determine to what extent auxiliary conditions affect thickness estimates. The purpose of this paper was not to illustrate the undoubted effects of unstirred layers on membrane processes, since a large amount of work is available on this topic (for a review, see Barry & Diamond, 1984). Nor was it to consider the effect of hydrodynamics on UWL thickness, as this also has been the subject of a comprehensive review (Pedley, 1983). The aim was to demonstrate that changes in the various auxiliary conditions can inadvertently be detected by the electrical method of UWL measurement and therefore misinterpreted as apparent changes in thickness. 2. Methods Section 2.1. T H E E X I S T I N G

MODEL

A prevalent experimental method for the estimation of the intestinal unstirred layer thickness involves rapidly changing the solute concentration of a solution perfused through the intestinal lumen and measuring the induced change in the transmural potential difference. Either a change to a mannitol-containing perfusate is used to induce an osmotic potential change or the perfusate sodium ion concentration is lowered to induce a diffusion potential change. Both methods assume that the change in potential difference across the intestinal membrane accurately reflects either the build-up or decay in concentration of solute at the intestinal membrane. The time course of the induced potential difference change should therefore provide information about the diffusion path length from the lumen across the UWL to the membrane. In particular, the time for one half of the induced potential difference change to occur is recorded and from this "half-time", the thickness of the unstirred layer is calculated. The method of calculation relies on a particular solution to the diffusion equation for the analogous case of heat diffusing from a source at constant temperature up to a lagged surface (Olson & Schultz, 1942). In physiological terms, this is equivalent to a constant concentration, achieved by perfusion of fresh solution, at the edge of the unstirred layer and impermeability of the intestine to the chosen solute. The calculation makes use of a particular property of the particular solution (Carslaw & Jaeger, 1959; Crank, 1975) in that the dimensionless combination of variables, Dt/ 62, where D is the diffusion coefficient of the selected solute, di=unstirred layer thickness, t=time, has a numerical value of 0"38 when one half of the induced potential difference change has occurred. Knowing the diffusion coefficient of the selected solute and the half-time (tl/2) allows 6 to be calculated (Dainty & House, 1966; Diamond, 1966). For this procedure to be valid, several auxiliary conditions must apply: (i) An initial step change in luminal concentration should occur. (ii) The intestine should be totally impermeable to the solute.

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(iii) Solute c o n c e n t r a t i o n should be linearly related to the induced potential difference change. (iv) T h e diffusion coefficient within the U W L should be k n o w n a n d should n o t v a r y with distance f r o m the membrane. (v) T h e dimensions o f the U W L should n o t be altered by changes in the solute c o n c e n t r a t i o n as it varies at the membrane. (vi) There should be no transport o f solute t h r o u g h the U W L by means other than diffusion. M a n y o f these requirements are clearly inapplicable to the small intestine. F o r this reason, physiologically m o r e a p p r o p r i a t e auxiliary conditions were included in the diffusion e q u a t i o n to investigate their effects on U W L thickness estimates.

2.2. THE PROPOSED MODEL A d o p t i n g the c o n v e n t i o n s o f the present model, the U W L is assumed to have a distinct b o u n d a r y at the U W L : bulk phase interface (see Fig. 1) with a well-mixed bulk phase (luminal solution) c o n c e n t r a t i o n o f Co and a n o t h e r b o u n d a r y at the

Intestinal lumen

i I

Unstirred water layer ( U W L )

Intestinal membrane

( bulk solution )

,qcoe°,;i'o:'o,:o.

Serasalor intracellular space

Permeobility ~/ coefficient =K

c ~ . = : ~'c8, ®

I-O

C onc enttahan

"'2,,

c~,=

c

Convection velocity, V

q,,

(

)

(~0

FIG. I. The unstirred layer model with associated terms as used for numerical simulation of diffusion proceeding in accordance with physiological auxiliary (initial and boundary) conditions in the presence of convection caused by fluid absorption. The velocity of fluid movement is determined by the concentration of solute at x = 6. In this example, mannitol diffusion is simulated with the luminal concentration Co given a fixed value of 1 and the transmembrane concentration G fixed at zero. Diffusion proceeds [line (a)] until a steady state is reached [line (b)]. When membrane permeability is zero, C~..~ would equal Co. In the presence of permeability C~ will be less than Co and the system represents diffusion across a composite medium, the UWL of thickness 6trwt and diffusion coefficient DuwL and the membrane of thickness &. and diffusion coefficient, D.,, In addition, the concentration C,.., of solute just within the membrane is a function of the partition coefficient (fl) and the concentration Cs., of solute just within the aqueous phase at the UWL:membrane interface. The permeability coefficient (K) represents flg~'and the concentration gradient is Cs.,- Ct.

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UWL:membrane interface. Immediately within the unstirred aqueous phase next to the membrane, the solute concentration is Ca, and in the aqueous phase immediately outside the membrane, the solute concentration is CI and is fixed. The concentration within the UWL varies within the range Co and CI. The precise meaning of C, is relatively unimportant since it is a fixed quantity outside of the UWL. It can be viewed as the aqueous concentration of solute just beyond the membrane so that the solute concentration gradient across the membrane is (Ca-G). The membrane adjoining the UWL therefore represents many types of membrane of various thicknesses and solute diffusion coefficients. The sole constraint at the UWL:membrane boundary is that net solute flux out of the UWL should equal solute flux into the membrane (see below). The concentration within the UWL is given by:

OC

_ _

t)=D 02C2 _ _

t)+ V(t) -~x (x, t), OC

(1)

the first term on the right-hand side of the equation representing diffusion with a diffusion coefficient D and the second term representing bulk convection with velocity, V(t), that results from fluid absorption by the intestinal membrane. Consequently, solute builds up or decays at the UWL:membrane interface through diffusion down the concentration gradient but also by convection caused by fluid movement. In this model it is assumed that fluid removed by the membrane is replaced by fluid entering the UWL from the bulk phase. In addition, the boundary conditions:

D ~ (6, t) = K(Ca.,- C,) + V(t)(1 - or)Ca.,

(2)

ox

apply. This eqn (2) states that solute leaves the U W L by diffusion through the membrane and by convection across the membrane. The first term on the righthand side represents diffusion through the membrane driven by the difference in the concentration CI, the fixed concentration outside of the UWL and Ca.,, the concentration at the membrane:UWL interface. The second term allows for convective carriage of solute across the membrane, depending on V(t), the velocity of convection, the concentration Cs., and a factor (1 - or) allowing for reflection of solute back into the UWL. These separate terms are explained in greater detail below. In addition, rapid but not instantaneous initial change in the solute concentration is allowed for by the following initial conditions: t=0;

x=0;

Co=0

or

1

t>0;

x=0;

Co=exp(-Ot)

or

[l-exp(-0t)]

where0>0.

(3)

To initiate diffusion into the UWL, Co becomes 1 if the luminal solution change is instantaneous or rises to a value of 1 with a time course determined by 0, the rate constant in the exponent. Conversely, where diffusion out of the UWL is simulated, Co either becomes zero instantaneously or decays with the rate of decay determined by O.

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These initial and boundary conditions provide a more general representation of the unstirred layer equation, which defaults to the simpler representation if fluid velocity and membrane permeability are zero and bulk solution change is instantaneous. Equation (l) with conditions (2) and (3) was solved numerically using finite difference approximations (Crank & Nicolson, 1947; Crank, 1975) for diffusion into ("wash-in") and diffusion out of ("wash-out") the UWL of the commonly used solutes, mannitol and sodium ion. When performing an experiment to assess UWL thickness, it is usual to change the transmucosal potential difference either by perfusing a mannitoi-containing solution or a solution deficient in sodium ion. When using mannitol, mannitol first washes into the UWL to alter the potential difference and subsequently is washed-out from the UWL by perfusion of a mannitol-free solution. In contrast, sodium ion is first washed out of the UWL by lowering the bulk solution concentration and then washed back into the U W L when the bulk solution concentration is restored. The normal cycle of physiological measurement is therefore "wash-in" followed by "wash-out" of mannitol, if mannitol is used, but "wash-out" followed by "wash-in" of sodium ion, if sodium ion is selected as the perturbing solute. Consequently, there are four physiological situations which were simulated, each with a particular pair of boundary conditions outside the two ends of the UWL. For mannitol "wash-in", C0=l.0 and G = 0 . 0 for t > 0 , which represents the sudden elevation of the luminal concentration, with sink conditions beyond the membrane. To simulate mannitol "wash-out", Co is restored to zero, after the steady state was achieved for the "wash-in" simulation. Mannitol "wash-out" therefore represents exit from the UWL ofmannitol which entered during the "wash-in" phase. Conversely, for sodium ion, "wash-out" is simulated by setting Co to zero for t > 0, with C~ fixed at l-0. This allows for the luminal fluid to act as a sink with a source of sodium ion just beyond the membrane. After a steady state is achieved, "washin" is simulated by restoring Co to 1.0. Initial conditions in the absence of permeability are Cx,0= 1.0 prior to "wash-out" and Cx,0=0.0 prior to "wash-in" for both sodium ion and mannitol diffusion simulations. Permeability requires different initial conditions for the concentration of solute within the UWL for both mannitol and sodium ion, and these are described in greater detail below.

2.3. T R E A T M E N T OF G E N E R A L I Z E D A U X I L I A R Y (INITIAL A N D B O U N D A R Y ) CONDITIONS

2.3.1. Membrane permeability Membrane permeability was incorporated into the boundary condition [eqn (2)] by making flux through the membrane proportional to the concentration difference across it and a permeability coefficient (K):

D OC (g,t)=K(Ca,-C~); a-x '

K=fl Dm 6,--~

(4)

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where D,,,= diffusion coefficient within the membrane, ~,, is the membrane thickness, /3 = membrane: aqueous phase partition coefficient, Caa is the aqueous concentration immediately to the left and Cm is that immediately to the right of the membrane. Consequently, this representation of flux through the membrane applies only to a diffusing substance and not to one whose uptake is governed by a carrier-mediated process at the membrane. In order to relate the aqueous and membrane phase concentrations, a partition coefficient is included in the description of this boundary. The present treatment of the permeability condition adopts the convention (eqns 2.6-2.8 in Schultz, 1980) that the partition coefficient is identical at both membrane interfaces. Flux through the membrane is therefore related to the term C8.,-C~, the aqueous concentrations just outside the membrane (see Fig. 1). The permeability coefficient (K) therefore incorporates the membrane thickness, 6,,, the membrane diffusion coefficient, D,,,, and the partition coefficient/3. In this way, K generalises many types of membranes, and the sole constraint on the system is the fixed concentration at C~. The term "membrane" as applied to the small intestine is essentially an ambiguous term since it can refer to the cell membranes of individual epithelial cells or, more commonly, the multicellular structure composed of individual cells with intervening paracellular spaces. In the present model, the term membrane refers to the composite multicellular structure so that CI represents the concentration of solute in the serosal space. This is either a source or a sink concentration, consistent with the blood supply delivering or removing solute. However, the model applies with equal force to a cell membrane, and in this case Ct would refer to the intracellular concentration. In both cases, the same constraint applies, i.e. the net amount of solute leaving the UWL equals the amount entering the "membrane". In the experimental circumstances that are simulated here, the term membrane is used synonymously with the more familiar term of intestinal mucosa as used in the physiological literature. The flux condition at the UWL:membrane boundary requires that solute leaving the UWL equals the transcellular solute flux into the cells, determined by cellular permeability to the solute, plus solute moving past the membrane through the paracellular pathway, determined by fluid absorption. Solute leaving the cells by diffusion into the serosal space and convected past the cells by fluid absorption is carried away by the blood. Consequently, "sink" conditions can be imposed, to correspond with this situation, by making Ct equal to zero. When the solute was mannitol, Ct was fixed at 0.0 and Co at 1.0, for diffusion into the UWL. This constraint imposes "sink" conditions on the concentration of mannitol immediately beyond the membrane, with the luminal fluid acting as a source. Diffusion out of the UWL was then modelled by making Co equal to zero. When sodium chloride diffusion out of the UWL was simulated, CI was given a value of 1, implying a source beyond the membrane and the luminal concentration, Co, was set at zero, making the luminal fluid a sink. Diffusion into the UWL of sodium chloride is simulated when the luminal concentration is restored to a value of 1. By assigning values to Co and Ct of zero or 1, Cs., is expressed as a fraction of Co, the luminal concentration which drives solute into the UWL or Cs.0, which drives solute out of the UWL when Co is lowered. Consequently, the interracial concentration is

AUXILIARY CONDITIONS ON INTESTINAL DIFFUSION

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expressed as a non-dimensional fraction which can be given appropriate values when the boundary concentrations are given dimensions. Membrafie permeation requires an initial gradient in the concentration, Cx.0, within the UWL which exists prior to the start of the simulated process. This is evident when one considers the simulation of a sodium ion "wash-out" and "washin" cycle in the absence of permeability and contrasts this with the same events in the presence of permeability. In the absence of permeability, removal of sodium ion from the bulk phase results in diffusion of sodium ion out of the UWL, until the UWL is free of sodium ion. Restoration of bulk-phase concentration causes sodium ion to diffuse back into the U W L until bulk-phase and UWL concentrations are equal. In contrast, in the presence of significant membrane permeability to sodium ion, removing sodium ion from the lumen still causes it to diffuse out of the UWL, but now sodium ion can also enter the UWL from the source beyond the membrane. The final steady state is therefore a gradient across the UWL where sodium ion exiting from the UWL into the bulk phase is replaced by sodium ion entering the UWL through the membrane. This steady-state distribution at the end of a "washout" process is by definition the initial condition within the UWL prior to a "washin" process. On restoration of the bulk-phase sodium ion concentration, the concentration profile from Co to C~ is "saw-toothed" with both Co and C~ at I-0 [Fig. 2(c)]. In addition, the steady-state concentration at the UWL:membrane interface, C6,~, is less than I. Immediately after Co becomes 1, diffusion proceeds and the concentration gradient within the UWL disappears until the final steady state at the end of "wash-in" is a uniform concentration within the UWL, Cx.~ = 1.0, with Co and Ci = 1-0. This "wash-out" and "wash-in" cycle in the presence of permeability is shown in Fig. 2, which emphasizes that the two end-points of the simulations are a gradient and uniform concentration within the UWL and not the "end-points" of a full or empty UWL which occur in the absence of permeability. Initial conditions therefore differ for build-up and decay processes and their formulation depends on the process being simulated. This pre-existing concentration gradient can be calculated if D/6 and K are known (Jacobs, 1967) or it can be derived by first simulating solute exit from the UWL.

2.3.2. Fluid movement through the unstirred layer As fluid next to the membrane is absorbed, it is replaced by fluid from the luminal solution. Fluid absorption decreases linearly with sodium chloride concentration (Curran, 1960) and decreases linearly with osmotic pressure gradients imposed by increasing concentrations of solutes such as mannitol (Smyth & Wright, 1966). The treatment of fluid absorption depends therefore on the type of solute used. Where a reduction in luminal sodium ion concentration was modelled, the fluid absorption term was reduced as the sodium ion concentration at the membrane declined. Convection (fluid absorption) was therefore made to depend on the solute concentration at the membrane and a proportionality constant A [eqn (5)]. As sodium ion concentration is restored in the luminal phase, sodium ion concentration increases at the

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UWL (o)

UWL A

co

c~

(b)C o

C~

"Wash-out"

1 0

,

!

f- ~ ,

1

ic) col

ic,

C°nientrati°n

~/

"Wash-in" / ~

/v~/ 1

1

Bulk phase: Membrane Bulk phase: Membrane UWL interface UWL interface FIG. 2. Diagram showing that initial conditions, C.~.ofor solute distributed through the unstirred layer differ between "wash-in" and "wash-out" simulations when significant membrane permeability is present. At (b), simulation of "wash-out" begins with the initial condition that C,.0 is constant everywhere within the U W L "Wash-out" proceeds until a steady state is reached (d) when solute leaving the U W L is replaced by solute entering the U W L through the membrane. "Wash-in" begins (c) when the bulk concentration is elevated to Co = I but the initial condition is that Cx.0 is a gradient and not uniform everywhere as before. "Wash-in" ends (a) when Co= C~ = C~, the concentration is uniform everywhere within the UWL and provides the initial conditions for the next "wash-out" process.

membrane and fluid absorption (convection) also increases.

v(t) =AC~.,.

(5)

When mannitol diffusion was simulated, as increasing osmolarity reduces fluid absorption, convection was made to vary [eqn (6)] with (1 - C8.,). This means that when there is no mannitol at the UWL:membrane interface, fluid absorption is maximal and when mannitol concentration increases, fluid absorption and hence convection across the UWL decreases.

V(t) = A(I - Ca.,).

(6)

Fluid absorption influences the accumulation of solute in two ways. First, as solute accumulates, it influences the rate of fluid absorption and therefore the velocity of convection throughout the unstirred layer [eqns (5) and (6)]. This, in turn, will affect the time-course of solute accumulation. Second, fluid absorption will carry material past the membrane:UWL boundary, depending on the convection velocity, V(t), and the concentration of solute at the membrane, C~.,, which will also be the concentration of solute in fluid leaving the system, and cr, the reflection coefficient [eqn (2)]. The ability of fluid movement to transport material past the membrane is termed

AUXILIARY

CONDITIONS

ON

INTESTINAL

DIFFUSION

279

"solvent drag" or "fluid entrainment" and represents the rate of loss from the UWL of solute entrained in the fluid (cf. eqn 22, p. 419, Davson, 1970). The reflection coefficient represents the extent to which the solute is reflected back into the UWL. If cr is zero, there is no reflection and the fluid leaving the UWL has the same concentration as C6.,. Where reflection is present and cr is greater than zero, fluid leaving the system has a concentration of solute which is less than C8,,, the reflected solute adding to the concentration at Cs.,. A reflection coefficient of 1 represents complete filtration with all solute extracted from the fluid. In this case, the concentration C6., of a solute such as mannitol could theoretically be higher than the luminal solution concentration. As fluid movement declines with increasing mannitol concentration, this extreme case of complete filtration with a rise in C~., to values above either Co or C~ does not occur. 3. Results

Output from the simulations was in the form of a concentration profile across the UWL for each increment in the non-dimensional variable, Dt/62, where D is the solute diffusion coefficient within the unstirred layer, 6 is the thickness of the unstirred layer, and t is the time after setting up the initial concentration conditions. Since the half-time is of most interest, the variable Dtj/2/6 ~ is presented as the dependent variable, with other factors of interest presented as the independent variable, in non-dimensional form. In this way Dtu262 is presented as a number which is also a correction factor since one can see how far the value of Dtt/2/62 deviates from the ideal value of 0.38 for the simple case and the extent to which this value needs correction. In effect, the numerical values for Dtu262 for the present, more general model become the values which must be inserted into the simple Diamond formula to obtain a correct estimate of 6, for any given value of the boundary condition under investigation. 3.1, T H E E F F E C T

OF MEMBRANE

PERMEABILITY

Values for Dt,/2/62 were calculated for a UWL adjoining a membrane with permeability to the solute ranging from zero to infinity. This range of permeability coefficient can be compactly expressed if the membrane permeability coefficient K is divided by the sum of the UWL permeability coefficient (D/6) and the membrane term (K). This ratio is zero when K is zero and has a limiting value of 1 when K becomes large. At zero membrane permeability, the numerical value of t,/2 obtained is 0-38, when D and 6 are assigned a value of 1, corresponding exactly to the analytical solution value (Carslaw & Jaeger, 1959: 101-102). As permeability increases, t~/2 becomes smaller, indicating that the time to reach one half of the final concentration change is arrived at earlier than in the absence of permeability (Fig. 3). Membrane permeability attenuates the extent to which concentration change at the membrane matches concentration change in the bulk solution. Increasing permeability steepens the final concentration gradient achieved across the UWL so that the extent of concentration

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0.4

0"3

( °'~,/ ~ ~ - / 0-2 Apporenl holf-lime 0"1

0"0

I

t

l

I

I

I

I

I

0-5 Permeobili'ty

I

I-0 ---~

FIG. 3. The effect of m e m b r a n e permeability (K) on the apparent half-time for diffusion across the unstirred layer. Half-time is expressed as the dimensionless variable Dt~/2/~ 2 and permeability is expressed as K / ( K + D / S ) .

change at the membrane is progressively reduced. Membrane permeability reduces the apparent half-time of the wash-in and wash-out processes to an equal extent for both mannitoi and sodium ion, and for both solutes, i.e. the half-time at a given permeability was identical for both sodium ion and mannitol "wash-in" and "washout."

3.2. T H E E F F E C T O F T H E R A T E O F C H A N G E O F T H E B U L K S O L U T I O N S O L U T E CONCENTRATION

The effect on half-time of less than instantaneous concentration change in the bulk solution was investigated. The rate constant for bulk solution change was chosen so that the associated half-time spanned the range for UWL half-times by an order of magnitude in either direction. The rate of bulk solution change altered tt/2 between predictable limits of almost instant change having no discernible effect to the other extreme where t~/2 is entirely determined by a slow rate of change in the bulk solution. The half-time for solute build-up or decay at the membrane for instantaneous bulk ,"real". concentration change is termed ,~/2 , the bulk solution concentration change halftime [(In 2)/0] is referred to as t~/2; the apparent half-time for membrane solute change when the bulk solution change is not instantaneous is termed t ap ~,,[Fsince this is measured by experiment. When ,'"°°~/2is an order of magnitude larger than t°~/2, overestimation ol- It~2 ...,c,r' is of the order of 10% [Fig. 4(a)]. Conversely when bulk and apparent half-times are equal, •,"rc, ~/2 r' is overestimated by a factor of ten. Permeability at the membrane does not alter the underlying relationship between t~/2 and t ~ . When the membrane permeability constant (K) was made to be ten times D/6, the unstirred layer term, i.e. KS~D= 10, such that the membrane concentration change was less than 5% of the bulk solution concentration change, the relationship between t°/2 and t ~ remained unchanged. As 0 becomes large, the ratio

AUXILIARY

(o)

CONDITIONS

ON INTESTINAL

DIFFUSION

281

(b)

I0 9 8

7 6 5 t*"

2

\

l 4

f t*P~ ½ -I } I 0 -I

I

1

1

I

I 234

1

t

56

t

t

t

-2

789

~gpp t/z

to

t

2

,oI,, l

FIG. 4. (a) The effect o f rate o f change of bulk-phase concentration on the apparent U W L half-times. In this representation, the apparent (experimentally observed) half-time t ~ and the real half-time t ~ I are presented as a ratio which expresses the extent o f overestimation o f t~=/~2~by a non-instantaneous change in bulk-phase solute concentration with a half-time of t~/2. o The slowness o f the bulk solution change t°/2 is expressed relative to the apparent half-time tT~ as the ratio t~/2/h/z. ,vp o Values are presented for zero ( A ; KS/D =0) and high ( • ; K 6 / D = 10) membrane permeability. (b) An approximate linearizing plot of values in (a). The natural logarithm of the ratio o f the overestimation o f real half-time 0 (t app i / 2 - t real I/2) to real half-time (t~/2I) is plotted against the natural logarithm of t app I/2/tl/2. When the latter ratio exceeds 2, Y= 1-20- 1.232X; for a ratio less than 2, Y=2.30-2.89X.

"real" of t~'~' to t,/2 approaches a limiting value of unity (there is no overestimation). The +"real'" and normalextent of overestimation can therefore be expressed as ~app ,/2 minus ,i/2 ~,app/,"r=l'" -- 1) is the basis of ized by dividing through by ,"real" ,/2 . This resulting term ~,1/2/,I/2 the linearizing plot [Fig. 4(b)] since the natural logarithm of this term is linearly related to natural logarithm (tr~'/t~/2). G o o d fits to the data are obtained when t~y~/t°l/2 is greater than two. Inspection of the linearizing plot [Fig. 4(b)] indicates that there is a steepening of the relationship when (t~P/t°j/2) is less than 2 and a o,,real" second line provides better estimates of ,i/z from t ~ . The change in apparent halftime for a given value of 0 was the same for mannitol and sodium ion "wash-in" and "wash-out."

3.3. T H E E F F E C T OF F L U I D M O V E M E N T

As expected, the introduction of fluid absorption into the UWL model altered the half-times for build-up and decay of solute at the membrane (Fig. 5). However, whether fluid absorption accelerated or delayed a process was dependent on the chosen solute. With mannitol, diffusion into the UWL was assisted by fluid absorption and hence half-times were shorter than in the absence of convection. Conversely, exit of mannitol was delayed and half-times increased during wash-out of mannitol

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0.5~

/"

/e e / e / °

o-4

(o,,,. 1

/e

q

'

~

0"3

0.2 0

I

t

I

0.02

0.O4

O-06

L

I

0"O80'lO

I

I

0'12

0"14

Fluid velocity ( "--~" ) FiG. 5. The effect of fluid velocity on the apparent half-time for diffusion across the unstirred layer. Half-time is expressed in dimensionless form for the simulated cases of mannitol and sodium ion buildup and decay processes, fluid velocity is expressed as the dimensionless variable (I/6/D). Values for the reflection coefficient were or= I for mannitol, ( r = 0 for sodium ions. (O), Mannitol decay; ( ~ ) , sodium ion build-up; ( & ) , sodium ion decay; (©), mannitol build-up.

against the direction of convection. In contrast, fluid absorption did not cause identical changes when diffusion of sodium ion was simulated. In this case, on lowering the luminal concentration, the decay of sodium ion at the membrane occurred earlier in the presence of convection than in its absence. The half-time for decay was less than the half-time for build-up, in contrast to mannitol diffusion. This is because of the complicating aspect of reflection at the intestinal membrane. In the presence of significant transport of material out of the UWL across the membrane by convection, fluid movement delays the build-up of sodium ion at the membrane because solute is also lost in the fluid stream. Sodium ion concentration at the membrane also decays rapidly because loss of solute occurs by diffusion out of the UWL against the flow of convection but also by fluid entrainment past the membrane. Consequently the half-times for build-up and decay for sodium ion are not the same as those for mannitol. For the purposes of simulation, a reflection coefficient of one was assigned to mannitol and a value of zero to sodium ion. These represent upper limits since the true reflection coefficient for mannitol is probably tess than 1 and above zero for sodium ion. Less extreme values for the reflection coefficients would obviously reduce the differences in half-time for "wash-in" and "wash-out" processes for both solutes. When the convection velocity is given appropriate dimensions, numerical values for velocity required to alter thickness estimates significantly are close to average values for intestinal fluid absorption. For example, the half-time for mannitol diffusion out of the UWL is increased by 10% when V&/D is 0-05 (Fig. 5). For an

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unstirred layer of 0.05 cm and for a solute diffusion coefficient of 0.9 x 10-5 c m 2 s e c - I , the fluid velocity can be calculated to be 0.9 × 10-5 cm see-~. This represents an absorption rate of 32 ~tl hr -I cm -2 intestinal area which is close to average values for the proximal jejunum. Since the presented non-dimensional half-times can also serve as correction factors, the simulation suggests a way of correcting unstirred layer thickness estimates for fluid movement. If the half-time for "wash-in" and "wash-out" process are measured experimentally, the ratio of these qualities allows correction factors to be selected which can be inserted into the simple Diamond formula. 4. Discussion

Experimental observations consistently confirm (Dugas et aL, 1975; Lewis & Fordtran, 1975; Winne, 1979; Ortiz et al., 1982) the validity of the concept (Dietschy et al., 1971) of an unstirred fluid layer separating luminal fluid from the underlying mucosal surface of the intestine. Estimation of its thickness by potential difference measurements at present makes use of a simple formula derived from a particular solution (Diamond, 1966) to the diffusion equation. However, the particular solution requires auxiliary conditions which are unlikely to be valid for the small intestine. The present simulations consider the effect on estimates of U W L thickness of fluid absorption, the speed of bulk solution change and membrane permeability to the solute. Analytical expressions for the diffusion equation with these auxiliary conditions are available, for example, non-instantaneous bulk-phase change (Carslaw & Jaeger, 1959: 104), boundary permeability (Carslaw & Jaeger, 1959: 114-115) and diffusion with convection (Jacobs, 1967:126; Pedley, 1983). However, combinations of the auxiliary conditions or concentration-dependent change in them can make analytical solutions cumbersome to use and for this reason the method of numerical solutions was preferred. This simulation process indicated that changes in the auxiliary conditions within the physiological range have significant effects on estimates of intestinal unstirred layer thickness. Less than instantaneous change in the solute concentration of the bulk solution overestimates unstirred layer thickness. As an externally imposed variable, this is the easiest to correct by making the solute change as rapid as possible. This is achieved in oitro by the well-known expedient of high rates of stirring of luminal solutions. Correction is more difficult in oivo but high rates of fluid perfusion through intestinal loops will reduce half-times for concentration change of the bulk phase. An important additional feature might be the use of as wide an entry and exit cannula as possible, through a short segment. This would allow a rapid change of bulk solution whereas high flow through a small cannula would induce a mixing artefact. The present simulations indicate how rapid bulk solution change must be before it has little effect on UWL estimates. The problem with using rapid stirring or perfusion rates to reduce bulk solution mixing times is that these manoeuvres will also thin the unstirred layer. The present simulation studies indicate that unstirred layer estimates could be corrected for bulk solution mixing times for any given flow rate, provided some means is found for detecting the rate of luminal solute concentration change.

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The other factors which were considered are properties of the intestinal mucosa rather than of the perfusion system. Permeability of the membrane to the selected solute reduces the half-time and consequently the simple Diamond formula underestimates UWL thickness when permeability is present. Using the Diamond formula, apparent reductions in UWL thickness (shorter half-times) have been seen in senescence (Holland & Dadufalza, 1983), exposure to alcohol (Thomson, 1984) and intestinal disease (Read et al., 1977). In each case, increased intestinal permeability has also been detected (Worthington et al., 1978; Bjarnason et al., 1983; Lin & Hayton, 1983). Where sodium ion is used as the solute, zero permeability is clearly an invalid assumption since there are several routes by which sodium ions enter mucosal cells. Similarly, intestinal permeability to mannitol almost equals permeability to calcium ion (Favus et al., 1983) and cannot therefore be considered negligible. Whether this degree of permeability seriously compromises UWL estimates is at present difficult to assess. Permeability coefficient data in the literature are inevitably composites of a UWL and a membrane term. To answer the question satisfactorily, UWL-corrected permeability estimates are required. An answer might be obtained by measuring the UWL by methods that are unaffected by permeability and to compare these estimates with results obtained by the present method. One approach might be to measure the time lag, i.e. the interval of time between the start of luminal solution change and the onset of the membrane potential difference change, as well as the half-time. The other intrinsic property of the intestinal mucosa which affected UWL estimates was fluid absorption. Since this transports solute to the membrane and beyond by convection, it is not surprising that half-times were altered. Values for fluid absorption that are typical for the small intestine produced significant changes in half-times, the direction of which depended on the selected solute. It is appropriate here to distinguish between the effect of convection on unstirred layer thickness estimates and its effect on concentration at the membrane. Much work is available on the extent to which convection prevents the full expression of an osmotic gradient across a membrane (Pedley & Fischbarg, 1980; for a review, see Turnheim, 1984). Similarly, previous work has emphasised how hydrodynamics affects unstirred layer thickness (Pedley, 1983). The present simulation considers how convection alters the estimate of unstirred layer thickness rather than the thickness itself. It indicates the likely extent of under or overestimation of UWL thickness given the prevailing convection velocity (which can be estimated from fluid absorption measurements) and the selected solute. In conclusion, the method of numerical solutions allows the modelling of diffusion across the unstirred layer with auxiliary conditions that reflect circumstances more likely to occur in the perfused small intestine than those required by the simple model. If the simple formula is used in situations where the more realistic auxiliary conditions should apply, thickness estimates will be less accurate and changes in the auxiliary conditions will always be misconstrued as changes in unstirred layer thickness. In past studies of intestinal unstirred layer thickness and its alteration under various circumstances, this feature of the calculation based on the simple Diamond formula has gone largely unrecognised. Simulation alone does not determine which factors may have contributed to an apparent change in thickness but it does

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emphasise the inherent ambiguity in the interpretation of apparent thickness changes. From the experimental point of view, it is evident that relevant auxiliary conditions have also to be measured before changes in measured half-time can be attributed with confidence to changes in unstirred layer thickness. The authors acknowledge the assistance o f the Scottish H o m e and Health D e p a r t m e n t for vacation studentships for L. Sood, M. M c G r e g o r and N. Sattar. Thanks are also due to M r J. Beck o f the consultancy staff o f the University C o m p u t i n g Service and to Professor R. Whitehead o f the Natural Philosophy D e p a r t m e n t for their forebearance and helpful discussions. Thanks are also due to Mrs F. M c G a r r i t y for her secretarial resilience. The authors gratefully acknowledge the encouragement and criticism o f Professor J. Crank.

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BJARNASON,l., PETERS,T. J. & VEALL,N. (1983). A persistent defect in intestinal permeability in coeliac disease demonstration by a StCr-labelled EDTA absorption test. Lancet i, 323-325.

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