# Intestinal villi and crypts density maximizing nutrient absorption

Martin Garic, Rohan Vernekar, Dacil Idaira Yanez Martin, Stéphane Tanguy, Clément de Loubens, Claude Loverdo

<figure><img src="image_1.png" /></figure>

## To cite this version:

Martin Garic, Rohan Vernekar, Dacil Idaira Yanez Martin, Stéphane Tanguy, Clément de Loubens, et al.. Intestinal villi and crypts density maximizing nutrient absorption. 2025. hal-05156983

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Intestinal villi and crypts density maximizing
nutrient absorption

Martin Garic<sup>a</sup>, Rohan Vernekar<sup>b</sup>, Dácil I. Yánez Martín<sup>c</sup>, Stéphane Tanguy<sup>c</sup>, Clément de Loubens<sup>b</sup>, and Claude Loverdo<sup>a,1</sup>

<sup>a</sup>Sorbonne Université, CNRS, Laboratoire Jean Perrin, LJP,
F-75005 Paris, France

<sup>b</sup>Univ. Grenoble Alpes, CNRS, Grenoble INP, LRP, 38000 Grenoble, France

<sup>c</sup>Univ. Grenoble Alpes, CNRS, UMR 5525, VetAgro Sup, Grenoble INP, TIMC, 38000 Grenoble, France

July 8, 2025

Abstract

The villi and crypts of the gastrointestinal tract increase the effective
surface area of the intestinal mucosa, potentially enhancing nutrient ab-
sorption. It is commonly assumed that this is their primary function,
and that a higher villi density necessarily leads to improved absorption.
However, when villi are packed too closely together, diffusion can be hin-
dered, potentially offsetting this benefit. In this work, we investigate
the relationship between the density of these structures and the overall
efficiency of absorption. In three different simplified geometries, approxi-
mating crypts, leaf-like villi, and finger-like villi we calculate analytically
the concentration profile and the absorption flux, assuming that there is
only diffusion between these structures while the lumen is well mixed.
When plotting the absorption flux per unit of gut length as a function of
the structures' density, we observe that there is a density maximizing ab-
sorption. We study numerically this optimum. It depends weakly on the
absorption properties of the given nutrient, so that a geometry optimal
for one nutrient is close to optimum for another nutrient. Physiological
data from various animal species align with this predicted optimal range
and potentially reflect evolutionary selection for efficient nutrient uptake,
supporting the model's validity.

The density of villi and crypts, structures on the inner surface of the gas-
trointestinal tract, may affect absorption. It has been argued that increasing
the density of these structures increases surface area and thus absorption but
actually there is a trade-off: when these structures are closer, fewer nutrients
penetrate between them, leading to a lower local concentration and thus decreas-
ing absorption. In three geometries approximating the real ones, we calculate
the total nutrient absorption per unit of length of the digestive tract, and find
the distance between these structures that maximizes absorption. Physiological data in various animals fall within the optimal range, suggesting evolutionary adaptation. This might explain why diseases with modified villi geometries are associated with lower nutrient absorption.

The gastrointestinal tract's main function is absorbing nutrients, critical for sustaining metabolic activities and overall health [1]. Propelled by the stomach into the small intestine, the chyme is transported by various patterns of muscle contractions [2, 3]. These patterns optimize enzymatic digestion, breaking down large nutrients into smaller ones that can be absorbed by the intestinal wall.

The inner surface of the small intestine is characterized by the presence of small projections (of the .1 – 1mm size range) known as villi. These structures are lined with epithelial cells through which nutrient molecules are transported towards nearby blood and lymph capillaries. Across different species, positions along the digestive tract and stages of development, villi exhibit morphological variations, ranging from elongated finger-shaped structures to leaflet formations [4]. Their size and density also vary from species to species [5, 6]. They can be more or less long and more or less thin. After traversing the small intestine, the chyme continues its journey through the large intestine. This organ lacks villi but possesses crypts [7]. Similar to villi, these crypts vary in depth, width, and density between species [8, 9]. These morphological differences may play a key role in optimizing absorption across species.

A widely accepted idea is that villi increase the absorptive surface area, thereby enhancing nutrient uptake [10–12]. However, there exists a trade-off: increasing the density of villi (resp. crypts), increases the surface area for absorption; but this reduces the intervillous space (resp. crypt width), which decreases the amount of nutrients that enter these spaces, reducing absorption [13]. The question of optimal distance microstructures, or in other words their density, has not been studied quantitatively yet. If better absorption has been selected by evolution, villi geometry will be close to optimal. This study aims to test this evolutionary hypothesis and provide a theoretical framework for understanding the observed variations in villi and crypt structures across species [14].

It has also been proposed that villi enhance mixing and flow near the intestinal wall [13, 15–18]. For instance, their muscular contraction driven movement may enhance local mixing and reduce boundary layer thickness. However, the extent of this flow remains debated, especially in presence of mucus [19–21]. Crypts, in particular are generally thought to be filled with a denser mucus [22, 23]. In this work, we explore a limiting case in which only diffusion occurs between villi and crypts, neglecting any convective contribution.

In this article, through analytical calculations, we examine the interplay between these structures' configurations and absorption efficiency. We first present a simplified model for absorption between leaf-like villi, in which symmetries allow us to solve this system in 2D, similar to [24, 25]. We then study the case of finger-like villi using a 3D model. Thirdly, we conduct a similar study for crypts. Finally, we compare our results for the optimal density with physiological data for various species.
# 1 General Model

In the present study, we assume that the villi are not moving relative to each other. In particular, this means that there is no fluid flow caused by their movement in the intervillous space. We also assume that convective mixing in the lumen does not extend to the intervilli space. While the extent of flow near the epithelial surface remains debated, several studies suggest the presence of an unstirred water layer thicker than the villi height [19–21, 26], which would strongly limit convective transport between villi. In the colon, a dense mucus layer several hundred micron thick covers the epithelium [27], resulting in a predominantly diffusive transport. In this study we assume the lumen is well-mixed, while there is only diffusive transport between villi and within crypts.

Due to this localized diffusion regime, we model the system using the steady-state diffusion equation (also called the Laplace equation) where C is the concentration profile close to these structures. We model the system as steady-state because we assume that diffusion time-scales are much shorter than transit time-scales in the gut. The longest villi are of order 1mm so the diffusion time-scale for glucose between villi is $T \approx L^2/(2D) \approx (10^{-3})^2/(2 \times 6 \times 10^{-10}) \approx 833$ seconds [28] which is less than 15 minutes, much smaller than the residence time in the gut. Secondly, we assume that the length scales for longitudinal concentration gradients along the digestive tract are much larger than the typical size of the villi or crypt structures studied. We also consider that mixing is efficient in the lumen, and thus the luminal concentration is fixed at a concentration $c_0$. This gives the following equations:

$$ \nabla^2 C = 0 \qquad (1) $$

$$ C(y = h) = c_0 \qquad (2) $$

Lastly, we consider that the epithelial walls absorb imperfectly. For this, we use the Robin boundary condition:

$$ D(\vec{n} \cdot \nabla C) + kC = 0 \qquad (3) $$

Where D is the molecular diffusion coefficient, $\vec{n}$ is the normal vector to the wall pointing outward, and k is a positive absorptivity constant. When k approaches zero, the condition tends towards reflection of nutrient particles by the structures' walls, whereas when k tends towards infinity, it approaches perfect absorption.

The archetypal geometries of the structures studied in this article are presented in figure 1. We calculate nutrient absorption for the three geometries: leaf-like villi, finger-like villi, and colonic crypts. Equation 1 is solved for each of the three geometries using Equations 2 and 3 as boundary conditions. We find concentration profiles, quantify absorption for each geometry and discuss which geometries maximize absorption.

# 2 Results

## 2.1 Leaf-like villi

We first look at absorption between two leaf-like villi, which is the simplest of the three cases. The symmetry allows us to reduce the system from 3D to 2D. As
<figure><img src="image_3.png" /><figcaption>a. Inner surface of a rat ileum</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>b. Sketch of leaf-like intestinal villi.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>c. Sketch of finger-like intestinal villi.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>d. Sketch of colonic crypts.</figcaption></figure>

Figure 1: Structures on the inner surface of the gastrointestinal tract. (a) Experimental image of the inner surface of a rat ileum. The blue arrow represents the longitudinal axis, and the magenta dashed lines represent where the tube has been cut open for imaging. A zoom shows that in this sample, the villi are leaf-like. (b,c,d) Sketches of the three studied geometries in this paper. The parameters are not to scale. *h* corresponds to the length or depth of the villi and crypts. *e* corresponds to the width of the villi or the distance between crypts. *2R* corresponds to the intervillous distance or the crypt diameter. The space characterized by *h* and *R* is the space where particles diffuse freely.
illustrated in Figure 1b, we consider villi as stationary rectangular projections, and the space between two villi as analogous to a well. In our system, h is the villi height, R is half of the intervillous distance, and e is the villi width.

To non-dimensionalize the equations, we introduce dimensionless units: $\tilde{x} = x/R$, $\tilde{y} = y/h$, $\xi = R/h$, $\Theta = D/(kh)$, and $c = C/c_0$. Here, $\xi$ is the rescaled intervillous width, and $\Theta$ is the ratio of the absorption time scale to the diffusion time scale for a given nutrient, which we call the rescaled inverse surface absorptivity. Consequently, the Equations (1-3) become:

$$ \frac{\partial^2 c}{\partial \tilde{x}^2} + \xi^2 \frac{\partial^2 c}{\partial \tilde{y}^2} = 0 \qquad (4) $$

$$ c(\tilde{x}, 1) = 1 \qquad (5) $$

$$ \Theta \vec{n} \left( \frac{1}{\xi} \frac{\partial c}{\partial \tilde{x}} \vec{e}_x + \frac{\partial c}{\partial \tilde{y}} \vec{e}_y \right) + c = 0 \qquad (6) $$

We solve equation 4 with the boundary conditions of equations 5 and 6 using separation of variables. We obtain the following solution (details in SI section A):

$$ c(\tilde{x}, \tilde{y}) = \sum_{n=1}^{\infty} \frac{4 \sin(\mu_n) \cos(\mu_n \tilde{x}) (\Theta \frac{\mu_n}{\xi} \cosh(\frac{\mu_n}{\xi} \tilde{y}) + \sinh(\frac{\mu_n}{\xi} \tilde{y}) (2\mu_n + \sin(2\mu_n)) (\Theta \frac{\mu_n}{\xi} \cosh(\frac{\mu_n}{\xi}) + \sinh(\frac{\mu_n}{\xi})}}{(7) \qquad (7) $$

where $\mu_n$ is the n-th positive root of the equation:

$$ \Theta \frac{\mu_n}{\xi} = \frac{1}{\tan(\mu_n)} \qquad (8) $$

Figure 2 illustrates the concentration distribution between two villi for $\Theta = 1$ and $\xi = 1$. Consistent with expectations, the concentration diminishes with depth between the villi, due to particle absorption by the walls. A reduction in $\Theta$ intensifies absorption, resulting in a steeper concentration gradient (SI Figure S1). We recover the results from article [24], where similar assumptions and this geometry were used. In addition, we study the absorbing flux, taking into account absorption at the tip of the villi, and discuss the optimal spacing between them.

The rate of absorption is given by the concentration flux through the villi walls:

$$ J = -D \iint_S \vec{n} \cdot \nabla C(x, y) d\vec{S} \qquad (9) $$

This measure gives information on the efficiency of the intervillous space in absorbing particles diffusing in this space. A higher flux means more absorption. The expression of the dimensionless flux for one intervillous space is:

$$ \tilde{J} = \sum_{n=1}^{\infty} \frac{8 \sin(\mu_n)^2 (\Theta \frac{\mu_n}{\xi} \sinh(\frac{\mu_n}{\xi}) + \cosh(\frac{\mu_n}{\xi}))}{(2\mu_n + \sin(2\mu_n))(\Theta \frac{\mu_n}{\xi} \cosh(\frac{\mu_n}{\xi}) + \sinh(\frac{\mu_n}{\xi}))} \qquad (10) $$

With $\tilde{J} = J/(Dc_0)$ the dimensionless flux. This calculation is done for one intervillous space. What matters is nutrient absorption per unit length. To be able to compare different villi densities, we introduce the notion of flux per unit
<figure><img src="image_7.png" /><figcaption>Figure 2: Contour plot of the rescaled nutrient concentration within an intervillous space for the leaf-like villi geometry ((7)). In this figure, the rescaled intervillous width $\xi = 1$ and the rescaled inverse surface absorptivity $\Theta = 1$. The colors correspond to the rescaled concentration within this space (the lumen concentration is set to 1). Absorption between villi results in lower concentrations the lower the position. Left-right symmetry is a result of the symmetry of the system.</figcaption></figure>

length (flux density) where the unit length is one period $(2R + e)$ where $e$ is the villi width. In this case, the flux per unit length is:

$$j = \frac{J + kc_0e}{2R + e} \qquad (11)$$

Where $kc_0e$ is the flux on the villi tip. In the absence of villi (flat gut walls), the flux per unit length would be $j = kc_0$ under the assumption of perfect mixing in the lumen. We use this information to non-dimensionalize this flux. Using $\tilde{e} = e/h$, the dimensionless flux per unit length becomes:

$$\tilde{j} = \frac{\Theta \tilde{J} + \tilde{e}}{2\xi + \tilde{e}}, \qquad (12)$$

where $\tilde{j} = j/(kc_0)$.

The flux depends on several parameters. The smaller the value of the inverse rescaled absorptivity $\Theta$ is, the larger the absorption. For a given nutrient it will be fixed by the nutrient's diffusion coefficient and the epithelial cell absorption properties. The greater the value of $h$ is, the greater the absorption is. We assume $h$ is set by other constraints such as resources needed to build longer
villi or diffusion times to the bottom of these structures. Also, the smaller the villi width $\tilde{e}$, the better the absorption, as there can be more intervillous space per unit length. But $\tilde{e}$ needs to have some minimal value to accommodate blood and lymph circulation [29]. The remaining parameter that can be varied to maximize absorption is the distance between villi, R, which corresponds to $\xi$ in dimensionless units.

Figure 3 illustrates the flux density as a function of $\xi$ and $\Theta$, shown for $\tilde{e} = 0.1$, which is a physiological value for some animals. The flux landscape shows that for each value of $\Theta$, there exists a value of $\xi$ for which the flux is maximal. This means that there is an intervillous space, i.e., a villi density, that maximizes flux. The maxima are shown with the green line. Changing $\tilde{e}$ slightly shifts the optimum, see Supplementary Figure S2 for other $\tilde{e}$ values. It is observed that the optimal value of $\xi$ does not change much in the physiological range of $\Theta$. This is biologically relevant: different nutrients will have different $\Theta$, and our results show that a villi density optimal for a nutrient with a given $\Theta$ will be close to optimal for another one.

<figure><img src="image_8.png" /></figure>

Figure 3: Leaf-like villi log flux density per unit of gut length $\tilde{j}$ ((12)) as a function of $\xi$ (the rescaled intervillous width) and $\Theta$ (the rescaled inverse surface absorptivity) for $\tilde{e} = 0.1$ (the rescaled villi width). There exists a value of $\xi$ that maximizes flux at a given $\Theta$. The green line represents the maximum flux for each $\Theta$. The two black dashed vertical lines correspond to the edges of physiological values for $\Theta$, details in SI section E. For $\tilde{e} = 0.1$, the optimal value of $\xi$ ranges from 0.01 to 0.03 in this range of $\Theta$.
## 2.2 Finger-like villi

We now look at absorption between finger-like villi. We model the villi as evenly spaced cylinders of height $h$ on a triangular lattice such that the radius of a villus is $e/2$ and the shortest distance between the sides of two villi is $2R'$ (consistent with the 2D model as seen in Figure 1c). We have repeating hexagons centered on the villi at the nodes of the triangular lattice of side length $R' + e/2$. To solve this problem using cylindrical coordinates, we approximate the hexagons as circles of radius $R + e/2$ keeping the surface of the circle equal to the surface of the hexagon (see figure S3). The dimensionless units for the system are: $\tilde{r} = r/R$, $\tilde{z} = z/h$, $\xi = R/h$, $\tilde{e} = e/h$, $\Theta = \frac{D}{kh}$, and $c = C/c_0$. Using cylindrical coordinates, we reduce the system to an axisymmetric one, where Equations (1-3, 9) give the following dimensionless flux within an intervillous space (details in SI Section 2):

$$ \tilde{J} = \sum_{n=1}^{\infty} \frac{\tilde{e}\pi}{\left(\int_{\frac{\tilde{e}}{2\tilde{\xi}}}^{1+\frac{\tilde{e}}{2\tilde{\xi}}} r f_n(r)^2 dr\right) \left(\Theta \frac{\mu_n}{\tilde{\xi}} \cosh\left(\frac{\mu_n}{\tilde{\xi}}\right) + \sinh\left(\frac{\mu_n}{\tilde{\xi}}\right)\right)} \\ \times \left(\int_{\frac{\tilde{e}}{2\tilde{\xi}}}^{1+\frac{\tilde{e}}{2\tilde{\xi}}} r f_n(r) dr\right) \\ \times \left(-\frac{Y_1(\mu_n(1+\frac{\tilde{e}}{2\tilde{\xi})))}{J_1(\mu_n(1+\frac{\tilde{e}}{2\tilde{\xi})))} J_0(\mu_n \frac{\tilde{e}}{2\tilde{\xi}}) + Y_0(\mu_n \frac{\tilde{e}}{2\tilde{\xi}})\right) \\ \times \left(\Theta \frac{\mu_n}{\tilde{\xi}} \sinh\left(\frac{\mu_n}{\tilde{\xi}}\right) + \cosh\left(\frac{\mu_n}{\tilde{\xi}}\right)\right), \quad (13) $$

with $\mu_n$ as the solution to the following transcendental equation:

$$ \Theta \frac{\mu_n}{\tilde{\xi}} = - \frac{Y_1(\mu_n(1 + \frac{\tilde{e}}{2\tilde{\xi}))) J_0(\mu_n \frac{\tilde{e}}{2\tilde{\xi}}) - J_1(\mu_n(1 + \frac{\tilde{e}}{2\tilde{\xi}))) Y_0(\mu_n \frac{\tilde{e}}{2\tilde{\xi}})}{Y_1(\mu_n(1 + \frac{\tilde{e}}{2\tilde{\xi}))) J_1(\mu_n \frac{\tilde{e}}{2\tilde{\xi}}) - J_1(\mu_n(1 + \frac{\tilde{e}}{2\tilde{\xi}))) Y_1(\mu_n \frac{\tilde{e}}{2\tilde{\xi}})} \quad (14) $$

and $f_n(\tilde{r})$ is:

$$ f_n(\tilde{r}) = - \frac{Y_1(\mu_n(1 + \frac{\tilde{e}}{2\tilde{\xi})))}{J_1(\mu_n(1 + \frac{\tilde{e}}{2\tilde{\xi})))} J_0(\mu_n \tilde{r}) + Y_0(\mu_n \tilde{r}), \quad (15) $$

where $\tilde{J} = J/(Dc_0h)$. In the previous equations, $J_n(x)$ and $Y_n(x)$ are Bessel functions of the first and second type, of order $n$. To account for villi density, we introduce the flux per unit surface (in comparison to the flux per unit length in the previous geometry), $j$, where the unit surface is a circle of radius $R + e/2$. In this case, the new dimensionless flux is:

$$ \tilde{j} = \frac{\Theta \tilde{J} + \pi \left(\frac{\tilde{e}}{2}\right)^2}{\pi \left(\xi + \frac{\tilde{e}}{2}\right)^2}, \quad (16) $$

where $\tilde{j} = j/(kc_0)$. We obtain a flux per unit surface that uses the same parameters as the flux per unit length for the case of leaf-like villi. This will allow us, in a later section, to compare the results between geometries.
## 2.3 Crypts

Crypts can be modeled as 3D wells of radius $R$ and height $h$. For symmetry simplifications, we assume that the crypts are positioned on a triangular lattice such that the shortest distance between the sides of two crypts is $e$. Solving the system using Equations (1-3, 9) in cylindrical coordinates gives the following dimensionless flux in a crypt (details in SI Section 3):

$$ \tilde{J} = \sum_{n=1}^{\infty} \frac{4\pi\xi J_1^2(\mu_n)(\Theta\frac{\mu_n}{\xi}\sinh(\frac{\mu_n}{\xi}) + \cosh(\frac{\mu_n}{\xi}))}{\mu_n(J_0^2(\mu_n) + J_1^2(\mu_n))(\Theta\frac{\mu_n}{\xi}\cosh(\frac{\mu_n}{\xi}) + \sinh(\frac{\mu_n}{\xi}))}, \quad (17) $$

with $\mu_n$ a solution to:

$$ \Theta \frac{\mu_n}{\xi} = \frac{J_0(\mu_n)}{J_1(\mu_n)}. \quad (18) $$

with hexagons centered on the crypts of side length $R + e/2$ (Figure 1d), the dimensionless flux per unit surface is:

$$ \tilde{j} = \frac{\Theta \tilde{J} + \frac{3\sqrt{3}}{2}(\xi + \tilde{e}/2)^2 - \pi\xi^2}{\frac{3\sqrt{3}}{2}(\xi + \tilde{e}/2)^2}. \quad (19) $$

## 2.4 Comparing the three models

In the previous three subsections, absorption was quantified in the intestines for three archetypal geometries: leaf-like villi, finger-like villi, and crypts. To compare the three models effectively, similar dimensionless parameters are introduced: $\xi$ is the intervillous distance (resp. crypt radius) rescaled by its depth, $\tilde{e}$ is the villi width (resp. inter-crypt distance), and $\Theta$ is the rescaled inverse surface absorptivity. According to [30–33] we estimate physiological $\Theta$ values to be in the range [1–100] (more details in SI section 5). We find that the optimum has a weak dependence on $\Theta$ in the range 1–100. The optimal intervillous distance and crypt radius are a function of $\tilde{e}$. By solving the following equation numerically, we can determine the villus or crypt density that maximizes absorption.

$$ \frac{\partial J(\xi, \tilde{e}, \Theta)}{\partial \xi} = 0 \quad (20) $$

Figure 4 shows this function for all three geometries for $\Theta = 10$. As $\tilde{e}$ increases, the optimal value of $\xi$ also weakly increases. The two villi geometries have similar optima. The optimum for crypts is found at a much higher $\xi$.

# 3 Comparison with physiological data

The objective of this study is to determine if the villi and crypt geometry is adapted to maximize absorption. We calculate the structure density that maximizes absorption through different types of structures. $\xi$ denotes the rescaled intervillous width or crypt radius, while $\tilde{e}$ represents the rescaled villi width or the distance between two crypts.

Our theoretical analysis shows that for a single intervillous space, it is beneficial to have an infinitely wide gap to maximize absorption. However, a smaller gap allows for a higher density of structures per unit of gut length. We show that
<figure><img src="image_9.png" /><figcaption>Figure 4: Plot of the rescaled intervillous width maximizing absorption per unit of gut length as a function of the rescaled inter-structure distance for the three geometries introduced earlier. The shaded regions correspond to 95% of the maximal absorption for each geometry. Data points show physiological data from various species, as recorded in Table 1. The green markers correspond to crypts, the blue markers correspond to leaf-like villi, the orange markers correspond to finger-like villi and the purple markers to a mix of finger- and leaf-like villi. Here $\Theta = 10$, in the middle of the physiological range for $\Theta$, but similar results are obtained for other values of $\Theta$ in Supplementary Figure S10.</figcaption></figure>

the density maximizing the absorption flux per unit of gut length comes from this trade-off. These results are illustrated in Figure 4, which also compares our model's predictions with physiological data for various species, as summarized in Table 1, see SI section D for details of the methods used to collect this information.

Most physiological geometries fall within the predicted region for 95% of the maximal absorption. We captured the similarity between leaf-like and finger-like villi, as well as the fact that the crypt width is much larger than the distance between villi.

# 4 Conclusion

In this study, we examine the role of micro-structures, intestinal villi, and colonic crypts, in maximizing intestinal absorption. While it is usually assumed that any increase in surface increases absorption [10–12], we actually quantify a crucial trade-off: higher villi and crypt densities increase total absorption surface area, but also decrease nutrient penetration in a given intervillous space or in a given crypt [13].

We assume that nutrients are well mixed in the lumen, but that their transport within the intervillous spaces and colonic crypts occurs solely by diffusion. In three simplified geometries, representing leaf-like villi, finger-like villi, and crypts, we solve analytically the diffusion equation with semi-absorbing boundary conditions. We obtain the absorption flux per unit of gut length.
<table><caption>Table 1: Physiological data on villi and crypt length, width, and spacing. Every length is in µm, except the horse data which is in arbitrary units. The second column corresponds to the section of the small intestine for villi, where D denotes Duodenum while J denotes Jejunum. The deviation around the mean lengths is also shown in the data. The shapes of the villi are shown in the penultimate column. When the shape is unknown due to the images only showing a 2D section of the villi, the description shows "Unclear". When the villi are finger- and leaf-like on a single image, the description shows "Both". Data is plotted in Figure 4 to show the connection with theoretical predictions.</caption><thead><tr><th colspan="8">Villi Measurements</th></tr><tr><th>Species</th><th>Section</th><th>Length (h)</th><th>Width (e)</th><th>Intervillous space (2R)</th><th>ε̃</th><th>ξ</th><th>Villi shape</th><th>Source</th></tr></thead><tbody><tr><td>Human</td><td>J</td><td>360 ± 10</td><td>160 ± 5</td><td>20 ± 5</td><td>0.44</td><td>0.028</td><td>Leaf</td><td>[34]</td></tr><tr><td>Rat</td><td>D</td><td>390 ± 34</td><td>70 ± 6</td><td>10 ± 5</td><td>0.18</td><td>0.013</td><td>Unclear</td><td>[35],[36]</td></tr><tr><td>Mouse</td><td>J</td><td>405 ± 8</td><td>75 ± 10</td><td>15 ± 10</td><td>0.18</td><td>0.019</td><td>Both</td><td>[37]</td></tr><tr><td>Piglet</td><td>D</td><td>340 ± 15</td><td>100 ± 6</td><td>30 ± 8</td><td>0.29</td><td>0.044</td><td>Finger</td><td>[38]</td></tr><tr><td>Pheasant</td><td>J</td><td>460 ± 10</td><td>52 ± 7</td><td>26 ± 7</td><td>0.11</td><td>0.028</td><td>Unclear</td><td>[39]</td></tr><tr><td>Horse</td><td>J</td><td>775 ± 20</td><td>400 ± 5</td><td>80 ± 10</td><td>0.52</td><td>0.052</td><td>Finger</td><td>[40]</td></tr><tr><td>Chicken</td><td>D</td><td>1000 ± 200</td><td>90 ± 6</td><td>32 ± 8</td><td>0.09</td><td>0.016</td><td>Leaf</td><td>[41]</td></tr></tbody></table><table><thead><tr><th colspan="8">Crypt Measurements</th></tr><tr><th>Species</th><th>Depth (h)</th><th>Inter-crypt distance (e)</th><th>Crypt diameter (2r)</th><th>ε̃</th><th>ξ</th><th></th><th>Source</th></tr></thead><tbody><tr><td>Human</td><td>500 ± 50</td><td>100 ± 5</td><td>120 ± 10</td><td>0.2</td><td>0.24</td><td></td><td>[42]</td></tr><tr><td>Mouse</td><td>350 ± 50</td><td>50 ± 20</td><td>60 ± 10</td><td>0.14</td><td>0.17</td><td></td><td>[43]</td></tr></tbody></table>

Our hypothesis is that villi/crypt geometry may have been selected to maximize absorption. The absorption flux per unit of gut length depends on the dimensionless parameters Θ, ε̃, and ξ. Θ is the rescaled inverse surface absorptivity and a decrease of Θ means that the epithelial surface would better absorb nutrients, so Θ should be as low as possible. ε̃ and ξ are two lengths rescaled by *h* the villi height / crypt depth. A larger *h* enables more absorption, but is likely limited by physiological factors that we did not explore. ε̃ is *e*/*h*, with *e* the thickness of the structure (villi width or distance between crypts). Smaller ε̃ enables denser structures with the same nutrient penetration, so it increases absorption. But it has to have some finite value to be able to perform its physiological functions, like blood and lymph transport. Thus the parameter that may be optimized is ξ = *R*/*h*, with *R* the intervilli distance, or the crypt width.

We found numerically that for a given (ε̃, Θ), there is one value of ξ maximizing absorption. Additionally, the optimal value of ξ depends only weakly on Θ, thus the optimal geometry is similar for different nutrients. We measured physiological geometries for various animals using experimental images in the scientific literature. Most of the data is in the region predicted to have an absorption within 95% of the maximal absorption efficiency. The model correctly predicts a larger ξ for crypts than for villi. This suggests that intestinal structure density has been selected to maximize absorption efficiency.

For the physiological data, we relied on measurements from published experimental studies. A key limitation of this approach is that tissue fixation for imaging can alter the native geometry of the samples, potentially affecting
dimensional accuracy. Additionally, during digestion, physiological factors such as luminal pressure may cause dynamic changes in tissue shape that are not captured in fixed samples. However, experimental studies suggest that longitudinal distension is relatively limited under physiological conditions [44]. Despite these modifications, the flatness of the optimum suggests that such variations are unlikely to move measurements significantly away from the optimal range.

One limitation in this study is the fact that we approximated the hexagons in the finger-like geometry as cylinders to solve the equations (see schematic SI section B). This approximation may be less adequate at high villi density. Additionally, we assumed both that the lumen is well-mixed and that there is only diffusion in the intervillous space and crypts. The gradient of concentration induced by luminal mixing may extend to the intervillus space for large rescaled intervillous widths / crypt widths $ξ$. However, in the limit of very large $ξ$, the absorption will be the same as for a smooth surface. We found that the optimum for absorption is for $ξ \ll 1$, thus in the regime where our approximation should be valid. We also neglected secretion and absorption, in particular of water, which could result in a net flow that would modify transport. Finally, we also assumed that the structures are static. For the crypts, it is likely a good approximation. For the villi, there are contractions of the underlying muscles which might move the villi. Th will create flow and pumping between them [17, 45, 46], and the purely diffusive approximation between villi may fail, though the presence of mucus may make convective transport less efficient.

While the most common idea in the literature on villi and crypts is that increased surface area always enhances absorption [10, 11], some studies, such as [13], have suggested that overly dense structures may actually hinder diffusion. However, there was no quantitative model like ours that explicitly calculates the optimal geometry.

Our results underscore the principle that biological systems are optimized for functional efficiency. The study suggests that villi density is not arbitrary but has evolved to maximize nutrient uptake [14]. There exist diseases that modify villi and crypt geometries and that are associated with weaker nutrient absorption [47]. There could be other biological parameters affecting absorption, but our results show that geometry itself plays an important role.

LRP is part of the LabEx Tec21 (ANR-11-LABX-0030) and of the PolyNat Carnot Institute (ANR-11-CARN-007-01). The authors thank Agence Nationale de la Recherche for its financial support of the project TransportGut, ANR-21-CE45-0015.
# Supplementary material

## A Leaf-like geometry calculations

Here we solve the Laplace equation in the leaf-like villi case. Starting from equation 1 of the main text and with the rescaled $\tilde{x} = x/R$, $\tilde{y} = y/h$, $c = C/C_0$, $\xi = R/h$ and $\Theta = D/(kh)$, the differential equation to solve is:

$$ \frac{\partial^2 \tilde{c}}{\partial \tilde{x}^2} + \xi^2 \frac{\partial^2 \tilde{c}}{\partial \tilde{y}^2} = 0 \qquad (1) $$

With the following boundary conditions:

$$ c(\tilde{x}, 1) = 1 \qquad (2) $$

$$ \Theta \left( \xi^{-1} \frac{\partial c}{\partial \tilde{x}} \tilde{e}_x + \frac{\partial c}{\partial \tilde{y}} \tilde{e}_y \right) \vec{n} + c = 0, (\tilde{x}, \tilde{y}) \in S, \qquad (3) $$

where $S$ is either $\tilde{x} = -1$, $\tilde{x} = 1$ or $\tilde{y} = 0$. We solve the Laplace equation using separation of variables. We can write $c$ as the product of two functions such that:

$$ c(\tilde{x}, \tilde{y}) = f(\tilde{x})g(\tilde{y}) \qquad (4) $$

Substituting equation 4 into equation 1 yields:

$$ f''(\tilde{x})g(\tilde{y}) + \xi^2 f(\tilde{x})g''(\tilde{y}) = 0 \qquad (5) $$

$$ \frac{f''(\tilde{x})}{f(\tilde{x})} = -\xi^2 \frac{g''(\tilde{y})}{g(\tilde{y})} = -\lambda \qquad (6) $$

Where $f'(\tilde{x})$ is the derivative of $f(\tilde{x})$ with respect to $\tilde{x}$, $g'(\tilde{y})$ is the derivative of $g(\tilde{y})$ with respect to $\tilde{y}$ and $\lambda$ a constant which must be positive; otherwise, non-physical solutions are found. We define $\mu$ such that $\mu^2 = \lambda$.

We first solve for $f(\tilde{x})$:

$$ f''(\tilde{x}) + \mu^2 f(\tilde{x}) = 0 \qquad (7) $$

$$ f(\tilde{x}) = C \cos(\mu \tilde{x}) + D \sin(\mu \tilde{x}) \qquad (8) $$

We then use the boundary equation from equation 3 to find $C$ and $D$. The left-right symmetry implies that $D = 0$. For an arbitrary non-zero $C$ we find:

$$ \tan(\mu) = \frac{\xi}{\Theta \mu} \qquad (9) $$

There exists an infinite number of solutions $\mu_n$ to equation 9, where $n$ is a positive integer. So, we define $f_n(\tilde{x})$ as:

$$ f_n(\tilde{x}) = \cos(\mu_n \tilde{x}) \qquad (10) $$

Where $\mu_n$ is the nth solution to eq 9 and $g_n$ such that $c_n(\tilde{x}, \tilde{y}) = f_n(\tilde{x})g_n(\tilde{y})$. Each solution $f_n(\tilde{x})$ form an orthogonal basis because integration over the space $[-1, 1]$ of $\cos(\mu_n x) \cos(\mu_m x) = 0$ if $n \neq m$. The full solution to the differential
equation is a superposition of these solutions that form an orthogonal basis: $c(\tilde{x}, \tilde{y}) = \sum_{n=1}^{\infty} c_n(\tilde{x}, \tilde{y})$. We now solve for $g_n(\tilde{y})$ using equation 5:

$$ \xi^2 g_n''(\tilde{y}) - \lambda g_n(\tilde{y}) = 0 \qquad (11) $$

$$ \frac{\mu_n}{\xi} g_n(\tilde{y}) = B_n \cosh\left(\frac{\mu_n}{\xi} \tilde{y}\right) + A_n \sinh\left(\frac{\mu_n}{\xi} \tilde{y}\right) \qquad (12) $$

We then use the boundary conditions 3 in $\tilde{y} = 0$ to find the constants $A_n$ and $B_n$:

$$ -\frac{\Theta}{\xi} \mu_n A_n + B_n = 0 \qquad (13) $$

$$ B_n = \Theta \frac{\mu_n}{\xi} A_n \qquad (14) $$

We then obtain $g_n(\tilde{y})$ for a non-zero $A_n$:

$$ g_n(\tilde{y}) = \left( \Theta \frac{\mu_n}{\xi} \cosh\left(\frac{\mu_n}{\xi} \tilde{y}\right) + \sinh\left(\frac{\mu_n}{\xi} \tilde{y}\right) \right) A_n \qquad (15) $$

Then,

$$ c_n(\tilde{x}, \tilde{y}) = A_n \cos(\mu_n x) \left( \Theta \frac{\mu_n}{\xi} \cosh\left(\frac{\mu_n}{\xi} \tilde{y}\right) + \sinh\left(\frac{\mu_n}{\xi} \tilde{y}\right) \right) \qquad (16) $$

And:

$$ c(\tilde{x}, \tilde{y}) = \sum_{n=1}^{n=\infty} A_n \cos(\mu_n x) \left( \Theta \frac{\mu_n}{\xi} \cosh\left(\frac{\mu_n}{\xi} \tilde{y}\right) + \sinh\left(\frac{\mu_n}{\xi} \tilde{y}\right) \right) \qquad (17) $$

We now calculate $A_n$. To do so we use the final boundary condition (equation 2) $c(\tilde{x}, 1) = 1$.

$$ c(\tilde{x}, 1) = 1 = \sum_{n=1}^{n=\infty} A_n \cos(\mu_n x) \left( \Theta \frac{\mu_n}{\xi} \cosh\left(\frac{\mu_n}{\xi}\right) + \sinh\left(\frac{\mu_n}{\xi}\right) \right) \qquad (18) $$

Substituting $\tilde{A}_n = A_n(\Theta \frac{\mu_n}{\xi} \cosh(\frac{\mu_n}{\xi}) + \sinh(\frac{\mu_n}{\xi}))$ we obtain the following equation:

$$ 1 = \sum_{n=1}^{n=\infty} \tilde{A}_n \cos(\mu_n x) \qquad (19) $$

We recognize a generalized Fourier series. Multiplying both sides of equation 19 by $\cos(\mu_n x)$ and integrating for $x$ between $-1$ and $1$, we find that:

$$ \tilde{A}_n = \frac{\int_{-1}^{1} \cos(\mu_n x) dx}{\int_{-1}^{1} \cos^2(\mu_n x) dx} = \frac{4 \sin(\mu_n)}{2\mu_n + \sin(2\mu_n)} \qquad (20) $$

So the full expression of $c(\tilde{x}, \tilde{y})$ is:

$$ c(\tilde{x}, \tilde{y}) = \sum_{n=1}^{\infty} \frac{4 \sin(\mu_n) \cos(\mu_n \tilde{x}) \left( \Theta \frac{\mu_n}{\xi} \cosh\left(\frac{\mu_n}{\xi} \tilde{y}\right) + \sinh\left(\frac{\mu_n}{\xi} \tilde{y}\right) \right)}{(2\mu_n + \sin(2\mu_n)) \left( \Theta \frac{\mu_n}{\xi} \cosh\left(\frac{\mu_n}{\xi}\right) + \sinh\left(\frac{\mu_n}{\xi}\right) \right)} \qquad (21) $$

This expression is plotted figure S1.
<figure><img src="image_10.png" /><figcaption>Figure S1: Contour plots of the rescaled concentration within an intervillous space for the leaf-like villi geometry like in figure 1 of main text for various values of the rescaled intervillous space $\xi$ and the rescaled inverse surface absorptivity $\Theta$</figcaption></figure>

The expression of nutrient concentration flux integrated over one intervillous space is the following:

$$J = -D \iint_S \vec{n} \nabla C(x, y) d\vec{S} \qquad (22)$$

In dimensionless units this equation becomes:

$$\tilde{J} = -\left(\xi \int_{-1}^{1} \frac{\partial c}{\partial \tilde{y}}|_{\tilde{y}=0} d\tilde{x} - \frac{1}{\xi} \int_{0}^{1} \frac{\partial c}{\partial \tilde{x}}|_{\tilde{x}=1} d\tilde{y} + \frac{1}{\xi} \int_{0}^{1} \frac{\partial c}{\partial \tilde{x}}|_{\tilde{x}=-1} d\tilde{y}\right) \quad (23)$$

With $\tilde{J} = J/(Dc_0)$. The first term corresponds to the flux at the bottom of the intervillous space while the second and third terms correspond to the flux on each side of the intervillous space. Integrating each term gives the following expression of the rescaled flux:

$$\tilde{J} = \sum_{n=1}^{\infty} \frac{8 \sin(\mu_n)^2 \left(\Theta \frac{\mu_n}{\xi} \sinh\left(\frac{\mu_n}{\xi}\right) + \cosh\left(\frac{\mu_n}{\xi}\right)\right)}{\left(2\mu_n + \sin(2\mu_n)\left(\Theta \frac{\mu_n}{\xi} \cosh\left(\frac{\mu_n}{\xi}\right) + \sinh\left(\frac{\mu_n}{\xi}\right)\right)\right)} \quad (24)$$

As explained in the main text, this expression is then used to calculate the flux density per unit of gut length, shown figure S2.
<figure><img src="image_11.png" /><figcaption>Figure S2: Log flux density per unit of gut length $\tilde{j}$ for the leaf-like geometry as a function of $\xi$ and $\Theta$, like figure 3 of the main text, but for several values of $\tilde{e}$.</figcaption></figure>

## B Finger-like geometry calculations

Here we solve the Laplace equation in the finger-like villi case. We approximate the hexagonal repeating geometry to a cylindrical one by keeping a constant surface area as seen in the figure S1. The differential equation 1 in this geometry with the rescaled parameters $\tilde{r} = r/R$, $\tilde{z} = z/h$, $c = C/C_0$, $\xi = R/h$ and $\Theta = D/(kh)$ is:

$$ \frac{1}{\tilde{r}} \frac{\partial c}{\partial \tilde{r}} + \frac{\partial^2 \tilde{c}}{\partial \tilde{r}^2} + \xi^2 \frac{\partial^2 \tilde{c}}{\partial \tilde{z}^2} = 0 \qquad (25) $$

with the following boundary conditions:

$$ c(\tilde{r}, 1) = 1 \qquad (26) $$

$$ \Theta \left( \xi^{-1} \frac{\partial c}{\partial \tilde{r}} \tilde{e}_r + \frac{\partial c}{\partial \tilde{z}} \tilde{e}_z \right) \tilde{n} + c = 0, (\tilde{r}, \tilde{z}) \in S, \qquad (27) $$

$$ \tilde{n} \cdot \nabla c \left( 1 + \frac{\tilde{e}}{2\xi}, \tilde{z} \right) = 0 \qquad (28) $$

where $S$ is either $\tilde{r} = \tilde{e}/2$ or $\tilde{z} = 0$. We solve the Laplace equation using separation of variables. We can write $c$ as the product of two functions such that:

$$ c(\tilde{r}, \tilde{z}) = f(\tilde{r})g(\tilde{z}) \qquad (29) $$
<figure><img src="image_12.png" /><figcaption>Figure S3: Sketch of how the hexagon is converted to a circle. We keep an equivalent surface such that $R_1 = \sqrt{\frac{3\sqrt{3}}{2\pi}} R_2$.</figcaption></figure>

Substituting equation 29 into equation 25 yields:

$$ \frac{1}{\tilde{r}} f'g + f''g + fg'' = 0 \qquad (30) $$

$$ \frac{1}{\tilde{r}} \frac{f'}{f} + \frac{f''}{f} = -\xi^2 \frac{g''}{g} = -\lambda \qquad (31) $$

Where $f'(\tilde{r})$ is the derivative of $f(\tilde{r})$ with respect to $\tilde{r}$, $g'(\tilde{z})$ is the derivative of $g(\tilde{z})$ with respect to $\tilde{z}$ and $\lambda$ is chosen positive; otherwise, non-physical solutions are found.

We first solve for $f(\tilde{r})$:

$$ \frac{1}{\tilde{r}} f'(\tilde{r}) + f''(\tilde{r}) + \lambda f(\tilde{r}) = 0 \qquad (32) $$

As in the previous case, $\lambda > 0$ for physical solutions, and we define $\mu$ such that $\lambda = \mu^2$.

$$ f(\tilde{r}) = C J_0(\mu \tilde{r}) + D Y_0(\mu \tilde{r}) \qquad (33) $$

Using equation 28 we find:

$$ C = -D \frac{Y_1(\mu(1 + \frac{\tilde{e}}{2\xi}))}{J_1(\mu(1 + \frac{\tilde{e}}{2\xi}))} \qquad (34) $$

We then use the boundary equation from equation 27:

$$ \Theta \frac{\mu_n}{\xi} = - \frac{Y_1(\mu_n(1 + \frac{\tilde{e}}{2\xi}))J_0(\mu_n \frac{\tilde{e}}{2\xi}) - J_1(\mu_n(1 + \frac{\tilde{e}}{2\xi}))Y_0(\mu_n \frac{\tilde{e}}{2\xi})}{Y_1(\mu_n(1 + \frac{\tilde{e}}{2\xi}))J_1(\mu_n \frac{\tilde{e}}{2\xi}) - J_1(\mu_n(1 + \frac{\tilde{e}}{2\xi}))Y_1(\mu_n \frac{\tilde{e}}{2\xi})} \qquad (35) $$

There exist an infinite number of solutions $\mu_n$ to equation 35, where $n$ is a positive integer. So, we define

$$ f_n(\tilde{r}) = - \frac{Y_1(\mu_n(1 + \frac{\tilde{e}}{2\xi}))}{J_1(\mu_n(1 + \frac{\tilde{e}}{2\xi}))} J_0(\mu_n \tilde{r}) + Y_0(\mu_n \tilde{r}) \qquad (36) $$
Where $\mu_n$ is the nth solution to eq 35. We now solve for $g_n(\tilde{z})$:

$$ \xi^2 g_n''(\tilde{z}) - \lambda g_n(\tilde{z}) = 0 \qquad (37) $$

$$ g_n(\tilde{z}) = B_n \cosh\left(\frac{\mu_n}{\xi}\tilde{z}\right) + A_n \sinh\left(\frac{\mu_n}{\xi}\tilde{z}\right) \qquad (38) $$

We then use the boundary conditions to find $A_n$ and $B_n$:

$$ -\Theta \frac{\mu_n}{\xi} A_n + B_n = 0 \qquad (39) $$

$$ B_n = \Theta \frac{\mu_n}{\xi} A_n \qquad (40) $$

We then obtain $g_n(\tilde{z})$ for a non-zero $A_n$:

$$ g_n(\tilde{z}) = \left( \Theta \frac{\mu_n}{\xi} \cosh\left(\frac{\mu_n}{\xi}\tilde{z}\right) + \sinh\left(\frac{\mu_n}{\xi}\tilde{z}\right) \right) A_n \qquad (41) $$

Then,

$$ c_n(\tilde{x}, \tilde{y}) = A_n \left( - \frac{Y_1(\mu_n(1 + \frac{\tilde{e}}{2\xi}))}{J_1(\mu_n(1 + \frac{\tilde{e}}{2\xi}))} J_0(\mu_n \tilde{r}) + Y_0(\mu_n \tilde{r}) \right) \left( \Theta \frac{\mu_n}{\xi} \cosh\left(\frac{\mu_n}{\xi}\tilde{y}\right) + \sinh\left(\frac{\mu_n}{\xi}\tilde{y}\right) \right) \qquad (42) $$

And:

$$ c(\tilde{x}, \tilde{y}) = \sum_{n=1}^{n=\infty} A_n \left( - \frac{Y_1(\mu_n(1 + \frac{\tilde{e}}{2\xi}))}{J_1(\mu_n(1 + \frac{\tilde{e}}{2\xi}))} J_0(\mu_n \tilde{r}) + Y_0(\mu_n \tilde{r}) \right) \left( \Theta \frac{\mu_n}{\xi} \cosh\left(\frac{\mu_n}{\xi}\tilde{y}\right) + \sinh\left(\frac{\mu_n}{\xi}\tilde{y}\right) \right) \qquad (43) $$

We now calculate $A_n$. To do so we use the final boundary condition (equation 26) $c(\tilde{r}, 1) = 1$.

$$ c(\tilde{r}, 1) = \sum_{n=1}^{n=\infty} A_n \left( - \frac{Y_1(\mu_n(1 + \frac{\tilde{e}}{2\xi}))}{J_1(\mu_n(1 + \frac{\tilde{e}}{2\xi}))} J_0(\mu_n \tilde{r}) + Y_0(\mu_n \tilde{r}) \right) \left( \Theta \frac{\mu_n}{\xi} \cosh\left(\frac{\mu_n}{\xi}\tilde{y}\right) + \sinh\left(\frac{\mu_n}{\xi}\tilde{y}\right) \right) \qquad (44) $$

Substituting $\tilde{A}_n = A_n(\Theta \frac{\mu_n}{\xi} \cosh(\frac{\mu_n}{\xi}) + \sinh(\frac{\mu_n}{\xi}))$, we obtain the following equation:

$$ 1 = \sum_{n=1}^{n=\infty} \tilde{A}_n \left( - \frac{Y_1(\mu_n(1 + \frac{\tilde{e}}{2\xi}))}{J_1(\mu_n(1 + \frac{\tilde{e}}{2\xi}))} J_0(\mu_n \tilde{r}) + Y_0(\mu_n \tilde{r}) \right) \qquad (45) $$

We recognize a generalized Fourier series. It is possible to find $\tilde{A}_n$ as:

$$ \tilde{A}_n = \frac{\int_{\frac{\tilde{e}}{2\xi}}^{1+\frac{\tilde{e}}{2\xi}} r f_n(r) dr}{\int_{\frac{\tilde{e}}{2\xi}}^{1+\frac{\tilde{e}}{2\xi}} r f_n(r)^2 dr} \qquad (46) $$

So the full expression of $c(\tilde{r}, \tilde{z})$ is:

$$ c(\tilde{r}, \tilde{z}) = \sum_{n=1}^{\infty} \frac{\left( \int_{\frac{\tilde{e}}{2\xi}}^{1+\frac{\tilde{e}}{2\xi}} r f_n(r) dr \right) \left( - \frac{Y_1(\mu_n(1+\frac{\tilde{e}}{2\xi}))}{J_1(\mu_n(1+\frac{\tilde{e}}{2\xi}))} J_0(\mu_n \tilde{r}) + Y_0(\mu_n \tilde{r}) \right) \left( \Theta \frac{\mu_n}{\xi} \cosh\left(\frac{\mu_n}{\xi}\tilde{y}\right) + \sinh\left(\frac{\mu_n}{\xi}\tilde{y}\right) \right)}{\left( \int_{\frac{\tilde{e}}{2\xi}}^{1+\frac{\tilde{e}}{2\xi}} r f_n(r)^2 dr \right) \left( \Theta \frac{\mu_n}{\xi} \cosh\left(\frac{\mu_n}{\xi}\tilde{y}\right) + \sinh\left(\frac{\mu_n}{\xi}\tilde{y}\right) \right)} \qquad (47) $$
<figure><img src="image_13.png" /><figcaption>Figure S4: Contour plots of the rescaled concentration within an intervillous space for the finger-like villi geometry like in figure 1 of main text for various values of the rescaled intervillous space $\xi$ and the rescaled inverse surface absorption $\Theta$. The left side corresponds to the villi wall while the right side corresponds to the center of the intervillous space.</figcaption></figure>

It is plotted figure S4.

The expression of nutrient concentration flux integrated over one intervillous space is the following:

$$J = -D \iint_S \vec{n} \nabla C(r, z) d\vec{S} \qquad (48)$$

Integrating each term gives the following expression for the dimensionless flux:

$$\tilde{J} = \sum_{n=1}^{\infty} \frac{\tilde{e}\pi \left( \int_{\frac{\tilde{e}}{2\tilde{\xi}}}^{1+\frac{\tilde{e}}{2\tilde{\xi}}} r f_n(r) dr \right) \left( -\frac{Y_1(\mu_n(1+\frac{\tilde{e}}{2\tilde{\xi}}))}{J_1(\mu_n(1+\frac{\tilde{e}}{2\tilde{\xi}}))} J_1(\mu_n \frac{\tilde{e}}{2\tilde{\xi}}) + Y_1(\mu_n \frac{\tilde{e}}{2\tilde{\xi}}) \right) \left( \Theta\left(\frac{\mu_n}{\tilde{\xi}}\right) \sinh\left(\frac{\mu_n}{\tilde{\xi}}\right) + \cosh\left(\frac{\mu_n}{\tilde{\xi}}\right) \right)}{\left( \int_{\frac{\tilde{e}}{2\tilde{\xi}}}^{1+\frac{\tilde{e}}{2\tilde{\xi}}} r f_n(r)^2 dr \right) \left( \Theta\left(\frac{\mu_n}{\tilde{\xi}}\right) \cosh\left(\frac{\mu_n}{\tilde{\xi}}\right) + \sinh\left(\frac{\mu_n}{\tilde{\xi}}\right) \right)} \quad (49)$$

It is plotted figure S5. As explained in the main text, this result is used in the calculation of the flux per unit of gut length, plotted in figure S6.
<figure><img src="image_14.png" /><figcaption>Figure S5: Non-dimensionalized flux for one intervillous space for the finger-like villi geometry for a wide range of ξ and Θ in log-scale.</figcaption></figure>
<figure><img src="image_15.png" /><figcaption>Figure S6: Log flux density per unit of gut length $\tilde{j}$ for the finger-like geometry as a function of $\xi$ and $\Theta$ for several values of $\tilde{e}$.</figcaption></figure>
C Crypt geometry calculations

Here we solve the Laplace equation in the colonic crypt case. Starting from equation 1 of the main text and with the rescaled parameters $\tilde{r} = r/R$, $\tilde{z} = z/h$, $c = C/C_0$, $\xi = R/h$, $\Theta = D/(kh)$, the differential equation to solve is:

$$ \frac{1}{\tilde{r}} \frac{\partial \tilde{c}}{\partial \tilde{r}} + \frac{\partial^2 \tilde{c}}{\partial \tilde{r}^2} + \xi^2 \frac{\partial^2 \tilde{c}}{\partial \tilde{z}^2} = 0 \qquad (50) $$

With the following boundary conditions:

$$ c(\tilde{r}, 1) = 1 \qquad (51) $$

$$ \Theta(\xi^{-1} \frac{\partial c}{\partial \tilde{r}} \tilde{e}_r + \frac{\partial c}{\partial \tilde{z}} \tilde{e}_z) \vec{n} + c = 0, (\tilde{r}, \tilde{z}) \in S \qquad (52) $$

where S is either $\tilde{r} = 1$ or $\tilde{z} = 0$. We solve the cylindrical Laplace assuming c is the product of two functions such that:

$$ c(\tilde{r}, \tilde{z}) = f(\tilde{r})g(\tilde{z}) \qquad (53) $$

Substituting equation C4 into equation C1 yields:

$$ \frac{1}{\tilde{r}} f'(\tilde{r}) g(\tilde{z}) + f''(\tilde{r}) g(\tilde{z}) + \xi^2 f(\tilde{r}) g''(\tilde{z}) = 0 \qquad (54) $$

$$ \frac{1}{\tilde{r}} \frac{f'(\tilde{r})}{f(\tilde{r})} + \frac{f''(\tilde{r})}{f(\tilde{r})} = -\xi^2 \frac{g''(\tilde{z})}{g(\tilde{z})} = -\lambda \qquad (55) $$

Where $f'(\tilde{r})$ is the derivative of $f(\tilde{r})$ with respect to $\tilde{r}$, $g'(\tilde{z})$ is the derivative of $g(\tilde{z})$ with respect to $\tilde{z}$ and $\lambda$ is chosen positive otherwise non-physical solutions are found. We define $\lambda = \mu^2$

We first solve for $f(\tilde{r})$:

$$ \frac{1}{\tilde{r}} f'(\tilde{r}) + 1 f''(\tilde{r}) + \lambda f(\tilde{r}) = 0 \qquad (56) $$

We recognize a Bessel differential equation:

$$ f(\tilde{r}) = A J_0(\mu \tilde{r}) + B Y_0(\mu \tilde{r}) \qquad (57) $$

Finiteness of the solution implies that $B = 0$. Using the imperfect absorption boundary condition:

$$ \Theta \xi^{-1} \frac{\partial c}{\partial \tilde{r}}|_{\tilde{r}=1} + c(1, z) = 0 \qquad (58) $$

We find for a non-zero A:

$$ \Theta \frac{\mu}{\xi} = \frac{J_0(\mu)}{J_1(\mu)} \qquad (59) $$

There exists an infinite number of solutions $\mu_n$, where n is a positive integer. So, for a non-zero B we have $f_n(\tilde{r})$:

$$ f_n(\tilde{r}) = J_0(\mu_n \tilde{r}) \qquad (60) $$

Where $\mu_n$ is the nth solution to eq C10. We now solve for $g(\tilde{z})$:

$$ g''(\tilde{z}) - \xi^2 \lambda g(\tilde{z}) = 0 \qquad (61) $$
$$g(\tilde{z}) = A \cosh(\frac{\mu_n}{\xi} \tilde{z}) + B \sinh(\frac{\mu_n}{\xi} \tilde{z}) \quad (62)$$

We then use the boundary conditions to find A and B:

$$-\Theta \frac{\mu_n}{\xi} B + A = 0 \quad (63)$$

$$A = \Theta \frac{\mu_n}{\xi} B \quad (64)$$

We then obtain $g_n(\tilde{z})$ for a non-zero B:

$$g_n(\tilde{z}) = B \left( \Theta \frac{\mu_n}{\xi} \cosh\left(\frac{\mu_n}{\xi} \tilde{z}\right) + \sinh\left(\frac{\mu_n}{\xi} \tilde{z}\right) \right) \quad (65)$$

Then,

$$c_n(\tilde{r}, \tilde{z}) = A_n J_0(\mu_n \tilde{r}) \left( \Theta \frac{\mu_n}{\xi} \cosh\left(\frac{\mu_n}{\xi} \tilde{z}\right) + \sinh\left(\frac{\mu_n}{\xi} \tilde{z}\right) \right) \quad (66)$$

And:

$$c(\tilde{r}, \tilde{z}) = \sum_{n=1}^{n=\infty} A_n J_0(\mu_n \tilde{r}) \left( \Theta \frac{\mu_n}{\xi} \cosh\left(\frac{\mu_n}{\xi} \tilde{z}\right) + \sinh\left(\frac{\mu_n}{\xi} \tilde{z}\right) \right) \quad (67)$$

We now calculate $A_n$. To do so, we use the final boundary condition $c(\tilde{r}, 1) = 1$. The equation becomes: ù

$$1 = \sum_{n=1}^{n=\infty} \tilde{A}_n J_0(\mu_n \tilde{r}) \quad (68)$$

Where $\tilde{A}_n = A_n(\Theta \frac{\mu_n}{\xi} \cosh(\frac{\mu_n}{\xi}) + \sinh(\frac{\mu_n}{\xi}))$. It is possible to find $\tilde{A}_n$:

$$\int_{0}^{1} r J_{0}(\mu_{n} r) d r = \sum_{n=1}^{n=\infty} \tilde{A}_{n} \int_{0}^{1} r J_{0}(\mu_{n} r) J_{0}(\mu_{n} r) d r \quad (69)$$

$$\tilde{A}_n = \frac{2 J_1(\mu_n)}{\mu_n (J_0(\mu_n)^2 + J_1(\mu_n)^2)} \quad (70)$$

So, the complete expression of $c(\tilde{r}, \tilde{z})$ is:

$$c(\tilde{r}, \tilde{z}) = \sum_{n=1}^{n=\infty} \frac{2 J_1(\mu_n) J_0(\mu_n \tilde{r}) \left( \Theta \frac{\mu_n}{\xi} \cosh\left(\frac{\mu_n}{\xi} \tilde{z}\right) + \sinh\left(\frac{\mu_n}{\xi} \tilde{z}\right) \right)}{\mu_n (J_0(\mu_n)^2 + J_1(\mu_n)^2) \left( \Theta \frac{\mu_n}{\xi} \cosh\left(\frac{\mu_n}{\xi}\right) + \sinh\left(\frac{\mu_n}{\xi}\right) \right)} \quad (71)$$

This expression is plotted figure S7

The expression of nutrient concentration flux integrated over one intervillous space is the following:

$$J = -D \iint_S \vec{n} \nabla C(r, z) d\vec{S} \quad (72)$$

In dimensionless units this equation becomes:

$$\tilde{J} = -2\pi \left( \int_{0}^{1} \left. \frac{\partial c}{\partial \tilde{r}} \right|_{\tilde{r}=1} d\tilde{z} - \xi^2 \int_{0}^{1} \left. \tilde{r} \frac{\partial c}{\partial \tilde{z}} \right|_{\tilde{z}=0} d\tilde{r} \right) \quad (73)$$
<figure><img src="image_16.png" /><figcaption>Figure S7: Contour plots of the rescaled concentration within an crypt for the crypt geometry like in figure 1 of main text for various values of the rescaled crypt width $\xi$ and the rescaled inverse surface absorptivity $\Theta$ are used.</figcaption></figure>

With $\tilde{J} = J/(Dc_0h)$. The first term is the absorption on the walls and the second term is the absorption at the bottom of the crypt. Integrating each term gives the following expression of the rescaled flux:

$$ \tilde{J} = \sum_{n=1}^{\infty} \frac{4\pi J_1^2(\mu_n)(\Theta \frac{\mu_n}{\xi} \sinh(\frac{\mu_n}{\xi}) + \cosh(\frac{\mu_n}{\xi}))}{\mu_n(J_0^2(\mu_n) + J_1^2(\mu_n))(\Theta \frac{\mu_n}{\xi} \cosh(\frac{\mu_n}{\xi}) + \sinh(\frac{\mu_n}{\xi}))} \quad (74) $$

It is plotted figure S8. As explained in the main text, this result is then used to calculate the flu per unit of gut length, plotted figure S9.
<figure><img src="image_17.png" /><figcaption>Figure S8: Non-dimensionalized flux for one crypt for the crypt-like geometry for a wide range of ξ and Θ in log-scale.</figcaption></figure>
<figure><img src="image_18.png" /><figcaption>Figure S9: Log flux density per unit of gut length $\tilde{j}$ for the crypt geometry as a function of $\xi$ and $\Theta$ for several values of $\tilde{e}$.</figcaption></figure>
<table><caption>Table 2: Summary of sources and corresponding figures or tables used for extracting geometrical measurements of villi and crypts. D stands for duodenum, J for jejunum</caption><thead><tr><th>Species</th><th>Section</th><th>Source</th><th>Direct measures or images within the source</th></tr></thead><tbody><tr><td>Human</td><td>villi J</td><td>[34]</td><td>Figures 3 and 7</td></tr><tr><td>Rat</td><td>villi D</td><td>[35]</td><td>Table 1 and Figure 1</td></tr><tr><td>Rat</td><td>villi D</td><td>[36]</td><td>Figure 1</td></tr><tr><td>Mouse</td><td>villi J</td><td>[37]</td><td>Figure 1 and results section text</td></tr><tr><td>Piglet</td><td>villi D</td><td>[38]</td><td>Figures 1 and 2</td></tr><tr><td>Pheasant</td><td>villi J</td><td>[39]</td><td>Figures 3 and 5</td></tr><tr><td>Horse (au)</td><td>villi J</td><td>[40]</td><td>Figures 5 and 7</td></tr><tr><td>Chicken</td><td>villi D</td><td>[41]</td><td>Figures 1 and 2</td></tr><tr><td>Human</td><td>crypts</td><td>[42]</td><td>Figures 3 and 5</td></tr><tr><td>Mouse</td><td>crypts</td><td>[43]</td><td>Figure 2</td></tr></tbody></table>

## D Methods for collecting physiological data

Here we present briefly the methods used to collect data for Table 1 of the main text. We use published physiological data. When the quantities we are interested in are already measured in the source article, we directly use the data. In other cases, we exploit the microscopy images, measuring in ImageJ [48], and converting the pixels into real distances using the scale bars. In table 2, we give the list of sources, with for each source where we took the relevant information from.

For finger-like villi there is a conversion to make. We measure the villi diameter *e* and the distance between two villi $2R_2$, villi arrangement is approximately hexagonal. As shown in figure S3, we approximate it by a cylindrical geometry, with $R_1 = \sqrt{\frac{3\sqrt{3}}{2\pi}}R_2$ giving the same surface, and $R_1 = e/2 + R$. As *e* is directly measured, this expression gives *R* which is reported in the table from the main text.

## E Estimation of the rescaled inverse surface absorptivity

The rescaled inverse surface absorptivity is defined as $\Theta = D/(kh)$ where *D* is the molecular diffusion coefficient, *k* is a reactivity constant, or permeability according to [33], and *h* is the villi height or crypt depth. As seen in the main text it is important to know the value of $\Theta$ to get the optimal structure density.

Let us take glucose and mannitol as examples. They have similar diffusion coefficients, about $6 \cdot 10^{-10} \text{m}^2/\text{s}$ [28]. Glucose is an important nutrient and we expect the gut epithelia to absorb it very well, while mannitol is an example of sugar that is poorly absorbed (about 20% is absorbed [49]) and is used as a reference to compare with drugs absorption. An overestimate of the permeability of glucose is $k = 10^{-5} \text{m/s}$ [33]. It is an overestimate because this measure is for a surface with villi, but the measure is renormalized for the projected surface, not the total epithelium surface. For mannitol on a flat surface, $k = 5 \times 10^{-9} \text{m/s}$
[30]. The villi height *h* varies from 300 – 1000 µm with most values closer to the lower bound (see Table 1 of main text). Using these parameters, we can calculate the rescaled inverse surface absorptivity $\Theta = D/(kh)$. We find $\Theta \approx 0.2$ as an underestimate for glucose and $\Theta \approx 400$ for mannitol.

We expect most relevant nutrients to have the same order of magnitude for the diffusion coefficients and to have an absorbtion constant between the one for glucose and mannitol. So, we estimate the physiological range for $\Theta$ to be between 1 and 100.

## F Absorption by unit length

<figure><img src="image_19.png" /><figcaption>Figure S10: Plot of the rescaled intervillous distance or crypt radius maximizing absorption per unit length as a function of the rescaled villi width or interecrypt distance for the three geometries for several values of $\Theta$, the rescaled inverse surface absorptivity. The shaded regions correspond to 95% of the optimum for each geometry.</figcaption></figure>

## G Experimental images

To illustrate the different geometries of the villi, we took images of the inner structures of the rat small intestine.

An adult Male Wistar rat (350g) was euthanized via intraperitoneal injec- tion of pentobarbital (180 mg/kg). A midline abdominal incision was made to access the small intestine. The first 7 cm distal to the stomach (corresponding to the duodenum and jejunum) and the final 4 cm proximal to the cecum (cor- responding to the ileum) were dissected and immersed in Tyrode buffer at room temperature (Ph=7.4). The 7 cm segment was then divided into two portions representing the duodenum and jejunum. All three segments were flushed with the Tyrode buffer to remove intestinal contents. They were then cut along the mesenteric border to expose the lumen. Tissues were pinned in a Petri dish pre- viously coated with PDMS (polydimethylsiloxane), and images were captured using a Basler acA4112-20uc camera equipped with 0.75x lenses from Opto En- gineering. The pictures were edited to enhance contrast using Adobe Lightroom Classic CC.

In the ileum, the villi are more leaf-like, whereas in the duodenum, they have a more intermediate shape.
<figure><img src="image_20.png" /><figcaption>Figure S11: An image of the inner surface of a rat duodenum</figcaption></figure>
<figure><img src="image_21.png" /><figcaption>Figure S12: Zoom of an image of the inner surface of a rat duodenum</figcaption></figure>
<figure><img src="image_22.png" /><figcaption>Figure S13: An image of the inner surface of a rat jejunum</figcaption></figure>

<figure><img src="image_23.png" /><figcaption>Figure S14: An image of the inner surface of a rat ileum, used in figure 1 of the main text</figcaption></figure>
<figure><img src="image_24.png" /><figcaption>Figure S15: Zoom of an image of the inner surface of a rat ileum, used in figure 1 of the main text</figcaption></figure>
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