Application of Mathematical Modeling and Numerical Simulation of Blood Biomarker Transport in Paper-Based Microdevices

Abstract

This study introduces a novel mathematical model tailored to the unique fluid dynamics of paper-based microfluidic devices (PBMDs), focusing specifically on the transport behavior of human blood plasma, albumin, and heat. Unlike previous models that depend on generic commercial software, our custom-developed computational incorporates the Richards equation to extend Darcy’s law for more accurately capturing capillary-driven flow and thermal transport in porous paper substrates. The model’s predictions were validated through experimental data and demonstrated high accuracy in both two- and three-dimensional simulations. Key findings include new analytical expressions for uniform paper wetting after sudden geometric expansions and the discovery that plasma and albumin preferentially migrate along paper edges—a phenomenon driven by surface tension and capillary effects that varies with paper type. Additionally, heat transfer analysis indicates that a one-minute equilibration period is necessary for the reaction zone to reach ambient temperature, an important parameter for assay timing. These insights provide a deeper physical understanding of PBMD operation and establish a robust modeling tool that bridges experimental and computational approaches, offering a foundation for the optimized design of next-generation diagnostic devices for biomedical applications.

1. Introduction

Over the past 20 years, significant technological advancements have been made in PBMDs, driven by the development of new paper types and innovative chemical and biochemical techniques. PBMDs are typically constructed by creating hydrophilic channels in a paper substrate, which are separated by hydrophobic barriers (e.g., wax printing or 3D-printed filaments) [1,2]. Many of the results generated by these microdevices rely on colorimetric techniques, which involve a color change caused by the interaction of a chemical or biological compound in the sample with a specially designed indicator [3,4] that can be quantified using image-processing techniques [5]. The temperature is a key variable that influences the flux velocity and the sensitivity of colorimetric reactions [6].

The potential of PBMDs using human bodily fluids for medical analysis and their integration into the commercial healthcare market is well recognized. Recently, development blood separation papers, such as MF1, LF1, Fusion 5, and GX have played a crucial role in various biomedical applications [6,7]. These papers utilize capillary action to separate blood components, such as erythrocytes, plasma, serum, and cells, based on size and affinity. PBMDs have been designed to conduct immunoassays for detecting bacteria, viruses, proteins, and small molecules, serving biopharmaceutical analysis and clinical diagnostics, among other applications [8]. Blood, composed of approximately 45% cellular elements—primarily red blood cells—and 55% plasma [9], exhibits non-Newtonian behavior, with viscosity decreasing as shear stress increases [10]. In PBMDs, where blood flow velocities are low, this shear-thinning property becomes significant, affecting device performance and clot formation [11]. Blood coagulation involves a transition from a viscoelastic fluid to a solid state, marked by the gel point (GP) [11]. The time to reach this point (TGP) increases under low shear conditions, common in PBMDs, potentially leading to clot formation within 6 min [10]. Understanding these rheological and coagulation dynamics is crucial for the design and operation of PBMDs to ensure optimal performance.

The integration of mathematical modeling approaches with experimental data and computational simulations has proven invaluable in elucidating the underlying physical and chemical processes governing the behavior of these miniaturized systems with different geometries [12,13]. This synergistic approach facilitates the optimization of device design, enhances performance prediction, and accelerates the development and commercialization of microfluidic technologies. Accurately predicting liquid front propagation in paper substrates is crucial for dimensioning PBMDs. Current numerical simulations for PBMDs predominantly rely on commercial software packages [14,15]. These simulations typically apply Navier–Stokes equations for non-turbulent incompressible flows as mathematical models, while incorporating Darcy’s law to characterize porous media (paper) behavior and capillary pressure dynamics [16]. Our comprehensive literature review reveals a significant gap in mathematical modeling and numerical simulation of paper-based microfluidic devices (PBMDs) for medical applications. This deficiency can be attributed to two key factors: the absence of an integrated computational simulation methodology throughout the PBMD development process, and the lack of specific non-Newtonian mathematical models that accurately represent human biological fluids. Among the limited existing research, one recent study employs a three-dimensional heat conduction model using COMSOL to simulate the thermal behavior of a DNA amplification chamber [17].

In this paper, we present a comprehensive mathematical model characterizing the transport of human plasma, albumin, and heat in paper strips used for paper-based microfluidic device (PBMD) construction. Our model addresses how surface tension, pore geometry, and liquid–solid contact angle collectively determine capillary-driven flow dynamics and influence fluid front travel distance within the paper’s porous matrix—factors critical for optimal PBMD design and dimensioning. The plasma flow model in unsaturated porous media builds upon the Richards equation integrated with Darcy’s flux equation. While this foundation typically assumes Newtonian fluid behavior similar to water, with flow dynamics primarily determined by porous medium properties, we extend the analysis to compare model performance between Newtonian fluids (water) and non-Newtonian fluids (human blood plasma). Our approach uniquely links these distinct flow characteristics to the transport dynamics within unsaturated paper strip matrices. Compared to the classical Navier–Stokes equations, which model fluid flow in fully saturated—or, at best, two-phase continuous media (fluid–air)—our use of the Richards equation introduces key innovations specifically suited for unsaturated porous systems like paper strips used in microdevices. While Navier–Stokes assumes free-flow conditions and typically neglects capillary effects intrinsic to porous materials, the Richards equation incorporates both saturation-dependent permeability and capillarity through a non-linear extension of Darcy’s law. This allows for accurate modeling of capillary-driven flow where air and fluid phases coexist—a condition typical in PBMDs. To accommodate the unique characteristics of thin, fibrous paper strips, we developed a custom numerical solver specifically designed for microscale applications. This solver dynamic wetting fronts with computational efficiency is suitable. This solver efficiently handles dynamic wetting fronts. We implement two- and three-dimensional computational simulations using this custom-developed software based on our adapted mathematical model. To validate our model, we benchmarked the computational results against experimental data from peer-reviewed literature, demonstrating strong agreement. This integrated approach establishes the Richards equation—combined with custom simulation tools and experimental validation—as a more suitable and accurate framework for modeling flow in paper-based microdevices. The paper concludes with an in-depth analysis and discussion aimed at bridging the gap between experimental research and computational modeling, thereby accelerating PBMD development for medical applications. This work targets biomedical engineers, computational modelers, and microfluidic device designers involved in paper-based point-of-care diagnostics, as well as applied mathematicians and clinical researchers studying fluid transport in porous media.

This paper is organized as follows: Section 2 presents the mathematical model for fluid flow, temperature, and solute transport in a paper strip used in a microdevice. Section 3 introduces the studied cases along with their corresponding initial and boundary conditions. Section 4 describes the computational procedure and the main parameters of the numerical method used to solve the mathematical model. Section 5 presents the results for the two cases studied, including comparisons with experimental data and mesh refinement studies. Finally, conclusions are drawn, and the results and their applications to the design and fabrication of microdevices are discussed.

2. Mathematical Model

This section presents the governing equations for fluid flow, energy (temperature), and solute transport in an unsaturated porous medium. The originality of our model lies in its direct derivation from the fundamental equations governing water flow in unsaturated porous media, such as soil. Rather than relying on empirical or semi-empirical approaches, the model builds upon established physical principles commonly used in soil physics, thereby providing a more rigorous and mechanistic framework for describing fluid, heat, and solute transport in porous materials used in microdevices. A detailed analysis of these equations, as well as the assumptions made to derive the system presented here, is available in the Supplementary Material accompanying this paper, which can be accessed via the following link: https://docs.google.com/document/d/1x0SW4nh4IlPwi8A2sKKJ4ZztO4nK6DB4/edit?usp=drive_link&ouid=115870385999688068704&rtpof=true&sd=true (accessed on 5 June of 2025).

2.1. Mathematical Model for Fluid Flow in Paper Strips and Microdevices

In order to mathematically model the fluid flow ($J_{flc}$) in a paper strip, the Darcy’s equation is frequently used [10,12], which, in terms of the x, y, and z coordinates, is given by the following:

$$J_{flc,x}=K_{s,x}{\displaystyle \frac{\partial H}{\partial x}}; J_{flc,y}=K_{s,y}{\displaystyle \frac{\partial H}{\partial y}}; J_{flc,z}=K_{s,z}{\displaystyle \frac{\partial H}{\partial z}}$$

(1)

where $K_{s,x}, K_{s,y}, K_{s,z}$ represent the saturated hydraulic conductivity in the x, y, and z directions. The Darcy’s equation model assumes that a liquid infiltrating an unsaturated porous medium, such as a paper strip, behaves primarily as a Newtonian fluid. Any deviations, such as non-Newtonian behavior, are attributed to the specific properties of the porous medium itself. In Equation (1) the dependent variable is the potential head ($H$) of the paper strip. The $H$ is composed of four components that affect the energy of the fluid in the porous media:

$$H=y\prime +h+s+a$$

(2)

It is assumed that the gravitational force acts along the y-axis. The $y\prime$ quantifies the gravitational force field in the y-axis force influencing the fluid’s energy as it moves through the porous medium along the y-axis. If we consider that the paper strip possesses an initial moisture content in equilibrium with the ambient relative humidity, we can assume that $y\prime$ aligns with the control domain of the problem, i.e., $y=y\prime$. For a more detailed discussion on this matter, please refer to the Supplementary Materials accompanying this paper. In Equation (2) $h$ is the pressure head or matric potential head. This quantifies the effect of capillarity and adsorption on the energy of the fluid under unsaturated conditions, $s$ quantifies the effect on the energy of the fluid in the paper strip due to the presence of solutes, and $a$ represents the air pressure component that quantifies the impact of the air pressure within the pores on the energy driving fluid flow through the paper strip. If the effects of the solute concentration and the air pressure within the porous media ($s+a$) are neglected, the conservation equation for non-steady three-dimensional fluid flow in a paper strip can be represented by the following equation:

$${\displaystyle \frac{\partial \theta (h)}{\partial t}}={\displaystyle \frac{\partial }{\partial x}}\left(K(h){\displaystyle \frac{\partial h}{\partial x}}\right)+{\displaystyle \frac{\partial }{\partial y}}\left(K(h){\displaystyle \frac{\partial h}{\partial y}}\right)+{\displaystyle \frac{\partial K(h)}{\partial y}}+{\displaystyle \frac{\partial }{\partial z}}\left(K(h){\displaystyle \frac{\partial h}{\partial z}}\right)$$

(3)

This is known as the Richards equation, and it is an advection–diffusion type equation characterized by having the volumetric fluid content ($\theta (h)$) and the pressure head as its dependent variables. In this equation, the saturated hydraulic conductivity in the Darcy’s equation (Equation (1)) was replaced by a more general variable called the unsaturated hydraulic conductivity, $K(h)$, which is a function of the pressure head. Solving this equation requires the incorporation of constitutive relationships that link hydraulic pressure with the volumetric fluid content and the hydraulic conductivity of the unsaturated porous medium. Under unsaturated conditions—when the paper is dry or its moisture is below saturation—capillarity forces and fluid surface tension play a crucial role. This region, known as the capillary fringe, is characterized by a sharp increase in both lateral and vertical flow as saturation is approached [18]. Within this unsaturated region, the volumetric fluid content is described by the fluid retention curve of the porous medium ($\theta (h)$), which defines the relationship between the volumetric content and the matric potential head ($h$) of the porous media. Various empirical and theoretical functions have been formulated to model the fluid retention behavior for $K(h)$ and $\theta (h)$ within the capillary fringe. In this paper, the solution to this equation is derived using the van Genuchten model [19], as is presented in the following equation:

$$K(h)=K(S_e)=K_s{S}_e^l{\left[1-{\left(1-S_e^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$m$}\right.}}\right)}^m\right]}^2; S_e={\displaystyle \frac{{\theta }_{flc}(t)-{\theta }_{flc,r}}{{\theta }_{flc,s}-{\theta }_{flc,r}}}; m=1-{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$n$}\right.}, n>1$$

(4)

where $S_e$ is the effective saturation (volumetric rate), $l$ is the pore-connectivity, $m$ is a fitting parameter, ${\theta }_{flc,s}$ is the saturated volumetric content, ${\theta }_{flc,r}$ is the retained volumetric content, and ${\theta }_{flc}(t)$ is the volumetric fluid content in the paper strip at time t. The van Genuchten model presents unsaturated hydraulic conductivity as a function of the fluid content $K(S_e)$ to emphasize that this conductivity directly depends on the volumetric fluid content within the porous medium at a given time t. However, as will be demonstrated in Equation (5), the volumetric fluid content at any time t is itself a function of the pressure head (h). Consequently, while the model expresses conductivity in terms of fluid content, it is fundamentally—or, if preferred, indirectly—dependent on the pressure head. The retention curve equation ($\theta (h)$), for the paper strip and its reciprocal ($h(S_e)$) are calculated using the following equations:

$${\theta }_{flc}(t)=\theta (h)={\displaystyle \frac{{\theta }_{flc,s}-{\theta }_{flc,r}}{{\left[1+{\left(-\alpha h\right)}^n\right]}^m}}+{\theta }_{flc,r}; h(S_e)=-{\displaystyle \frac{1}{\alpha }}{\left({\displaystyle \frac{1-S_e^{1/m}}{S_e^{1/m}}}\right)}^{1/n}$$

(5)

where $\alpha$ is a fitting parameter. An equation based in the volumetric content as dependent variable can be obtained from Equation (3) applying a change of variable to the gradients that accompany the hydraulic conductivity and using the follow equivalence $D(\theta )=K(h){\displaystyle \frac{dh}{d\theta }}$. Considering the above, and if the gravitational effects are neglected, the formulation based on the volumetric content as a dependent variable is as follows:

$${\displaystyle \frac{\partial {\theta }_{flc}}{\partial t}}={\displaystyle \frac{\partial }{\partial x}}\left(D(\theta ){\displaystyle \frac{\partial {\theta }_{flc}}{\partial x}}\right)+{\displaystyle \frac{\partial }{\partial y}}\left(D(\theta ){\displaystyle \frac{\partial {\theta }_{flc}}{\partial y}}\right)+{\displaystyle \frac{\partial }{\partial z}}\left(D(\theta ){\displaystyle \frac{\partial {\theta }_{flc}}{\partial z}}\right) .$$

(6)

In this paper, the effective diffusion coefficient $D(\theta )$, which depends on the volumetric content of the paper strip is adjusted using the following exponential function [20]:

$$D(\theta )=D_{ff}{e}^{\lambda {\left(\theta \right)}^{{\alpha }_{ff}}}-{\vartheta }_2$$

(7)

where the symbols ϑ₂, λ, and α_ff are adjustment coefficients. The diffusion coefficient in the paper strip ($D_{ff}$) during wet-out in this paper is obtained from the Washburn equation, which is usually used to calculate the wet advancing front in the paper strip of the microdevice [12]:

$$L^2={\displaystyle \frac{\gamma \cdot d_p\cdot t}{4\cdot \mu }} \Rightarrow D_{ff}={\displaystyle \frac{\gamma \cdot d_p}{4\cdot \mu }}.$$

(8)

where $L$ is the distance of the fluid front (m), t is time (s), $\gamma$ is the surface tension, $d_{p,i}$ is the pore diameter (m), and $\mu$ is the solution viscosity. Equations (6)–(8) were used for the 2D simulations, while Equations (3)–(5) were used for the 3D case. To simulate the flow of the fluid component of whole blood, it is assumed that the plasma has not yet reached the gel point (GP), so the blood is treated as a fluid governed by the properties of the porous medium (Equations (4) and (5)). For more detailed explanations of the equations, please refer to the Supplementary Materials of this paper.

2.2. Mathematical Model for the Temperature in the Paper Strip Used for Microdevices

The energy equation was considered only in the 3D simulation. The temperature (T) is a variable that influences the flow rate and the sensitivity of the colorimetric reaction that occurs in the paper strip. The governing transient diffusion equation in three dimensions is as follows:

$$C_{p,eff}\left(\theta \right){\displaystyle \frac{\partial T}{\partial t}}={\displaystyle \frac{\partial }{\partial x}}\left(k_{eff}\left(\theta \right){\displaystyle \frac{\partial T}{\partial x}}\right)+{\displaystyle \frac{\partial }{\partial y}}\left(k_{eff}\left(\theta \right){\displaystyle \frac{\partial T}{\partial y}}\right)+{\displaystyle \frac{\partial }{\partial z}}\left(k_{eff}\left(\theta \right){\displaystyle \frac{\partial T}{\partial z}}\right)$$

(9)

Terms on the right side represents the heat transport by conduction. The properties $C_{p,eff}\left(\theta \right)$ and $k_{eff}\left(\theta \right)$ are the volumetric heat capacity and the apparent thermal conductivity of the paper strip, respectively. For unsaturated conditions, these are calculated with the equations:

$$k_{eff}\left(\theta \right)=(1-\epsilon )k_{paper}+{\theta }_{flc}{k}_{flc}+{\theta }_{air}{k}_{air}; C_{p,eff}\left(\theta \right)=(1-\epsilon )C_{p,paper}+{\theta }_{flc}\cdot C_{p,flc}+{\theta }_{air}\cdot C_{p,air}$$

(10)

where $\epsilon$ is the porosity, $k_w$, $k_{air}$, and $k_{solid}$ are water, air, and solid thermal conductivity, respectively, $C_{p,flc}$, $C_{p,air}$, and $C_{p,paper}$ are the fluid, air, and solid heat capacity, respectively. If ${\theta }_{flc}$ is less than ${\theta }_{flc,s}$, then ${\theta }_{air}=1-{\theta }_{flc}$, otherwise ${\theta }_{air}=0$, and ${\theta }_{flc}=\epsilon$. For the microdevice, areas without paper or exposed to the atmosphere were assumed to have the thermal properties of the wax used to insulate the paper and channel the fluid. More detailed explanations can be found in the Supplementary Materials.

$$k_{eff}=k_{wax};\hspace{1em}\hspace{1em}\hspace{1em}{\left(\rho C_p\right)}_{eff}={\rho }_{wax}{C}_{p,wax}$$

(11)

2.3. Mathematical Model for Solute Transport in the Paper Strip Used for Microdevices

For the solute transport in the paper strip we assume that no adsorption occurs and that the coefficients in the advection-diffusion terms do not depend on the solute concentration ($C_{sol,i}$). In PBMDs, adsorption phenomena are often neglected due to the generally low-affinity interactions between clinically relevant biomarkers—such as albumin or small metabolites—and cellulose-based paper. These interactions are typically weak and reversible, particularly when the paper is untreated or has been surface passivated. Additionally, PBMDs operate over short timescales and at low solute concentrations, which further limits the cumulative effects of adsorption. From a modeling perspective, incorporating adsorption would necessitate the inclusion of additional kinetic parameters, such as adsorption and desorption rate constants, which are specific to the material and often difficult to measure with precision. As such, omitting adsorption simplifies the model without significantly compromising its first-order predictive capabilities. However, this simplification may lead to slight overestimations of solute front velocity and concentration uniformity, particularly for proteins or analytes that exhibit moderate affinity for paper fibers. While such effects could influence quantitative biomarker detection, especially near detection zones, the overall impact on flow dynamics or thermal distribution is expected to be minimal. The solute conservation equation in the paper strip is:

$${\displaystyle \frac{\partial C_{sol,i}}{\partial t}}=D_{sol,x}{\displaystyle \frac{{\partial }^2{C}_{sol,i}}{\partial x^2}}+D_{sol,y}{\displaystyle \frac{{\partial }^2{C}_{sol,i}}{\partial y^2}}+D_{sol,z}{\displaystyle \frac{{\partial }^2{C}_{sol,i}}{\partial z^2}}-{\nu }_{p,x}{\displaystyle \frac{\partial C_{sol,i}}{\partial x}}-{\nu }_{p,y}{\displaystyle \frac{\partial C_{sol,i}}{\partial y}}-{\nu }_{p,z}{\displaystyle \frac{\partial C_{sol,i}}{\partial z}}$$

(12)

where, $C_{sol,i}$ is the concentration of the i-th (i) solute (dye in our 2D case and albumin in the 3D case), ${\nu }_{p,x}, {\nu }_{p,y}$ and ${\nu }_{p,z}$ are the pore velocity. The terms $D_{sol,x}$, $D_{sol,y}$, $D_{sol,z}$ represent the effective dispersion coefficients in the x, y, and z directions, respectively. These are obtained by summing the solute diffusion coefficient ($D_{eff}$) and the hydrodynamic dispersion coefficient ($D_{hy}$) for each coordinate:

$$D_{sol,x}=D_{eff,x}+D_{hy,x}; D_{sol,y}=D_{eff,y}+D_{hy,y}; D_{sol,z}=D_{eff,z}+D_{hy,z}$$

(13)

A typical value of $D_{eff}$ for water is 0.0416 cm³/h [19]. According to experiments [21,22] the coefficient of hydrodynamic dispersion is often linearly proportional to the pore velocity in the respective coordinate (${\nu }_{p,x}, {\nu }_{p,y}, {\nu }_{p,z}$). The hydrodynamic dispersion coefficient can be calculated as follows:

$$D_{hy,x}={\lambda }_{p,x}{\nu }_{p,x}; D_{hy,y}={\lambda }_{p,y}{\nu }_{p,y}; D_{hy,z}={\lambda }_{p,z}{\nu }_{p,z}$$

(14)

In this equation ${\lambda }_{p,x}$, ${\lambda }_{p,y}$, and ${\lambda }_{p,z}$ are the dispersivity for each coordinate [21]. More detailed explanations can be found in the Supplementary Materials of this paper.

3. Studied Cases: Geometry, Materials, and Initial and Boundary Conditions

This paper presents two fluid flow simulations. The first is a two-dimensional transport of water and solute in strips with variable widths, while the second is a three-dimensional simulation of blood transport in a paper-based microdevice used for detecting albumin.

3.1. Two-Dimensional Simulation (2D)

In Figure 1, the paper strips used for this simulation are shown. The water flows through five strips (E, A, B, C, D), three of which (A, B, C) have expansions at different heights. The widths of strips E and D along the x-axis are constant, measuring 5.8 mm and 1.2 mm, respectively. Strips A, B, and C have variable widths along the x-axis, ranging from 1.2 mm in the narrowest section to 5.8 mm in the widest section. The length of the strips along the y-axis is L_strip = 17 mm. Figure 1 also shows the areas corresponding to each paper strip, calculated using the previously presented data. The strip D has the smallest area (20 mm²), while the one with the largest area among those with variable width is the strip C (79 mm²). Simulation results were validated against experimental data obtained using nitrocellulose paper strips with a nominal thickness of 0.12 mm. To prevent inter-strip fluid transfer, impermeable barriers were placed between adjacent strips. The paper properties used in the simulations will be provided later.

Initial and Boundary Conditions

Initial conditions in the paper strip domain are specified in terms of the fluid content $\theta (x,y,0)={\theta }_r(x,y,0)=0$. A zero initial condition for solute concentration in the paper strip domain is used $C_{sol,i}(x,y,0)=0$.

A constant flow of water ($J_w$) and solute ($J_{sol}$) was prescribed at the y = 0 boundary:

$$J_w(x,0,t)={\left.-D_{eff,s}\left({\displaystyle \frac{\partial \theta }{\partial y}}\right)\right|}_{y=0};\hspace{1em}\hspace{1em}\hspace{1em}{J}_{sol}(x,0,t)={\left.-D_{sol,y}\left({\displaystyle \frac{\partial C_{sol}}{\partial y}}\right)\right|}_{y=0}$$

(15)

The other boundaries were considered impermeable to both water and solute:

$$J_w(0,y,t)=J_w(x_{\max ,i},y,t)=J_w(x,L_{strip},t)=0;J_{sol}(0,y,t)=J_{sol}(x_{\max ,i},y,t)=J_{sol}(x,L_{strip},t)=0.$$

(16)

3.2. Three-Dimensional Simulation (3D)

Figure 2 shows the microdevice used in this simulation. The microdevice has dimensions of 14 × 4 × 6 mm along the x, y, and z axes. The blood injection zone and the viewing area for the reaction zone have dimensions of 2 × 2.2 × 2 mm and 3 × 1.7 × 3 mm along the x, y, and z axes, respectively. These two areas are open to the atmosphere, as shown in Figure 2A. The reaction zone, measuring 3 × 0.2 × 3 mm, is connected to the separation membrane by a conduit measuring 1.25 × 0.2 × 3 mm. The separation zone, which connects the blood injection area to the reaction zone, measures 9.25 × 0.4 × 3 mm. The reaction zone is made from laboratory filter paper, while the separation zone is constructed with a blood separation membrane. The previously identified zones are surrounded by wax, which channels the blood flow through the papers. The properties used for these papers in the simulations will be provided later.

Initial and Boundary Conditions

To model plasma flow within the microdevice, the Richards equation, as presented in Equation (3), was employed in the computational simulation. A blood droplet with a total volume of 17.5 µL was considered, in which blood plasma—constituting approximately 57% of the total volume—served as the fluid phase transported through the porous paper matrix. Under these conditions, the initial plasma volume was estimated at 10 µL. Plasma, in contrast to whole blood, generally exhibits Newtonian behavior under most flow conditions, particularly in microfluidic environments characterized by low shear rates, such as those typical of capillary-driven paper flow. Furthermore, the high water content of plasma (approximately 92%) reinforces the assumption that its rheological properties closely resemble those of a Newtonian fluid. This characteristic facilitates the application of the Richards equation, allowing the use of well-established hydraulic relationships derived for aqueous media. Nevertheless, minor deviations may arise in regions where viscosity is more sensitive to variations, such as at elevated protein concentrations or under fluctuating temperature conditions. In such cases, the Newtonian assumption may lead to slight inaccuracies in predicting flow velocity and saturation front advancement, particularly if plasma exhibits mild shear-thinning behavior. In our simulations, any such deviations are accounted for by calibrating the parameters of the van Genuchten model—specifically, the n and α parameters from Equations (4) and (5), respectively.

The initial condition in the paper strip domain for the volume content is specified in terms of the pressure head $h(x,y,z,0)=h_r(x,y,z,0)$, where h_r corresponds to the pressure head obtained from Equation (5) with the retained fluid content in the paper strip. Microdevices are generally stored in controlled environments where the temperature remains uniform—for example, in cases where devices are kept under refrigeration. This justifies the use of homogeneous initial temperature conditions throughout the entire device in the simulation. A prescribed initial temperature was applied to the microdevice, $T(x,y,z,0)=T_{ini}$. A zero initial condition for solute concentration in the paper strip domain is used in the simulations, $C_{sol,i}(x,y,z,0)=0$.

Specific boundary conditions for the injection of a drop of blood on the surface of porous media are detailed as follows. In microdevices designed for detecting specific biomarkers (e.g., albumin), a blood sample is typically applied as a drop on a designated area in the paper strip. The blood drop is deposited from a perpendicular direction onto this delimited zone. When simulating this process, special considerations for these boundary conditions are necessary. These conditions must accurately represent the interaction between the liquid and the paper substrate, accounting for factors such as surface tension, capillary action, and the porous nature of the paper. The volume of the drop produces a hydrostatic pressure on the paper, which has a predefined hydraulic conductivity and initial moisture content. Additionally, it must be considered that the drop diminishes in volume as it infiltrates into the paper, resulting in a variable flow rate. This case corresponds to a dependent boundary condition that cannot be predefined because the paper–air interface is exposed to atmospheric influences. While the potential fluid flux across this interface is solely governed by external factors, the actual flux is also contingent upon the transient paper moisture state at the surface. Consequently, the boundary condition at the surface can dynamically transition between prescribed flux and prescribed head conditions, adapting to the prevailing circumstances. The flux at the boundary can be calculated limiting its absolute value at the surface by the following conditions [19]:

$$\left.\begin{array}{l}\left|-K(h){\displaystyle \frac{\partial h}{\partial y}}-K(h)\right|\le E\\ h_{\min }\le h\le h_{\max }\end{array}\right\} at y_{inj}=injection zone$$

(17)

where $y_{inj}$ is the injection zone surface at the y coordinate, E denotes the maximum potential rate of infiltration or evaporation under the prevailing atmospheric conditions, and h_min and h_max represent the minimum and maximum pressure heads, respectively, permissible at the surface given the current porous media conditions (paper humidity). The value of h_min is determined from the equilibrium relationship between soil water and atmospheric water vapor, while h_max is typically set to zero; however, a positive value of h_max signifies a shallow layer of ponded fluid that can accumulate on the paper surface during the drop injection prior to the initiation of surface runoff. The minimum pressure head at the surface h_A can be calculated from the air humidity (RH) as follows:

$$h_{\min }={\displaystyle \frac{R\cdot T}{MW\cdot g}}\ln (RH); where RH=\exp \left({\displaystyle \frac{h_{\min }\cdot {\rm M}W\cdot g}{R\cdot T}}\right)$$

(18)

where MW is the molecular weight of the fluid (0.018015 kg mol⁻¹, in this paper), g is the gravitational acceleration (9.81 m/s²), and R is the gas constant (8.314 J/mol K).

A modified version of the effective saturation (Equation (4)) is used to calculate the variation of the volumetric rate of the drop on the paper strip surface:

$$S_{e,in}={\displaystyle \frac{\theta {(t)}_d}{{\theta }_{d,i}}},$$

(19)

where ${\theta }_{d,i}$ is the initial drop volume and $\theta {(t)}_d$ is the volume of the droplet at time t that has not yet infiltrated the paper and remains on its surface. This equation is used to calculate the $K(h)$ value at the boundary using Equation (4), which is then applied in Equation (17) to calculate the flux at the injection zone surface. The infiltration rate and the drop volume of a droplet on a porous medium decrease rapidly as a result of decreasing head pressure and increasing saturation of the medium. Characteristic curves of water infiltration in a porous medium (soil) can be observed in reference [22] and for whole blood on a porous paper in reference [23]. Based on this latest reference, which shows parameterized infiltration curves for any blood drop volume, we have developed a curve for the decrease of plasma volume over the paper. Figure 3 presents the curve developed for our case. The maximum injection time for these curves (10 s) has been calculated using the methodology from the same reference [23], assuming that the blood has 50% hematocrit. In Figure 3, the polynomial fitted to the plasma drop volume curve ($\theta {(t)}_d$) is also shown. In the simulations, this polynomial is used in Equation (19) for calculating the effective saturation. Using a droplet volume decay function allows realistic simulation of blood infiltration into paper strips by accounting for the time-dependent reduction of fluid at the contact surface. This improves accuracy in modeling early-stage capillary flow and better aligns simulations with experimental data, which is crucial for predicting biomarker transport in paper-based biomedical devices.

Once the drop has been fully absorbed into the paper, there is no longer a boundary condition that allows additional volume to be incorporated. The mathematical model interprets this as a decrease in head pressure (h), which in turn leads to a reduction in the volumetric content of the paper, as described by Equation (5). In practice, once a section of the paper strip is wetted by liquid, it does not experience any further volume loss. After the drop has completely infiltrated the paper, the actual loss is the head potential, as there is no longer a blood injection boundary condition. To account for this, the equivalence in Equation (5) related to head pressure variation is decoupled for the areas where the paper has become saturated. For further details, please refer to the Supplementary Materials.

The temperature of a blood droplet infiltrating a porous medium varies rapidly due to its contact with the medium, which has its own thermal properties, as well as the intrinsic characteristics of blood. As a result, this boundary condition cannot be accurately represented by either a constant temperature or a constant heat flux. Temperature variations for the injection zone are modeled using a third-type (or Cauchy type) boundary condition to characterize the heat flux induced by droplet infiltration. This boundary condition incorporates three key parameters: the fluid’s specific heat capacity ($C_{p,flc}$), fluid flux ($J_{flc,y}$), and the microdevice’s initial temperature ($T_0$). The mathematical expression for this condition is given by:

$$k_{eff}(\theta ){\displaystyle \frac{\partial T}{\partial y}}=T_0\cdot C_{p,flc}\cdot J_{flc,y}$$

(20)

A prescribed temperature corresponding to the ambient temperature surrounding the device was applied to all outer edges of the microdevice.

A solute flux boundary condition was imposed at the injection zone, which depends on the albumin concentration ($C_{sol,0}$), plasma fluid flux ($J_{flc,y}$), and plasma content (${\theta }_{flc}$). This condition is expressed as follows:

$$J_{flc,y} \cdot {\left.C_{sol}(x,y,z,t)\right|}_{\begin{array}{l}{y}_{inj}=injection zone\\ t=injection time\end{array}}-{\theta }_{flc}(x,y,z,t)\cdot D_e\cdot {\left.{\displaystyle \frac{\partial C_{sol}(x,y,z,t)}{\partial y}}\right|}_{\begin{array}{l}{y}_{inj}=injection zone\\ t=injection time\end{array}}=J_{flc,y} C_{sol,0}\left(t\right)$$

(21)

The boundary albumin concentration $C_{sol,0}\left(t\right)$ is determined by two key parameters: the albumin infiltration velocity through the separation membrane (${\nu }_{p,y}$) and the total droplet infiltration time (10 s in this study). This value was derived using an analytical solution for a one-dimensional case under comparable boundary conditions obtained from reference [19]. A comprehensive description of this derivation is provided in the Supplementary Materials.

All boundaries, except for the injection zone, were subjected to a no-flux boundary condition for both plasma and albumin.

4. Computational Procedure

The coupled partial differential equation (PDE) system is solved numerically using the finite volume method (FVM) which discretizes the general transport equation for a diffusion transient model [24]. A summarized explanation is presented below. Our programmed code, written in Fortran, solves the PDE system using the generalized form for the transport equation that contains non-steady diffusion and linearized source terms for the variable $\phi$:

$${\displaystyle \frac{\partial (\rho \phi )}{\partial t}}=div\left(\Gamma \cdot grad\phi \right)+S_c+S_p\cdot \phi$$

(22)

The time derivative and advective terms of each PDE in the mathematical model are solved by treating them as source terms, which may either depend ($S_p$) on or be independent ($S_c$) of the variable $\phi$. Thus, time integration of the control volume ($i,j,k$) is carried out using an explicit Euler scheme as follows:

$${\displaystyle \frac{\partial \phi }{\partial t}}={\displaystyle \frac{{\phi }_{i,j,k}^{t+\Delta t}-{\phi }_{i,j,k}^t}{\Delta t}}$$

(23)

where $\Delta t$ is the time step, ${\phi }_{i,j,k}^t$ is the S_p term, and ${\phi }_{i,j,k}^{t+\Delta t}$ is the S_c term. The spatial advective terms are discretized by considering the value of the variable $\phi$ in the control volume ($i,j,k$), as well as the neighboring volumes $(i+1,j,k), (i,j+1,k)$, and $(i,j,k+1)$, in each coordinate direction:

$${\displaystyle \frac{\partial \phi }{\partial x}}={\displaystyle \frac{{\phi }_{i,j,k}^t-{\phi }_{i+1,j,k}^t}{\Delta x}}; {\displaystyle \frac{\partial \phi }{\partial y}}={\displaystyle \frac{{\phi }_{i,j,k}^t-{\phi }_{i,j+1,k}^t}{\Delta y}}; {\displaystyle \frac{\partial \phi }{\partial z}}={\displaystyle \frac{{\phi }_{i,j,k}^t-{\phi }_{i,j,k+1}^t}{\Delta z}}$$

(24)

At each time step, the system of discretized nodal equations for each primary variable (${\theta }_{flc}, T, C_{sol}$) is solved iteratively using a combination of the alternating tridiagonal matrix algorithm (TDMA) and the Gauss–Seidel method.

This procedure applies under-relaxation coefficients of 0.8 for each variable. The convergence criterion for the variable at each dimensionless time step ($\Delta t$ = 1 s) and for each control volume ($i,j,k$) is $\left|{\phi }_{i,j,k}^{t+1}-{\phi }_{i,j,k}^t\right|\le {10}^{-4}$. The selection of an under-relaxation coefficient of 0.8 and a convergence criterion of ${10}^{-4}$ per variable per time step ensures a stable and accurate numerical solution for simulating non-linear flow in unsaturated porous media using the Finite Volume Method. The under-relaxation value balances convergence speed with stability, particularly important for the Richards equation where properties vary with saturation. The tight convergence threshold guarantees that each variable reaches a precise solution at every time step, capturing subtle but critical changes in flow dynamics. Together, these parameters enhance the reliability of simulations in paper-based biomedical microdevices. For more details on the finite volume method, refer to [25]. The flowchart in Figure 4 outlines the architecture of the custom computational algorithm implemented in the simulations.

5. Results

5.1. Results for Two-Dimensional Diffusion of Water and Solute in Strips with Variable Widths

In this section, a 2D simulation of a paper network with varying widths is conducted, and the results are compared with the experimental data obtained from the reference [12]. Transport phenomena of water and solute were modeled using the Equations (6) and (12). The effective diffusion coefficient required for Equation (6) was calculated using Equations (7) and (8). All simulation parameters are summarized in

Table 1. Value used in the mathematical model for water and solute diffusion in 2D simulation.

$\gamma$ (N/m) **	$d_p$(m) *	$\mu$ (kg/s m) **	$D_{ff}$ (m2/s)	ϑ2	λ	αff
0.072	1.16 × 10−5	0.001	2.088 × 10−4	2.093 × 10−4	6 × 10−6	0.8
Dsol,x (m2/s)	Dsol,y (m2/s)	Jw,in+ (m/s)	Jsol,in (m/s)	$C_{sol,i}$
2 × 10−8	1 × 10−7	2.963 × 10−4	5 × 10−3	0

* [10]; ** water viscosity [12]; ⁺ capillary flow of water [26].

. The initial conditions for both water and solute concentration within the paper strip domain were set to zero. A constant injection flow was imposed as the boundary condition for water (J_w,in) and solute (J_sol,in) at the lower boundary of the strip (y = 0) in accordance with the formulation presented in Section 3.1 (Equation (15)).

Figure 5 presents a comparison between experimental data and numerical simulations of water and solute front propagation distances as a function of time in nitrocellulose paper strips over an 82 s period. As shown in Figure 5A, the calculated advancement fronts demonstrate consistent agreement with experimental trends across all tested strips. Data from strip E are omitted due to their similarity with strip D results. The maximum relative error RE = 0.11 (%RE = 11%) occurs in paper strip D, which has a constant width, while the smallest error is observed in paper strip C (RE = 0.06 or %RE = 6%), which widens along the y-axis at a shorter distance than strips A and B. In all cases, the velocity of the advancing water front decreases over time.

As described earlier, water flow was imposed in uniform along the entire width of the x-boundary at y = 0 for all paper strips. Therefore, the difference in the front velocity values between the strips is directly related to the cross-sectional area perpendicular to the fluid Flow. Among the five paper strips analyzed, strips A and D exhibit the greatest water front advancement at a given time. These strips are characterized by the absence of abrupt geometric expansions. Although they differ in flow cross-sectional areas, both exhibit similar front propagation behavior and distance. This observation is attributed to capillary-driven flow, which dominates in porous media such as paper, and is well captured by our mathematical model. Strip C has the slowest water-front velocity of all the strips, as expected, since it has the largest surface area of the three strips with expansion. The widening of the strip causes a change in the cross-sectional area that the fluid must fill, which, according to the continuity equation, results in a lower flow velocity. Figure 5B compares the calculated solute front results with the experimental data for strips A and B over a period of 40 min. The relative errors between these results are even smaller than those for water, with an RE of 0.04 (%RE = 4%) for both strips. As with water, the solute front velocity decreases over time. However, for the solute, the range of velocity variation is smaller than that of water (0.295–0.47 mm/s for strip A and 0.225–0.47 mm/s for strip B). Once again, the lowest solute front velocities are observed in the strip with the largest surface area (strip B). The surface area ($A_{surf}$), the average velocity of the advancing water and solute fronts ($v_{ave,\theta }$ and $v_{ave,C_{sol,i}}$), and the mean deviation of the front velocities ($\overline{X}{v}_{ave,\theta }$ and $\overline{X}{v}_{ave,C_{sol,i}}$) for each paper strip are presented in

Table 2. Average velocities and mean deviation for the advanced front of water and solute.

Paper Strip	A	B	C	D
$A_{surf}$, mm2	39	59	79	20
$v_{ave,\theta }$ mm/s	0.35	0.28	0.22	0.36
$\overline{X}{v}_{ave,\theta }$ mm/s	0.19	0.15	0.12	0.19
$v_{ave,C_{sol,i}}$ mm/s	0.37	0.32	-	-
$\overline{X}{v}_{ave,C_{sol,i}}$ mm/s	0.05	0.08	-	-

. Analysis of the data presented in Figure 5A,B reveals that in strips with variable width, the average velocity of both water and solute fronts decreases with increasing surface area ($A>B>C$). For these configurations, the mean deviation (data dispersion) of water measurements decreases with increasing surface area from 0.19 to 0.12. Although the average solute front velocities appear higher than those of water, this observation does not indicate that the solute moves faster than its carrier fluid. As shown in Figure 5A, the initial water-front velocities for Strips A and B exceed 0.5 mm/s during the first 10 s, while solute front velocities range between 0.4 and 0.47 mm/s for both strips during the same period. This is reflected in the mean deviation, which is significantly higher for water than for the solute, explaining the higher average velocities observed for solute front advancement.

Figure 6 shows the two-dimensional variation of the advancing front at two different times (32 s and 50 s) for the five studied paper strips (Figure 6A,B), along with two additional time points near the end of the simulation for strips B and C, where the fully developed advancing front can be observed after the expansion (Figure 6C,D). The paper strips with a constant width (strips E and D) show the greatest advancement distances, which remain the same for both times displayed. This behavior was consistent throughout the entire simulation. While the strip does not undergo expansion, the advancing front remains constant and flat, which in this study is associated with a fully developed front. When expansion begins in strips A, B, and C, the front is no longer flat and becomes convex as the water fills the spaces. As the front continues to advance along the strip, it eventually flattens again after covering a certain distance. These observations are supported by both experimental results presented in the reference [12] and our numerical simulations (Figure 6C,D). To determine the distance required for the flow to return to its fully developed form, a correlation will be established. The length of the entrance region in a microfluidic system experiencing an abrupt expansion within a homogeneous porous medium, driven by surface tension and capillarity (denoted as $y_{c,2D}$, in milimeters), depends on the Darcy number ($D{a}_{2D}$) and the characteristic length ($L_{c,2D}$), which, for the varying-width paper strips used in this study, corresponds to the mean width (3.5 mm). The Darcy number for our 2D case is defined as follows:

$$D{a}_{2D}={\displaystyle \frac{K_{p2D}}{L_{c,2D}^2}}$$

(25)

The permeability ($K_{p2D}$) can be calculated using the Kozeny–Carman model [27]:

$$K_{p2D}={\displaystyle \frac{{\epsilon }_{p2D}^3}{c_{s2D}{(1-{\epsilon }_{p2D})}^2{S}_{p2D}^2}}$$

(26)

where ${\epsilon }_{p2D}$ is the porosity, $c_{s2D}$ is an empirical constant related with the geometry and tortuosity of the media, and $S_{p2D}$ is the specific surface area (surface area of the paper strip/volume of the paper strip). The length of the entrance region is then calculated as follows:

$$y_{c,2D}\approx 1000\cdot D{a}_{2D}\cdot L_{c,2D}$$

(27)

The values of the variables used in Equations (25)–(27) are presented in

Table 3. Values used to calculate the large entrance region for the paper strip of different widths.

	${\mathit{L}}_{\mathit{c},2\mathit{D}}$, mm	${\mathit{\epsilon }}_{\mathit{p}2\mathit{D}}$	${\mathit{c}}_{\mathit{s}2\mathit{D}}$	${\mathit{S}}_{\mathit{p}2\mathit{D}}$, m−1	${\mathit{K}}_{\mathit{p}2\mathit{D}}$, m2	$\mathit{D}{\mathit{a}}_{2\mathit{D}}$	${\mathit{y}}_{\mathit{c},2\mathit{D}}$, mm
Strip B–C	3.5	0.785	6.5	8330	3.86 × 10−8	3.15 × 10−3	6.7

.

The length of the entrance region for the abrupt expansion of the paper strip is 6.7 mm. Therefore, beyond this section, the water flow in the strip should become constant again (flat front) across the entire wetting front. This behavior is illustrated in Figure 6C,D, where it regains its planar configuration approximately 6.7 mm downstream from the strip’s expansion. Paper-based medical microdevices are typically designed with a reaction zone embedded with a reagent and a chemical indicator that changes color based on the concentration of a biomarker (colorimetric reaction), triggered by a pH change. In laboratory studies of the colorimetric reaction, prior to its application in the microdevice, the volumes of reagents and biomarkers are thoroughly mixed in known proportions. To translate these studies into functional microdevices, particularly those using plasma separation paper where the plasma must diffuse through a paper strip, it is crucial to ensure that both the biomarker and the plasma carrying it fully reach and wet the reaction zone. In this context, the Equation (27) developed in this work is particularly useful. This formula allows for the calculation of the optimal length of the paper strip required to ensure that, following a sudden expansion, the plasma fully wets the colorimetric reaction zone in the microdevice designed for biomarker analysis.

Figure 7 shows the 2D contours of the solute front for strips A and B. These two strips were selected due to the availability of corresponding experimental data, enabling direct comparison with the results reported in reference [12]. The simulated results depicted in Figure 6 demonstrate behavior consistent with the experimental findings reported by Fu et al. [12] at identical time intervals (12 s and 40 s). Prior to entering the expansion zone, the fluid advancement fronts maintain planar profiles, exhibiting uniform displacement along the y-coordinate (Figure 7A,B). The solute flow in the strip exhibited significant perturbation upon encountering the abrupt expansion, resulting in reduced advancement along both lateral edges in the x direction when compared with the water-front propagation pattern post-expansion, as illustrated in Figure 6A,B. Following an abrupt expansion, the preferential solute transport along the longitudinal axis (corresponding to the y-coordinate in this investigation) can generate heterogeneous reagent distribution in the reaction zones of microfluidic platforms designed for blood biomarker detection. Consequently, an integrated analysis of both colorimetric response patterns and biomarker-laden plasma front propagation dynamics is imperative for optimizing microfluidic platforms where biomarker detection occurs via spatially constrained colorimetric reactions.

5.2. Three-Dimensional Simulation of Blood Transport in a Paper-Based Microdevice Used to Detect Albumin

This section presents a three-dimensional numerical simulation of a paper-based microfluidic device for albumin detection in whole human blood, building upon the device previously developed and validated by Yang et al. [6]. The mathematical framework incorporates boundary conditions for plasma flow, albumin transport, and heat transfer, as described in Section 3.1. Since the maximum times in both the experimental results presented in the reference [6], and in this study are less than 6 min, it is considered that the moving plasma has not yet reached the gel point (GP). For further details, please refer to the Supplementary Materials. Device specifications and geometrical parameters are detailed in Section 2.1 and Figure 2. The device operates through the following mechanism: upon deposition of a blood droplet in the injection zone, the sample contacts the separation membrane directly. The droplet volume on the membrane surface follows a temporal decay function, as shown in Figure 3. During this process, erythrocytes (red blood cells) are retained within the separation membrane, while plasma and its dissolved albumin continue to diffuse toward the reaction zone. The colorimetric reaction zone comprises a filter paper in direct contact with the separation membrane, which undergoes capillary-driven wetting by the transported plasma. For simulation purposes, plasma is modeled as the primary fluid phase wetting the paper matrix and carrying albumin, consistent with its role as blood’s liquid component. All relevant simulation parameters are compiled in

Table 4. Parameter values used in the mathematical model for the 3D simulation.

${\theta }_{flc,r}$	${\theta }_{flc,s}^x$	${\rho }_{plasma}$ (kg/m3) ++	$C_{p,plasma}$ (J/kg°C) ++	$k_{plasma}$ (W/m K) ++	$C_{p,wax}$ (J/m3K) +	${\rho }_{wax}$ (kg/m3) +
0.051	0.8	1025	3930	0.525	1850	856
$k_{wax}$ (W/m K) +	%albumin in blood *	${\rho }_{Alb}$ (kg/m3) **	$C_{p,Alb}$ (J/kg °C) **	$k_{Alb}$ (W/m K) **	$n$ xx	$l$ xx
0.4	3	3560	1100	0.496	5	−1
$D_{eff}$ (cm2/h) xxx	${\lambda }_{p,x}$ (cm) a	${\lambda }_{p,y}$ (cm) a	${\lambda }_{p,z}$ (cm) a	$C_0$ (kg/m3) ***	${\alpha }^{ \mathrm{x}\mathrm{x}}$ (1/cm)	${\epsilon }^{ \mathrm{x}}$
0.0416	100	30	30	3.68	0.05	0.8

* percentage of albumin in blood [9]; ** albumin properties [28]; *** albumin initial concentration used in reference [6]; ⁺ heat capacity of the wax [29]; ⁺⁺ plasma properties [30]; ^x value assumed based on the void volume (20 uL/cm²) of the GX separation membrane [31]; ^xx typical values used in the van Genchten model [19]; ^xxx effective diffusivity of the solute in paper calculated as the product of the free-water diffusivity and the tortuosity factor, as determined from the correlation reported in [19]; ^a dispersivity values of the solute along each coordinate direction were adjusted to reflect the characteristics of the separation membrane. The resulting values are consistent with the ranges reported for water flow in porous media, such as soils [19].

.

The computational program developed in FORTRAN 77 to solve this problem and generate the results presented in this section can be downloaded from the following link: https://drive.google.com/file/d/1URdrFDzuP2qdox-RpLZQ8YyUNE5eVkA7/view?usp=drive_link (accessed on 6 June 2025).

5.2.1. Mesh Analysis

To ensure the convergence of the results calculated by the mathematical model and the calculation algorithm, a mesh study was conducted. Five non-uniform meshes, refined in the region where the plasma flows (separation membrane and filter paper), were used. The study focused only on the flow of blood plasma along a central line of the separation membrane, in the y–z plane. The volumetric plasma content profile was calculated between the points P_1,mesh(2 mm, −2.4 mm, 3 mm) and P_2,mesh(11 mm, −2.4 mm, 0.3 mm). The results were obtained for a calculation time of 2 min, in order to observe the program’s behavior as the plasma content was still wetting unsaturated regions of the paper (separation membrane). The parameters used in this study are those in Table 4, and the initial and boundary conditions are those described in Section 3.2. The mesh grids used, along with the total number of control volumes (CVs) and the CVs in the region containing the device’s papers (separation membrane and filter paper), are presented in

Table 5. Control volumes (CVs) used for the mesh study and their corresponding percent relative errors (%RE) compared to the most refined mesh (mesh 1).

N° Mesh	N° of CVs in x,y,z	Total N° of CVs	N° CVs in Separation and Filter Paper Zone	%RE
1	188 × 42 × 68	536,928	119,070	0
2	148 × 28 × 53	219,632	43,747	9.9
3	158 × 33 × 53	276,342	54,612	7.8
4	168 × 33 × 53	293,832	58,220	2.0
5	196 × 38 × 58	431,984	93,420	0.81

. This table also shows the relative percentage errors (%RE) for each mesh. The %RE was obtained by comparing the volumetric plasma content values of each mesh (meshes two through five) to the most refined mesh 1, at intervals of x = 0.5 mm. The %RE decreases as the number of CVs is increased. The %RE for mesh 5 is only 0.81%, and it contains around 100,000 fewer CV than mesh 1, which reduces the calculation time by approximately 2 h and the size of the output files.

The results in Table 5 show a clear trend of convergence, with the error decreasing by an order of magnitude as mesh resolution improves. A logarithmic regression of %RE against the number of CVs in the porous region (separation and filtration zone) yielded a coefficient of determination R² = 0.717, indicating a moderate-to-strong logarithmic correlation and confirming the mesh convergence behavior. Additionally, the Root Mean Square Error (RMSE) of the volumetric plasma content profiles between mesh 5 and mesh 1 was calculated, yielding a value of 2.3 × 10⁻³. This low RMSE reinforces the negligible difference in the output profiles between the two meshes. The convergence of the mesh refinement towards a solution can be observed in Figure 8, for both the pressure head and the volumetric plasma content. This figure shows that the results from mesh 5 have the same trend as those from the most refined mesh 1, and their values are very close. Therefore, mesh 5 offers a favorable compromise between computational cost and solution accuracy. Given the sub-1% error and minimal profile deviation, mesh 5 was selected for all subsequent simulations in this study.

5.2.2. Results and Analysis for the 3D Case

Figure 9 shows the variation in plasma volumetric content and albumin concentration at point P_RX (11 mm, −2 mm, 3 mm), located at the center of the reaction zone on the filter paper, over a 6-min period. In Figure 9A, it is shown that the volumetric plasma content at the P_RX point does not vary with respect to the residual moisture content of the paper (0.051) during the first 2.4 min after the blood drop is placed. However, between 2.4 and 2.6 min, a rapid increase in plasma content is observed approaching the saturated value of 0.8, suggesting a rise of 0.749 in just 0.2 min. The estimated rate of change in plasma content during this period is 3.745 1/min. The saturation of the filter paper occurs in a matter of seconds from the arrival of the first plasma molecules. This wetting time of the reaction zone agrees with the experimental results of Yang et al. [6], and is consistent with the behavior of liquids flowing by capillarity in paper. It is worth noting that plasma is composed of approximately 92% water [9]. In Figure 9B, the albumin concentration at the P_RX point is observed. Albumin concentration remains approximately 0 kg/m³ until minute 3, marking a clear lag phase. Between 3 and 4 min, concentration increases rapidly until it saturates at 3.68 kg/m³. This implied rate of change is 3.68 kg/(m³ min). Unlike plasma, the increase in albumin concentration is more gradual, and more time is required for it to reach and completely saturate the colorimetric reaction zone, which suggests a sigmoidal or logistic-like curve rather than a sharp jump. Consequently, once the plasma has wetted the reaction zone, an additional minute or so is necessary for the actual albumin concentration in the blood to reach this area. Only then can appropriate colorimetric changes be observed to accurately correlate with the total albumin content in the blood.

Figure 9 shows the 2D and 3D distribution of plasma within the microdevice at three different time points. The 2D images, corresponding to the x,y plane at z = 3 mm, located at the center of the microdevice, reveal that the plasma volumetric content does not follow a symmetrical profile, unlike what would be expected in a pipe with forced flow. Capillarity causes a greater volume of plasma to preferentially advance adhered to the upper edge in the region where the blood separation paper (separation membrane) is located. This fluid behavior in porous paper materials can also be observed in the results reported in reference [32], where analyte transport in a lateral flow paper-based microdevice was simulated using the commercial software COMSOL Multiphysics 6.0. After 1 min, the advancing front of the plasma has not yet reached halfway across the separation membrane. At 3 min, the plasma has reached the reaction zone, although it has not yet fully wetted it. At this point, the fluid initially rises from the upper edge of the separation membrane towards the top of the filter paper and then begins to completely soak it. Finally, at 4 min, the filter paper is completely saturated. The 3D views in Figure 10 show the non-uniform distribution of plasma within the microdevice. At 3 min, when the plasma begins to wet the filter paper in the colorimetric reaction zone, capillarity causes the edges of this area to become wet first, while the center takes more time to saturate. After 4 min, although the entire reaction zone is damp, it is still observed that the lower central part of the separation membrane has not been completely wetted. Finally, at 6 min, the plasma has fully saturated both the separation membrane and the filter paper.

The distribution of albumin follows a similar trend to that of plasma (Figure 11); its concentration is higher at the upper edge of the separation membrane and along the lateral edges of the reaction zone (filter paper). At 4 min, although as shown in Figure 8, the PRX point is already saturated, and the reaction zone has not yet reached the total albumin concentration present in the blood that has entered the device (3.68 kg/m³). However, the albumin concentration in the plasma that has penetrated the device manages to wet the entire reaction zone, allowing the colorimetric reaction and the device to perform an accurate measurement. These findings align with those reported in reference [6], where total blood albumin quantification was successfully completed in under six minutes using a device comparable to that utilized in the present study.

Temperature is a critical factor influencing the stability of the chemical reagents involved in the colorimetric reaction. For this reason, microdevices are often stored under refrigeration until use. Colorimetric reactions are most effective within a specific temperature range, which enhances the visibility of color changes. For instance, the device developed in the reference [6] operates optimally at approximately 37 °C and includes a thermostatic chamber to maintain a stable temperature, along with a CMOS camera and a smartphone app to digitize the reaction zone’s coloration using the RGB (Red, Green, Blue) method. However, this solution presents practical limitations: it increases costs, requires additional equipment for transport, and demands a power source. If the reaction chamber was omitted, knowing the temperature at which the colorimetric reaction occurs in the microdevice would be essential to assign an appropriate error margin to the measurements. To illustrate the utility of our simulations in this regard, we analyze the behavior of a device without a thermostatic chamber under two different conditions: unrefrigerated storage and refrigerated storage. The initial and boundary conditions for the plasma and albumin are the same as those used in the previous simulations. In the first case, the device initially has a uniform temperature of 20 °C (293.15 K). It is then exposed to an external temperature of 27 °C (300.15 K), applied as a boundary condition) and a blood drop at 36.5 °C (309.65 K) is introduced in the injection zone. In the second case, the microdevice starts at an initial temperature of 5 °C and is exposed to an external temperature of 27 °C, along with a blood drop in the injection zone at 36.5 °C. The isotherms in the x,y plane with z = 3 mm, corresponding to the central plane of the microdevice, are shown in Figure 12 for three different time points, reflecting the results of both analyzed scenarios. After 11 s of exposure to the new environmental conditions and the blood drop, the reaction zone of the microdevice has increased its temperature by approximately 4 °C. After 20 s, the ambient temperature has managed to reach the center of the microdevice, decreasing the temperature of the blood and increasing the initial temperature of the device. Finally, after 60 s, the entire device, including the reaction zone, is close to the ambient temperature of 300 K. The isotherms of the initially refrigerated microdevice show minimal variation (Figure 12B) compared to the non-refrigerated case (Figure 12A). The thermal conductivity properties of the device materials, combined with its small size, result in the device reaching ambient temperature—including the colorimetric reaction zone—within 1 min. The volume of the blood droplet is too small to influence the device’s isotherms for more than 20 s. Based on these results, and to enhance portability, reduce costs, and simplify device use by eliminating the thermostatic chamber, it is recommended to develop additional calibration curves over a range of ambient temperatures, such as 15 °C to 35 °C, in addition to the calibration curve presented in the reference [6] (limited to 37 °C), including the error percentage associated with each curve. According to simulations conducted in this study, the microdevice should reach ambient temperature within approximately 1 min after being removed from storage. To eliminate the need for a camera and smartphone application to determine the RGB value, it is proposed to design a portable device equipped with a laser sensor capable of instantly providing the average RGB value in the reaction zone, preprogrammed with calibration curves obtained in the laboratory at various temperatures. This technology, available on the market, has significant potential for enabling the production of low-cost, low-power paper-based microdevices that are also portable and easy to use in the field.

The results of this study provide valuable insights that complement experimental observations. The mathematical model and simulations demonstrated the ability to reveal the spatial and temporal distribution of biomarkers within regions of the device where experimental measurements are limited due to technical constraints, such as sensor resolution or flow interference. Moreover, the simulations allowed the decoupling of effects that are typically superimposed in experimental data, such as the relative contributions of diffusion, advection, and biochemical reactions to solute transport. This capability enables a deeper understanding of the underlying transport mechanisms, which would otherwise be difficult to isolate and validate through experimental means alone.

6. Conclusions

The mathematical model in this work, based on Darcy’s flows and equations for an unsaturated porous medium, has been successfully used to simulate the flow of water and plasma from human whole blood in paper-based microdevices.

The simplified two-dimensional mathematical model that was used for the flow of water and dye in paper demonstrated good accuracy when compared to the experimental results. In the capillary-driven microfluidic flow through two-dimensional microporous media analyzed in this study, a fully developed profile appears as a flat front. The solute flow in the strip was visibly affected by abrupt expansion, resulting in a lower solute flow rate in the x direction compared to the water front. For successful translation of laboratory findings into paper-based microdevices, it is essential to ensure complete wetting of the reaction zone by both the biomarker and plasma. A mathematical expression was formulated to optimize device design by determining the necessary paper strip length. This approach ensures full wetting following abrupt expansion, which is critical for achieving reliable biomarker detection.

A boundary condition has been proposed for a blood drop entering the paper perpendicularly, which has enabled the accurate three-dimensional simulation of plasma and albumin transport in a microdevice, as detailed in the reference [6]. The results presented in this study are consistent with experimental findings regarding the time it takes for plasma and albumin to reach the reaction zone.

The three-dimensional simulation allows for the observation of capillary effects on plasma movement within the microdevice. The albumin concentration follows a similar trend to that of plasma, as it is contained within it. Only after 5 min does the albumin completely fill the colorimetric reaction zone, which aligns with experimental results that concluded that the coloration of the reaction zone stabilizes after 5 min. The statistical trends observed reinforce the conclusion regarding the sequential transport behavior of plasma and albumin in the reaction zone. Temporal lag analysis indicates that the plasma wets the reaction zone approximately 0.6 min before albumin begins to accumulate significantly, highlighting a clear delay between solvent arrival and solute transport. The steep gradient and rapid increase in plasma content suggest the presence of a sharp wetting front, which is characteristic of capillary flow saturation in porous media. In contrast, the more gradual increase in albumin concentration supports the notion of solute separation and delayed diffusion or advection mechanisms, necessitating additional time for the solute to reach detectable levels in the reaction zone. Overall, these dynamic behaviors align well with first-order capillary wetting models and confirm the mechanistic expectations described in the literature, including consistency with the experimental results reported by Yang et al. [6].

The simulations from this study indicate that the microdevice reaches ambient temperature within approximately one minute of removal from storage.

The results of this study demonstrate that the proposed mathematical model, based on Richards’ equation coupled with solute and temperature transport in unsaturated porous media, offers a robust framework for predicting fluid and biomarker dynamics in microfluidic paper-based devices. Beyond reproducing experimental trends, the simulations provided insights into spatial and temporal distributions in regions where direct measurements are limited, and allowed the decoupling of transport mechanisms such as advection, diffusion, and reaction. Compared to multiphase Navier–Stokes models, this approach presents significant advantages in terms of computational efficiency and practical implementation, particularly during early-stage design and optimization. While Navier–Stokes formulations offer high fidelity in modeling complex fluid behavior, they require extensive computational resources and detailed parameterization. In contrast, the Richards-based model is well suited for applications where capillary-driven flow and solute transport dominate, as in biomarker filtration or detection processes in unsaturated porous substrates. Overall, this modeling framework not only complements experimental observations but also extends their applicability by enabling predictive simulations under untested conditions. As such, it constitutes a valuable tool for guiding the rational design and performance optimization of point-of-care diagnostic devices.