Analytical Solutions for the Mathematical Model Describing the Formation of Liver Zones via Adomian’s Method

Abstract

The formation of liver zones is modeled by a system of two integropartial differential equations. In this research, we introduce the mathematical formulation of these integro-partial differential equations obtained by Bass et al. in 1987. For better understanding of this mathematical formulation, we present a medical introduction for the liver in order to make the formulation as clear as possible. In applied mathematics, the Adomian decomposition method is an effective procedure to obtain analytic and approximate solutions for different types of operator equations. This Adomian decomposition method is used in this work to solve the proposed model analytically. The stationary solutions (as time tends to infinity) are also obtained through it, which are in full agreement with those obtained by Bass et al. in 1987.

1. Introduction

The dark-red liver is the body’s largest single gland (1 to 1.5 kg). It is a metabolic middleman because it takes up and secretes more than 500 different kinds of molecules. The liver stores and releases glucose, keeping blood glucose levels relatively constant. The location of the liver reflects its middleman’s role (Figure 1). The gland lies between the diaphragm above and the stomach and intestines below (Figure 2). Glucose and many other molecules enter the liver through the hepatic portal vein, and their products go directly through the inferior vena cava to the heart and lungs and then into the systemic circulation. The liver takes its dome-like shape from the diaphragm, which covers its superior surface, called diaphragmatic surface. The sagittal fossa divides the liver into two great lobes, right and left. The right lobe is larger and displays two smaller quadrate and caudate lobes on its visceral surface, defined by gallbladder and inferior vena cava, respectively [1]. The hepatic veins drain into the inferior vena cava arising from the posterior part of diaphragmatic surface of the liver. Visceral peritoneum binds the liver to the diaphragm and to the posterior wall of the abdomen.

Figure 1 

Figure 2 

Although there is an extensive bare area on the diaphragmatic surface of the liver where the peritoneum does not reach, the connective tissue attaches this area directly to the diaphragm. Most of the blood to the liver (70–80%) comes from the portal vein, and the smaller percentage is supplied by the hepatic artery (Figure 3). All the materials absorbed via the intestines reach the liver through the portal vein, except the complex lipids which are transported mainly by lymph vessels. The position of the liver in the circulatory system is optimal for gathering, transforming, and accumulating metabolites and for neutralizing and eliminating toxic substances. This elimination occurs in the form of bile, an exocrine secretion of the liver which is important in lipid digestion. The basic structural component of the liver is the liver cell or hepatocyte. In light microscope, structural units called classic liver lobules can be seen. The liver lobule is formed of a polygonal mass of tissue about 0.7×2 mm in size (Figure 4).

Figure 3 

Figure 4 

In certain animals (e.g., the pig), lobules are separated from each other by a layer of connective tissue. In humans, it is difficult to establish the exact limits between different lobules since they are in close contact in most of their extent (Figure 5). In some regions, the lobules are demarcated by connective tissue containing bile ducts, lymphatic vessels, nerves, and blood vessels. These regions, located at the corners of the lobules and occupied by portal triads, are called portal spaces. The human liver contains 3–6 portal triads per lobule, each with a venule (a branch of the portal vein); an arteriole (a branch of the hepatic artery); a duct (part of the bile duct system); and lymphatic vessels. The venule is usually the largest of these structures, containing blood from the superior and inferior mesenteric and splenic veins. An arteriole contains blood from the celiac trunk of the abdominal aorta.

Figure 5 

The hepatocytes in the liver lobule are radially disposed and arranged like the bricks of a wall. These cellular plates are directed from the periphery of the lobule to its center and anastomose freely, forming a labyrinthine and sponge-like structure. The space between these plates contains capillaries, the liver sinusoids [2]. Portal and arterial blood mixes in the sinusoids and flows past hepatocytes, draining through a central vein from each lobule that leads ultimately to the hepatic veins. Bile from the lobules drains into the interlobular branches of the bile duct by way of bile canaliculi. The hepatic lobules act as endocrine and exocrine glands. In endocrine secretion, hepatocytes take up and secrete molecules into the sinusoids [1].

The liver has an extraordinary capacity for regeneration. Hence, the loss of hepatic tissue by surgical removal or from the action of toxic substances is restored. The liver performs its metabolic functions with the aid of various enzymes fixed inside liver cells. These liver cells line many capillaries (hepatic sinusoids) through which the total hepatic blood flow is manifolded, whereby exchange of substances between blood flow and cells is facilitated. The interplay of the unidirectional blood flow with local metabolism generates concentration gradients of blood-borne substances (such as oxygen) between the inlet and the outlet of the liver. The unidirectionality of that blood, that is, the blood flows form the portal triads to the central vein (Figure 5), has a major influence on the mathematical structure of the model, which appears to be capable of describing the formation of zones with a jump discontinuity at a certain distance along a capillary [3].

Several metabolic functions of the liver have been found to be organized in spatial zones arranged in relation to the direction of hepatic blood flow, in such a way that some enzymes act almost wholly upstream others [3]. Bass et al. [3] attributed such distributions of enzymes activities to distributions of cell types. For the simplest case of two enzymes, there are two corresponding cell types, each containing only one of the enzymes; separate metabolic zones occur when all cells of one type are located upstream all cells of the other type. Furthermore, it was reported in [3] that each cell type reproduces itself by division. The mathematical model was discussed in [3], but for convenience the main steps in its derivation are repeated in the next section. The mathematical model describing the formation of liver zones is a system of nonlinear integropartial differential equations. The objective of this paper is to apply Adomian’s decomposition method to the system in order to find its stationary solutions (as the time tends to infinity) and the general solutions (at any position 𝑥 and any time 𝑡) for arbitrary initial conditions.

2. Mathematical Modelling and Solutions

2.1. Mathematical Formulation

About 1100 milliliters of blood flows from the portal vein into the liver sinusoids (Figure 6) each minute, and approximately an additional 350 milliliters flows into the sinusoids from the hepatic artery, the total averaging is about 1450 mL/min. This amounts to about 29% of the resting cardiac output. As the many capillaries comprising the liver are similar and act essentially in parallel, Bass et al. [3] modelled a representative capillary lined with cells of two kinds. It was suggested to put the 𝑥-axis along the blood flow, with inlet at 𝑥=0 and outlet at 𝑥=𝐿 [3]. The density of cells of the first kind is defined by 𝜌₁(𝑥,𝑡) as a continuous representation of the number of cells of the first kind per unit length of capillary at time 𝑡 at the position 𝑥. The density 𝜌₂(𝑥,𝑡) of cells of the second kind is defined analogously. The total cell density 𝜌₁+𝜌₂ cannot exceed some fixed maximum density 𝜎 of cell sites, as division of the cell is limited by the familiar phenomenon of contact inhibition. The local rate of change 𝜕𝜌₁/𝜕𝑡 of the density of cells of the first kind is assumed to consist of a growth rate term proportional to 𝜌₁ (self-generation) and to the density of sites available, 𝜎−𝜌₁−𝜌₂, and of a death rate term proportional to 𝜌₁, with a coefficient 𝛽₁(𝑐)>0 dependent on the local concentration 𝑐 of a controlling blood-borne substance. In what follows, for definiteness, oxygen is taken as that substance. Then 𝜕𝜌₁𝜕𝑡=𝐾₁𝜌₁𝜎−𝜌₁−𝜌₂−𝛽₁(𝑐)𝜌₁, (1) with a constant coefficient 𝐾₁>0. A similar equation for 𝜌₂ is obtained from (1) by interchanging the suffices 1, 2. So 𝜕𝜌₂𝜕𝑡=𝐾₂𝜌₂𝜎−𝜌₁−𝜌₂−𝛽₂(𝑐)𝜌₂. (2) Let 𝑓 be the steady rate of blood flow through the capillary. If oxygen is transported in the 𝑥-direction predominantly by convection with the blood and used up by the two cell types at the rates 𝑘₁𝜌₁ and 𝑘₂𝜌₂ (with positive constants 𝑘₁, 𝑘₂), then changes in 𝑐 caused by changes in 𝜌₁ and 𝜌₂ are quasisteady. Therefore, 𝑐 satisfies 𝑓𝜕𝑐𝜕𝑥=−𝑘₁𝜌₁−𝑘₂𝜌₂. (3) If (3) is integrated, then 𝑐(𝑥,𝑡)=𝑐₀−1𝑓^𝑥₀𝑘₁𝜌₁(𝜉,𝑡)+𝑘₂𝜌₂(𝜉,𝑡)𝑑𝜉, (4) where 𝑐₀ is the steady oxygen concentration in the blood entering the liver. It is assumed that as oxygen concentration falls, the death rate of cells increases (𝑑𝛽₁(𝑐)/𝑑𝑐≤0,𝑑𝛽₂(𝑐)/𝑑𝑐≤0), though not necessarily equally for both cell types. It is assumed that 𝛽₁(𝑐) has the following form (similarly for 𝛽₂(𝑐)): 𝛽₁(𝑐)=𝜇₁+𝜈₁𝑐₀−𝑐, (5) where 𝜇₁=𝛽₁𝑐₀≥0,𝜈₁=−𝑑𝛽₁𝑑𝑐_𝑐0≥0. (6) Introducing (4) and (5) into (1) and (2), we arrive at the pair of equations 𝜕𝜌₁𝜕𝑡=𝜌₁𝐾₁𝜎−𝜌₁−𝜌₂−𝜇₁−𝜈₁𝑓^𝑥₀𝑘₁𝜌₁+𝑘₂𝜌₂𝑑𝜉,𝜕𝜌₂𝜕𝑡=𝜌₂𝐾₂𝜎−𝜌₁−𝜌₂−𝜇₂−𝜈₂𝑓^𝑥₀𝑘₁𝜌₁+𝑘₂𝜌₂𝑑𝜉. (7) If 𝐾₁𝜎≤𝜇₁, then 𝜌₁(𝑥,𝑡)→0 as 𝑡→∞ for all 𝑥, and similarly for 𝜌₂. Therefore, it is assumed that 𝐾₁𝜎>𝜇₁ and 𝐾₂𝜎>𝜇₂. For similar reasons, it is assumed that at least one of 𝜈₁ and 𝜈₂ is positive (say 𝜈₁). It is noted at once that unless the first cell type is inevitably to die out, its greatest possible specific growth rate 𝐾₁𝜎 must exceed its least possible specific death rate 𝜇₁. Similar remarks apply for the second cell type, and accordingly it is assumed in [3] that 𝐾₁𝜎>𝜇₁,𝐾₂𝜎>𝜇₂. (8) To obtain some preliminary heuristic ideas about the formation of zones in their model, Bass et al. [3] supposed that (7) admits solutions which at all finite times are everywhere positive and satisfy (𝜌₁+𝜌₂)<𝜎. For such solutions, (7) can be written in the form 𝜕𝜕𝑡ln𝜌₁=𝐾₁𝜎−𝜌₁−𝜌₂−𝜇₁−𝜈₁𝑓^𝑥₀𝑘₁𝜌₁+𝑘₂𝜌₂𝜕𝑑𝜉,𝜕𝑡ln𝜌₂=𝐾₂𝜎−𝜌₁−𝜌₂−𝜇₂−𝜈₂𝑓^𝑥₀𝑘₁𝜌₁+𝑘₂𝜌₂𝑑𝜉. (9) Multiplying the first equation in (9) by 𝐾₂ and the second by 𝐾₁, we have 𝜕𝜕𝑡ln𝜌^𝐾2₁=𝐾₁𝐾₂𝜎−𝜌₁−𝜌₂−𝜇₁𝐾₁−𝜈₁𝐾₁𝑓^𝑥₀𝑘₁𝜌₁+𝑘₂𝜌₂𝜕𝑑𝜉,𝜕𝑡ln𝜌^𝐾1₂=𝐾₁𝐾₂𝜎−𝜌₁−𝜌₂−𝜇₂𝐾₂−𝜈₂𝐾₂𝑓^𝑥₀𝑘₁𝜌₁+𝑘₂𝜌₂𝑑𝜉. (10)

Figure 6 

2.2. The Stationary Solutions

For such solutions, we can combine (10) in the form 𝜕𝜌𝜕𝑡ln^𝐾21𝜌^𝐾1₂𝐵=𝐴−𝑓^𝑥₀𝑘₁𝜌₁+𝑘₂𝜌₂𝑑𝜉, (11) where 𝐴=𝜇₂𝐾₁−𝜇₁𝐾₂,𝐵=𝜈₁𝐾₂−𝜈₂𝐾₁. (12) Suppose that 𝜇₂𝐾₁>𝜇₁𝐾₂, 𝜈₁𝐾₂>𝜈₂𝐾₁, or 𝜇₂𝐾₂>𝜇₁𝐾₁,𝜈₂𝐾₂<𝜈₁𝐾₁, (13) so that the constants 𝐴 and 𝐵 are positive. Since the integral in (11) is bounded above by (𝑘₁+𝑘₂)𝑥𝜎, the right-hand side of (11) is positive at all times for sufficiently small 𝑥, where 𝑥<𝐴𝑓𝑘₁+𝑘₂.𝜎𝐵 (14) Volterra’s argument [3] then applies: as 𝑡→∞, 𝜌^𝐾2₁/𝜌^𝐾1₂→∞, and with 𝜌₁ bounded above by 𝜎, 𝜌₂ must tends to zero. It is then plausible that for these values of 𝑥 in (14), 𝜌₁ will approach a stationary form determined from the first equations of (7) with 𝜌₂=0, namely [3, 4], 𝐾₁𝜎−𝜌₁−𝜇₁−𝜈₁𝑓^𝑥₀𝑘₁𝜌₁𝑑𝜉=0. (15) In order to solve this equation by Adomian’s decomposition method [5–13], we put the equation in the form 𝜌₁=𝑐₁𝐾₁−𝜈₁𝑘₁𝑓𝐾₁^𝑥₀𝜌₁𝑑𝜉, (16) where 𝑐_𝑖=𝐾_𝑖𝜎−𝜇₁,𝑖=1,2. (17) According to Adomian’s method, 𝜌₁ is assumed as 𝜌₁=^∞_𝑛=0𝜌_1𝑛. (18) Substituting (18) into (16), we obtain ^∞_𝑛=0𝜌_1𝑛=𝑐₁𝐾₁−𝜈₁𝑘₁𝑓𝐾₁^∞_𝑛=0^𝑥₀𝜌_1𝑛𝑑𝜉. (19) Let 𝜌₁₀=𝑐₁𝐾₁. (20) Then the solution can be elegantly computed by using the recurrence relation 𝜌_1(𝑛+1)𝜈=−¹𝑘₁𝑓𝐾₁^𝑥₀𝜌_1𝑛𝑑𝜉,𝑛≥0. (21) This gives 𝜌₁₁(𝑥)=−𝜈₁𝑘₁𝑓𝐾₁𝑐₁𝐾₁𝜌𝑥,₁₂(𝑥)=−𝜈₁𝑘₁𝑓𝐾₁²𝑐₁𝐾₁𝑥²,𝜌2!₁₃(𝑥)=−𝜈₁𝑘₁𝑓𝐾₁³𝑐₁𝐾₁𝑥³,⋮𝜌3!_1𝑛(𝑥)=−𝜈₁𝑘₁𝑓𝐾₁^𝑛𝑐₁𝐾₁𝑥^𝑛𝑛!,𝑛≥0. (22) According to (18), we obtain 𝜌₁ in the form 𝜌₁(𝑥)=𝜌^∗₁(𝑥)=𝑐₁𝐾₁−𝜈exp¹𝑘₁𝑓𝐾₁𝑥. (23) Set 𝜌₂ equal to zero and 𝜌₁ equal to 𝜌^∗₁, then (11) becomes 𝜕𝜌𝜕𝑡ln^𝐾21𝜌^𝐾1₂=𝐴−𝐵𝑐₁𝜈₁−𝜈1−exp¹𝑘₁𝑓𝐾₁𝑥. (24) We note that the right-hand side of (24) decreases with increasing 𝑥 and reaches zero at a value 𝑥=𝑥^∗ determined by 𝜈exp¹𝑘₁𝑥^∗𝑓𝐾₁=𝐵𝑐₁𝐾₁𝜈₁𝑐₂−𝜈₂𝑐₁. (25) Provided that 𝜈₁𝑐₂>𝜈₂𝑐₁. (26) The point 𝑥=𝑥^∗ determined by (25) lies in the interval (0,𝐿) of interest provided that 𝜈exp¹𝑘₁𝐿𝑓𝐾₁>𝐵𝑐₁𝐾₁𝜈₁𝑐₂−𝜈₂𝑐₁. (27) Under these conditions, it is then reasonable to suppose that, for 𝑥>𝑥^∗, the right-hand side of (11) will in fact be negative for sufficiently large values of 𝑡 [3]. Volterra’s argument then indicates that we can expect to find 𝜌₁→0 as 𝑡→∞ for 𝑥>𝑥^∗. Furthermore, we may also expect that, for 𝑥>𝑥^∗, 𝜌₂ will approach a stationary form determined from the second equation of (7) by 𝐾₂𝜎−𝜌₂−𝜇₂−𝜈₂𝑓^𝑥∗₀𝑘₁𝜌^∗₁𝑑𝜉+^𝑥_𝑥∗𝑘₂𝜌₂𝑑𝜉=0. (28) This equation can be solved by Adomian’s method; we rewrite the equation in the form 𝜌₂+𝜈₂𝑘₂𝑓𝐾₂^𝑥_𝑥^∗𝜌₂𝑑𝜉=𝐷, (29) where 𝜈𝐷=¹𝑐₂−𝜈₂𝑐₁𝐵. (30) We assume that 𝜌₂=^∞_𝑛=0𝜌_2𝑛. (31) Let 𝜌₂₀=𝐷, then the solution can be computed by using the recurrence relation 𝜌_2(𝑛+1)𝜈=−²𝑘₂𝑓𝐾₂^𝑥_𝑥^∗𝜌_2𝑛𝑑𝜉,𝑛≥0. (32) This gives 𝜌₂₁=−−𝜈₂𝑘₂𝑓𝐾₂𝐷𝑥−𝑥^∗,𝜌₂₂=−𝜈₂𝑘₂𝑓𝐾₂²𝐷𝑥−𝑥^∗2,⋮𝜌2!_2𝑛=−𝜈₂𝑘₂𝑓𝐾₂^𝑛𝐷𝑥−𝑥^∗𝑛𝑛!,𝑛≥0. (33) Therefore 𝜌₂=𝜌^∗₂(𝑥)−𝜈=𝐷exp²𝑘₂𝑓𝐾₂𝑥−𝑥^∗,𝑥>𝑥^∗. (34) So, as 𝑡→∞, the formation of liver zones can be described as follows: 𝜌₁=𝜌^∗₁(𝑥),𝜌₂=0,0≤𝑥<𝑥^∗,𝜌₁=0,𝜌₂=𝜌^∗₂(𝑥),𝑥^∗<𝑥≤𝐿. (35)

3. Analytical Solutions

In applied mathematics, Adomian’s decomposition method is an effective procedure to obtain analytic and approximate solutions for different types of equations. This method is used here to obtain a general solution for the system (7). Following Bass et al. [3], we define new variables 𝑡^′=𝑐₁𝑡,𝑥^′=𝜈₁𝑘₁𝑓𝐾₁𝑥,𝑣_𝑖𝑡^′,𝑥^′=𝐾₁𝑐₁𝜌_𝑖(𝑡,𝑥) (36) and new parameters 𝑘𝜃=²𝑘₁𝐾,𝛾=²𝐾₁𝐾,𝜆=¹𝑐₂𝐾₂𝑐₁𝜈,𝜂=²𝐾₁𝜈₁𝐾₂. (37) Then (7) becomes, on dropping at once the primes from the new independent variables, 𝜕𝑣₁𝜕𝑡=𝑣₁1−𝑣₁−𝑣₂−^𝑥₀𝑣₁(𝑡,𝜉)+𝜃𝑣₂(𝑡,𝜉)𝑑𝜉,𝜕𝑣₂𝜕𝑡=𝛾𝑣₂𝜆−𝑣₁−𝑣₂−𝜂^𝑥₀𝑣₁(𝑡,𝜉)+𝜃𝑣₂(𝑡,𝜉)𝑑𝜉, (38) with constant parameters 𝜃>0,𝛾>0,𝜆>0,𝜂≥0. (39) The spatial interval of interest is now [0,Λ], where Λ=(𝜈₁𝑘₁/𝑓𝐾₁)𝐿, and then we have [3] 𝜂<𝜆<1,ln1−𝜂𝜆−𝜂<Λ. (40) The stationary solutions become 𝑣₁=𝑒^−𝑥,𝑣₂=0,0≤𝑥<𝑥^∗,𝑣₁=0,𝑣₂=𝜆−𝜂𝑒1−𝜂^{−𝜂𝜃(𝑥−𝑥∗)},𝑥^∗<𝑥≤Λ, (41) where now 𝑥^∗=ln1−𝜂.𝜆−𝜂 (42) For searching analytical solutions, we firstly rewrite the system we want to solve as two separate integro-partial differential equations: 𝜕𝑣₁𝜕𝑡=𝑣₁1−𝑣₁−𝑣₂−^𝑥₀𝑣₁+𝜃𝑣₂𝑑𝜉,𝜕𝑣₂𝜕𝑡=𝛾𝑣₂𝜆−𝑣₁−𝑣₂−𝜂^𝑥₀𝑣₁+𝜃𝑣₂𝑑𝜉. (43)(44) According to the decomposition method, we assume that 𝑣₁=^∞_𝑛=0𝑣_1𝑛,𝑣₂=^∞_𝑛=0𝑣_2𝑛. (45) Equation (43) can be put in the following operator form: 𝐿_𝑡𝑣₁=𝑣₁−𝑣²₁−𝑣₁𝑣₂−𝑣₁^𝑥₀𝑣₁(𝑡,𝜉)+𝜃𝑣₂()𝐿𝑡,𝜉𝑑𝜉,_𝑡=𝜕.𝜕𝑡 (46) Applying the inverse operator 𝐿_𝑡⁻¹[⋅]=∫^𝑡₀[⋅]𝑑𝑡, on both sides of this equation, yields 𝑣₁−𝑣₁(𝑥,0)=𝐿_𝑡⁻¹𝑣₁−𝐿_𝑡⁻¹𝑣²₁−𝐿_𝑡⁻¹𝑣₁𝑣₂−𝐿_𝑡⁻¹𝑣₁^𝑥₀𝑣₁+𝜃𝑣₂𝑑𝜉. (47) Substituting (45) into (47), we obtain 𝑣₁=𝑣₁(𝑥,0)+𝐿_𝑡^∞−1_𝑛=0𝑣_1𝑛−𝐿_𝑡^∞−1𝑛_𝑛=0𝑘=0𝑣_1𝑘𝑣_{1(𝑛−𝑘)}−𝐿_𝑡^∞−1𝑛_𝑛=0𝑘=0𝑣_1𝑘𝑣_{2(𝑛−𝑘)}−𝐿_𝑡^∞−1_𝑛=0𝑣_1𝑛^𝑥₀^∞_𝑛=0𝑣_1𝑛+𝜃𝑣_2𝑛𝑑𝜉. (48) Now, the solution 𝑣₁ can be evaluated through the recursive scheme: 𝑣₁₀(𝑥,𝑡)=𝑣₁(),𝑣𝑥,0_1(𝑛+1)(𝑥,𝑡)=𝐿_𝑡⁻¹𝑣_1𝑛−𝐿_𝑡^−1𝑛_𝑘=0𝑣_1𝑘𝑣_{1(𝑛−𝑘)}−𝐿_𝑡^−1𝑛_𝑘=0𝑣_1𝑘𝑣_{2(𝑛−𝑘)}−𝐿_𝑡^𝑛−1_𝑘=0𝑣_1𝑘×^𝑥₀𝑣_{1(𝑛−𝑘)}+𝜃𝑣_{2(𝑛−𝑘)}𝑑𝜉,𝑛≥0. (49) By similar analysis, we can get the solution 𝑣₂ by the recursive scheme: 𝑣₂₀(𝑥,𝑡)=𝑣₂(),𝑣𝑥,0_2(𝑛+1)(𝑥,𝑡)=𝛾𝜆𝐿_𝑡⁻¹𝑣_2𝑛−𝛾𝐿_𝑡^−1𝑛_𝑘=0𝑣_1𝑘𝑣_{2(𝑛−𝑘)}−𝛾𝐿_𝑡^−1𝑛_𝑘=0𝑣_2𝑘𝑣_{2(𝑛−𝑘)}−𝛾𝜆𝐿_𝑡^𝑛−1_𝑘=0𝑣_2𝑘×^𝑥₀𝑣_{1(𝑛−𝑘)}+𝜃𝑣_{2(𝑛−𝑘)}𝑑𝜉,𝑛≥0. (50) For simplicity, we assume that the two types of the liver cells have the same distribution along the hepatic capillary at 𝑡=0; that is, 𝑣₁(𝑥,0)=𝑣₂(𝑥,0);thatis,𝑣₁₀(𝑥)=𝑣₂₀(𝑥)=𝑣₀(𝑥)say. (51) By this, we can get the first few terms of Adomian’s series from the recurrence relations (49) and (50) as follows: 𝑣₁₁(𝑥,𝑡)=𝑣₀1−2𝑣₀−(1+𝜃)𝐼₁(𝑥𝑣)𝑡,₂₁(𝑥,𝑡)=𝛾𝑣₀𝜆−2𝑣₀−𝜂(1+𝜃)𝐼₁(𝑥)𝑣𝑡,₁₂=𝑣₀1−3𝑣₀−(1+𝜃)𝐼₁(𝑥)1−2𝑣₀−(1+𝜃)𝐼₁(𝑥)−𝛾𝑣₀𝜆−2𝑣₀−𝜂(1+𝜃)𝐼₁(𝑥)−1+𝛾𝜃𝜆𝐼₁(𝑥)+21+𝛾𝜃𝐼₂(𝑥)1+𝛾𝜂𝜃(1+𝜃)𝐼₃(𝑥𝑡)²,𝑣2!₂₂=𝛾𝜆−3𝛾𝑣₀−𝛾𝜂(1+𝜃)𝐼₁(𝑥×)𝛾𝜆𝑣₀−2𝛾𝑣²₀−𝛾𝜂(1+𝜃)𝑣₀𝐼₁(𝑥−)𝛾𝑣²₀−2𝛾𝑣³₀−𝛾(1+𝜃)𝑣²₀𝐼₁(𝑥𝑣)−𝛾𝜂1+𝛾𝜃𝜆₀𝐼₁(𝑥)+2𝛾𝜂1+𝛾𝜃𝑣₀𝐼₂(𝑥)+𝛾𝜂(1+𝜃)1+𝛾𝜂𝜃𝑣₀𝐼₃(𝑥𝑡)²,2! (52) where 𝐼₁(𝑥)=^𝑥₀𝑣₀𝑑𝜉,𝐼₂(𝑥)=^𝑥₀𝑣²₀𝐼𝑑𝜉,₃(𝑥)=^𝑥₀𝐼₁(𝜉)𝑣₀𝑑𝜉. (53)

4. Remarks

Here, we indicate that at particular values of the parameters 𝛾, 𝜆, and 𝜂, the solutions 𝑣₁ and 𝑣₂ are equivalent. In order to do this, we prefer to put the solutions 𝑣₁ and 𝑣₂ in the form 𝑣₁(𝑥,𝑡)=𝛼₀(𝑥)+𝛼₁(𝑥)𝑡+𝛼₂(𝑥)𝑡²𝑣+⋯,₂(𝑥,𝑡)=𝛽₀(𝑥)+𝛽₁(𝑥)𝑡+𝛽₂(𝑥)𝑡²+⋯, (54) where 𝛼₀(𝑥)=𝛽₀(𝑥)=𝑣₀(𝑥),𝛼₁(𝑥)=𝑣₀1−2𝑣₀−(1+𝜃)𝐼₁(𝑥,𝛼₂)(𝑥)=1𝑣2!⁰1−3𝑣₀−(1+𝜃)𝐼₁(𝑥×)1−2𝑣₀−(1+𝜃)𝐼₁(𝑥)−𝛾𝑣₀𝜆−2𝑣₀−𝜂(1+𝜃)𝐼₁(𝑥)−1+𝛾𝜃𝜆𝐼₁(𝑥)+21+𝛾𝜃𝐼₂(𝑥)+1+𝛾𝜂𝜃(1+𝜃)𝐼₃(𝑥),𝛽₁(𝑥)=𝛾𝑣₀𝜆−2𝑣₀−(1+𝜃)𝐼₁(𝑥,𝛽₂)(𝑥)=12!𝛾𝜆−3𝛾𝑣₀−𝛾𝜂(1+𝜃)𝐼₁(𝑥×)𝛾𝜆𝑣₀−2𝛾𝑣²₀−𝛾𝜂(1+𝜃)𝑣₀𝐼₁(𝑥−)𝛾𝑣²₀−2𝛾𝑣³₀−𝛾(1+𝜃)𝑣²₀𝐼₁(𝑥𝑣)−𝜂1+𝛾𝜃𝜆₀𝐼₁(𝑥)+2𝛾𝜂1+𝛾𝜃𝑣₀𝐼₂(𝑥)+𝛾𝜂(1+𝜃)1+𝛾𝜂𝜃𝑣₀𝐼₃(𝑥). (55) Firstly, substituting 𝛾=𝜆=𝜂=1 into the original equations (38), we obtain 1𝑣₁𝜕𝑣₁𝜕𝑡=1−𝑣₁−𝑣₂−^𝑥₀𝑣₁+𝜃𝑣₂1𝑑𝜉,𝑣₂𝜕𝑣₂𝜕𝑡=1−𝑣₁−𝑣₂−^𝑥₀𝑣₁+𝜃𝑣₂𝑑𝜉. (56) We can combine these equations in the form 𝜕𝜕𝑡ln𝑣₁(=𝜕𝑥,𝑡)𝜕𝑡ln𝑣₂(𝑥,𝑡). (57) By integrating both sides with respect to 𝑡 from 0 to 𝑡, we get 𝑣ln¹(𝑥,𝑡)𝑣₀(𝑥)𝑣=ln²(𝑥,𝑡)𝑣₀(𝑥), (58) where we used the relation 𝑣₁(𝑥,0)=𝑣₂(𝑥,0)=𝑣₀(𝑥). Thus, 𝑣₁(𝑥,𝑡)=𝑣₂(𝑥,𝑡). Now, substituting 𝛾=𝜆=𝜂=1 into (55), we can easily observe that 𝛼₀(𝑥)=𝛽₀(𝑥),𝛼₁(𝑥)=𝛽₁(𝑥),𝛼₂(𝑥)=𝛽₂(𝑥), (59) which leads also to 𝑣₁(𝑥,𝑡)=𝑣₂(𝑥,𝑡).

5. Conclusion

In this paper, the Adomian decomposition method has been applied successfully to a system of nonlinear integro-partial differential equations describing the formation of liver zones. As time tends to infinity, the stationary solutions are obtained in exact forms by using Adomian’s method, where full agreement with those obtained in the literature has been achieved. Also, at any time of the liver regeneration process, the analytical solutions are obtained explicitly in series form. Finally, the current solutions may shed some light on the mathematical aspects of the formation of liver zones and also on describing the distribution of the two types of the liver cells.

Acknowledgment

The author wishes to express his thanks to Professor Mostafa El-Shahed for his kind help and very valuable suggestions.