A mathematical model of the formation of urine

1 Introducción

Junto al cerebro y el corazón, el riñón es un órgano fundamental en el sistema cardiovascular, ya que recibe aproximadamente el 20% de la sangre que el corazón expulsa hacia el organismo, en cada latido. Entre sus principales funciones se encuentran: regular la presión arterial, eliminar los desechos tóxicos a través de la orina y producir la hormona que estimula la formación de los glóbulos rojos [5].

Cuando no hay un funcionamiento adecuado del riñón, pueden presentarse enfermedades renales de diversas formas, desde infecciones y cálculos renales hasta enfermedades crónicas como la nefropatía diabética y la enfermedad renal crónica (ERC). Estas generan un problema de salud cada vez más importante a nivel global, afectando no solo la vida de los pacientes, sino también generando una carga significativa para los sistemas de salud debido a su alta prevalencia y costos asociados.

Por otra parte, los modelos hemodinámicos renales proporcionan una comprensión profunda sobre los patrones y comportamientos del flujo sanguíneo renal, lo que puede permitir detectar anomalías y disfunciones que de otra manera podrían pasar desapercibidas. Asimismo contribuyen a tener una perspectiva global e integral del problema, lo que coadyuva a avanzar en el diagnóstico, tratamiento y prevención de las afecciones renales.

Ahora bien, el torrente sanguíneo arrastra una gran cantidad de sustancias, muchas de la cuales entran al riñón y alguna proporción de estas pasa a formar parte de la orina. Sin embargo, la mayoría de los modelos nefrológicos consideran solo la concentración de una sustancia [7]. Estos modelos se han utilizado con fines terapéuticos o tratamiento de enfermedades vía control (cantidad de medicamentos) [1], y también con fines de diseño de riñones artificiales. En general el modelado del proceso de hemodiálisis se basa en ecuaciones cinético-hemodinámicas [3] o ecuaciones cinético-matemáticas [2].

Figura 1: Esquema de las etapas de la formación de la orina en la nefrona [7].

En este contexto, el propósito de este trabajo es analizar la hemodinámica renal modelando matemáticamente los procesos básicos de la producción de la orina, siguiendo las ideas expuestas en [7]. Para esto se asumen varias cosas: que el tejido renal es homogéneo, que ciertos parámetros fisiológicos son constantes y que la anatomía del riñón es simple. Además, se supone que el flujo sanguíneo es uniforme, se simplifican las interacciones químicas y se asume la ausencia de patologías. Con estos supuestos, el modelo resultante se puede resolver al menos numéricamente. En su mayoría este trabajo es expositivo, aunque se complementa con el desarrollo de los detalles que justifican las afirmaciones, y con los resultados de las simulaciones propias respectivas que no están en el artículo. El modelo que se considera está formado por: i) una EDO de primer orden con dos condiciones de frontera (el valor de una de ellas es también desconocido y debe determinarse como parte del modelo), ii) una EDP con una condición de frontera y su respectiva condición inicial, y iii) 8 EDOs adicionales con sus respectivas condiciones iniciales.

2 Formulación del modelo

La principal unidad operativa del riñón se llama nefrona, de la que hay aproximadamente un millón en cada riñón y cada nefrona es capaz de formar orina por sí misma. Es en esta donde se llevan acabo las 3 etapas para la producción de la orina (ver figura 1). En la primer etapa, la etapa de filtración glomerular, el glomérulo filtra el agua y otras sustancias del torrente sanguíneo.

Cada nefrona tiene un glomérulo cuya función es un filtrado inicial de la sangre. El glomérulo consta de una red de capilares envuelto por la cápsula de Bowman (una estructura en forma de copa). La presión arterial de la sangre en los capilares empuja el agua y los solutos pequeños hacia la cápsula de Bowman, a través de una membrana de filtrado. Con esto inicia el proceso de formación de la orina [7]. Aproximadamente el 20% de la sangre que entra a la nefrona, se filtra hacia la cápsula de Bowman. Tanto las plaquetas como las proteínas plasmáticas y células sanguíneas son demasiado grandes para pasar por el filtro, por lo que éstas continúan por la arteriola eferente, formando el 80% del líquido que no se filtró [5].

2.1 Modelo para la filtración glomerular.

Para modelar matemáticamente la dinámica del filtrado glomerular [7], suponemos que los capilares glomerulares se distribuyen en una región en forma de un tubo unidimensional con flujo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle q_1}

y que la cápsula de Bowman que lo rodea  también tienen la misma forma y flujo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle q_2}
(ver figura 2). Suponemos que el filtrado glomerular se da a través de una pared capilar de logitud Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle L}
(linea punteada de la figura 2). Dado que el flujo a través de los capilares glomerulares es proporcional a la diferencia de presión a través de la pared capilar, el modelo para la filtración glomerular es 

Figura 2: Modelo tubular del glomérulo [7].

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left\{ \begin{array}{ll}\frac{dq_1}{dx}= K_f(P_2 - P_1 + \pi _i \frac{Q_i}{q_1}), \quad 0<x<L, \\ q_1(0)= Q_i, \\ q_1(L)= Q_e, \end{array} \right.

(1)

junto con las relaciones

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): P_a - R_aQ_i = P_1,

(2)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): P_1 - P_e = R_eQ_e,

(3)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): P_2 -P_d = R_d Q_d,

(4)

donde

• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle P_1}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle P_2}
son las presiones hidrostáticas del fluido en los tubos 1 y 2 respectivamente (como se muestra en la figura 2, donde el tubo 1 esta en rojo y el tubo 2 en verde) y satisfacen las relaciones (2),(3), (4), donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle P_{i}}
representa las presiones, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle R_{i}}
las resistencias y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q_{i}}
los flujos. Aquí, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i=a,e,d }
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle a}
hace referencia a la arteriola aferente, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle e}
hace referencia a  la arteriola eferente y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle d}
hace referencia al túbulo descendente. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q_i}
 denota el flujo de entrada, es decir el flujo  en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle x=0}

. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q_i}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q_e}
son ambas desconocidas y deben determinarse. Esto hace que este problema esté bien definido y sea no  trivial.

• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle K_f}

es la tasa de filtración capilar.

• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \pi _i}

es la presión osmótica tanto de las proteínas suspendidas  como de  otras sustancias de alto peso molecular y depende de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle q_1}

.

2.1.1.1 Solución analítica implícita

Con unas simplificaciones algebraicas, la EDO en (1) se puede resolver por el método de separación de variables. La solución está definida implícitamente por

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{1}{ K_f(P_2-P_1)} \left[q_1- \frac{K_f Q_i \pi _i}{ K_f(P_2-P_1)} \ln \left( K_f(P_2-P_1) q_1+K_f Q_i \pi _i\right)\right]= x+ C.

(5)

Usando la condición inicial Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle q_1(0)= Q_i}

 y la condición final Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle  q_1(L)= Q_e,}
determinamos la constante de integración Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle C.}
Usando propiedades de logaritmo y haciendo  algunas manipulaciones algebraicas (ver [6]), se llega a  la siguiente  expresión

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{Q_e}{Q_i} + \alpha \ln \left[\frac{\frac{Q_e}{Q_i}- \alpha }{1- \alpha }\right]= 1- \frac{K_f\pi _i L}{\alpha Q_i},

(6)

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \alpha = \frac{\pi _i}{(P_1 - P_2)}.}

En resumen, hasta ahora tenemos la solución implícita para Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle q_1}

dada por (5) y  la de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q_e, Q_i}
dada por (6) y además tenemos la relación Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q_i=Q_e+ Q_d}

. Así que si damos un valor para Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q_d}

podemos resolver (6) para Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q_e}
 y obtener Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q_i}
o de igual forma resolver (6)  para Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q_i}
 y obtener Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q_e}

. También obtenemos Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle P_a} , vía la relación (2).

Aún con esto no podemos obtener una formula explícita para Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle q_1}

pero sí podemos resolver numéricamente, como lo explicaremos en la siguiente sección.

2.1.1.2 Solución numérica

Los valores típicos de los parámetros son Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle P_1 = 60, P_2 = 18, P_a = 100, P_e = 18, P_d = 14 - 18, \pi _i = 25 mm Hg} , con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q_i = 650, Q_d = Q_i - Q_e = 125 ml/min.}

Estos valores corresponden a una dinámica estacionaria del riñón, donde los parámetros permanecen constantes. La solución del modelo con estos parámetros se ilustra en la figura 3 (izquierda)   para Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle q_1}
obtenida cuando Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q_d=125,}
que se obtuvo dandole valor a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q_d}

, a las resistencias Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle R_a,R_e}

y a las presiones Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle P_e, P_d}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \pi _i}
especificadas y fijadas en niveles típicos.  Con esto, resolvemos (6)   para Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q_i}
 (utilizando un simple algoritmo de bisección) y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle  Q_e,}
a partir de ello, las presiones correspondientes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle  P_1, P_2, P_a}
se determinan con ayuda  de (2)-(4). Con los valores de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q_e}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q_i}
podemos darle solución la ecuación (5) por medio de bisección  para encontrar a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle q_1}
o bien aplicar un método como Runge Kutta para problemas de valor inicial. 

Si quisiéramos estudiar el funcionamiento del riñón en estado no estacionario, por ejemplo, en una situación donde variara la presión de la arteria aferente Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle P_a}

entonces esto haría variar todos los demás parámetros. Los flujos y las presiones varían en función de la presión arterial. Para entender algo de esta variación, se dan valores para Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q_d}
y para cada una de ellas se hace el procedimiento explicado anteriormente, obteniendo a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q_i,Q_e, P_1, P_2, P_a}
 correspondientes a cada Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q_d}
 y  se grafican (como se muestran en la figura 3 derecha) la tasa de flujo sanguíneo renal Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q_i}
y la tasa de flujo de filtración glomerular Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q_e}
como funciones de la presión arterial Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle P_a}

. Observamos que tanto Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q_i}

como Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q_d}
 crecen linealmente con respecto a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle P_a}

. Sin embargo, en la realidad (según los datos mostrados en la figura 4), la tasa de filtración glomerular permanece relativamente constante incluso cuando la presión arterial varía entre 75 y 160 mmHg, lo que sugiere que existe cierta autorregulación en la tasa de filtración y por tanto en las tasas de flujo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q_i.}

Este fenómeno lo estudiaremos en la siguiente sección.

(3) Gráfica de la Curva Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): q_1(x) correspondiente a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): Q_d=125 . Para este caso, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): Q_i=652.1 y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): Q_e=527.1 , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): x \in [0,L] (izquierda). Gráficas de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{Q_i}{5} y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): Q_d contra Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): P_a (derecha).

Figura 4: Gráficas reales del flujo sanguíneo renal con autorregulación y de la tasa de filtración glomerular [7].

2.2 Autorregulación

En la segunda etapa, que es la reabsorción, algunos nutrientes y agua del líquido filtrado que fluye por el túbulo renal se reincorporan al torrente sanguíneo (capilares peritubulares). Lo que retorna son grandes cantidades de: aminoácidos, vitaminas, agua, glucosa, parte de la urea, los iones Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle K^+} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Na^+} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle NaHCO_3}

(bicarbonato), Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Cl^-}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle HPO_4}

(fosfato).

Por lo que se concluyó en la sección anterior, es claro que el riñón necesita regular la tasa de filtración glomerular. La función principal del túbulo ascendente grueso es bombear iones de sodio Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Na^+}

fuera de éste hacia el espacio intersticial, siendo casi totalmente impermeable al agua. Si la velocidad de flujo  es lenta a través del túbulo, habrá una mayor reabsorción  y una concentración de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Cl^-}
baja al final del túbulo que esta conectado con la  mácula densa y el glomérulo. Por otro lado, cuando la velocidad de flujo es alta, hay una baja reabsorción y se produce una mayor concentración de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Na^+}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Cl^-}
en la mácula densa. Esto se resume en la tabla 1.

Tabla. 1 Relación del flujo, la reabsorción y la concentración de cloro en el túbulo ascendente.

Flujo del túbulo ascendente	Reabsorción	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): C(L, \tau )
Caso 1: lento	Alta	Baja
Caso 2: rápido	Baja	Alta

Tabla. 2 Acciones que se deben tomar en las arteriolas para aumentar o disminuir el flujo.

Flujo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle q_1}	Filtración	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): P_1	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): R_a	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): R_e
Caso 1: Aumentar	Aumentar	Aumentar	Disminuir	Disminuir
Caso 2: Disminuir	Disminuir	Disminuir	Disminuir	Aumentar

Para normalizar el flujo del túbulo ascendente ya sea porque este sea lento o rápido, las células de la mácula densa responden a la disminución de la concentración de NaCl (a través de un mecanismo no completamente conocido) liberando un vasodilatador que aumenta o disminuye la resistencia de las arteriolas aferentes. En el caso 1, es decir cuando el flujo es lento, las células yuxtaglomerulares liberan renina, una enzima que promueve la formación de angiotensina II, la cual constriñe las arteriolas eferentes. Esto resulta en un aumento simultáneo del flujo de filtrado a través del glomérulo. En cambio, en el caso 2, cuando el flujo es rápido, la concentración de NaCl en la mácula densa aumenta, se disminuye la resistencia de la arteriola aferente y aumenta la resistencia de la arteriola eferente, lo que disminuye la tasa de filtración y el flujo a lo largo del túbulo. Esto se resume en la tabla 2.

2.2.1 Modelado de la concentración de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Cl^-} para la autorregulación2.2.1 Modelado de la concentración de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Cl^-} para la autorregulación

Un modelo sencillo de las oscilaciones tubuloglomerulares [7], se centra en el papel de la concentración de cloruro en la rama ascendente gruesa. Para modelar matemáticamente esta dinámica, suponemos que la rama ascendente gruesa tiene forma de un tubo unidimensional de longitud Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle L}

 y radio constante, a través del cual el Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Cl^-}
es transportado y al mismo tiempo bombeado a través de las paredes del túbulo (es el Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Na^+}
el que se elimina activamente, pero como el Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Cl^-}
sigue pasivamente, el efecto es el mismo).  

Supondremos que la concentración de cloruro es uniforme en cada sección transversal. Si usamos una coordenada Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle y}

para describir cada sección transversal del túbulo   y dado los supuestos  que hemos mencionado, denotamos por Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle C(y, \tau )}
la concentración de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Cl^-}
en la sección transversal asociada con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle y}
en el momento Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tau }
y modelamos esa concentración con la siguiente ecuación diferencial parcial  de transporte

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{\partial C}{\partial \tau } + \phi (C(L,\tau -\overline{\tau }))\frac{\partial C}{\partial y}= -R(C), \quad y \in (0,L) ,\tau \in (0, \infty ), C(0,\tau )=C_0\equiv cte,

(7)

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle R(C)}

representa la tasa de eliminación de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Cl^-}
del tubo mediante el bombeo de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Na^+}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle  \phi \frac{\partial C}{\partial y}}
 la tasa  de cloro que pasa a través de cada sección transversal.  Consideramos las expresiones siguientes

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \quad R(C)=r_c C, \quad \phi (C)=F_{o} + F_{\delta } \tanh (\alpha (\overline{C}-C)),

con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \alpha }

constante, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle F_{\delta }}
una diferencia de presiones, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle  F_o}
el flujo de referencia de una persona normal  y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \overline{C}}
 la concentración de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Cl^-}
para la cual Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \phi }
tiene un punto de inflexión, constante. A esta ecuación de transporte le  asociamos la condición inicial Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle C(0,\tau )=C_0\equiv cte}

.

2.2.2 Análisis cualitativo de la solución del modelo para la autorregulación

Introducimos el cambio de variable Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle y =Lx, \tau = \frac{L}{F_{o}} t, C(y,\tau )=C_0 c(x,t)}

y definimos Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle  \overline{\tau }=\frac{L}{F_o}\overline{t}, K_1=\frac{F_{\delta }}{F_o}, K_2= C_0, \Omega = (0,1) \times (0,T)}
donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle  \overline{t}}
 es el periodo de retardo adimensional. 

De donde obtenemos la ecuación diferencial parcial retardada

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \begin{array}{ll}\frac{\partial c}{\partial t}+ F(c(1,t-\overline{t})) \frac{\partial c}{ \partial x}= -\mu c , \forall (x,t) \in \Omega , \\ c(x,0)=0, \quad \forall \quad x \quad \in (0,1] \\ c(0,t)=1, \quad \forall \quad t \quad \in [0,T]. \end{array}

(8)

con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle F(c)=1+K_1 \tanh (K_2 (\overline{c}-c))}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mu = r_c \frac{L}{F_{o}}}

. Para iniciar el estudio de la ecuación (8), vamos a obtener las soluciones en estado estacionario. Es decir le daremos solución a la ecuación

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): F(s(1)) \frac{d s}{dx}=- \mu s,

(9)

con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle s(0) = 1} . Claramente la solución de esta ecuación es

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): s(x)= e^{-kx},

(10)

con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k=\frac{\mu }{F(s(1))}} . El valor de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k}

debe ser determinado y tomando en cuenta que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle s(1) =e^{-k},}
el valor de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k}
debe satisfacer

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): kF(e^{-k})=\mu{.}

(11)

Dado que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle F}

es una función decreciente de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle c}
 ya que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle F'(c)<0,}
así existe un único valor de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k}
para el cual se cumple (11) y  por tanto, una solución única en estado estacionario.

Al analizar la estabilidad de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle s(x)}

se presentan  oscilaciones en la concentración de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Cl^-}
de la mácula densa, y por tanto en las demás variables como la tasa de flujo y la presión glomerular, estas podrían ser explicadas por un mecanismo similar al de  una bifurcación de Hopf. En este mecanismo, existen algunos parámetros que dentro de un rango producen soluciones oscilatorias que convergen al estado estacionario, mientras que para otro rango de valores de los parámetros resultan soluciones que oscilan sin acercarse, alrededor del estado estacionario. Podemos encontrar más sobre esto en [6].

2.2.3 Solución numérica de la concentración de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Cl^-} a través del tiempo2.2.3 Solución numérica de la concentración de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Cl^-} a través del tiempo

Se realizó un programa para aproximar numéricamente la solución del problema (8). Para ello realizamos los siguientes pasos:

1. Discretizamos en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (M+1) \times (N+1)} nodos ((Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle M \times N} ) rectángulos) a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Omega } . Para esto, consideramos una partición Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle x_j} en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle x} , donde

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \begin{array}{ll} hx=1/M,\\ x_j=j*hx, \quad j=0,1,...,M, \\ x_0=0, x_M=1, \end{array}

y una partición Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t_i}

en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t}

, donde

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \begin{array}{ll} ht=T/N, \\ t_i=i*ht, \quad i=0,1,...,N,\\ t_0=0, t_N=T. \end{array}

De lo anterior se tiene como resultado una malla con nodos o vértices Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n_{ij}=(x_j,t_i)} . Para cálculos posteriores necesitamos enumerar cada nodo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n_{ij}= (x_j, t_i)}

 y esta numeración se hará  recorriendo las rectas horizontales de abajo hacia arriba y los nodos en cada recta horizontal recorridos de izquierda a derecha, numerando cada intersección entre ellas. Al nodo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n_{ij}}
se le asocia el numero de nodo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k=(i-1)M+j, j=1,..,N}
e Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i=1,..,N.}

Esto se muestra en la figura 5.

Figura 5: Resultado del paso 1: Malla computacional.

2. Construimos una aproximación Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle c_{ij}} para Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle c(x_j, t_i)} , es decir, para la solución en cada nodo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n_{ij}= (x_j, t_i)} . De acuedo a las condiciones de frontera para toda Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t} conocemos el valor de la función en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle x=0,} entonces simplemente le asignamos a los nodos con esta característica el valor 1. Para todos los demás nodos se aproxima la solución mediante el método de Euler explícito, que se resume en el siguiente esquema

   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): c_{ij}=\left(1-\frac{s}{\rho }\right) c_{(i-1)j}+ \frac{s}{\rho } c_{(i-1)(j-1)},

   (12)

para Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i=1,2,...,N}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle j=1,2,...,M}
 con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \rho =(1/Nx)/(T/Nt)}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle s=F(c(1,t_i-\overline{t}))=F(c_{(i-itg)M}),}
  donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle itg}
es la parte entera de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \frac{\overline{t}}{Nt}}

.

3. Finalmente construimos una función Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \overline{c}} continua por pedazos, a partir de los valores de aproximación obtenidos en lo anterior. Esta función será la aproximación numérica de la solución para la ecuación (8) y graficándola obtenemos gráficas dadas en las figuras 6 y 7 .

Dado el dominio Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Omega = (0, 1) \times (0, T),}

con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle F(z) =1+ k_1 tanh(k_2(\overline{c}-z))}
y  los valores típicos de los parámetros Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle K_1=1, K_2=10, \overline{c}=e^{(-0.91)}, \mu=0.5,}
con las ideas anteriores,  hicimos un programa  para encontrar las soluciones numéricas de la concentración de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle  Cl^-}
a través del tiempo y con los datos siguientes  para el caso estable Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \gamma=0.1223,\overline{t}=0.2,\frac{k}{\mu }=1,}
(ver  figura 6) y para el caso inestable Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \gamma=3.1787, \overline{t}=0.2,\frac{k}{\mu }=1}
 (ver  figura 7).  La concentración de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Cl^-}
se representa en unidades adimensionales y  la concentración al inicio de la rama ascendente gruesa es 1. La variable Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t}
también es adimensional; para obtener segundos se debe  multiplicar por 15.5 . 

Una vez obtenida la concentración de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Cl^-}

  como función global es decir la superficie obtenida  para cada Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle x}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t,}
para nuestros propósitos  la grafíca de la concentración  de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Cl^-}
  que nos interesa es cuando estamos al final del asa ascendente de Henle, es decir en este caso para Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle x=1,}
gráficas a la derecha en las figuras  6 y 7. Esta será la concentración (en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle x=1}

) que vamos a monitorear para saber cuál debe ser la cantidad de flujo en la filtración glomerular Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q_d}

(la variable Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q_d}
 es adimensional, para hacerla dimensional se debe  multiplicar por Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle F_o=10}

). Si se da que en cierto tiempo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle c(1,t)>>s(1),}

se tendría que acelerar el flujo, al  contrario, si en  cierto tiempo se da que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle c(1,t)<<s(1)}
 se tendría  que disminuir el flujo. Esto se  hace modificando las resistencias de las arteriolas aferente y eferente, lo cual tienen como efecto la modificación respectiva de la presión, del flujo y de la concentración.  A este efecto  le llamamos autorregulación. 

Con las acciones que se hacen para los dos casos anteriores, se puede regular el flujo para que a medida que la presión aumente, el flujo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q_d}

se vuelva casi constante, como se muestra en la figura 8, que se obtuvo fijando la resistencia Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle R_a}
y variando la resistencia Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle R_e}

. Con esto observamos que aunque la presión aumente, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q_d}

se puede mantener acotada variando adecuadamente las resistencias.

(6) Caso estable de las soluciones oscilatorias del modelo de la concentración de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): Cl^- para la autorregulación, a la izquierda la superficie de la concentración de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): Cl^- para cada Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): x y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): t , a la derecha la curva que pertenece a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): x=1 para toda Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): t .

(7) Caso inestable de las soluciones oscilatorias del modelo de la concentración de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): Cl^- para la autorregulación, a la izquierda la superficie de la concentración de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): Cl^- para cada Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): x y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): t , a la derecha la curva que pertenece a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): x=1 para toda Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): t .

Figura 8: Grafica del flujo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): Q_d con el proceso de autorregulación.

2.3 Modelado de la reabsorción y la secreción

En esta etapa de secreción, se eliminan sustancias no deseadas hacia los túbulos renales, funcionando como un proceso inverso a la secreción. En la nefrona, el fluido dentro del asa de Henle fluye en direcciones opuestas en los segmentos ascendente y descendente, lo que permite la creación de un gradiente osmótico en la médula renal. Este gradiente facilita la reabsorción de agua. Además, debido a la conexión en el extremo inferior del asa de Henle, el flujo y la concentración de solutos que salen del tubo descendente deben coincidir con los que entran en el tubo ascendente.

Siguiendo a [7], consideramos que el asa de Henle consta de cuatro compartimentos: tres túbulos (rama descendente, rama ascendente y conducto colector (túbulo amarillo en la figura 1)) y un compartimento único para el intersticio y los capilares peritubulares (túbulo rojo en la figura 1). El lecho intersticial/capilar se trata como un tubo unidimensional que recibe líquido de los otros tres túbulos y lo entrega a las vénulas. En cada uno de estos compartimentos, seguimos el flujo de agua y la concentración de solutos.

Suponemos que el flujo en cada uno de los tubos sigue un patrón simple (positivo en la dirección Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle x}

positiva), con tasas de flujo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle q_d, q_a, q_c, q_s}
para los túbulos descendentes, ascendentes, colectores y los túbulos intersticiales, respectivamente. De igual manera, la concentración de soluto en cada uno de estos se representa como Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle c_d, c_a, c_c, c_s}

. Se considera que los túbulos son unidimensionales, con el flujo del filtrado glomerular ingresando a la rama descendente en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle x = 0} , pasando de la rama descendente a la ascendente en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle x = L} , dirigiéndose desde la rama ascendente al conducto colector en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle x = 0} , y finalmente saliendo del conducto colector en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle x = L} . Se asume que el intersticio/compartimento capilar se vacía en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle x = 0} , y que no hay flujo en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle x = L} . Para una mejor comprensión de lo anterior, ver figura 9.

Figura 9: Diagrama del modelo de cuatro compartimentos del asa de Henle [7].

El sistema de ecuaciones que resultan en nuestra modelación de la actividad en el asa de Henle es:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left\{ \begin{array}{ll}\frac{1}{k_d} \frac{dq_d}{dx}= P_s- \pi _s - P_d + 2RT(c_d - c_s), \quad q_d(0)=Q_i -Q_e, \\ \frac{d(q_d c_d)}{dx}= h_d (c_s - c_d), \quad c_d(0)= c_d0,\\ \frac{dq_a}{dx}=0, \quad q_a(L)=-q_d(L),\\ \frac{d(q_a c_a)}{dx}=-p, \quad c_a(L)=c_d(L), \\ \frac{1}{k_c} \frac{dq_c}{dx}= P_s- \pi _s - P_c + 2RT(c_c - c_s), \quad q_c(0)=-q_a(0), \\ \frac{d(q_c c_c)}{dx}= h_c (c_s - c_c), \quad c_c(0)=c_a(0), \\ \frac{dq_s}{dx}= - \frac{d}{dx}(q_d + q_a + q_c), \quad q_s(0)=q_c(L)-q_d(0), \\ \frac{d(q_s c_s)}{dx}=- \frac{d}{dx} (q_d c_d + q_a c_a + q_c c_c), \quad c_s(0)=c_c(L)-c_d(0), \end{array} \right.

(13)

2.3.1 Solución del modelo para la reabsorción y secreción

Con fines de simplificar la descripción de la solución del modelo, haremos una adimensionalización al sistema de ecuaciones (13), normalizando los flujos y las concentraciones de solutos, con

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): x= Ly, Q_j= \frac{q_j}{q_d(0)}, C_j= \frac{c_j}{c_d(0)}, j=d,a,c,s

y los parámetros adimensionales Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \rho _j= \frac{q_d(0)}{2LRT_cd(0)k_j}, \Delta P_j= \frac{P_j + \pi _s - P_s}{RT2c_d(0)}, H_j= \frac{Lhj}{q_d(0)}, j=d,c.}

La versión adimensional de (13) es

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{dQ_d}{dy}= \frac{1}{\rho _d} \left[- \Delta P_d + (C_d - C_s)\right], \quad Q_d(0)=1,

(14)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{d(Q_d C_d)}{dy}= -H_d (C_d -C _s), \quad C_d(0)= 1,

(15)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{dQ_a}{dy}=0, \quad Q_a(0)=Q_a(L),

(16)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{d(Q_a C_a)}{dy}=\frac{-pL}{q_a(0) c_d(0)}, \quad C_a(L)=C_d(L),

(17)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{dQ_c}{dy}= \frac{1}{\rho _c} \left[- \Delta P_c + (C_c - C_s)\right], \quad Q_c(0)=-Q_a(0),

(18)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{d(Q_c C_c)}{dy}= -H_c (C_c - C_s), \quad C_c(0)=C_a(0),

(19)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{dQ_s}{dy}= - \frac{d}{dy}(Q_d + Q_a + Q_c), \quad Q_s(0)=Q_c(L)-Q_d(0),

(20)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{d(Q_s C_s)}{dy}=- \frac{d}{dy} (Q_d C_d + Q_a C_a + Q_c C_c), \quad C_s(0)=C_c(L)-C_d(0).

(21)

La ecuaciones (14)- (21) tienen las siguientes soluciones implícitas, respectivamente

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \rho _d (Q_d -1) + \frac{1}{H_d}(Q_dC_d -1)= - \Delta P_d y,

(22)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): C_d(y) =F_1(Q_d,y)= \frac{1}{Q_d}(1+ \rho _d H_d(1-Q_d)- \Delta P_d H_dy),

(23)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): Q_a(y)=Q_a(0)= Q_a(L)=Q_a,

(24)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): C_a(y)= \frac{\frac{-pL}{q_d(0) c_d(0)}y + Q_a(0) C_a(0)}{Q_a},

(25)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \rho _c(Q_c - Q_c(0)) + \frac{1}{H_c}(Q_c C_c- Q_c(0)C_c(0))= - \Delta P_c y,

(26)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): C_c(y) =F_2(Q_c,y)= \frac{1}{Q_c}( Q_c(0)C_c(0)+ \rho _c H_c( Q_c(0)C_c(0)-Q_c)- \Delta P_c H_cy),

(27)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): Q_s(y)= -(Q_d + Q_a + Q_c) (y)+Q_c(L),

(28)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): C_s(y)=F_3 (Q_d,Q_a,Q_c,y)=\frac{ -(Q_d C_d + Q_a C_a + Q_c C_c )(y)+ Q_c(L) C_c(L)}{Q_s}.

(29)

La solución implícita del modelo de 8 ecuaciones (14)- (21) dice que si conocemos Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q_d(y),}

entonces Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle C_d}
está determinada, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q_a(y)}
está determinada si conocemos Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q_a}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle C_a(y),}
si conocemos Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q_a,}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q_c(y),}
entonces Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle C_c}
está determinada y finalmente si conocemos Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q_d,Q_a,Q_c,}
entonces Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q_s}
 y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle C_s}
están determinadas. En resumen si Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q_d}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q_c}
están determinadas, las seis  ecuaciones diferenciales restantes (15), (16),(17),(19),(20) y (21) tienen solución y por lo tanto  el sistema original de ocho ecuaciones estaría resuelto. Así que lo que hace falta es resolver  el siguiente  sistema de dos ecuaciones de primer orden en dos incógnitas.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left\{ \begin{array}{ll}\rho _d \frac{dQ_d}{dy}= - \Delta P_d + C_d - C_s, \\ Q_d(0)= 1, \\ Q_d(1)= -Q_a(1)=-Q_a, \\ \end{array} \right.

(30)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left\{ \begin{array}{ll}\rho _c \frac{dQ_c}{dy}= - \Delta P_c + C_c - C_s,\\ Q_c(0)= -Q_a(0)=-Q_a, \\ \end{array} \right.

(31)

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle C_c, C_s}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle C_d}
son funciones de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q_d}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q_c}

. Aunque hay tres condiciones iniciales para dos ecuaciones de primer orden, el número Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q_a}

también es desconocido, por lo que este problema está bien planteado. Con esto  observamos que la solución del modelo completo  de 8 ecuaciones  se reduce a resolver  el modelo de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 2 \times 2}
que en general  es difícil de resolver  tanto analítica como numéricamente.

En las siguientes secciones estudiaremos un poco del comportamiento de la solución de esta ecuación, considerando un caso particular o un caso límite que es cuando se considera ausencia de ADH en la formación de orina.

2.3.2 Solución del modelo en ausencia de ADH y de aldosterona

La hormona antidiurética (ADH) regula la permeabilidad al agua en los riñones, mientras que la aldosterona controla la permeabilidad al sodio. La falta de ADH, como en la diabetes insípida central, provoca la producción de grandes cantidades de orina diluida. La aldosterona, producida en la corteza suprarrenal, modula la cantidad de canales de sodio y la actividad de la ATPasa Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Na+/K+} , afectando la eliminación de sodio y potasio. Desórdenes como la enfermedad de Addison o el síndrome de Conn ilustran desequilibrios causados por niveles anormales de aldosterona. Vamos a considerar un caso en particular, suponemos que no hay ADH presente, entonces Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \rho _c =\infty }

y que no hay aldosterona presente, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle H_c = 0}

. En este caso se deduce que

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): Q_c = Q_c(0) = -Q_a, \quad \quad C_c = C_c(0) = C_a(0).

En otras palabras, no hay pérdida de agua o de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Na^+}

en el conducto colector por lo que la ecuación para Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q_c,}
 (31) ha sido reducida a una ecuación algebraica. Por lo que ahora, queda por determinar lo que ocurre en los túbulos descendente y ascendente. El flujo se rige por las siguientes ecuaciones

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left\{ \begin{array}{ll}\rho _d \frac{d Q_d}{dy}=f(Q_d, Q_a,y),\\ Q_d(0) = 1, \\ Q_d (1) = -Q_a, \\ Q_c(y)=-Q_a.\\ \end{array} \right.

(32)

con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle f(Q_d, Q_a,y)=C_d - C_s - \Delta P_d.}

De las ecuaciones (22) y (27) se deduce que

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): C_d= \frac{1}{Q_d}(1+ \rho _d H_d(1-Q_d)- \Delta P_d H_dy),\quad C_s= \frac{(P+ \Delta P_d H_d)(1-y)}{Q_d + Q_a}- \rho _d H_d.

Con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q_a}

 constante ya que el miembro ascendente es impermeable al agua y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle C_a}
una función linealmente decreciente de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle y}

.

Observamos que el problema (32), tiene asociadas dos condiciones de frontera, pero una de ellas está en términos de un parámetro desconocido Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q_a,}

que se debe determinar como parte del problema.

En el caso que nos ocupa (ausencia de ADH), adicionalmente consideraremos que el túbulo descendente sea bastante permeable al agua. Lo cual implicaría que debemos tomar Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \rho _d}

 pequeño en el problema (32). Pero entonces, la ecuación diferencial (32) es difícil de resolver tanto analítica como numéricamente ya que resulta ser  singular. Dado que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q_d}
es diferenciable e invertible, por el teorema de la función inversa, podemos  resolver este problema buscando una solución en la forma Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle y = y(Q_d, \rho _d)}
que satisface la ecuación diferencial

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left\{ \begin{array}{ll}f(Q_d, Q_a,y) \frac{dy}{dQ_d}= \rho _d,\\ Q_d(0) = 1,Q_d (1) = -Q_a(0). \\ \end{array} \right.

(33)

Para resolver este problema buscamos Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle y}

en función de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q_d}
como serie de potencias de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \rho _d.}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): y = y_0 + \rho _d y_1 + \rho _d^2 y_2 + O(\rho _d^2 ),

(34)

sustituyendo en (33), igualando y expandiendo en potencias de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \rho _d,}

se tiene que

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): y=1- \frac{Q_a + Q_d}{P Q_d - \Delta P_d H_d Q_a} \left[1- \Delta P_d (Q_d+H_d)\right]+ O(\rho _d).

(35)

Ahora determinamos Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q_a}

haciendo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle y = 0, Q_d = 1}
en (35), y despejando Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q_a}
para determinarla, obteniendo

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): -Q_a=1- \frac{P+ H_d \Delta P_d}{1- \Delta P_d} +O(\rho _d).

(36)

Ahora podemos graficar Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle y}

en función de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q_d}
y girar los ejes para ver Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q_d}
en función de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle y}

. Esto se representa en las figuras 10. Una vez que se determina Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q_d}

en función de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle y}

, podemos graficar la concentración Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle C_d}

 en función de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle y}

, dada en la figura 11.

Figura 10: Curva del flujo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): Q_d (izquierda) y la curva de la función Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): y (derecha) con los valores de los parámetros con los valores de los parámetros: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): P=0.09,\Delta P_d=0.15, H_d=0.1, \rho _d=0.12, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): H_c=0.

Figura 11: Curva de la concentración en el túbulo descendente Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): C_d con los valores de los parámetros: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): P=0.09,\Delta P_d=0.15, H_d=0.1, \rho _d=0.12, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): H_c=0.

A partir de las gráficas anteriores observamos que la función Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q_d}

es decreciente  ya que  hay  ausencia de ADH y  de aldosteorona en el tubo descendente, es decir no hay   regulación en el fujo de agua,  produciendo  una gran cantidad de orina diluída y un flujo rápido. En el caso de la función Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle C_d}
es creciente  ya que no hay una adecuada reabsorción, quedando gran cantidad de sodio en el túbulo descendente.

3 Conclusión

En este trabajo, se presenta un modelo para la hemodinámica renal, centrado en la producción de orina. El modelo se descompone en tres submodelos:

1. Filtración glomerular: Se modela mediante una ecuación diferencial de primer orden con dos condiciones de frontera desconocidas, mostrando que el flujo resultante es lineal con respecto a la presión arterial, lo cual no es del todo preciso en la realidad debido a un proceso de autorregulación renal que mantiene constante el filtrado a pesar de variaciones en la presión.

2. Autorregulación: Se utiliza una ecuación diferencial parcial de transporte con retardo para modelar la concentración de cloro Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Cl^-}

en el túbulo ascendente del asa de Henle. Se descubrió que, dependiendo de algunos rangos de parámetros, las oscilaciones en la concentración de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Cl^-}
pueden llevar a soluciones que convergen rápidamente al estado estacionario o a soluciones casi periódicas, lo que sugiere la posibilidad de una bifurcación en el sistema.

3. Reabsorción y secreción: Se aborda mediante un sistema de ecuaciones diferenciales ordinarias que modela el flujo de agua y sodio en el asa de Henle. Se demuestra que, en ausencia de ADH y aldosterona, se produce un flujo decreciente de agua, lo que resulta en una gran cantidad de orina diluida y un aumento en la concentración de cloro.

En conjunto, aunque el modelo presenta limitaciones, logra simular satisfactoriamente la dinámica de la eliminación de desechos metabólicos a través de la orina en situaciones específicas.

BIBLIOGRAFÍA

[1] Bazaev, N.A, Grinvald, V. M. Selischev, S. V. (2010). A Mathematical Model for a Biotechnological Hemodialysis system. Biomedical Engineering, Vol. 44(No.3), 79-84.

[2] Baigent, S. Unwin, R. Yeng, C. (2001). Mathematical Modelling of Profiled Hemodialysis: A Simplified Approach. Journal of Theoretical Medicine, Vol 3(2), 143-160. https://doi.org/10.1080/10273660108833070.

[3] Cavalcanti, S. Ciandrini, A. Avanzolini, G. (2006). Mathematical modeling of arterial pressure response to hemodialysis-induced hypovolemia. Computers in Biology and Medicine, Vol 36(2), 128- 144. https://doi.org/10.1016/j.compbiomed.2004.08.00.

[4] Cooper, J. (2000). Introduction to partial Differential Equations with MATLAB, (3ed). New York. Birkhauser.

[5] Escuela Universitaria de Enfermería. Universidad de Barcelona,. Sistema urinario: anatomía. https://www.infermeravirtual.com/files/media/file/103/Sistema

[6] Domínguez, F. (2024). Un modelo matemático del funcionamiento del riñon [Tesis de Maestría en Ciencias en Matemáticas Aplicadas, Universidad Juárez Autónoma de Tabasco].

[7] James, K. James, S. (2009). Renal Physiology, Mathematical Physiology Systems Physiology , (2ed),(821-850). Midtown Manhattan. Springer.

[8] Perko, L. (2008). Differential Equations and Dynamical Systems, (3ed). New York. Springer.